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CAPILLARY ELECTROPHORESIS FOR
THERMODYNAMIC AND KINETIC STUDIES
By

Carl l. D. Newman

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Chemistry

2006

ABSTRACT
CAPILLARY ELECTROPHORESIS FOR
THERMODYNAMIC AND KINETIC STUDIES
By
Carl l. D. Newman

Capillary electrophoresis (CE) is a technique in which the individual
components of complex mixtures can be separated based upon their physical
characteristics of size and electrical charge. This work details the theoretical and
instrumental development, simulation, validation, application, and limitations of
capillary electrophoresis for investigating interactions between solute molecules.

In order to understand the evolution of zone profiles in reactive CE, 3
series of stochastic (Monte Carlo) simulations were employed. These
simulations were used to systematically investigate the effects of equilibrium
constant, characteristic reaction lifetime, and electrophoretic mobility difference
on zone profiles. Statistical moments were used to calculate the figures of merit
velocity, plate height, and asymmetry of each zone profile. It was shown that the
equilibrium constants are directly related to zone velocity; whereas, kinetic rate
constants are directly related to zone plate height.

Using conclusions drawn from the simulation studies, a theoretical
description is developed of the effects of reaction on velocity and plate height in
capillary electrophoresis. The developed equations are validated with the
simulations performed previously. The equations for velocity and plate height are

also validated by investigating peptidyI-proline isomerizations that have been

studied previously. These isomerizations exemplify first-order reactions and are
notorious for slow rate constants. Calculated values for equilibrium constants
and rate constants are shown to be in good agreement with previously published
vmues.

The plate height model is also applied to the reversible, second-order
inclusion of dansyl phenylalanine by B-cyclodextrin. Although equilibrium or
binding constants with B-cyclodextrin have been reported previously, rate
constants have remained elusive. The plate height model is used to calculate
equilibrium constants and rate constants from the effects of B—cyclodextrin
concentration on velocity and plate height, respectively. The effect of
temperature on equilibrium constant is used to calculate molar enthalpy and
molar entropy. The calculated thermodynamic quanitities are in agreement with
previously published values. However, it was concluded that the rate constants
for this reaction are too rapid for accurate evaluation via the plate height model.
In light of this unsuccessful application, the influence of the physical properties of
the instrumentation and of the solutes on the maximum calculable rate constants
is discussed.

Finally, avenues of future development are discussed. Potential
theoretical developments include expanding the existing theoretical equations to
include mathematical descriptions of solute zone evolution for reactions that are
far from steady state. Potential instrumental developments include applications
of the plate height model investigate solute-stationary phase interactions via

chip-based, open-tube capillary electrochromatographic separations.

To my family and in memory of my grandparents:
George Kraemer
Milo Newman
Evelyn Newman

‘I

uN-n.

AKNOWLEDGEMENTS

There are so many individuals to whom I owe thanks. My progress has
been aided by so many wonderful people in a multitude of ways, that it is difficult
to decide where to begin. So, I will attempt to do this semi-chronologically . . .

My parents and my family for their love and unwavering support. They
were there for me as l vacillated between undergraduate universities and
degrees. As I pursued my education from Massachusetts to Arizona to Florida
and finally to Michigan under the auspices of higher education, they were a
constant connection to home. Moreover, as my interests varied from music to
linguistics to anthropology to chemistry, they served as constant reminders of
who I am and from where I come.

My undergraduate professors: Dr. Walter Birkby, Dr. Glen Doran, and Dr.
Thomas \fickers. Their patience and guidance facilitated my growth and
independence as a young scientist. Through their tutelage I discovered that
scientificresearch can be a viable and rewarding path and was inspired to
pursue graduate education.

My graduate advisor, Vicki McGuffin. Vlfithout her wisdom, expertise, and
guidance this work not be possibie. Our conversations during my tenure at
Michigan State University have shaped me into not only a better scientist, but a
better author and person as well. I am forever in her debt for the confidence,

independence, and intellectual development that I now enjoy. I have immense

gratitude and. respect for Vicki personally and professionally and am forever
grateful to her.

My Ph.D. committee for their efforts, input and guidance. Dr. Merlin
Bruening was not only second reader for my dissertation but also brought a level
of insight and thoughtfulness that has helped me to gain further understanding of
this work. Dr. Gary Blanchard for his moral and intellectual support. His door
was always open to me. Dr. Brian Teppen and Dr. Thomas Pinnavaia for their
thoughtfulness and commitment to my progress through the Ph.D. program.

My friends for the‘emotional and moral support they have provided me
over the years. I am grateful to my Americorps teammates who are to me a
second family cf loving siblings. I am grateful to Jason Stotter, Nate Lynd, and
Sam Howerton. These are cherished friends and companions for drinking, cigar
smoking, and philosophizing. Our conversations regularly helped to remind me
that chemistry can indeed be fun and that it is possible to remain both a chemist
and a child at heart. I am thankful to Melissa Meaney and Amber Hupp for
bringing new life and excitement to the group. And I am forever grateful to
Haruko Yabe, my friend, confidant, and fiance. Her love, support, and
confidence in my abilities nourished and sustained me through times of joy,
frustration, and self-doubt.

Lastly I thank the many students, faculty, and staff at‘ MSU and within the
Department of Chemistry without whom my success would not be possible.
Kathy Severin for her advice and assistance with the numerous departmental

instruments that were used during the course of my reasearch. Paul Reed, Tom

vi

Carter, and Tom Atkinson of Computing Services for their valuable'assistance
and insight with numerous software, hardware, and network questions. Glen
Wesley and Tom Bartlett of the Machine Shop for their craftsmanship and
assistance in the fabrication of numerous components for the instrumentation I
developed during the course of my research. Scott Sanderson and Dave
Cedarstaff of Instrument Services for their invaluable assistance and advice in
the assembly and maintenance of the instrumentation I developed. The
Graduate School and Department of Chemistry for financial support. Lastly, the
Council of Graduate Students (COGS) and the COGS faculty advisor Marylee
Davis for facilitating my growth socially and as a member of the University
community. During graduate school it is easy to forget that we do not live within
departmental vacuums. My time with COGS and the advice from Marylee helped
to sustain. within me the philosophy I lived while in Americorps - that we should all

do what we can to make each others’ lives a little better.

vii

‘I met a traveler from an antique land
Who said: "Two vast and trunkless legs of stone

Stand in the desert... Near them, on the sand, ~

Half sunk a shattered visage lies, whose frown,
And wrinkled lip, and sneer of cold command,
Tell that its sculptor well those passions read

Which yet survive, stamped on these lifeless things,
The‘hand that mocked them and the heart that fed;
And on the pedestal these words appear:

My name is Ozymandius, King of Kings,

Look on my works, ye Mighty, and despair!
Nothing beside remains. Round the decay
Of that colossal wreck, boundless and bare
The lone and level sands stretch far away.

Ozymandius, Percy Bysshe Shelley

viii

 

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

CHAPTER 1

1.1

'1.2

1.3
1.5

INTRODUCTION AND BACKGROUND

CAPILLARY ELECTROPHORESIS

1.1.1

1.1.2 Physical Contributions to Separation and Broadening

1.1.3

1.1.4

History

1.1.2.1 Electrical Double-Layer
1.1.2.2. Mass Transport
1.1.2.2.1 Diffusion
1 1 2.2.2 Migration
1 .1 .2.2.2.1 Electroosmotic Migration
1.1.2.2.2.1 Electrophoretic Migration
1.1.2.3 Influence of Electrical Conductivity
1 1.2. 4 Column Contributions to Broadening
..2 4.1 Diffusion
..2 4. 2 Column Geometry

1.1
1.1
1.1..2 4. 3 Thermal Gradients

A 1.1.2.5 Extra-Column Contributions to Broadening

1.1.2.5.1 Injection

1.1 .2.5.2 Detection
Chemical Contributions to Separation
1.1.3.1 Acid-Base Equilibria
1.1.3.2 Other Equilibria
Experimental Methods for Reactive
Capillary Electrophoresis

1.1.4.1 Methods for Estimating Equilibrium Constants

1.1.4.2 Methods for Estimating Rate Constants

THERMODYNAMIC AND KINETIC THEORY

1.2.1 Thermodynamic Calculations

1.2.2

Kinetic Calculations

CONCLUSIONS
REFERENCES

CHAPTER 2 STOCHASTIC SIMULATIONS OF REACTIVE

2.1
2.2

2.3

CAPILLARY ELECTROPHORESIS

INTRODUCTION
SIMULATION METHODS

2.2.1
2.2.2

Simulation Conditions

Simulation Output and Calculations

RESULTS AND DISCUSSION

xiii

42

52
53
58

2.3.1 Effects of Rate Constant 58

2.3.2 Effects of Equilibrium Constant 65
2.3.3 Effects of Electrophoretic Mobility Difference 77
2.4 CONCLUSIONS 86
2.5 REFERENCES 88

CHAPTER 3 THEORETICAL PLATE HEIGHT MODEL FOR
REACTIVE CAPILLARY ELECTROPHORESIS

3.1 INTRODUCTION 92
3.2 THEORY 93
3.2.1 Development of the Plate Height Model 93
3.2.2 Evaluation of the Plate Height Model 98
3.3 SIMULATION METHODS 111
3.3.1 Simulation Conditions 111
3.3.2 Simulation Output and Calculations 112
3.4 RESULTS AND DISCUSSION 113
3.4.1 Comparison of Simulation and Theory 113
3.4.1 Propagation of Random Error 123
3.5 CONCLUSIONS ' 132
3.6 REFERENCES ' 139
CHAPTER 4 EXPERIMENTAL METHODS
4.1 INTRODUCTION 140
4.2 EXPERIMENTAL SYSTEMS 140
4.2.1 Capillary Preparation 140
4.2.2 Capillary Electrophoresis Systems 141
4.2.2.1 CE System with Diode Array Absorbance
Detection System 141
4.2.2.2 CE System with Laser-Induced Fluorescence
Detection System 141
4.3 DATA TREATMENT AND ANALYSIS 145
4.3.1 Mathematical Functions 145
4.3.2 Statistical Moments ' ‘149
4.3.3 Figures of Merit 149
4.4 CONCLUSIONS 150
4.5 REFERENCES , 151

CHAPTER 5 THEORETICAL PLATE HEIGHT MODEL FOR
THERMODYNAMIC AND KINETIC STUDIES OF
REVERSIBLE, FIRST-ORDER REACTIONS

5.1 INTRODUCTION 152
5.2 THEORY 153
5.2.1 Velocity and Equilibrium Constant » 156
5.2.2 Plate Height and Characteristic Reaction Lifetime 156
5.3 . SIMULATION METHODS 157
5.3.2 Simulation Conditions 157

5.4 EXPERIMENTAL METHODS
5.4.1 Chemicals
5.4.2 Experimental System
5.4.3 Data Analysis
5.4.3.1 Theoretical Plate Height Model
5.4.3.2 ChromVlfin
5.5 RESULTS AND DISCUSSION
5.5.1 Simulation Studies
5.5.1.1 Accuracy of Theoretical Plate Height
Model and ChromVIfin
5.5:2 Experimental Studies
5.5.2.1 Effects of Electric Field Strength
5.5.2.1.1 Evolution of Reactive
Zone Profiles
5.5.2.1.2 Identification of lsomers
5.5.2.1.3 Comparison of Rate Constants
5.5.2.2 Effects of Temperature
5.5.2.2.1 Electrophoretic Mobility
5.5.2.2.2 Equilibrium Constant
5.5.2.221 Molar Enthalpy and
Molar Entropy
5.5.2.2.3 Rate Constants
f 5.5.2.231 Molar Activation Energy
5.6 CONCLUSIONS
5.7 REFERENCES

CHAPTER 6 THEORETICAL PLATE HEIGHT MODEL FOR
THERMODYNAMIC AND KINETIC STUDIES OF
REVERSIBLE, SECOND-ORDER REACTIONS

6.1 INTRODUCTION

6.2 THEORY

6.2.1Velocity and Equilibrium Constant
6.2.2Plate Height and Rate Constant

6.3 EXPERIMENTAL METHODS

6.3.1Chemicals
6.3.2Experimental System
6.3.3Data Analysis

6.4 RESULTS AND DISCUSSION

6.4.1Velocity and Equilibrium Constant

6.4.1.1 Effects of Electric Field Strength and
Concentration on Velocity
6.4.1.2 Effects of Temperature and Electric

Field Strength on Equilibrium Constant
6.4.1.2.1 Molar Enthalpy and Molar Entropy
6.4.2Plate Height and Rate Constant
6.4.2.1 Effects of Electric Field Strength and

xi

158
158
158
158
158
159
159
159

162
166
166

166
170
170
172
172 .
178

181
181
185
188
190

192
195
197
197
198
198
198
201
201
201

- 201

207
208
208

Concentration on Plate Height 212

 

6.4.2.2 Effects of Temperature and Electric
Field Strength on Rate Constants 216
6.5 CONCLUSIONS 220
6.6 REFERENCES 224
CHAPTER 7 LIMITATIONS OF THE PLATE HEIGHT MODEL
7.1 INTRODUCTION 226
7.2 THEORY 227
7.3 CAPILLARY ELECTROPHORESIS 229
7.3.1 Effect of Electrophoretic Mobility Difference on kmax in
Reactive CE 229
7.3.2 Effect of Electric Field Strength on kmax in Reactive CE 232
7.3.3 Effect of Equilibrium Constant on kmax in Reactive CE 232
7.3.4 Effect of Diffusion Coefficient on kmax in Reactive CE 232
7.4 CAPILLARY ELECTROCHROMATOGRAPHY 239
7.4.1 Effect of Electrophoretic Mobility Difference on km.x CEC 240
7.4.2 Effect of Electric Field Strength on kmax in CEC 240
7.4.3 Effect of Equilibrium Constant on kmax in CEC 245
7.4.4 Effect of Diffusion Coefficient on kmax in CEC 248
7.5 CONCLUSIONS 248
CHAPTER 8 CONCLUSIONS AND FUTURE DIRECTIONS
8.1 INTRODUCTION 253
8.2 EVOLUTION OF REACTIVE ZONE PROFILES 254
8.3 THEORETICAL DEVELOPMENT 255
8.4 INSTRUMENT DESIGN 256
8.5 EXPERIMENTAL DESIGN 257
8.6 REFERENCES 258

APPENDIX VALIDATION OF INSTRUMENT CONTROL
PROGRAMS AND CAPILLARY ELECTROPHORESIS .
POWER SUPPLY . , 260

xii

 

TABLE 1.1:

TABLE 5.1:

TABLE 5.2:

TABLE 5.3:

TABLE 6.1:

TABLE 6.2:

TABLE OF TABLES

Reproducibility of velocity, plate height, and skew.

" Average and standard deviation calculated from four
independent simulations where k. = k, = 1 s" (17 = 0.5 s)

with 1x10‘5 s time increment. Other simulation conditions

as given in Figure 2.3.

b Average and standard deviation calculated from four
independent simulations where kg = kr = 1 s’1 (1: = 0.5 s),

one each at 1x10‘2, 1x10'3, 1x104, and 1x10'5 5 time
increments. Other simulation conditions as given in

FIGURE 2.3. 57

Damkohler numbers for simulation profiles in FIGURE 2A. 165

Comparison of Equilibrium Constants and Rate Constants of
Isomerization. Forward and reverse rate constants for Ala-Pro
and Phe-Pro isomerization at 10 °C calculated by plate

height model in the long time regime (Da > 5), ChromWIn

in the short time‘regime (Da < 1), and reported values

from [11] and [16].

Buffer conditions in this work: 1.0 x 10'2 M NazB4O7 (pH 9.3)
Buffer conditions for [11]: 7.0 x 10‘2 M NazB4O7 (pH 9.5)

Buffer conditions for [16]: 1.0 x 10'1 M NazB407 (pH 8.4). 171

Thermodynamic and Kinetic Parameters of Isomerization.
Thermodynamic changes in molar enthalpy (AH) and molar
entropy (AS) calculated from Equation 5 as well as molar
activation energy for the forward and reverse reactions (AEF
and AE,*) for Ala-Pro and Phe-Pro. 182

Calculated Equilibrium Constants for BCD Inclusion of DnsF.
Equilibrium constants for BCD inclusion of DnsF calculated
from separations as a function of electric field strength and
temperature. 209

Calculated Diffusion Coefficients of Free and Bound DnsF.

Diffusion coefficients of free and bound DnsF with (DA and Dc)
and without (D'A and DC) corrections for temperature and

xiii

ionic strength at 283, 285, 288, and 293 K. 213

‘ TABLE 6.3: Calculated Rate Constants for bCD Inclusion of DnsF. Average
rate constants for association (kuvg) and dissociation (kd,a,,g) of
SOD inclusion of DnsF calculated from separations

at 334, 368, and 418 Vlcm at 283, 285, and 288 K. 219

xiv

FIGURE 1.1:

FIGURE 1.2:

FIGURE 1.3:

FIGURE 1.4:

FIGURE 1.5:

FIGURE 2.1A:

FIGURE 2.18:

TABLE OF FIGURES

Schematic diagram of the electrical double-layer
structure with hydrogen ions (H+), buffer ions (+,-),
and solvent molecules (open) oriented towards the
negatively charged silica surface. The curved, solid
line represents the decay of electrical potential from

‘ the silica surface into the bulk solution This is a

simplified representation of the surface potential as
it intimates a uniform distribution of charge. 5

Effect of limiting ionic conductivity on effective

diffusion coeff' cient for |z.|- - 1 (El), |z.|= 2 (O),

|zi|= (A), |z.|= -4 (X), and |z.|= 5 (O) Calculation
conditions: 0.}: 1 .00x105 cm2,/s e= 78.,5 T=
298.15K, l=0.,02M |z,-|= |zk|=1, ,7211 (Na+)= 510. 11
cm2 0'1 equiv, 1M°(Cl)= 76.340m Q1equiv'19

Effect of limiting ionic conductivity on effective

diffusion coefficient for l- - 0 002 M (O), I- - 0. 005 M

(A), |=0.,02M(l:l) l=0.005M(X), and5l=0.2M(O).
Calculaiton conditions. D.z° = 1. 00 x 10'5 cm 2Is, 8 = 78. 5,
T= 298.15 K, |z.|= |z)|= |zk|- - 1, 2A,” 1=(Na+) 50.11 cm2
0'1 equiv, 121° (CI- -=) 76. 34 cm201equiv111

Effect of ionic strength on electrophoretic mobility
forlzil1(D) lzil= 2(0) IZiI= (A) lzil= 400 and
|z.|= 5 (O). Calculation conditions: p.31“ - 1. 00 x 10"
cmNs a=1.0,0x101°m n= 0.997cP, e= 78.,5

T= 298.,15K l=0.02M, |z,-|=1, of (Na+)= 5.19x10‘1
cmst 17

Effect of pH on observed electrophoretic mobility

of phosphate. Caalculation conditions: pKa1- =2 23,
pKaz= 7.21 pig: =12.,32 po=-288x10 3cm2Ns

m- =1. 18 x 10 acmst, 112' — 3. 73 x 10“ cmst,

n3 - 0.00 cmst 28

Flow chart of the stochastic simulation of reactive
capillary electrophoresis adapted from [55]. 49

Schematic diagram of reactive capillary electrophoresis
simulation process. 51

FIGURE 2.2:

FIGURE 2.3

FIGURE 2.4A:

FIGURE 2.48:

FIGURE 2.4C:

FIGURE 2.5:

FIGURE 2.6A:

Illustration of the relationship between the distribution
of molecules within the capillary and the resulting
zone profile. 54

Simulated CE profiles at multiple detector locations of

2, 4, 6, 8, and 10 cm. Simulation conditions: L = 100
cm; R... = 50 um; N = 2000; t = 1x105 s; om = 1x10‘5
cmzls; v0 = 0.2 cm/s; V = 25 kV; 0A = —1 x 10‘1 cmst;
p3=—2x10"1cm;2Ns K- 1 (A) k, k,= 1051,t=
0.058;=(B)kf kr=k151, 1:: 0.5S;=(C)kf 0=.66s'1,
t=0.176$;=(D)kf=.033S'1,1:=1.5ZS;(E)kf=

0.1s1, 1:= 5s; (F)k;= k.= 0.06651, 1: 7.;58s (G)k;
=k,= 0.033s ,‘t= 15.155 60

Effect of characteristic lifetime 1: on apparent zone
velocity (A) at detector locations of 2 cm (0), 4 cm

(A), 6 cm (El), 8 cm (0), and 10 cm (X). Simulation
conditions as given in Figure 2.3. 62

Effect of characteristic lifetime 1: on plate height (B) and
skew (C) at detector locations of 2 cm (0), 4 cm (A), 6
cm (CI), 8 cm (0), and 10 cm (X). Simulation conditions
as given in Figure 2.3. 66

Effect of characteristic lifetime t on plate height (B) and
skew (C) at detector locations of 2 cm (0), 4 cm (A), 6
cm (III), 8 cm (0), and 10 cm (X). Simulation conditions
as given in Figure 2.3. 67

Simulated CE profiles at multiple detector locations

of 2, 4, 6, 8. and 10 cm. Simulation conditions: L =

100 cm; Rm = 50 pm; N = 2000; t = 1x10‘5 s; 0,.1 =

1x10‘5 cmzls; v0 = 0.2 cm/s; v = 25 kV; in = —1 x 10"1
cn12/\I;s [13"- —2x10" cmst;1:= 0. 5 s; (A) K= 1000

k;= 1.998s ,=k, 0.00251;(B)K=1100, kf= 1.98 1s"

k.= 0.102s (,C)K= 10, in: 1818s k.= 0.";182s

(D)K=1k;1s’1,k,=1s'1,(E)K=0.1kf=0.182s'1,

k,= 1.818s'1' ,(F)K= 0.,01 k}: 002s" ,k.=1.98s1;

(G)K= 0.001, in: 0."002s ,k,=1.998s". 59

Effect of equilibrium constant K on apparent zone
velocity (A) at detector locations of 2 cm (0), 4 cm (A), 6
cm (El), 8 cm (0), and 10 cm (X). Simulation conditions
as given in Figure 2.5. 72

FIGUREZSB:

FIGURE 2.6C:

FIGURE 2.7:

FIGURE 2.8A:

FIGURE 2.88:

FIGURE 2.8C:

FIGURE 3.1:

Effect of equilibrium constant K on plate height (B) and
skew (C) at'detector locations of 2 cm (<>), 4 cm (A), 6
cm (III), 8 cm (0), and 10 cm (X). Simulation conditions
as given in Figure 2.5. 75

Effect of equilibrium constant K on plate height (B) and
skew (C) at detector locations of 2 cm (0), 4 cm (A), 6
cm (El), 8 cm (0), and 10 cm (X). Simulation conditions
as given in Figure 2.5. 76

Simulated CE profiles at multiple detector locations
of 2, 4, 6, 8, and 10 cm. Simulation conditions: L =
100 cm; R... = 50 pm; N = 2000; t = 1 x 10'5 s; D... =
1 x 10'5 cmzls; v0 = 0.2 cmls; v = 25 W; K = 1; k; =
k. = 1 s"; (A) 0A =1.5 x 10‘1 cmst, (is = —4.5 x 10‘1

-cm2Ns; (B) m = 1.0 x 10‘1 cmst, (is = —4.0 x 10*1

cm’Ns; (C) in = 0.5 x 10‘5 cmst, [La = —3.5 x 10*1

‘ cmst; (D) m = 0.0 x 10*1 cmst, (is = —3.0 x 10‘1

cmst; (E) m = —0.5 x 10‘1 cmst, (is = —2.5 x 10"
cmst; (F) m = -1.0 x 10‘1 cmst, (is = —2.0 x 10"
cmst. 79

Effect of electrophoretic mobility difference Au. on
apparent zone velocity (A) at detector locations of 2 cm
(0), 4 cm (A), 6 cm (Cl), 8 cm (0), and 10 cm (X).
Simulation conditions as given in Figure 2.7. 81

Effect of electrophoretic mobility difference Art on plate
height (B) and skew (C) at detector locations of 2 cm
(0), 4 cm (A), 6 cm (El), 8 cm (O), and 10 cm (X).
Simulation conditions as given in Figure 2.7. 84

Effect of electrophoretic mobility difference Au on plate
height (8) and skew (C) at detector locations of 2 cm
(0), 4 cm (A), 6 cm (El), 8 cm (0), and 10 cm (X).
Simulation conditions as

given in Figure 2.7. 85

Effect of electroosmotic velocity and electric field
strength on plate height contributions from diffusion
(----) and mass transfer (- -), as well as on total plate
height (—). Calculation conditions: (1.0 = 8.00x10“1
cmst, ILA = —1.00 x 10*1 cmst, [.13 = -2.00 x 10‘1
cmst, om = 1.00 x 10'5 cm2/s, K = 1, ice = k8). = 1 s".

1 00

FIGURE 3.2A:

FIGURE 3.28:

FIGURE 3.2C:

FIGURE 3.3:

FIGURE 3.4A:

Effect of electroosmotic velocity on plate height for RAB
=l<BA=0.01 8'1,t=5OS(--);kAe=kaA=0.184,17:

5 s (----); kAe = kBA =1s'1,1: = 0.5 s (—); RAB = k3,, =10 s",
c: 0.05 s (---); kAB = km: 100 s", 1:= 0.005 s (—--—).
Calculation conditions: it... = 8.00 x 10"1 cmst, (ix =
-1.00 x 10‘1 cmst, us = —2.00 x 10‘1 cmst, 0... =

1.00 x 10'5cm2/s, K= 1. 103

Effect of electroosmotic velocity on plate height for Au

= 5.00 x 10‘1 cmZNs (— —); An = 2.00 x 10“1 cmst (--~);
Ap. = 1.00 x 10*1 cmst (—); An = 5.00 x 10'5 cmZNs

(— - —); An = 2.00 x 10'5 cmst (— - - —). Calculation
conditions: it... = 8.00 x 10‘1 cmst, (in + inn/2 =

-1.50 x 10" cmZNs, o... = 1.00x10’5 cmzls, K = 1, king =
k3,. = 1 s". 106

Effect of electroosmotic velocity on plate height for K

= 100 (-—); K = 10 (----); K =1 (—); K = 0.1 (— - -); K =
0.01 (— - ; —). Calculation conditions: p.90 = 8.00 x 10‘1
cmZNs, ILA = —1.00 x 10"1 cmst, [1.3 = —2.00 x 10‘1
cmZNs, o... = 1.00 x 10‘5 cmzls, RAB = k... = 1 s". 108

Effect of equilibrium constant on velocity for Au =

5.00 x 10'5 cmZNs, (ix/us = 2.00 H; Ap. = 2.00 x 10'5
cmst, (ix/us = 1.25 (— —); Au = 0.00 x 10*1 cmst,

(ix/us = 1.00 (----); An = 2.00 x 10" cmst, (ix/us = 0.333
(- - -); Au = 5.00 x 10" cmst, (ix/us = 0.167 (— . - —).
Calculation conditions: it... = 8.00 x 10*1 cmZNs, P-A =
—1.00 x 10‘1 cmst, E = 250 Vlcm, om = 1.00 x 10'5
cmzls,K=1,kA3=k3A=1s'1. 110

Effect of characteristic reaction lifetime on plate height

from theory without diffusion (----), with diffusion (—),

and from simulation (El). Theoretical conditions: vo =

0.2 cm/s; in = —1.00 x 10‘1 cmst; (.9 = —2.00 x 10‘1

cmst; E = 250 Vlcm; Dm = 1.00x10'5 cmzls; K = 1;

0.033 s'1 (1: = 15.15 s) 5 RAB = km 510 s'1 (1: = 0.05 s).

Simulation conditions: N = 2000; t = 1.00x10'5 s; z =

10 cm; all other conditions same as theoretical conditions.
1 15

xviii

FIGURE 3.48:

FIGURE 3.4C:

FIGURE 3.5A:

FIGURE 3.58:

FIGURE 3.6A:

Effect of mobility difference on plate height from theory
without diffusion (----), with diffusion (—), and from
simulation ([3). Theoretical conditions: vo = 0.2 cm/s;
—1.00 x 10‘1 cmst 5 Au 5 -6.00 x 10‘1 cmst; E = 250
Vlcm; 0... =1.00 x 10'5 cm2/s; K = 1; RAB = k3,. =1 s"

(17 = 0.5 s). Simulation conditions: N = 2000; t =

1.00 x 10'5 s; z = 10 cm; all other conditions same as
theoretical conditions. 1 18

Effect of equilibrium constant on plate height from theory
without diffusion (----), with diffusion (—), and from
simulation (III). Theoretical conditions: vo = 0.2 cm/s;

PA = —1.00 x 10‘1 cmZNs; us = —2.00 x 10*1 cmst; E =
250 Vlcm; D... = 1.00 x 10'5 cmzls; 0.001 5 K 5 1000;

k)“; = kaA = 1 s'1 (I = 0.5 s). Simulation conditions: N

= 2000; t = 1.00 x 10'5 s; z = 10 cm; all other conditions
same as theoretical conditions. 120

Comparison of theoretical velocity with simulation

velocity for simulations with varying equilibrium constant
(El) and varying mobility difference (A) on the primary ‘
y-axis. The diagonal solid line represents a linear
relationship between theoretical and simulation velocities.
Percent error‘(X) is shown on the secondary y—axis. The
horizontal solid line represents zero percent error.
Simulation and theoretical conditions as described in
Figures 3.4A and 3.4B. 122

Comparison of theoretical plate height with simulation
plate height for simulations with varying equilibrium
constant (III), varying rate constants (I), and varying
mobility difference (A) on the primary y-axis. The
diagonal solid line represents a linear relationship
between theoretical and simulation plate heights.

Percent error (X) is shown on the secondary y-axis.

The horizontal solid line represents zero percent error.
Simulation and theoretical conditions as described in
Figures 3.4A - 3.4C. 125

Individual and cumulative contributions to propagated
percent error in equilibrium constant for 0.01 5 K 5 100:
0K2 (—); (aK/avA)2 on? (— -); (aK/av3)2 ova? (-~-); (aK/av)2
of (— - —). Calculation conditions: v0 = 0.2 cmls; E = 250
Vlcm; up, = -1.00 x 10"1 cmst; 0.3 = —2.00 x 10“1 cmst;
ch, ova, and (IV = 1.00 %. 128

xix

FIGURE 3.68:

FIGURE 3.7A:

FIGURE 3.78:

FIGURE 4.1A:

FIGURE 4.18:

FIGURE 5.1:

FIGURE 5.2A:

Effects of mobility difference on propagated percent error
in equilibrium constant for Au = 1.00x10"1 cmst (—);

Ap. = 5.00 x 10'5 cmst (— —); An = 2.00 x10'5 cmst (W);
and A0 = 1.00x10'5 cmst (- - -). Calculation conditions:
v0 = 0.2 cm/s, E = 250 Vlcm, (in + uB)/2 = -1.50 x 10*1
cmst; ovA, ova, and 0,, = 1.00 %. 130

Individual and cumulative contributions to propagated
percent error in reaction lifetime for 0.05 5 t 5 500 s:

of (—); (at/6H)2 0H2 (- -); (at/dAV)2 om? (~--); (at/6V)2

0,,2 (— - —); (Gt/60m)2 on“? (— - - —); 2(01/6H)(dT/6V) oHov
(—). Calculation conditions: vo = 0.2 cmls; E = 250 Vlcm;
HA = -1.00x10" cmst; (is = —2.00x10‘1 cmst; K = 1;
0;), 00m, UV, and 0m = 1.00 %. , 134

Effects of equilibrium constant on propagated percent
error in reaction lifetime for 0.05 5 ‘t 5 500 s: K = 10 (— —);
K=2(--~-);K=1(—);K=0.5(—-—);K=0.1(---—).
Calculation conditions: vn = 0.2 cmls; E = 250 Vlcm; “A =
-1.00x10‘1 cmst; (1.3 = -2.00x10“ cm2Ns; 0H, 00..., 0v.
and 0H,, = 1.00 %. 136

Schematic Diagram of the Agilent G1600A capillary
electrophoresis system. Separation system components:
Pt electrodes (E), :l: 0 - 30 kV high voltage power supply (V),
inlet vial (l), fused-silica capillary (C), outlet vial (O), and
ground (G). Detection system components: deuteriUm

light source (S), focusing lens (L), and UV-vis diode

array detector (D). 143

Schematic diagram of the capillary electrophoresis system
built in-house. Separation system components: Pt
electrodes (E), d: 0 - 30 kV high voltage power supply (V),
inlet vial (I), fused-silica capillary (C), outlet vial (O), and
ground (G). Detection system components: HeCd laser (S),
focusing lens (L), excitation optical fibers (EX), emission
optical fibers (EM), optical filters (F), monochromators (M),
and photomultiplier tubes (P). 147

Illustration of the cis-trans isomerization for Alanine-Proine
(top) and. Phenylalanine-proline (bottom). 1 55

Simulated CE profiles. Simulation conditions: N = 2000
molecules; t = 1 x 10'5 s; Lid = 100 cm; Last = 2, 4, 6, 8,
10 cm, id = 100 pm; V = 25 kV; veo = 0.20 cmls; D =

FIGURE 5.28:

FIGURE 5.3A:

' FIGURE 5.38:

FIGURE 5.4:

FIGURE 5.5A:

FIGURE 5.58:

1 x 10'5 cmzls; 0A = —1.00 x 10‘1 cmst; us = —2.00 x 10‘1
cmst; K = 1. 161

Effect of Damkohler number on the accuracy of rate
constants calculated by Chrolen (X) and by plate
height model (0) for profiles in Figure 2A. 164

Effect of separation time on electropherogramzs of MO
and Ala- Pro. Separation conditions: 1 0 x 10'2 M
Na2B407 buffer (9. 3),10 °C, L(o(= 34 cm, Lda= 26 cm,
i.d.= 75 pm, (A) 7500 V, (B) 500 V. UV absorbance
detected at 210 nm. 168

Effect of separation time on electropherogramzs of MO
and Ala-Pro. Separation conditions: 1. 0 x 10’2 M

Na28407 buffer (9. 3), 10 C, Lim= 34 cm, Lda= 26 cm,
i..=d 75 pm, (A) 7500 V, (B) 500 V. UV absorbance
detected at 210 nm. 169

Effect of temperature on electropherograzms of MO and
Ala-Pro. Separation conditions: 1 .0 x 10'2 M NazB407
buffer (9 3), 10 °C, Ltd: 34 cm, Ldei= 26 cm, I d 75 pm,
7500 V. UV absorbance detected at 210 nm. 174

Effect of temperature on (A) pm.» (0), pas.» (X), (up

(A). 30d (3) Poms-FF (O) Hal's-FF (X) We (A) separation

conditions same as for FIGURE 5. 4. Regression

parameters (slope, intercept, R2 ): pm.» (- 3. 54 x 10,

8. 82 x1010. 9995); UasAp (3. 53 x 1011, 8. 63 x104,

0.9998); (nip-(-344 x 10‘, 8.45 x 10", 0.9999); um.»

(-3.20 x 1011, 7.77 x 104, 0.9998); (ta-s.» (-3.07 x 10‘6,

7.29 x 10", 0.9994); up): (-3.03 x 10", 7.18 x 10‘, 0.9996).
176

Effect of temperature on (A) [1W (0), ”as” (x ), “AP
(A) and (B) ”trans-PP (0) use ..p (X) |le (A) Separation
conditions same as for FIGURE 5. 4. Regression
parameters (slope, intercept, R2 ): um...» (- 3. 54 x 1011,
8.82x10‘1,0.;9995)p.c,s.)i(i:i(3.53x1011,8.63x10‘1
0.9998); uxp (-344 x 101‘, 8.45 x 10", 0.9999); um.»
(-3.20 x 1011, 7.77 x 10", 0.9998); (imp (-3.07 x 1011,
7.29 x 104, 0.9994); ups (-3.03 x 10‘, 7.18 x 10", 0.9996).
177

xxi

FIGURE 5.6A:

FIGURE 5.68:

FIGURE 5.6C:

FIGURE 6.1:

FIGURE 6.2:

FIGURE 6.3:

_ FIGURE 6.4:

FIGURE 6.5A:

FIGURE 6.58:

Effect of temperature on calculated equilibrium constants
for (A) Ala-Pro and (O) Phe-Pro. Separation conditions
as described in text. 180

Effect of temperature on calculated forward rate constants,
for (A) Ala-Pro and (O) Phe-Pro. Separation conditions as
described in text. 184

Effect of temperature on calculated reverse rate constants
for (A) Ala-Pro and (O) Phe—Pro. Separation conditions
as described in text. 187

Molecular structures of dansyl amide, dansyl phenylalanine,
and B-cyclodextrin. 200

Effects of electric field strength and BCD concentration on
electropherograms of DnsA and DnsF. Separation
conditions: Ln = 59.85 cm, L6,. = 38.05 cm, 285 K, 1.0 x
10“1 M Na28407-10H20 buffer (pH 9.3), V = 25 W, [800]
= 0 - 10 x 10'2 M. Fluorescence detection conditions:
Ami. = 325 nm, Item). = 570 nm. 204

Effects of electric field strength and BCD concentration on
electrophoretic velocity at 334 (O), 368 (A), and 416 (El)
Vlcm. Separation coanditions: L(o.= 59. 85 em, de= 38. 05
cm, 285 K, 1 .0 x 10'3 M NazB407-10H20 buffer (pH 9 3),

\/= 25 kV, [BCD]= 0- 10 x 10'2 M. Fluorescence detection .
conditions: Ami. = 325 nm, lam). = 570 nm. 206

Effect of temperature on calculated equilibrium constants
for 800 inclusion of DnsF. Experimental conditions same

as given for FIGURE 6.2. 211

. Effects of electric field Strength and BCD concentration on

plate height at 334 (O), 368 (A), and 416 (III) Vlcm.
Separation conditions. L(o(= 59. 85 cm, Lam= 38. 05 cm,
285 K, 1.0 x 10'3 M NazB407-10H20 buffer (pH 9. 3), V- -
25 kV, [BCD]= 0- 10 x 102 M. Fluorescence detection
conditions: )3..ch 325 nm, Aemu- - 570 nm. 215

Effects of electric field strength and BCD concentration on

, plate height corrected for diffusion at 334 (O), 368 (A),

and 416 (El) Vlcm. Separation conditions: Lu - 59. 85 cm,
Ld.(= 38. 05 cm, 285 K, 1.0 x 10'3 M Na23407-10H20 buffer

xxii

FIGURE 7.1:

FIGURE 7.2:

FIGURE 7.3:

FIGURE 7.4:

FIGURE 7.5:

FIGURE 7.6:

(pH 9.3), V = 25 kV, [BCD] = 0 - 10 x 10'2 M. Fluorescence
detection conditions: Ami. = 325 nm, A6...“ = 570 nm. 218

Effect of electrophoretic mobility difference on maximum
calculable rate constants via capillary electrophoresis.
Calculation conditions: E = 500 Vlcm, u... = 6.0 x 104
cm2Ns, (IIA'I'IIBVZ = -1.5 x 10‘1 cm2Ns, o = 1.0 x 10'5 cm2/s,
K =1,10'2 < k <10'1 s", and Au =1.0 x 10'5 cm2Ns (El),
2.0 x 10'5 cm2Ns (A), 5.0 x 10'5 cm2Ns (0), 1.0 x 10"
cm2Ns (I), 2.0 x 10‘1 cm2Ns (A), 5.0 x 10*1 cm2Ns (e).

n = 5 (- -). ‘ 231

Effect of electric field strength on maximum calculable rate
constants via capillary electrophoresis. Calculation
conditions: it... = 8.0 x 10‘1 cm2Ns, (u = -1.0 x 10'5 cm2Ns,
pa = -2.0 x 101 cm2Ns, 0 =1.0 x 10'5 cm2/s, K =1,10'2 <

k < 104 s", and E = 100 Vlcm (O), 200 Vlcm (A), 500 Vlcm
(I), 1000 Vlcm (O), 1200 Vlcm (A), 1500 Vlcm (El).

h = 5 (- -). 234

Effect of equilibrium constant on maximum calculable rate
constants via capillary electrophoresis. Calculation
conditions: E = 500 Vlcm, u... = 8.0 x 10'4 cm2Ns, u). =
-1.0 x 10'5 cm2Ns, (is = -20 x 10'5 cm2Ns, o = 1.0 x 10'5
cm2/s, 10'2 < k < 10‘1 s", and K = 0.01 (0), 0.1 (A), 1 (I),
10 (O), 100 (A). h = 5 (- -). 236

Effect of diffusion coefficient on maximum calculable rate
constants via capillary electrophoresis. Calculation
conditions: E = 500/V/cm u... = 8.0 x 10‘1 cm2Ns, 0A =
-1.0 x 10°5 cm2Ns, (is = -20 x 10'5 cm2Ns, K = 1, 10'2 <

k < 101 s", and o = 1.0 x 10*3 cm2ls (a), 2.0 x 10"5 cm2/s
(A), 4.0 x 10‘6 cm2ls (I), 6.0 x 10'6 cm Is (0), 8.0 x 10"
cm Is (A), 1.0 x 10'5 cm2ls (El). n = 5 (- -). 238

Effect of electrophoretic mobility difference on maximum
calculable rate constants via capillary electrochroma-
tography. Calculation conditions: E = 500 Vlcm, (too =

8.0 x 10*1 cm2Ns, 0 :10 x 10'5 cm2/s, K =1,10'2 < k
<101 s", and u), :10 x 10‘1 cm2Ns (El), 5.0 x 10'5
cm2Ns (A), 0.0 x 10'5 cm2Ns (O), -5.0 x 10'5 cm2Ns
(A), -10 x 10‘1 cm2Ns (I). n = 5 (- -). 242

Effect of electric field strength on: maximum calculable
rate constants via capillary electrochromatography.

xxiii

FIGURE 7.7:

FIGURE 7.8:

FIGURE A.1A:

FIGURE A.18:

FIGURE A.2A:

Calculation conditions: pa, = 8.0 x 10“ cm2Ns, in =

-'1.0 x 10'5 cm2Ns, 0 =1.0 x 10‘5 cm2ls, K =1,10'2 < k

< 101 s", and E = 100 Vlcm (0), 200 Vlcm (A), 500

V/cm (I), 1000 Vlcm (O), 1200 Vlcm (A), 1500 Vlcm
(El). n = 5 (- -). - 244

Effect of equilibrium constant on maximum calculable

rate constants via capillary electrochromatography.
Calculation conditions: E = 500 Vlcm, u... = 8.0 x 10‘1
cm2Ns, m = -1.0 x 10'5 cm2Ns, D = 1.0 x 10'5 cm2ls,

10'2 < k < 10‘1 s", and K = 0.01 (e), 0.1 (A), 1 (I), 10
(O), 100 (A). h = 5 (- -). 247

Effect of diffusion coefficient on maximum calculable

rate constants via capillary electrochromatography.
Calculation conditions: E = 500/ Vlcm 0.90 = 8.0 x10“11
cm2Ns, ox = -1.0 x 10'5 cm2Ns, K =1,10‘2 < k <10:1 s",
and o = 1.0 x 10‘11 cm2/s (O), 2.0 x 1011 cm2/s (A), 4.0

x 10'6 cm2/s (El), 6.0 x 1045 cm2/s (0), 8.0 x 1015 cm2/s

(A), 1.0 x 10'5 cm2/s (I). n = 5 (- -). 250 ,

Typical screen view of the step-IogicZ program. The
toggle switch allows the user to choose either voltage

or current control of the HVPS. The digital controls ,
labeled “T1” - “T6” allow the user to specify the elapsed
time at which voltage or current changes occur. The
buttons labeled “set T4” - “set T6” allow the user to
choose additional (> 3) voltage steps. The digital controls
labeled “kV1' - “kV6” and “uA1” - “uA6” allow the user to
specify the magnitude of the applied voltage or current.
The digital control labeled “T max” allows the user to

. specify the maximum time for voltage or current application.

The digital indicator labeled “Time” shows the elapsed
experiment time. The digital indicator labeled “Output”
shows the voltage output to the HVPS. 262

Schematic of the step-logi62 program. The schematic

shows the data flow as logical cascade from T1 through

T6 to the output channels that interface with the HVPS.
264

Typical screen view of the PUnCC program. The dialog
box labeled “Input Comments” allows the user to add
comments regarding each experiment to the data file.
The toggle switch allows the user to monitor either the
current or voltage from the HVPS. The digital controls

xxiv

FIGURE A.2B:

FIGURE A.3:

FIGURE A.4:

FIGURE A.5:

FIGURE A.6:

labeled “Buffer size”, “Scan rate", and “Scans to read at
time” allow the user to specify the number of scans stored
in the data buffer, the number of scans recorded each
second, and the number of scans written to file, respectively.
The button labeled “New file” allows the user to choose to
store the data as a new file or to append the data to an
existing file. The digital controls labeled “Input limits” allows
the user to specify the board gain. The digital indicators
labeled “Scan backlog” and “Time Elapsed” show the
number of scans in the buffer and the elapsed time,
respectively. The button labeled “STOP” allows the user

to stop the experiment. The waveform charts and digital
indicators show the real-time response from the detectors
and the monitored current or voltage from the HVPS. 266

Schematic of the PUnCC program. The schematic shows
the data flow of specified inputs from the screen view to the
data acquisition loop and the data file. The schematic also
shows the interface to the step-logic2 program as a sub-
routine (labeled “V or I Step”) within the data acquisition
loop. 268

Programmed voltage versus the output voltage (El) and
percent error of the output voltage (X). 271

Programmed voltage versus the measured current (El)
and percent error of the measured current (X). 273

Programmed current versus the measured voltage ([3)
and percent error of the measured voltage (X). 275

Input voltage versus the voltage measured by PUnCC (III),

DMM (A), and percent error of the voltage measured by
PUnCC (x ). 278

XXV

CHAPTER 1: INTRODUCTION AND BACKGROUND

1.1 CAPILLARY ELECTROPHORESIS
1.1.1 History

Electrophoresis is the separation of charged molecules or particles by
differential migration in an electric field. The first reported use of electrophoresis
as a separation technique was in the 1930s by Tiselius for the separation of
constituent proteins from human serum [1,2]. This pioneering work eventually
earned Tiselius the NObel Prize in 1948.

Electrophoresis in a glass capillary was first reported by Hjerten in 1967
[3], but did not gain popularity until some 14 years later when Jorgenson and
Lukacs [4] demonstrated the efficiency of the technique for separating small
biological molecules. In the intervening years, capillary electrophoresis (CE) and
related techniques have become a staple for fast and efficient separations of
charged and neutral solutes. Applications of CE include trace analysis of metals
from environmental samples and protein separations for the Human Genome
Project [5]. A recent review by lssaq [6] discusses a broad range of solutes for
which separation by CE has become standard operating procedure.

The separation efficiency of CE is a result of differential and concurrent
solute migration within the bulk solution under the influence of an applied electric
field. Physical origins of this unique separation mechanism are derived from the
formation of an electrical double-layer at the interface of the bulk solution and the

capillary walls, combined with the mass transfer response of solute and bulk

components in an electric field. Chemical origins of the separation originate from
acid-base equilibria, complexation equilibria, and wall adsorption. This
introductory chapter discusses some of the origins of the physical and chemical
phenomena that contribute to separation in CE.

1.1.2 Physical Contributions to Separation and Broadening

- 1 .1 .2.1 Electrical Double-Layer

The interaction of a charged solid with an electrolyte solution results inthe
formation of an electrical double-layer at the interface [7]. For a fused silica
capillary in aqueous solution, the walls carry a negative charge from the
deprotonation of surface silanol‘groups. Water molecules orient their dipoles to
the charged surface while some cations adsorb to negatively charged sites on
the capillary wall [8,9]. Hydrogen ions are bound covalently to the surface but
other cations are bound ionically. This organized, initial layer of water molecules
and hydrogen ions and other specifically adsorbed ions constitutes the inner
Helmholtz plane (IHP). A second, less organized layer of water molecules and
electrolyte ions is oriented adjacent to the IHP and is known as the outer
Helmholtz plane (OHP).

The region bounded by the IHP and the OHP is referred to as the compact
portion of the double-layer. Because all of the negative surface charge is not
compensated by specific adsorption, some cations are electrostatically attracted
towards the OHP. The resulting concentration of cations is thus large close to
the OHP but approaches the concentration of the bulk with increasing distance

from the capillary wall. This region of decreasing concentration from the OHP

and extending into the bulk is referred to as the diffuse portion of the double-
Iayer. A schematic diagram of the structure of the electrical double-layer is
shown in FIGURE 1.1.

The compact and diffuse portions of the double-layer are characterized by

an effective thickness known as the Debye length [10,11]

5= 28KBT (1)
1000NA e2l

where c is the dielectric constant of the buffer, k3 is Boltzmann’s constant, T is
the absolute temperature, NA is Avogadro’s number, e is the charge of an
electron, and l is the ionic strength. For small surface potentials, 6 describes the
distance (y) into solution at which the surface potential (We) has decayed by ~

37% [7,11]

‘I’(y)-‘PceXP(-%) (2)

' 1.1.2.2 Mass Transport

Transport phenomena may be generally considered as the response of a
molecule or ion to an applied gradient. For an electrolyte solution, the response
of the concentration of the i111 solute with an electrical charge 2 (Cu) to a gradient

may be described mathematically by

a .
11?ch - —v in - (3)

where Jr; is defined as the flux of the i,z1h species,

4.40.2), <4)
X

FIGURE 1.1:Schematic diagram of the electrical double-layer structure with
hydrogen ions (H+), buffer ions (+,-), and solvent molecules (open)
oriented towards the negatively charged silica surface. The curved,
solid line represents the decay of electrical potential away from the
silica surface and into the bulk solution. This is a simplified
representation of the surface potential as it intimates a uniform

distribution of charge.

ompact

C

(9+ :1

$5.1“ 9

013,501:

0 .6
223° 38

o

O 0

FIGURE 1.1

 

G +'.'+'. 1'". 29751311
{ .11. + . C p] + '0.
+

9 +
/O H H1" H+ .

 

and the summation index X refers to the diffusion, migration, convection and
thermal components of flow. This expression describes the decrease of
concentration of i,z in a unit volume with time. Unwanted convective flow can
arise in CE from pressure differences between the ends of the column.
Unwanted thermal flow can arise from temperature gradients that develop as a
result of poor heat dissipation. The use of narrow diameter capillaries can
effectively mitigate both of these unwanted contributions to flow. This is because
pressure differences and axial temperature gradients decrease with the square of
the internal diameter of the capillary. Thus, for electroosmosis in narrow
diameter capillaries, thermal and convective contributions to flow are negligible
with respect to the contributions from diffusion and migration [8,12]. The effects
of temperature on transport are discussed further in a later section.
1.1.2.2.1 Diffusion

Diffusion is the mass transport response to a concentration gradient. Flow
from diffusion may be described by
(le)dif ' ”01.2 V CL2 (5)
where Du is the diffusion coefficient of the i,z‘11 species. Diffusion coefficients are
a measure of the ability of a molecule to drift randomly through a given medium.
Smaller molecules tend to have larger diffusion coefficients than larger molecules
because they experience less frictional resistance (viscous drag) to movement.
The Stokes-Einstein equation illustrates the dependence of diffusion coefficient

at infinite dilution (D550) on molecular size and viscosity [13]

 

D.0 _ kBT
1'2 61mmz

(6)

where n is the solution viscosity and rLz is the solvated radius of the i,zth species.
Diffusion coefficients of electrolytes also depend upon the electrical properties of
the medium. The diffusion of ions at near-infinite dilution in an otherwise uniform
electrolyte solution has been discussed by Onsager [14]. Onsager’s treatment of

the confluence of diffusion and electrical conduction shows that at low total ionic

strengths, the-effective diffusion coefficient of an ion in vanishly small amounts is

6 .
Di,z -D€z 1-2'—8°—1—’;1i(1-(/q(u;z))z,2~fi (7)
(ET-)4

where q(p;z) is a complex function of the mobilities and valencies of the ions

present in solution. The function q(pi,z) is simplified to

, Izil lzil’ti Izkllit (8)

11811434 rrlxi+lzl>i1lzl2~i+lzklxi

where j and k represent ions of the main electrolyte and lo the limiting ionic
conductivity. FIGURE 1.2 illustrates the effects of analyte valence (1 < |z;| .< 5)
and limiting ionic conductivity (50 < >411 < 150 cm2 0'1 equiv‘1) on Du. It is
apparent that Du decreases with increasing 14° and that the extent of deviation
from D; 1° is exacerbated as |z;| increases. FIGURE 1.3 illustrates the effects of
ionic strength (0.002 < I < 0.2 M) and limiting ionic conduCtivity (50 < 14° < 150
Cm2 0’1 equiv'1) on Du. Once again, it is apparent that Dr,z decreases with

increasing 24°. It is also apparent that the extent of deviation from D;,z° is

FIGURE 1.2: Effect of limiting ionic conductivity on effective diffusion coefficient
for |z;| = 1 (El), |z;| = 2 (O), lzrl = 3 (A), |z;| = 4 (X), and |z.| = 5 (O).
Calculation conditions: D;,z° = 1.00 x 10'5 cm2ls, e = 78.5, T =
298.15 K, I = 0.02 M, (2,) = (2..) = 1, 3,9 (Na+) = 50.11 cm2 0"

equiv", N? (CH = 76.34 cm2 0" equiv"

FIGURE 1.2

of.

A7235 76 «ES >._._>_._.ODDZOO 0.20. Oz_._.=>=r_
of. o: oo on

— — tr —

 

om
ooimoo

. comma

- cameo

- comma

r comes )
m

. comes a
St...

cosmos (

. comma

.. comes

 

 

 

.. 8-8.5

 

 

coefficient
I.) = 5(0).
8.5, T =
cm2 0'1

- moimo._.

iNEIIOlzlzlEIOO NOISszllO EIAILOEHdEI

FIGURE 1.3: Effect of limiting ionic conductivity on effective diffusion coefficient
for l = 0.002 M (O), l = 0.005 M (A), I = 0.02 M (El), l = 0.05 M (X),
and l = 0.2 M (O). Calculation conditions: Dr,z° = 1.00x10'5 cm2/s, 5
= 78.5, T = 298.15 K, |z,| = |sz = |zk| = 1, If (Na+) = 50.11 cm2 0"

equiv'1, 33° (OH = 76.34 cm2 0'1 equiv'1

10

FIGURE 1.3

p-238 cease E26328 020. 02.52:
02 02 o: 8 on 8

 

 

I
\
\

I
)

 

 

( < <
) > > ‘I x x < < ’\ ’\ <
‘I > ) ) ) x < ’\
J 1.. _I. J J J \l \I
.I. .J .I. ..J 1..
“H'H'N'M'I I I I I I I I I I I I I F“— E

U

'I'I'i'. .I. .'. .|. .' | l | I . l . l4 s
. . . . . . .l . l
‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ x a

11

 

J.N3IO|:I:IEIOO NOISn:I:I|CI EIAIiOElea

 

exacerbated as I increases; however, the effect of l on Dr,z is not as great as the
effect of |z;|. Thus, the effective diffusion coefficient is more strongly influenced
by electronic properties intrinsic to the solute than by those of the bulk solution.
1.1.2.2.2 Migration

1 .1 .2.2.2.1 Electrocsmotic Migration

Migration is the mass transport response to an applied external force. In
electrophoresis, migration is induced by an electric field. Application of an
electric field tangential to the capillary walls induces migration of cations in the
diffuse portion of the double layer towards the cathode. The flow of free solvent
molecules arises from the minimization of viscous forces within the bulk solution
responding to the solvated ion migration. This bulk flow originating from the
response to an electric field is referred to as electroosmotic flow (EOF).

Ions closest to the capillary wall do not migrate because the frictional force
resulting from attraction to the capillary wall is too great. With increasing
distance from the wall, the effect of this friction decreases until it is exactly equal
to the force of EOF. Beyond this distance, the EOF force becomes increasingly
. greater than the retarding friction, such that a plane of shear exists within the
diffuse double-layer between the stagnant and moving regimes. The potential
drop from the wall to the shear plane is called the zeta potential (2;) and is related
to both 0 and EOF velocity [8,15]. Dependence of C on the double-layer
thickness is thus a dependence on the ionic strength of the solution. As the ionic
strength decreases, fewer charged Species are available in the double-layer to

induce sufficient movement of the bulk to overcome. the retarding effects of

12

friction. Consequently, the plane of shear and 8 are extended further from the
wall, thereby decreasing the zeta potential and EOF [16].

1 The velocity of electroosmotic flow (vosm) is a function of the electric field
strength (E), the vacuum permittivity (so), the dielectric constant (8), and viscosity
(1)) of the double-layer as well as the zeta potential (2;) [15,17]

E880:
TI

 

(9)

VoSm ‘

It is important to note that although t;, e, and n are all dependent upon ionic
strength, the f; is the most sensitive to changes in | [11].

This expression for EOF Velocity describes the effects producing and
impeding bulk flow. The principal components creating flow are the coupled E, 7;,
and 8 terms, while Tl describes the friction inhibiting flow. Thus, the migration of
the bulk solution in response to the application of an electric field is determined
by the extent to which the ions within the double—layer are influenced by the
I applied field, the intermolecular friction within the double-layer, and the distance
of the plane of shear from the capillary wall. As the effective double-layer
thickness decreases and the potential at the plane of shear increases, the
greater will be the force of attraction experienced by solvated electrolytes to an
applied electric field, and the greater the EOF velocity. However, this bulk
migration velocity is impeded by intermolecular friction of solvent molecules in
the double-layer and by electrostatic attraction of ions to the capillary walls.

The EOF profile is different from the pressure-driven laminar flow profile
observed in liquid chromatography (LC) because it is generated near the

capillary walls. The velocity of EOF increases with decreasing electrolyte

13

concentration from the shear plane into the bulk, where it remains fairly uniform.
In contrast, laminar flow velocity increases with the square of the distance from
the wall to its maximum at the center of the capillary. Therefore, while the
laminar flow profile is parabolic, the EOF profile is plug-like. This means that
electrophoretic separations facilitate more narrow solute zones than
chromatographic separations.
1 .1 .2.2.2.2 Electrophoretic Migration

The velocity of the i,z1h species (viz) when acted upon by an electric field is
described by
viz = plz E (10)
where Mu is the electrophoretic mobility of the species [9]. Assuming the volume
occupied by the solvated species is spherical, themobility at infinite dilution ((5,211)
may be estimated by

O Zi e

(11)

From Equations 10 and 11 it is shown that the rate at which an ion travels
in response to an electric field is proportional to the coupling of the applied field
to the charge on the ion. However, this movement is inhibited by the drag of the
solvated ion and the intermolecular friction between solvent molecules. It is
worthy to note that substitution of Equation 11 into Equation 6 yields the well
known Einstein relation
11:12 R T

0
D12 - zF

 

(12)

14

‘.A
J.-

- Ad
‘5

‘-
Ir .H

Similar to diffusion coefficients, electrophoretic mobility is dependent upon
temperature and the solution electrical properties. Li et al. have adapted the
detailed conductivity theory of Pitts to describe this relationship by [18-20]

If 1

(fl

1+ 5.03x109aJ(—1—)
i ET I

where a is the sum of the solvated radii of the analyte and its counter-ion, and q

o 1.40x1062i

. - .o- .82_-5_i_ . 8
“12‘2"” int/E)21z (8.0% [1111/91]

 

(13)

 

 

is a function of the analyte and counter-ion mobilities and valencies

. lzil H.412.
IZ‘WII Ilili‘i’ulzillriz

FIGURE 1.4 illustrates the effects of analyte valence (-1 < z; < -5) and

 

 

q- : (14>

 

 

 

ionic strength (10'3 < l < 10" M) on the variation of In. with ionic strength.
Although there is a complicated dependence of [1.1.1 on z; and l (Equation 13), the
effect of 2; on the deviation of mi,z from the infinite dilution value is greater than
that of l. However, even at z; = -5 the deviation of pi; from (5,211 is still less than
0.01%.

Individual solutes within a mixture migrate according to their respective
mobilities-under the influence of an electric field, inevitably forming solute zones.
Each zone travels at a net velocity (vm) determined by the electroosmotic flow of
the buffer electrolyte (v03...) from Equation 9 and the electrophoretic flow of the i111
solute (vi) from Equation 10

15

FIGURE 1.4: Effect of ionic strength on electrophoretic mobility for |z;| = 1 (III), |z;|
= 2 (<>), [2;] = 3 (A), |z;| = 4 (X), and |z;| = 5 (0). Calculation
conditions: (55° = 1.00 x 10" cm2Ns, a = 1.00 x 10111 m, n = 0.997
cP, e = 78.5, T = 298.15 K, I = 0.02 M, |sz = 1, p,° (Na+) = 5.19x10‘1

cm2Ns

16

as Ieozmmew 0.29

_..o 3.0 50.0

 

 

 

FIGURE 1.4

 

 

 

 

 

 

 

.f. . . . . . _ . . . . . i . . oo+mood
. mormoo._.

u u u u u u art u Am mo-m_oo.~
6 IV 6 o o 6 o Aw. mo-m_oo.m

. mormooé

a. a c h a. 4.. a. momooo
mo..Moo.©

> ) > > > ) > r moimoofi

0 o o o O o o to moflooo
morwood

 

17

02111 / (02111 " 21111) X 00L

Finally, if it is assumed that solutes have negligible interactions with the
capillary walls, the mass transport of the i,zth species along the column (Equation
3) may be simplified to

a a a2
—C =v —C +D. —-C.
at tz net 3X tz tza 2 1,2

(16)
Equation 16 indicates that the physical contributions to solute transport
are dependent upon properties of the individual solute molecules as well as
properties of the bulk solution. Each solute will share contributions to transport
from the electric field and the viscosity of the buffer solution. However, the
diffusion coefficient, charge, and hydrated radius of a solute will contribute
uniquely to its movement within the system.
1.1.2.3 Influence of Electrical Conductivity
The electric field, and hence the consequential electroosmotic and
electrophoretic flow processes, is dependent upon the conductivity (1c) of the

medium it traverses. Further, the conductivity of the solution within the capillary

is a function of the charge and concentration of ions 0) in the solution [17]
(-1: Elam (17)
l

where the summation includes all buffer and sample ions. Therefore, zones will
have contributionsito conductivity from buffer as well as solute ions (K5) and may
be expressed as the sum of the buffer and sample conductivities.

For neutral solutes, Ki is zero and the electric field in the solute zone will
be the same as in the buffer. However, Ki is not zero for charged solutes and the

electric field in the zone will vary from the electric field in the buffer to

18

compensate for the variances in conductivity, such that a constant current is
maintained [8,21]. Zone conductivities greater than the buffer conductivities will
result in a diminution of the electric field in the zone. Conversely, zone
conductivities less than the buffer conductivity will result in an amplification of the
electric field in the zone. The electric field at any point along the column (x) may

therefore be expressed as [8,22].
E -—1-j+l=gog,ic, (18)
’1 xx ax

where j is the current density (assumed invariant with column position), xx is the
local conductivity, and the summation is the local diffusion of ions between
zones. This local electric field strength has direct influence on the electroosmotic
(vow), electrophoretic (vi), and net (v.5) velocities, as given in Equations 3, 10,
and 16, respectively.

The above discussion assumes that the contribution of the capillary
surface to the passage of current is negligible. At low pH values this assumption
is generally valid, because the charge of the capillary surface is minimized.
However, under alkaline conditions it has been suggested that the capillary
surface also contributes to conductivity [17]. This is because as the number of
ionized silanol groups increases, the ability of the capillary surface to conduct
current is also increased.

When the conductivity of the solution is uniform along the column, the
electric field strength may be assumed independent of column position. Also,
because the. electric field is applied along the column axis and the resultant

Migration of ions in response to that field is axial, the only sources of radial

19

transport are the result of thermal and concentration gradients as well as
migration in response to the charged silica surface. When the applied electric
field is sufficiently large (~30 W), the magnitude of these sources of radial
transport is negligible with respect to axial migration. Further, while the passage
of current through a conductor generates 'heat, the use of narrow diameter
capillaries in CE mitigates the generation of convective flow in response to
temperature variations in solution. The temperature gradient from the center of

the capillary to the walls is described by [23,24]

2
AT- 011° - (19)
16K

 

where clc is the column innerdiameter, K is the thermal conductivity.- of the
solution, and Q is the volumetric rate at which heat is generated as described for
eleCtrophoretic separations by [23,24]
0 - E2K¢ (20)
where K is the conductivity of the buffer electrolyte and 4) is the total column
porosity, which is equal to unity for CE.
1.1.2.4 Column Contributions to Broadening

The preceding sections discuss mass transfer processes occurring on the
column. Although the migration processes discussed in Section 1.1.2.2.2
contribute positively to separation and increase resolution, diffusion and thermal
gradients contribute negatively to separation and decrease resolution.
Ultimately, separation efficiency is limited by resolution and, hence, by all of the

sources that contribute to zone broadening. These sources of broadening can

20

arise either from column processes or from extra-column processes. Extra-
column contributions to broadening will be discussed later. Column contributions
to broadening of importance to CE include diffusion, thermal gradients, and
capillary geometry.
1.1.2.4.1 Diffusion

The contribution to broadening from axial diffusion (02.13) is described by
the Einstein equation [13]
0%” -20th (21)
where our; represents the average distance traveled by a randomly moving
molecule. For separation systems, 02.,“ contributes to symmetric broadening of
the zone.
1.1.2.4.2 Column Geometry

Broadening can also occur in CE as a result of the spatial geometry of the
capillary. Kasicka et al. have shown» that capillary coiling can lead to
inhomogeneity in the distribution of electrical current density within the capillary
[25]. The expression developed to describe the contribution of coiling to zone

broadening is

02 _, Lia d3 (22)

°°1 165,20,

 

where red is radius of the coil. Equation 22 shows that o2maris negligible when
ran >>- dc, as is the case for most conventional-scale CE separations. However,

for microchip CE systems, dc can approach the magnitude of r00... thus resulting

21

 

in significant additional broadening. The critical value of rear. (rem) below which

capillary coiling has a greater contribution to zone broadening than diffusion is

-.11°t_dc_ (23)
4 D t

1cat
12

Equation 23 shows that for fixed capillary dimensions and migration times,
molecules with larger diffusion coefficients are influenced less by coiling than
those with smaller diffusion coefficients.
1.1.2.4.3 Thermal Gradients

Broadening can also arise in CE from Joule heating. Although a well-
known concept, the effects of Joule heating on mass transfer are complicated. A
simplified approach is to first consider the contribution of thermal flow to mass

transport
(le)therm - —C(z Du VT . (24)

where D*i,z is the thermal diffusion coefficient of the i,zth species. This means
that changes in velocity can arise as a result of thermally induced convective flow
from regions of higher temperatures to regions of lower temperatures. Thus,
additional zone broadening can occur from convective perturbations caused by
temperature variations within the column. Moreover, thermal gradients can lead
to additional broadening as a result of thermally induced variations in molecular
properties such as mobile-phase diffusion coefficient (Equations 6 and 7) and
electrophoretic mobility (Equation 13) as well as bulk properties such as dielectric
constant, viscosity, and ionic strength. These potential complications

notwithstanding, it has been shown that the effects of Joule heating are not

22

 

significant contributions to zone broadening (< 5%) when separations are
performed in capillaries with internal diameters s 75 pm with electric field

strengths 5 2000 Vlcm [26-28].

1.1.2.5 Extra-Column Contributions to Broadening

In addition to mass transfer processes, zone broadening has contributions
from extra-column sources. Although these extra-column sources contribute
(sometimes significantly) to the measured width of individual zones, they do not
contribute positively to the separation. Extra-column sources of zone broadening
of importance to CE include broadening from injection and detection.
1.1.2.5.1 Injection

The contribution to broadening from a finite injection volume (02“) is
approximated by the equation developed by Sternberg for a rectangular injection

profile [26,29]

'2 2
0.2. -fitJfli‘i (25)
'"J 12 12

where Vin; is the volume of the injection zone,1lrn) the injection length and, A the
cross-sectional area of the capillary.

For hydrodynamic injections, laminar flow is used to introduce the sample
via application of a temporary pressure gradient along the length of the capillary.
Thus, the length of the hydrodynamic injection zone is described by the Hagen-
Poiseuille equation [30]

.. (26)
' n] 16 '0 Ltot

23

where AP is the induced pressure gradient, r is the capillary radius, tan; is the
injection time, n is the solution viscosity, and LM is the total capillary length.
Hydrodynamic injections may be performed by directly applying pressure at the
capillary inlet or a vacuum at the capillary outlet. However, these types of
injections are usually performed by siphoning action, from which AP can be
calculated by

AP-pgAh (27)
where p is the solution density, 9 is the gravitational acceleration constant, and
Ah is the height difference between the inlet and outlet solution levels.

For electrokinetic injection, electroomotic and electrophoretic flow are
used to introduce the sample via application of a temporary potential gradient
along the length of the capillary. Thus, the length of the injection zone is
described by the electroosmotic and electrophoretic mobilities and the injection
time

(“so + Hobs)Vtinj
Ltot

 

(28)

.linj "
1.1.2.5.2 Detection

For any detector, a finite volume is in contact with the transducer for a
finite amount of time and the resulting output signal represents the average
response. Detectors that monitor flowing systems have unique characteristics ‘

that contribute to zone broadening. These types of detectors (e.g. absorbance

and fluorescence) have finite spatial boundaries that define a rectangular window

24

and the response of the detector is uniform along the length of the window (Ider).

The broadening from such detectors for a capillary flowcell is described by [29]

(29)

where 1V6... is the detection volume and dd... is the inner diameter of the flowcell.
For most CE applications, detection occurs along the length of the separation
capillary, such that, diet is the inner diameter of the column.
1.1.3 Chemical Contributions to Separation
1.1.3.1 Acid-Base Equilibria

Many solutes separated in CE are subject to acid-base equilibria.
Consequently, the observed electrophoretic mobility (pm) of these solutes is pH
dependent and may be described as a function of the solute acid dissociation
constant (K,) values and the pH of the buffer solution
Ilobs ' Zi‘iz Cli,z (30)

where a is the average fraction of solute i molecules existing with charge state z
n
B" [H*|
“i,z - N n (31)
1+ 2 811 [H]
n
where n is the summation index, N is the total number of exchangeable protons,

and 8 is the cumulative acid formation constant

 

N
B" - r1 1 (32)
n

25

8’1.

FIGURE 1.5 illustrates the effect of pH on the observed electrophoretic
mobility of phosphate. It is apparent that as pH increases [robs exhibits four
plateau-like regions of increasing magnitude connected by three inflection
regions.

The corresponding plateau regions occur when (rob. is representative of a single,
dominant phosphate species. The plateaus at pH < 1, pH ~ 5, pH ~ 10, and pH >
14 represent the mobilities of the triprotic, diprotic, monoprotic, and fully
dissociated phosphate species, respectively. The inflections occur when pH is
equal to pK.1, pKa2, and pK.‘1 for phosphate.

1.1.3.2 Other Equilibria

A logical extension of the effect of acid-base equilibria on (tabs in CE is the
effect of other types of chemical interactions solute molecules may undergo
during separation. These generally include interactions with the capillary wall or
other stationary phase [31-40] and buffer constituents [39-41]. In particular, CE
has been used extensively to evaluate second and higher order reactions such
as complex formation of proteins with other proteins as well as with DNA and

enzymes [42-50].
1 “The interactions of solute molecules with other mobile-phase molecules
are routinely exploited in CE to achieve increased resolution as well as to study
the selectivity of those interactions. Investigations of biomolecular interactions
are often designed to determine binding constants between the molecules of
interest. Affinity capillary electrophoresis, the Hummel—Dreyer method, vacancy

affinity capillary electrophoresis, vacancy peak method, and frontal analysis are

26

"

FIGURE 1.5: Effect of pH on observed electrophoretic mobility of phosphate.
Calculation conditions: pKa1 = 2.23, pKa2 = 7.21, pKa3 = 12.32, [to =
-2.88 x 10'3 cm2Ns, [1.1 = 1.18 x 10'3 cm2Ns, p2 = 3.73 x 10*1

cm2Ns, p3 - 0.00

27

FIGURE 1.5

3.

 

N_.

or

o

\I oo+wod

. . vormoo

- mormo._.

r mormm._.

.. morwocm

r mormmN

 

 

morwod

(SNzwo) AJJ'IIBOW oar/\aasao

28

frequently used to perform these investigations. Busch and coworkers have
recently reviewed the general principles and limitations of these methods [51,52].
Similar approaches have been developed for investigations of selective binding
to chiral selectors such as cyclodextrins and crown ethers [53-59]. Capillary
electrophoresis has also been used to investigate intramolecular processes such
as isomerization and denaturation of proteins [60-65].

All of these experimental methods can be described as types of reactive
separations. Consequently, the velocity of the reactive zone is dependent upon
the extent of reaction (thermodynamic equilibrium constant), whereas zone
broadening is dependent upon both the extent and rates of reaction (kinetic rate
constant) [66]. Some estimates of equilibrium constants and rate constants have
been performed via computer modeling .of electropherograms [49,61,64];
however, most have been Calculated from peak parameters such as peak height,
peak area, and migration time [47-50,54-60,63].

1.1.4 Experimental Methods for Reactive Capillary Electrophoresis
1.4.1 Methods for Estimating Equilibrium Constants

Applications of CE typically used to determine equilibrium constants for
binding include affinity capillary electrophoresis (ACE), the Hummel-Dreyerl (HD)
method, the vacancy peak (VP) method, and vacancy affinity capillary
electrophoresis (VACE) [51,52]. The ACE and HD methods are similar
experimentally, in that one of the interacting species is added to the buffer and
the other is injected. However, the two methods differ in the parameter

measured to calculate the equilibrium constant. In ACE, the change in the net

29

 

electrophoretic mobility is used to calculate the equilibrium constant; on the other
hand, the HD method uses the area of the vacancy peak to calculate the
equilibrium constant. The VACE and VP methods are also experimentally
similar. For these methods, both interacting species are added to the buffer and
neat buffer is injected. Likewise, the two methods differ in the parameter
measured to calculate the equilibrium constant. In VACE, the change in the net
electrophoretic mobility is used to calculate the equilibrium constant, Whereas the
VP method uses the area of the vacancy peak to calculate the equilibrium
Constant. 1

Although each of the methods discussed above can be used to estimate
equilibrium constants, they have different capabilities. Whereas the ACE method
is capable of determining only the equilibrium constant, each of the other
methods can be used to determine the number of binding sites as well as the
equilibrium constant [41]. Accurate determination of equilibrium constants can
only be obtained with the HD, VP, and VACE methods when the electrophoretic
mobilities of the complex and the species not monitored are equal. Conversely,
1 accurate determination of equilibrium constants can only be obtained with the
ACE method when the electrophoretic mobility of each of the reacting species
differs from that of the complex [52].
1.4.2 Methods for Estimating Rate Constants

Of the aforementioned methods, ACE has been extended to calculate rate
constants by matching experimental and simulated zone shapes [49]. More

recently, CE methods have been developed to evaluate rate constants as well as

30

equilibrium constants of reactions represented by zones that are overlapping or
connected by a plateau. In nonequilibrium capillary electrophoresis of
equilibrium mixtures (NECEEM),-reactants and products are introduced to the
column as an equilibrium mixture and separation is achieved with a buffer that
contains none of the reacting species. Under these conditions, the observed
zones have the general shape of two Gaussian-like zones connected by a
plateau. One zone represents the complex, one zone represents the completely
dissociated reactant, and the plateau represents the decay of the complex
[67,68]. On the other hand, for electrophoretically mediated microanalysis
(EMMA), reactants are injected onto the column as discrete zones, often
separated by a buffer. Mixing of the zones is achieved by electromigration as
both reactants and products are transported to the detector. Because reactants
undergo mixing and then separation in EMMA, the reactions are considered
irreversible and treated with a Michaelis-Menton model [69,70]. In each of the
above methods, experimentally measured parameters such as zone area, height,
and migration time are used to calculate binding constants. Additionally, rate
constants are evaluated from distorted or splitting zones by matching simulated
zones to the experimental data.

Moreover, although each of these methods can be used to estimate [rate
constants, they are not necessarily complimentary. The EMMA method is only
applicable for calculation of Michaelis-Menton constants for irreversible, second-
order reactions. Both the ACE and NECEEM methods can be used to calculate

rate constants for reversible, second-order reactions. However, calculation of

31

.‘7

 

rate constants by either method is inherently limited by the accuracy of implicit
assumptions of zone shape. Limitations of the ACE and NECEEM methods
differ in several important attributes. The accuracy of rate constants calculated
by the ACE method is limited by a priori knowledge of equilibrium constant as
well as electrophoretic mobilities of the free and bound species. The accuracy of
rate constants calculated by the NECEEM method is limited by the implicit
assumption that the observed migration time is less than or equal to the rate of
dissociation (ts 1/kd).

Capillary electrophoresis has also been used to investigate first-order
reactions such as protein isomerization [60-64]. For these types of
investigations, evaluation of equilibrium constants and rate constants is typically
confined to the time regime in which the isomers are detected individually (ts kd).
Moreover, the ability to calculate rate constants for isomerization is predicated
upon the ability to match simulated electropherograms to experimental data.
This means that the accuracy of the rate constants is inherently limited not only
by the mathematical models used to generate the simulated data, but also by the

computational methods employed to perform the requisite calculations.

1.2 THERMODYNAMIC AND KINETIC THEORY
1.2.1 Thermodynamic Calculations
The effect of temperature on equilibrium constant is described by standard

Gibbsian thermodynamic equations. By measuring the equilibrium constants (K)

32

 

‘1'

 

at different temperatures (T). the changes in molar free energy (AG), molar

enthalpy (AH) and molar entropy (AS) can be calculated via [13]

-AG
I K - —
n RT (33)
AG - AH — TAS (34)

where, R is the molar gas constant. Thus a plot of In K vs 1/T is used to evaluate
AH from the slope and AS from the intercept.
1.2.2 Kinetic Calculations

The effect of temperature on rate constants is described by the classical
Arrhenius model (Equation 35) or by the Eyring model (Equation 36) [13]. While
the underlying principles behind these two approaches are related, there are
some fundamental differences that distinguish them. Whereas both assume the
presence of a high-energy transition state, the equations developed by Eyring
presume that the transition state is an activated complex that exists in a quasi-
thermodynamic equilibrium with the reactants and products. Additionally, the
actiVation energy (AE5) in the Arrhenius expression is most closely related to the
internal free energy, whereas the activation free energy (AG2) in the Eyring
expression is representative of the free energy of the system. Finally, the
Arrhenius expression assumes that the pre-exponential factor (In A) is
independent of temperature. By measuring rate constants (k) at different
temperatures, the activation energy (AE5) is calculated via the Arrhenius

equafion

:t
Ink - lnA—é%— (35)

33

 

Altematively,'the molar activation enthalpy (AH*) and molar activation entropy

(A81) are calculated via the Eyring equation

 

 

k h —AG*
l a:
r(ken) RT (36)
A61 . AH2 — TASt (37)

where h is the Planck constant, v is the transmission coefficient, and kg is the
Boltzmann constant. A plot of In k vs 1/T is used to evaluate AEJ from the slope
and A from the intercept of Equation 35. Similarly a plot of In (le) vs 1/T is used
to evaluate AHt from the slope and AS‘ from the intercept of Equation 36.

In addition to the assumptions in the Eyring equation discussed above,
this application of transition state theory presumes that the kinetic pathways from
each of the ground states to the transition state represent elementary steps.
Moreover, this application of transition state theory assumes that all molecules
' that achieve the transition state cross the high-energy barrier (v = 1), and that a
Maxwell-Boltzmann distribution exists between each of the ground states and the
transition state (kgT/h) [71-73]. However, these latter assumptions introduce
significant potential error into the slope, such that little certainty should be

assigned to calculated values for As.

1.3 CONCLUSIONS
' Although CE does not share the historical depth of liquid and gas
chromatography, it has achieved prominence as a standard practice for analytical

separations. Much of this success is attributable to the rapid advancement of

genomics, proteomics, and biotechnology fields. The preceding sections
illustrate physical and chemical foundations that contribute to processes of
separation and broadening in CE. Additionally, it is the unique separation
mechanism of CE that enables the calculation of true thermodynamic equilibrium
Constants as well as true kinetic rate constants for molecular interactions.

The work described herein provides a physical and mathematical
description of how the thermodynamic and kinetic properties of reaction influence
zone shape in CE. Chapters 2 and 3 constitute the simulation and theoretical
development for evaluating thermodynamic and kinetic parameters in reactive
‘ CE. In Chapter 2, stochastic (Mente Carlo) simulations are used to illustrate the
effects of thermodynamic equilibrium constant, characteristic reaction lifetime,
and electrophoretic mobility on the shape, velocity, plate height, and asymmetry
of reactive zones. In Chapter 3, the mathematical relationships between reaction
and zone broadening in CE are developed and evaluated under the rubric of
Giddings’ theoretical plate height model. Chapter 4 describes the experimental
methods used to evaluate the thermodynamic and kinetic parameters of
reversible, first-order and second-order reactions with the plate'height model
developed in Chapter 3. Chapters 5 and 6 constitute the experimental validation
and application of the plate height model. Chapter 5 illustrates the experimental
validation of the plate height model by thermodynamic and kinetic evaluation of
, previously investigated reversible, first-order reactiOns that are known to be
kinetically slow. Chapter 6 illustrates an attempt to apply the plate height model

to a reversible, second-order system that was not previously kinetically

35

characterized. In Chapter 7, the physical limitations of the plate height model are
explored with respect to the effects of electrophoretic mobility, diffusion
coefficient, equilibrium constant, and electric field strength on the maximum.
calculable rate constants. Chapter 8 summarizes the results and implications of
the simulation, theoretical, and experimental work discussed herein as well as

potential future applications of the plate height model for reactive CE.

36

1O
11

12

13

14

15

16

1 .5 REFERENCES

Tiselius, A. Thesis: Nova Acta Regiae Societatis Scientiarum
Uppsaliensis, ser. IV, vol 7, Almqvist & Wlksell, Upsala, Sweden, 1930

Tiselius, A. Trans. Faraday Soc. 1937, 33, 524

Hjerten, S. Chromatogr. Rev. 1967, 9, 122

Jorgenson, J. W.; Lukacs, K. D. Anal. Chem. 1981, 53, 1298

Fung, Y.-S.; Tung, H.-S. Electrophoresis, 1999, 20, 1832

lssaq, H. J. Electrophoresis, 2000, 21, 1921

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40

73 Benson, S. W. The Foundations of Chemical Kinetics, McGraw Hill, New
York, 1960

41

CHAPTER 2: STOCHASTIC SIMULATIONS OF

REACTIVE CAPILLARY ELECTROPHORESIS

2.1 INTRODUCTION

Although theoretical models have been widely developed for reactive
separations in chromatography, there have been relatively few advances in
theoretical understanding of reactive capillary electrophoresis. The models that
have been develOped generally fall into one of two categories: those that solve
mass balance equations and those that approach the system stochastically.
Equations of mass balance can be solved analytically or by numerical methods to
generate zone profiles in distance or time with the corresponding statistical
moments. These models have been reviewed by Jeng and Langer [1]. One
common assumption is that mass transfer processes are at equilibrium and the
system response is. limited by reaction processes [26]. Conversely, another
common assumption is that reaction processes are at equilibrium and the system
response is limited by mass transfer processes [6-8]. Other mass balance
models treat mass transfer and reaction process explicitly, but assume that axial
dispersion is negligible or that it can be modeled independently of the reaction
process by an effective dispersion term [8-12]. Mass balance models of
transport without reaction in electrophoretic systems have been developed to
address specific modes of separation, including capillary zone electrophoresis
(CE) [13-18], moving boundary electrophoresis [19], isotachophoresis [17,20—24],

and isoelectric focusing [2529], as well as unified models of electrophoretic

42

separations [30-33]. Mass balance models of transport with reaction in
electrophoretic systems have beendeveloped to investigate irreversible enzyme-
substrate interactions [34] and reversible interactions including isomerization,
enantiomerization, and complexation [3538].

Unlike the mass balance models, stochastic models do not require the
solution of complex differential equations and the associated simplifying
assumptions. Rather, stochastic models utilize the fundamental equations of
motion for diffusion, convection, and other transport processes. The solutions to
these equations of motion are less complex than the solutions to equations for
mass and charge balance and require few, if any, simplifying assumptions.
Stochastic. models use the statistical behavior of individual molecules to develop
probability distribution functions, from which zone profiles and statistical
moments are generated. These models have been extended to Monte Carlo and
random walk applications [39-41]. Another stochastic method applies
fundamental equations of motion to individual molecules and follows the
trajectories of the individual molecules through the system [4247]. The
complexity of stochastic simulations can range from. models that are one-
dimensional with a fixed simulation time increment [48-50] to models that are
three-dimensional with a variable simulation time increment [42,43,46,47,51-53].

The work presented in this chapter uses the three-dimensional stochastic
simulation developed by Hopkins and McGuffin [54] and adapted to include
reactive separations by Krouskop [55]. The simulation is capable of modeling

separation systems in which solute molecules undergo irreversible and reversible

43

reactions with first order and pseudo-first order kinetics. The simulations
discussed herein examine the effects of equilibrium constant, kinetic rate
constants, and electrophoretic mobility on the shape and statistical moments of
zone profiles in reactive capillary electrophoresis separations. Insight gained
from these simulations will be valuable to understand and evaluate

thermodynamic and kinetic information from experimental data.

2.2 SIMULATION METHODS

The simulation program was written in the FORTRAN 90 programming
language and executed on a 32-processor Silicon Graphics Origin 3400
computer. The program utilizes algorithms for mass transfer and reaction to
simulate the spatial as well as temporal distribution of molecules during
separation [44.54.55]. Simulated mass transfer processes include diffusion,
convection by laminar and electroosmotic flow, electrophoretic migration, and
stationary phase partition or adsorption. Simulated reaction processes include
irreversible as well as reversible first order and pseudo-first order kinetic
reactions occurring in the mobile and stationary phases. These processes are
’ applied to each molecule at each time increment (t) until the totalsimulation time
(T) is reached. At any specified time or spatial position, the molecular distribution
and corresponding solute zone profile may be examined and characterized.

Simulation of molecular diffusion is performed with a three-dimensional

extension of the Einstein-Smoluchowski equation [56,57]. During each time

44

increment (t), the radial distance (p) that each molecule travels is selected

randomly from the probability distribution

(_ 2
P - —p—— exp J’— (1)
P 2
(4ant)V \4Dmt

 

where Dm is the diffusion coefficient of the molecule in the mobile phase. This
method of simulating the diffusion process provides a variable diffusion step
derived from a normal (Gaussian) distribution in which the direction of travel is
randomized in three-dimensional space. This algorithm has been validated over
the range of diffusion coefficients from 10‘1 to 10'10 cm2/s [46,54,55].

Molecular convection in the mobile phase occurs from electroosmotic flow.
The axial distance (2) that a molecule travels during each. time increment is
described by
z = vt (2)
where v is the linear velocity. The radial velocity profile may be assumed to be

flat or may be described by the Rice-Whitehead equation [58]

v- vo(1- '°("R) ] (3)

 

10(me)

where R is the radial coordinate of the molecule, Rm is the radius of the mobile
phase, x" is the Debye length, and to is the zero-'order modified Bessel function
of the first kind. The maximum electroosmotic velocity (v0) may be specified as
an input parameter or calculated from the Helmholtz-Smoluchowski equation [59]

eCV

41mL (4)

 

V0-

45

where V is the applied voltage, L is the column length, C is the zeta potential of
the interface between the mobile and stationary phases, and e and n are the
relative permittivity and viscosity of the mobile phase, respectively. The
algorithm for electroosmotic convection has been validated for linear velocities
over the range of 0.01 to 1.0 cm/s [54].

The electrophoretic migration velocity vp is described by

V
VI, - EL— (5)

where the electrophoretic mobility M of each molecule is corrected for the ionic

strength of the mobile phasegby a modified Onsager equation [54,60]

 

:1 =uo -(0.23uOIZI+31.3x10'9 [2])1:fl ' (6)

where Ito is the mobility at zero ionic strength and I is the ionic strength. The
algorithm for migration has been validated for electrophoretic mobilities over the
range of +10‘3 to —10‘3 cm2Ns [54].

First-order and pseudo-first-order irreversible or reversible chemical
reactions can be modeled to occur in the mobile phase, the stationary phase, or
both. It is possible to simulate systems ranging in complexity from a single
irreversible reaction to three sequential reversible reactions with one reversible
side reaction [55]; To simulate the reaction process, each molecule, upon

entering the reacting phase, is assigned a lifetime 0.)

it--i(—1log(x) . (7)

46

where k is the rate constant and X is a random number between zero and unity.
Any molecule that remains in the reacting phase for an amount of time equal to
or greater than the assigned lifetime is converted to the new species and
assigned the appropriate diffusion coefficient and electrophoretic mobility, which
are used in Equations 1 and 5, respectively. The algorithm for reaction has been
. validated for kinetic rate constants over the range of 1.0 x 10'8 to 1.0 x 1010 s’1
and for equilibrium constants over the range of 1.0 x 10'18 to 1.0 x 1018 [55]
FIGURE 2.1A is a flow chart of the stochastic simulation program. Input
parameters for the program fall into three general categories: system
parameters, molecular parameters, and simulation parameters. System
parameters include capillary dimensions, applied voltage, detection locations,
and electroosmotic velocity. Molecular parameters include the number and
charge of solute species, electrophoretic mobility, diffusion coefficient, rate
constant, and equilibrium constant. Simulation parameters include the number of
molecules, simulation time step, total simulation time, and the frequency at which
data are written to file.
FIGURE 2.18 is a schematic representation of the simulated processes of
migration, diffusion, and reaction for individual molecules in reactive capillary
electrophoresis. Open and shaded spheres represent the individual reacting
species. Each species exhibits a unique electrophoretic mobility, mobile phase
diffusion coefficient, and rate constant. Each sphere represents a random, 3-
dimensional diffusion step. The solid line extending from the center of each

sphere to the edge represents the magnitude and direction of each diffusion step.

47

FIGURE 2.1A: Flow chart of the stochastic simulation of reactive capillary

electrophoresis adapted from [55].

48

FIGURE 2.1A

 

INPUT PARAMETERS

I

INITIAL SPATIAL DISTRIBUTION

I

r—v TOTAL TIME

I

MOLECULE NUMBER

I

MOLECULE REACTS?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

YES

 

NO

 

 

REACTION

 

 

V
DIFFUSION 4

 

 

 

 

 

 

 

 

 

I ELECTROOSMOTIC FLOW I *
~ $ MOLECULAR DISTRIBUTION

STATISTICAL MOMENTS
ELECTROPHORETIC FLOW T

I

MOLECULE DETECTED? ‘
l YES

NO TEMPORAL DISTRIBUTION

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

INCREMENT MOLECULE H—

l YES

' OUTPUT INTERVAL?

No]

 

 

 

 

 

 

 

 

 

 

 

 

INCREMENT TOTAL TIME

I .

 

 

 

 

 

END OUTPUT

 

 

 

 

 

 

49

FIGURE 2.1B: Schematic diagram Of reactive capillary electrophoresis

simulation processes.

50

FIGURE 2.1B

 

 

 

 

 

 

 

 

 

A 2,2“.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

51

The dashed lines extending between spheres represent the cumulative
convection and migration steps.
2.2.1 Simulation Conditions

Capillary electrophoresis separations are simulated for the reversible

reaction
A B (8)

The physical relevance of this type Of reaction extends not only to first-order
reactions but also to pseudo-first-order reactions. lsomerization and
enantiomerization are examples Of chemically interesting, reversible first-order
reactions that can be described by these simulations. More broadly, reactions
normally considered to be second order, such as complex formation, host-guest
inclusion, and monoprotic acid-base equilibria, can be modeled as pseudo-first-
order reactions when one of the reactants (i.e. ligand, host, hydrogen ion) is in
sufficient excess as to be assumed constant. More complex reactions, such as
polyprotic acid-base equilibria, are a natural extension of this work with the
existing simulation algorithms.

For the simulations described in this chapter, the kinetic rate constants (k),
equilibrium constant (K), and electrophoretic mobility of species A (UA).and B ([13)
are varied. Each simulation begins with 2000 molecules distributed as species A
and B in equilibrium proportions. The processes of diffusion, convection,
electrophoretic migration, and reaction are then applied to each molecule in time

increments of 1.0 x 10'5 s. Separation occurs within a 100 cm x 100 um id

52

capillary column with an applied potential Of 25 kV, an electroosmotic velocity of
0.2 cm/s, and a flat flow profile. The resultant time distribution Of molecules is
written to the output file at distances Of 2, 4, 6, 8, and 10 cm along the column.
2.2.2 Simulation Output and Calculations

Output arrays are used to generate zone profiles and calculate statistical
moments at each of the fixed detector locations along the column. Zone profiles
at each detector are generated by binning the temporal distribution of molecules
that reach each detector. FIGURE 2.2 illustrates the relationship between the
temporal distribution of molecules within the capillary at 10 cm (top) and the
representative zone profile (bottom). The first statistical moment (M1) describes
the mean, the second statistical moment (M2) describes the variance, and the
third statistical moment (M3) describes the asymmetry of the solute zone. The

statistical moments are calculated as

.gti .
M1(t) - EN— (9)
N
.3 (ti _ MI)n
Mn(t) - FLT— (10)

where N is the total number of molecules and t; the time for each molecule to
reach a specified detector location. These statistical moments are then used to
calculate the Damkohler number as well as figures of merit such as velocity,
plate height, and skew for the zone profiles. The Damkohler nUmber (Da)
describes the ratio of total time for reaction to the characteristic lifetime for

reaction (1:) [61-63].

53

FIGURE 2.2: Illustration of the relationship between the distribution of

molecules within the capillary and the resulting zone profile.

54

FIGURE 2.2

 

 

 

 

 

 

 

 

 

45

40~

100100030
(OMNNPF'

SH'IFIOS'IOW :IO USGWIIN

55

In

 

O

60.9 61 .2 61 .4 61 .7 61 .9 62.2 62.4

60.7

TIME (s)

In reactive CE, the Damkohler number is given by [64]

Da-&(t_) , (11)

T

where ‘C = 1/(kf + k,). The apparent zone velocity (v) is calculated as

v-—z— (12)
M,

where z is the distance to the detector location. Plate height (H) and skew (S)

are calculated by

H-I—AZ—z— (13)

3-23. (1.)

M?

which describe the symmetric and asymmetric broadening about the mean. In
order to verify the reproducibility of the calculated figures of merit, replicate
simulations have been performed with varying time increments. TABLE 2.1
shows the average and standard deviation for the velocity, plate height, and
skew. For simulations with the same time increment (1.0 x 10'5 s), the relative
standard deviation is 0.03% for velocity and 4% for plate height, which is
consistent with previous results [46,47,55]. The relative standard deviation for
skew is quite high (~ 60%) because Of the very small asymmetry of the simulated
zone profiles. It is evident that the accuracy and precision are statistically
indistinguishable for simulations with different time increments (1.0 x 10'2 s t s

1.0 x10'ss).

56

TABLE 2.1

Reproducibility of Velocity, Plate Height, and Skew of

Simulated Zone Profiles.

 

 

 

 

 

 

 

Simulation
Time Velocity (cm/s) Plate Height (cm) Skew
Increment (5)
10-5 . 1.62x10" 1.06x10'3 6.30x10’2
(:t 4.47x10'5) (:l: 4.22x10'5) (:l: 4.01x10‘2)
10.2 _ 10-5 b 1.62x1 0'1 1.10x10‘3 5.12x10’2
(:1: 6.91x10‘5) (:l: 5.13x10'5) (a: 3.99x10'2)

 

‘ Average and standard deviation calculated from four independent simulations

where kg = k, = 1 s'1 (1: = 0.5 s) with 1.0 x 10'5 s time increment. Other simulation

conditions as given in FIGURE 2.3.

b Average and standard deviation calculated from four independent simulations
where k. = k, = 1 s" (1: = 0.5 s), one each at 1.0 x 10’2, 1.0 x 10'3, 1.0 x 10*, and

1.0 x 10'5 s time increments. Other simulation conditions as given in FIGURE

2.3.

57

 

2.3 RESULTS AND DISCUSSION
2.3.1 Effects of Rate Constant

The effects of kinetic rate constants on zone profiles, velocity, plate height,
and skew were studied by varying the forward (kg) and reverse (kg) rate constants
from 0.033 to 10 s‘1 for a fixed equilibrium constant Of unity. For each of these
simulations, the mobility of species A (MA) was —1.0 x 10“ cm2Ns, the mobility of
species B ([13) was —2.0 x 10“ cm2Ns, and the mobile-phase diffusion coefficient
for both A and B was 1.0 x 10'5 cm2/s.

The zone profiles are shown as a function of kinetic rate constants in
FIGURE 2.3. For values Of kg = kr s 0.1 5", two poorly resolved zones are
observed at early detector locations and merge into a single, broad zone at later
detectors. For values of kg = kr > 0.1 s“, a single zone is observed at all detector
locations. It is also apparent that broadening decreases with increasing rate
constant and that the broadening is symmetric about a fixed mean time. This
dependence of zone profiles on rate constant can be described in terms of the
Damkohler number (Da).

When the total time (M1) is small with respect to 1: (Da < 1), most
molecules have not had sufficient time to react and the initial zones of species A
and B migrate separately at their respective velocities [64]. As the total time
increases (1 s Da 5 5), molecules spend time traveling as both species A and B
and the zones begin to coalesce. The coalescence of reactive zones is manifest
as the formation of a plateau between the zones of the individual species. As

molecules continue to react and spend more time traveling as both species A

58

FIGURE 2.3:

Simulated CE profiles at multiple detector locations of 2, 4,
6, 8, and 10 cm. Simulation conditions: L = 100 cm; Rm = 50
um; N = 2000; t = 1.0 x10'5 s; Dm = 1.0 x10'5 cm2/s; vo = 0.2
cmls; v = 25 kV; 11.. = —1.0 x 10“ cm2Ns; 113 = —2.0 x 10“
cm2Ns; K = 1; (A) kg = k,=10 s", 1 = 0.05 s; (B) k1: k. =1
s‘1,1: = 0.5 s; (C) k. = k, = 0.66 s", 1 = 0.76 s; (D) k1: k. =
0.33 s“, 1 = 1.52 s; (E) kg = k, = 0.1 s“, 1 = 5 s; (F) 11. = k. =

0.066 s", 1 = 7.58 s; (G) k1: k. = 0.033 s",1: =15.15 s.

59

FIGURE 2.3

cm

3 m2:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_ _- ..j

, 1.
Iii-Til 11
141qu 4
4 4 4 4;.
1 .1. s s
< < < < <

 

 

60

and B, the plateau becomes more prevalent until all of the molecules have
converged into a single zone. Finally, when the total time is sufficiently large with
respect to ‘5 (Da 2 20), steady state for the reaction is achieved and all of the
molecules migrate as a single zone containing the proper proportions of both
species A and B. The same trends for evolution of the reactive zones are
Observed in FIGURE 2.3 for fixed values of total time and varying rate constant
or characteristic lifetime (I).

The graph of apparent velocity as a function of characteristic lifetime and
distance is shown in FIGURE 2.4A. It is evident that zone velocity is invariant
with rate constant and with detector location. Although the zones are only at
steady state for rate constants greater than 0.1 s", their velocities are the same
for all rate constants. Velocity is independent of rate constant for these profiles
becauSe species A and B are initially distributed in equilibrium proportions.
These proportions do not change as the reaction progresses, so each species
contributes to the zone velocity consistently at each detector location for all rate
constants.

The graph of plate height as a function of characteristic lifetime and
distance is shown in FIGURE 2.48. The plate height is relatively constant with
distance for zones at steady-state (kg = kr > 0.1 5"). Conversely, plate height
increases with distance for zones not at steady state (kg = .k, s 0.1 s") and
eventually converges to the steady-state value. At steady state, the plate height
is linearly dependent upon the characteristic lifetime. When 1: is small, all

molecules spend the appropriate steady-state proportion of time traveling as

61

FIGURE 2.4: Effect of characteristic lifetime 1? on apparent zone velocity
(A) and plate height (B) at detector locations of 2 cm (0), 4
cm (A), 6 cm (El), 8 cm (0), and 10 cm (X). Simulation

conditions as given in FIGURE 2.3.

62

FIGURE 2.4A

oo_.

E mzfime: O:w_mm5<m<zo

or

_.

v.0

5.0

 

 

 

_..o

 

 

(Slwo) ALIOO'IEIA

No 4

63

FIGURE 2.48

00_.

Amv m=>:._.m“=1_ O_._.w_mw._.0<m<IO
0w _. _..0

— h —

_.0.0

 

 

 

_.000.0

1 _.00.0

1 ..0.0

1 _..0

 

 

(we) .LHOIEIH ELV‘Id

species A and B. Consequently, the velocity of all molecules is near the mean
value and the broadening is minimized. When ‘5 is large, molecules have not had
sufficient time to react and spend a disproportionate amount of time traveling as
either species A or B. This leads to nonequilibrium broadening that is directly
related to the characteristic lifetime.

The graph of skew as a function of characteristic lifetime and distance is
shown in FIGURE 2.40. The skew is essentially zero and is invariant with t or
distance. Skew is independent Of rate constant because species A and B are
initially distributed in equilibrium proportions with an equilibrium constant of unity.
Thus, there are equal numbers of molecules traveling as species A and B at
each detector location for all rate constants. When the equilibrium constant is
. not unity, the numbers of molecules traveling as species A and B are not equal
and skew is observed, as discussed in the following section.

2.3.2 Effects of Equilibrium Constant

The effects Of equilibrium constant (K) on zone profiles, velocity, plate
height, and skew were studied by varying K between 0.001 and 1000 for a fixed
characteristic lifetime (1:) of 0.5 s. For each Of these simulations, the mobility of
species A (p.11) was -1.0 x 10“ cm2Ns, the mobility Of species B (Ila) was -2.0 x
10“ cm2Ns, and the mobile-phase diffusion coefficient for both A and B was 1.0
x 10'5 cm2/s.

The zone profiles are shown as a function of equilibrium constant in
FIGURE 2.5. While the total time to react varies for each value of K, the

Damkohler number is greater than 20 at each detector location. Consequently,

65

FIGURE 2.4: Effect Of characteristic lifetime 1 on skew (C) at detector
locations of 2 cm (0), 4 cm (A), 6 cm ([3), 8 cm (0), and 10

cm (X). Simulation conditions as given in FIGURE 2.3.

66

FIGURE 2.4C

E mzfimej DEED/EEO
2: 2 F to So

— — - . F .l

 

 

 

 

 

Maps

67

FIGURE 2.5:

Simulated CE profiles at multiple detector locations of 2, 4,
6, 8, and 10 cm. Simulation conditions: L = 100 cm; Rm = 50
em; N = 2000; t = 1.0 x 10'5 s; cm = 1.0 x 10'5 cm2/s; v0 = 0.2
cmls; v = 25 kV; 11. = —1.0 x 10‘4 cm2Ns; 113 = —2.0 x 10“
cm2Ns; 1 = 0.5 s; (A) K = 1000, I11: 1.996 s", k. = 0.002 s";
(B) K = 100, k. = 1.96 s“, k. = 0.02 s“; (C) K = 10, kg = 1.616
s'1,kg= 0.162 s"; (D) K = 1, k1: 1 s", k, = 1 s"; (E) K = 0.1,
k1: 0.182 s“, k. = 1.818 s"; (F) K = 0.01, k1: 0.02 s", k, =

1.96 s"; (G) K = 0.001, kg = 0.002 s", k. = 1.996 s".

68

FIGURE 2.5

 

 

 

"= '<<
1
I <<
<-< <
{I _
<—==
<< <
4
<<

 

 

 

 

 

 

 

 

69

4O 6O 80

TIME (s)

20

each simulation is representative of reactions at steady state at all detector
locations. From these profiles, several trends are immediately evident. The time
to reach each detector changes with the value of K. The profiles for K equal to
unity exhibit the greatest symmetric broadening. The direction and magnitude of
asymmetric broadening is dependent upon K. Positive asymmetry (fronting) is
observed when K > 1 whereas, negative asymmetry (tailing) is observed when K
< 1.

The graph of apparent velocity as a function of equilibrium constant and
distance is shown in FIGURE 2.6A. It is evident from these data that the velocity
for each zone is sigmoidally dependent upon K but invariant with distance. The
apparent velocity is equivalent to the weighted average of the velocity of species
A and B (VA and v3), respectively.

vA1I-Kvl3
1+K

(15)

When K is much greater than unity, molecules spend a large fraction of time as
species B and the zone velocity is similar to that of a zone of pure species B.
Conversely, the zone velocity is more similar to that of pure species A when K is
much smaller than unity. Finally, when K is equal to unity, molecules spend
equal fractions of time traveling as both species A and B and the velocity of the
solute zone is the numerical average of VA and V.;. This dependence Of velocity
on K is supported by previous expressions for the apparent electrophoretic

mobility Of reactive solute zones in CE [65,66].

The graph of plate height as a function of equilibrium constant and

70

FIGURE 2.6: Effect of equilibrium constant K on velocity (A) at detector
locations of 2 cm (0), 4 cm (A), 6 cm ([1), 8 cm (0), and 10

cm (X). Simulation conditions as given in FIGURE 2.5.

71

FIGURE 2.6A

000 _.

00_.

._.Z<._.wZOO 555535an
0.. _. _..0

n — —

_.0.0

-

F000

 

 

 

v.0

 

 

N.0

(Slum) AilOO'IEIA

72

distance is shown in FIGURE 2.68. From this graph, it is evident that plate
height is dependent upon K but independent of distance. The dependence of
plate height on K has similar origins as the dependence of velocity, as discussed
above. When molecules travel exclusively as species A (K << 1) or as species B
(K >> 1), the velocity is identical for each molecule. Hence, there is no
contribution to broadening from differences between the velocities Of species A
and B and the plate height is minimized to that resulting from diffusion alone.
When molecules spend a significant fraction of time traveling as both speciesA
and B, the velocity for each molecule has some statistical variability. This
variability causes the plate height to reach a maximum value when K = 1.

The graph of skew as a function of equilibrium constant and distance is
shown in FIGURE 2.6C. From these data, it is evident that the magnitude of
. skew is dependent upon K and upon distance. The magnitude of skew
decreases with increasing distance because the symmetric contributions to
broadening from diffusion increase throughout the separation. The trends of
negative skew (fronting) for values of K > 1 and positive skew (tailing) for values
Of K < 1 result because VA is greater than v3. When K > 1, molecules within the
steady-state zone travel a greater fraction of time as species B than as species
'A. The manifestation of this temporal distribution Of molecules traveling at
different velocities is a leading portion (fronting) to the zone. A following portion
(tailing) to the zones arises similarly for K < 1. Local maxima for skew, which
occur at approximately K = 0.1 and K = 10, are the result of molecules traveling

significant (non-equal) fractions of time as both species A and B. However, skew

73

FIGURE 2.6: Effect of equilibrium constant K on plate height (B) and skew
(C) at detector locations Of 2 cm (0), 4 cm (A), 6 cm (El), 8
cm (0), and 10 cm (X). Simulation conditions as given in

FIGURE 2.5.

74

FIGURE 2.68

000 _.

._.Z<._.wZOO EDEmZEOm.
00w 0_. F _..0 _.0.0 _.00.0

 

 

 

- - _ — . L .. — Foocco

1 _.00.0

(wo) iHOIElH ELLV'Id

 

 

_.0.0

75

FIGURE 2.60

._.Z<._.wZOO EDEmEEOw
000—. 00_. 0_. _. _..0 _.0.0 . F000
- Fl

- - — —

 

 

 

MEDIS

 

 

 

 

76

approaches zero when molecules travel exclusively as species A (K << 1) or as
species B (K >> 1), or when molecules travel equally as both species (K = 1).
2.3.3 Effects of Electrophoretic Mobility Difference

The effects Of the difference between the electrophoretic mobility of
species A (11A) and species B ((13) on zone profiles, velocity, plate height, and
skew were studied by varying Au = [1A — [1.3 between 1x104 and 6x10‘4 cm2Ns for
a fixed number average Of (1A and us Of -1.5x10“ cm2Ns. The equilibrium
constant for each of these simulations is unity and the kinetic rate constants (kg
and k.) are 1 s’1 (t = 0.5 s).

The zone profiles are shown as a function of electrophoretic mobility
difference (Ag) in FIGURE 2.7. For each value Of Au, the Damkohler number is
greater. than 20 at each detector location. Consequently, each simulation is
representative of reactions at steady state at all detector locations. From these
profiles, it is evident that the peak mean is unchanging, while the extent of
symmetric broadening increases with Au.

The graph of apparent velocity as a function of electrophoretic mobility
difference and distance is shown in FIGURE 2.8A. The velocity is invariant with
Au and distance because each of the reactions is at steady state and the
equilibrium constant is unity. As discussed for FIGURE 2.6A, the velocity of the
steady-state zone is the weighted average Of the velocities of species A and B

according to Equation 15. However, the equilibrium constant is unity for each of

77

FIGURE 2.7:

Simulated CE profiles at multiple detector locations of 2, 4,
6, 8, and 10 cm. Simulation conditions: L = 100 cm; R1.1 = 50
11111; N = 2000; t = 1.0 x 10'5 s; O... =10 x 10'5 cm2/s; v0 = 0.2
cmls; v = 25 W; K =1;kf= kg=1s'1;(A)l~lA =1.5 x 10“
cm2Ns, pa = —4.5 x 10" cm2Ns; (B) NA = 1.0 x 10" cm2Ns,
113 =—4.0 x 104 cm2Ns; (C) 11. = 0.5 x 10'5 cm2Ns, 113 = —3.5
x 10‘4 cm2Ns; (D) 111 = 0.0 x 10* cm2Ns, 1113 = —3.0 x 10“
cm2Ns; (E) 111 = —0.5 x 104 cm2Ns, 1111 = —2.5 x 104 cm2Ns;

(F) 11. = -1.0 x 10‘4 cm2Ns, 118 = —2.0 x 10“ 6111st.

78

FIGURE 2.7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

79

40 60 80

TIME (s)

20

FIGURE 2.8: Effect of electrophoretic mobility difference Au on apparent
zone velocity (A) at detector locations of 2 cm (0), 4 cm
(A), 6 em'(I:I), 6 cm (O), and 10 cm (X). Simulation

conditions as given in FIGURE 2.7.

80

FIGURE 2.8A

_.00.0

b

62.5502?th E4502 9551.6050me

_.000.0

 

 

f

 

 

 

_..0

N0

(Slum) ALIOOTEIA

81

these simulations, such that the zone velocity at steady state is the number
average of VA and v3. Velocity in CE is linearly dependent upon electrophoretic
mobility. Thus, because the number average Of (IA and lie was held constant, the
number average of VA and vs was also held constant. Therefore, velocity is
independent Of Ap. for these simulations strictly because the equilibrium constant
is unity.

The graph of plate height as a function of electrophoretic mobility
difference and distance is shown in FIGURE 2.88. The plate height is invariant
with distance because each of these zones is representative Of reactions at
steady state. However, plate height increases with Au because of nonequilibrium
broadening as discussed fOr the characteristic lifetime (1:) in FIGURE 2.48. It is
noteworthy that this broadening is linearly proportional to 1, but proportional to
the square of Au. These relationships are apparent from the slopes in FIGURES
2.48 and 2.88. It is also noteworthy that the observed dependence of plate
height on 1: and Ap. is closely related to what would be expected from a simple,
two-velocity random walk model, wherein the plate height is proportional to
2 (V2 - v1)2 1/ (v1 + v2). As a consequence of this relationship, a greater extent
of broadening is observed for greater differences in the velocities of species A

and B.

The graph of skew as a function of electrophoretic mobility difference and
distance is shOwn in FIGURE 2.80. The skew is invariant with distance because

each of these zones is representative of reactions at steady state. The skew is

82

FIGURE 2.8: Effect of electrophoretic mobility difference Au on plate
height (8) and skew (C) at detector locations of 2 cm (0), 4
cm (A), 6 cm (El), 8 cm (0), and 10 cm (X). Simulation

conditions as given in FIGURE 2.7.

83

FIGURE 2.88

AmZNEvaozwmquE >._._1=mO_>_ 0_._.m_m_OIn_Om._.Ow4m
_.00.0 _.000.0

1 . 500.0

 

 

_.00.0

 

l
._
O.
0

 

 

 

_..0

(um) J.HOI3H ELLV'Id

84

FIGURE 2.86

AmZNEOV wozmmwnEE >.:.=mO_>_ O_._.mmOIn_Om._10w1_m
_.00.0 500.0

F . - p . u - . Fl

 

 

'

a . 1 H

 

 

II

 

 

<

 

 

 

 

85

MBMS

small and independent of Au because species A and 8 are initially diStributed in
equilibrium proportions with an equilibrium constant of unity. Thus, there are
equal numbers of molecules traveling as species A and B for all values of Au.
When the equilibrium constant is not unity, the numbers of molecules traveling as
species A and B are not equal and skew is Observed, as discussed in the

previous section.

2.4 CONCLUSIONS

These simulations show that the effects of reaction in CE are manifest in
the shape of zone profiles and can be evaluated via the velocity, plate height,
and skew as calculated from the statistical moments of those zones. The zone
profiles suggest that molecules introduced as a single zone in equilibrium
proportions Of species A and B undergo an initial period of separation prior to
reaction and coalesce towards a single zone containing both species as the
reaction approaches steady state. Once at steady state, molecules continue to
travel as a single zone and the width of that zone is influenced by the amount of
time required to achieve steady state as well as the difference between the
electrophoretic mobilities Of the reacting species.

The figures of merit for these zone profiles demonstrate that the velocity at
steady state is dependent upon the equilibrium constant and the individual
mobilities, but independent of the kinetic rate constants. The extent of zone
broadening is influenced by the equilibrium constant, kinetic rate constants, and

mobilities of the reacting species. Plate height increases when not at steady

86

state, remains constant at steady state, and is greatest when the equilibrium
constant is unity. Conversely, the skew is greatest for equilibrium constants of
0.1 (tailing) and 10 (fronting), exhibiting an inflection point with no asymmetry
when the equilibrium constant is unity. While the equilibrium constant has
different effects on plate height and skew, both can be expected to increase with
electrophoretic mobility difference. Thus, the zone profiles in reactive CE arise
from the confluence of contributions from reaction (1, K) and from separation (11,1
and [13). The significance of these trends is that velocity and plate height are
statistically viable avenues for evaluating equilibrium constants and rate
constants, respectively, from experimental data. However, because of its poor
reproducibility, skew should not be used for any calculations, but only as a

means to establish if K is less than or greater than unity.

87

10
11

12

13
14
15
16
. 17

18

19

20

2.5 REFERENCES
Jeng, C. Y., Langer, S. H., J. Chromatogr. 1992, 589, 1
Klinkenberg, A., Chem. Eng. Sci. 1961, 15, 255
Magee, E. M., Ind. Eng. Chem. Fundam. 1963, 2, 32
Hattori, T., Murakami, Y., J. Catal. 1968, 12, 166
Schweich, 0., \fIllermaux, J., Ind. Eng. Chem. Fundam. 1976, 17, 1
Binous, H., McCoy, 8., J. Chem. Eng. Sci. 1992, 47,4333

Schweich, D., \fIllermaux, J., Sardin, M., AIChE J. 1980, 26,477

, VIllermaux, J, The Chromatographic Reactor, in Percolation Process:

Theory and Applications, Roderigues, A. E., Tondeur, D. Eds., Sijthoff en
Noordhoff, Alpen aan den Rijn, The Netherlands, 1981

1 Gore, F. E., Ind. Eng. Proc. Des. Devel. 1967, 6, 10

Kocirik, M., J. Chromatogr. 1967, 30, 459
Chu, C., Tsang, L. C., Ind. Eng. Chem. Proc. Des. Devel. 1971, 10, 47

Carta,G., Mahajan, A. J., Cohen, L. M., Byers, C. H., Chem. Eng. Sci.
1992, 47. 1645

Reijenga, J. C., Kenndler, E., J. Chromatogr. A 1994, 659, 403
Reijenga, J. C., Kenndler, E., J. Chromatogr. A 1994, 659,417

Cann, J. R., Goad, B. W., J. Biol. Chem. 1965, 240, 1162

‘ Ermakov, S. V., Righetti, P. G., J. Chromatogr. A 1994, 667, 257

Dose, E. V., Guiochon, G. A., Anal. Chem. 1991, 63, 1063

Mikkers, F. E., Everaerts, F. M., Verheggen, T. P. E. M., J. Chromatogr.
1979, 169, 1

Cam, J. R., Goad, B. W., J. Biol. Chem. 1965, 240, 148

Vacik, J., Fidler, V., In Analytical lsotachophoresis, Everaerts, F. M., Ed.,
Elsvier: Amsterdam, The Netherlands, 1981

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22
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27

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33

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36
37

38

Moore, G. T., J. Chromatogr. 1975, 106, 1

Shimao, K., Electrophoresis 1986, 7, 121

Shimao, K., Electrophoresis 1986, 7, 297

Gebauer, P., Bocek, P., J. Chromatogr. 1984, 299, 321
Cann, J. R., Stimpson, D. I., Biophys. Chem. 1977, 7, 103
Stimpson, D. l., Cann, J. R., Biophys. Chem. 1977, 7, 115

Palusinski, O. A., Allgyer, T. T., Mosher, R. A., Bier, M., Saville, D.,
Biophys. Chem. 1981, 13, 193

Bier, M., Mosher, R. A., Palusinski, O. A., J. Chromatogr. 1981, 211, 313
Shimao, K., Electrophoresis 1987, 8, 14

Bier, M. Plusinski, O. A., Mosher, R. A., Graham, A., Saville, D. A., In
Electrophoresis ’82, Stathokos, 0., Ed., Walter de Gruyter & 00.: Berlin,
Germany, 1983

Bier, M., Palusinski, O. A., Mosher, R. A., Saville, o. A., Science 1983,
219.1281 - ’

Bier, M., Palusinski, O. A., AIChE J. 1986, 32, 207

Palusinski, O. A., Graham, A., Mosher, R. A., Bier, M., Saville, D. A.,
AIChE J. 1986, 32, 215

Patterson, D. H., Harmon, B. J., Regnier, F. E., J. Chromatogr. A 1996,
732, 119

Thunecke, F., Kalman, A., Kalman, F., Ma, S., Rathore, A. S., Horvath,
Cs., J. Chromatogr. A. 1996, 744, 259 '

Moore, A. W., Jorgenson, J. W., Anal. Chem. 1995.67, 3464
Trapp, O.. Schurig, V., J. Chromatogr. A 2001, 911, 167

Avila, L. Z., Chu, Y.-H., Blossey, E. C., Whitesides, G. M., J. Med. Chem.
1993, 36, 126

89

39

40

41

42
43
44
45
46

47
48

49

50
51

52

53

55
56

Felinger, A., Cavazzini, A., Remelli, M, Dondi, F., Anal. Chem. 1999, 71 ,
4472

Cavazzini, A., Remelli, M., Dondi, F., Felinger, A., Anal. Chem. 1999. 71,
3453

Dondi, F., Munari, P.. Remelli, M., Cavazzini, A., Anal. Chem. 2000, 72,
4353

Schure, M. R., Anal. Chem. 1988, 60, 1109

Schure, M. R.. Lenhoff, A. M., Anal. Chem. 1993, 65, 3024

McGuffin. V. L., Wu, P., J. Chromatogr. A 1996, 722, 3

McGuffin, V. L., Krouskop, P. E., Wu, P., J. Chromatogr. A 1998, 828, 37
McGuffin, v. L., Krouskop, P. E., Hopkins, D. L., In Unified '
Chromatography, ACS Symposium Series 748, Parcher, J. F., Chester, T.
L. Eds.. American Chemical Society. Washington. DC, 2000

McGuffin, V. L., Electrophoresis 2001, 22, 3709

Weber, S. G., Anal. Chem. 1984, 56, 2104

Giddings, J. C., Dynamics of Chromatography, Marcel Dekker: New York,
1965

Giddings, J. C., J. Chem. Educ. 1958, 35, 588

Betteridge, D., Marczewski, C. 2., Wade, A. P., Anal. Chim. Acta 1984,
165, 227

Wentzell, P. D., Bowridge, M. R., Taylor. E. L., MacDonald, 0., Anal.
Chim. Acta 1993, 278, 293

McGuffin, V. L., Wu, R, Hopkins, D. L., In Proceedings of the lntemational
Symposium on Chromatography, Hatano. H., Hanai, T., Eds., World
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Hopkins, D. L., MycGuffin, V. L., Anal. Chem. 1998, 70, 1066

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90

57

58
59

60

61

62
63

65
66

Reid, R. C., Prausnitz, J. M.. Sherwood, T. K., The Properties of Gases
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Rice, C. L., Whitehead, R., J. Phys. Chem. 1965, 69, 4017

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91

CHAPTER 3: THE THEORETICAL PLATE HEIGHT
MODEL FOR REACTIVE CAPILLARY

ELECTROPHORESIS

3.1 INTRODUCTION

In Chapter 2, the effects of reaction and separation on zone profiles and
figures of merit in reactive capillary electrophoresis were investigated via
stochastic (Monte Carlo) simulations. It was shown that velocity and plate height
of the reactive zones are reasonable avenues for evaluating equilibrium constant
and rate constants, respectively, from experimental data. This chapter details
the theoretical development of velocity and plate height for capillary
electrophoresis separations In which solute molecules undergo reactive
separations. The development is consistent with chromatographic theory and the
resulting equations for velocity and plate height are functionally equivalent to
Chromatographic analogs. This is not the first attempt to illustrate unifying
fundamental concepts of chromatographic and electrophoretic separations.
Datta and Kotamarthi have previously used a Taylor dispersion approach to
describe the contribution to plate height from electrokinetic broadening in
capillary electrophoresis [1]. Plate height expressions have also been developed
_by McEldoon and Datta for capillary electrochromatOgraphy in which all
contributions to broadening are attributed to diffusion and resistance to mass

transfer in the mobile phase [2]. The mobile-phase resistance to mass transfer

92

terms were developed using the generalized dispersion theory of Aris and
illustrated a dependence on the Chromatographic retention factor. Rathore and
Horvath have applied the chromatographic construct of retention factor to
capillary electrophoresis and capillary electrochromatography [3]. However, in
electrophoretic systems, this formalism lacks any thermodynamic or kinetic
significance and is only useful as a peak locator [3,4].

The equations described in this chapter are developed under the rubric of
Giddings’ generalized non-equilibrium theory for chromatography [5] and are
validated with an independently developed stochastic (Monte Carlo) simulation
[6-11]. These equations show that velocity is directly dependent on the
equilibrium constant, whereas plate height is inversely proportional to rate
constant. Hence, these equations provide a novel approach to extract
thermodynamic and kinetic data from experimental zone profiles in reactive

capillary electrophoresis.

3.2 THEORY

3.2.1 Development of the Plate Height Model

The migration of solute molecules in electrophoretic systems is influenced by
direct interaction with a mobile-phase or stationary-phase additive. Broadening
of the solute zone is also influenced by these interactions and may be accurately
described by application of the plate height model. While this model has
Chromatographic origins, the general premise is equally applicable to

electrophoretic separation techniques [1,2].

93

The van Deemter equation describes plate height (H) as

H-A+%+Cv (H

where v is mobile-phase velocity in liquid chromatography or the zone velocity in
capillary electrophoresis. The contributions to zone broadening from multiple
paths (A), longitudinal diffusion (B), and resistance to mass transfer (C) are
assumed to be independent and additive. For open-tubular separations like
capillary electrophoresis, the contributions to zone broadening are reduced to
longitudinal diffusion and resistance to mass transfer. The contribution from
longitudinal diffusion is described with the mobile-phase diffusion coefficient (D...)
by

B - 20... (2)
Similarly, the contribution from mass transfer can be expressed in terms of an

effective dispersion coefficient (Dc) [5] as

c-%%z - (9

In order to describe zone broadening from mass transfer, there must be
consideration Of the rates of solute transfer between two or more states, the
equilibrium distribution of solute molecules, and the velocity of solute molecules
in each state. To begin, a general mass-balance statement is necessary

CT-gq (0

where CT is the total solute concentration and C; represents the concentration of
solute in state i. At steady state, each possible state will contain a fraction )0 of

the total concentration

94

Ci"
- CT (5)

 

)0

where the steady-state concentration of i (Cf) is related to C; via a non-

equilibrium departure term :1
Ci - cg“(1 + s.) (6)

Each positive displacement from equilibrium for one state is balanced by

negative displacement from equilibrium for another state. such that
ZCIEI = EXISI = 0 (7)
I I

For the specific case of reaction between two states,

"AB
A Z 8 (3)

I‘BA
km and RM are the rate constants for transfer from species A to B and from '
species 8 to A, respectively. The mass-transfer term arising from species A (r11)
is equal to the sum of all reaction paths over which the concentration (CA) may
be changed
rA - CB kBA - CA kAB _ (9)

At steady state, the rates of enrichment and depletion of species A are equal.

Hence, by substitution of Equation 6 into Equation 9 for CA and CB
rA - kAB CA(8B - 8A) (10)
The mass-transfer term arising from flow for species A is given by

2
d CA _dCA+V dCA

11
"‘Adzz dt Ach I)

95

where the contribution from diffusion to mass transfer is assumed negligible. At

steady state, Equation 11 can be rewritten as

ac; dc};
— 12
dz ”A dz ( )

 

I'A = -V
where the steady-state zone velocity (v) is the weighted average Of the individual
velocities of species A and 8 (VA and v3, respectively)
V=ZXIVi-XAVA+XBVB (13)

l

Equations 10 and 12 may be combined to relate mass transfer from non-

equilibrium to mass transfer from flow, such that

kABCM6B-6A) ”(11A -v)d:zA _ (14)

 

Equation 7 may be substituted into Equation 14, recalling that O; is proportional

to CT (Equation 5), and followed by rearrangement to yield

-xe (VA - v) dIan' .. "XB (VA - V) dln cT

 

 

.. 15 '
kAB (XA +X3) CIZ kAB CIZ ( )

and by analogy
-xA (VB -V) dlnCT a 'XA (VB -V) dInCT (16)

 

 

EB -
kBA (XA + X8) dZ kBA dZ
The effective dispersion coefficient (Dc) is equivalent to the weighted average of

zone broadening from the concerted effects of non-equilibrium and flow [5]

(17)

 

Z
D __ . . .

Upon substitution of Equations 15 and 16, Equation 17 is equivalent to

96

.. AVXA XS VA _ Avxi XB VB
kAB kBA

 

 

Be (18)

where Av = V)», - V.; > 0. Further substitution of R113 = K kBA, where K is the
equilibrium constant, results in the simplified expression

_AVZXiXB

19
kBA ( )

DC

Equations 3 and 19 may then be substituted into Equation 1 to express the

contributions to zone broadening from axial diffusion and mass transfer as

 

 

20m + 2sz xi x3 _ 20... + 2Av2K

H -
V "BA V V kBA v(1+ K)3

(20)

where 301 = 1/(1 + K), Xe = Kl(1 + K), and D... represents the weighted average of
the mobile-phase diffusion coefficients of species A and B (0.1.x and Dma,

respectively)

Dm-ExiDmI-XADmAHCBDmB (21)
I

Equation 20 can be rewritten specifically for electrophoretic separations by

making the appropriate substitutions for the velocity terms

11- En - 12(11... +9...) ' (22)
Av =1 EAp. - E(11A -113) (23)
where E is the electric field strength, I1 is the net mobility of the zone, (1..., is the
electroosmotic mobility, u... is electrophoretic mobility, (IA and [1.3 are the

electrophoretic mobilities of species A and 8, respectively, and Au is the mobility

difference between species A and B. Substitution of Equations 22 and 23 into

97

Equation 20 yields the following expression for plate height for electrophoretic

separations

20... + 2KAp2E

H' 3
E“ kBA11(1+K)

(24)

 

The final plate height equation for reactive capillary electrophoresis in
Equation 24 has the general form of H = BlE + CE, which is analogous to the
chromatographic expression for plate height in Equation 1. It is interesting to
note that Equation 20 is a generalized form of plate height from diffusion and
mass transfer and is equally applicable to chromatographic as well as
electrophoretic separations. Although, the contributions to broadening from
multiple paths and Joule heating are neglected in this equation, additional terms
may be added where appropriate. It is also noteworthy that the second term in
Equation 20 resembles the form of a two-velocity random walk, sztlv, where 1: =
mm; + k3..) is the characteristic reaction lifetime. When v3 = 0, the mass-
transfer term in Equation 20 reduces to that for retention via single-site
adsorption or partition in liquid chromatography [5,12]. Similarly, .when v3 = 0,
the mass-transfer term in Equation 24 describes the contribution in capillary
electrochromatography from interactions at the wall or at a particle surface.

3.2.2 Evaluation of the Plate Height Model

The evaluation of Equations 20 and 24 as a function Of electroosmotic
velocity or electric field strength, shown in FIGURE 3.1, is noticeably similar to its
Chromatographic analogs. As the electroosmotic velocity or field strength

increases, plate height passes from a diffusion-limited region to a mass-transfer

98

FIGURE 3.1:

Effect of electroosmotic velocity and electric field strength on

I plate height contributions from diffusion («--) and mass

transfer (— —), as well as on total plate height (—).
Calculation conditions: pea = 8.00 x 10“ cm2Ns, (111 =
—1.00 x 10“ cm2Ns, 113 = —2.00 x 104 cm2Ns, o... = 1.00 x

10'5 cm2/s, K = 1, km; = kg). = 1 s".

99

FIGURE 3.1

E62 EOzmmB Sum 2583

 

 

 

 

82 82 com _ ace 8... com o
6260 LEOO.__m> oFoszOEOmd
ed ed to No o
o

- Bee
. «8.0
1 Bee
. Bod

 

 

 

000.0

(um) .LHOlE-IH EILV‘ch

100

limited region with a minimum value (Hman), where the contributions of diffusion
and mass transfer are equal. The field strength at Hm... (E0...) is determined by
calculating the first derivative of H with respect to E and equating it with zero,

such that

 

0... kg... (1+ K)3
K A112

 

Eopt = (25)

Equation 25 illustrates the dependence Of E0... and, hence, Hm... on the
competing effects of diffusion and mass transfer. The value of E0... increases
with the mobile-phase diffusion coefficient and the rate of transfer from Species B
to A. Moreover, plate heights Obtained below E... are linearly dependent upon
D... whereas, plate heights. obtained above E0... are inversely proportiOnal to k3...
Thus, the magnitude of E required to achieve mass-transfer limited plate heights
increases with increasing rate constants. The value of E0... decreases linearly,
however, as An increases. Thus. the magnitude of E required to achieve mass-
transfer limited plate heights increases with decreasing Au.

FIGURE 3.2A is a graph of plate height as a function of electroosmotic
velocity for a range of rate constants (kAB = kg). = k, 0.01 s k s 100 s"). From this
graph, it is shown that H... decreases whereas E0... increases with increasing k.
Also, when H is greater than 5 times the contribution from diffusion (H 2 5 B/E),
the plate height increases linearly with electric field strength. Values of plate
height within this region are dominated by mass-transfer processes and are
inversely proportional to k. It is also shown that the range of mass-transfer-

limited plate heights increases with decreasing rate constant and that the range

101

FIGURE 3.2A: Effect of electroosmotic velocity on plate height for kAB = k3).
= 0.01 s", 1 = 50 s (— —); k... = k... = 0.1 s". 1 = 5 s (m); k... =
k... =1 s",1 = 0.5 s (—); k... = k... =10 s", 1 = 0.05 s (—.—);
k... = k... = 100 s", 1 = 0.005 s (— - . —). Calculation
conditions: .1,... = 6.00 x 10“ cm2Ns, .1,. = —1.00 x 10“
cm2Ns, .13 = -—2.00 x 10“ cm2Ns, o... = 1.00 x 10'5 cm2ls, K

=1.

102

FIGURE 3.2A

EEov Eood> {
to Se

5 -

50.0

 

 

 

 

 

5000.0

500.0

50.0

5.0

_..0

(um) LHOIaH 31in

103

of accessible rate constants increases with increasing electric field strength. This
means that the ability to experimentally access a broad range Of rate constants is
facilitated by increasing the applied voltage, decreasing the column length, or
both.

FIGURE 3.28 is a graph of plate height as a function of electroosmotic
velocity for a range of mobility differences (2.00 x 10'5 5 Au 5 5.00 x 10“ cm2Ns).
From this graph, it is evident that plate height increases with increasing Au. It is
also shown that mass-transfer-limited plate heights are achieved at decreasing
magnitudes of electric field strength as Au increases. This means that the range
of accessible rate constants increases with increasing Au.

FIGURE 3.20 is a graph of plate height as a function of electroosmotic
velocity for a range of equilibrium constants (0.01 s K s 100). From this graph, it
is evident that plate height achieves a maximum value when K is unity. It is also
apparent that the plate height is nearly symmetrically distributed for values of K
other than unity. Accordingly, the magnitude of K has less effect on the range of
accessible rate constants than either electric field strength or mobility difference.

FIGURE 3.3 is a graph of velocity as a function of equilibrium constant for
a range of mobility differences (2.00 x 10'5 5 Au 5 5.00 x 10" cm2Ns). From this
graph, it is evident that when K << 1 the zone velocity is Similar to that of species
A, and when K >> 1 the zone velocity is similar to that of species B. It is also .
shown that as the magnitude of Au increases, so does the range over which
velocity is linearly related to K. This means that the range of accessible

equilibrium constants increases with Au.

104

FIGURE 3.28:

Effect of electroosmotic velocity on plate height for Ap. = 5.00
x 104 cm2Ns (— —); Au = 2.00 x 10‘4 cm2Ns (----); Au = 1.00 x
10‘4 cm2Ns (——); Au = 5.00 x 10'5 cm2Ns (— -—); Au = 2.00 x
10‘5 cm2Ns (— . - —). Calculation conditions: .11,... = 8.00 x 10“
cm2Ns, (.1. + (13)/2 = -1.50 x 104 cm2Ns. D... = 1.00 x 10'5

cm2/s, K =1. k... = k... =1 s".

105

FIGURE 3.28

Eeov Eood>
to Be

50.0

 

 

 

500.0

I
‘-.
O

 

 

(Luv) LHOIaH Elin

106

FIGURE 3.2C:

Effect of electroosmotic velocity on plate height for K = 100
(— —); K = 10 (----); K =1 (—); K = 0.1 (—-—); K = 0.01 (— - . —).
Calculation conditions: .1... = 8.00 x 10“ cm2Ns, .1.. = —1.00 x
10“ cm2Ns. .13 = —2.00 x 10‘4 cm2Ns, D... = 1.00 x 10'5

cm2/s, kAB = kBA = 1 8-1.

107

FIGURE 3.2C

EEov EOO._m_>

to :3 Sod

 

 

 

500.0

1 50.0

108

1 50

(um) iHOIEIH Eli‘rfld

l
‘—.
O

 

 

IghtforK=100
(= 0.01 (... '
,5 “As—100x
= 1.00 x10

FIGURE 3.3: Effect of equilibrium constant on velocity for Au = 5.00 x 10‘5
cm2Ns, Int/Ito = 2.00 (—); Au = 2.00 x 10'5 cm2Ns, .1./.1.. =
1.25 (— —); Au = 0.00 x 10“ cm2Ns, .1./.1.. = 1.00 (m); Au =
2.00 x 10*1 cm2Ns, .1./.1.. = 0.333 (— - —); Au = 5.00 x 10‘4
cm2Ns, .1./113 = 0.167 (— - - —). Calculation conditions: .1.. =
8.00 x 10“ cm2Ns, .1.. = —1.00 x 104 cm2Ns, E = 250 Vlcm,

0... =1.00 x 10'5 cm2ls, K =1,kAB = k... =1 s".

109

FIGURE 3.3

 

._.Z<._.wZOO EDEmZEOM

 

000V 00.. 0.. _. _..0 5.0 50.0
_ _ _ p . 00.0
I. I I I. I 1 00.0
I!
/
O/O
. 1 0.3.0
/0
IIIIII I I I 1 /
I: oo
/. / 1 m_..0

 

 

 

 

 

0N0

= 5.00x10‘5
cm2Ns, III/P8 ‘
1.00 1-); All =
= 500 X 104
= 250 V/cm,

All
conditions: [lee =

II

 

, E

(ser) ALIOO'IEIA

110

 

3.3 SIMULATION METHODS

The derived equations for velocity and plate height (Equations 13 and 24,
respectively) are validated with a three-dimensional stochastic (Monte Carlo)
simulation developed by Hopkins and McGuffin [10] and adapted to include
reactive separations by Krouskop [11]. The simulation program was written in
the FORTRAN 90 programming language and executed on a 32-processor
Silicon Graphics Origin 3400 computer. The program utilizes algorithms for
mass transfer and reaction to simulate the spatial and temporal distribution of
molecules during separation [7.10.11]. Simulated mass-transfer processes
include diffusion, convection by electroosmotic flow, and electrophoretic
migration. Simulated reaction processes include irreversible as well as reversible
first order and pseudo-first order kinetic reactions occurring in the mobile phase.
These processes are applied to each molecule at each simulation time increment
until the total simulation time is reached. At any Specified time or spatial position,
the molecular distribution and corresponding solute zone profile may be
examined and characterized. The algorithms used to simulate mass transfer and
reaction have been described in detail previously in Chapter 2.
3.3.1 Simulation Conditions

Capillary electrophoresis separations are simulated for the reversible
reaction shown in Equation 8, where the kinetic rate constants (kAn, kBA),
equilibrium constant (K), and electrophoretic mobility of species A ((1)1)‘and B ([13)
are varied. Each simulation begins with 2000 molecules distributed as species A

and B in equilibrium proportions. The processes Of diffusion, convection,

111

electrophoretic migration, and reaction are then applied to each molecule in time
increments of 1.00 x 10‘5 s until the total simulation time of 100 s has elapsed.
Separation occurs within a 100 cm x 100 um i.d. open-tubular capillary column
with fixed electroosmotic velocity of 0.2 cmls and electric field strength of 250
Vlcm. The resultant time distribution Of molecules is written to the output file at
distances of 2, 4, 6, 8, and 10 cm along the column.

3.3.2 Simulation Output and Calculations

Output arrays are used to generate Zone profiles and calculate statistical
moments at each of the fixed detector locations along the column. The statistical

moments are calculated as

N

 

2h

11.-.... <29
N
_2(t1-M1)"

NI" ..= M N ' (27)

where N is the total number of molecules and t. is the time for each molecule to
reach a specified detector location. The first (M1) and second (M2) statistical
moments describe the mean and variance, respectively, of the solute zone.
These statistical moments are then used to calculate figures of merit such as

Velocity and plate height for the zone profiles. The apparent zone velocity (v)

and plate height (H) are calculated as

v-_Z_ A . 26
M1 ‘ ( )

112

H - M (29)

where z is the distance to the detector location.

3.4 RESULTS AND DISCUSSION
3.4.1 Comparison of Simulation and Theory

Simulations are performed to validate the proposed model and to examine
the independent effects of equilibrium constant, kinetic rate constants, and
electrOphoretic mobility in reactive capillary electrophoresis [6]. The time
distribution of molecules at each detector is used to generate zone profiles and to
calculate statistical moments via Equations 26 and 27. The statistical moments
are then used to calculate velocity and plate height via Equations 28 and 29,
respectively.

Simulated and theoretical (with and without diffusion) values of plate
height are shown in FIGURE 3.4A as a function of characteristic reaction lifetime
(0.005 s 1 5 15.15 s). The data show that as 1: increases, plate height passes
from a regime in which it is independent of ‘C (diffusion limited), to a regime in
Which it is linearly related to t (mass-transfer limited). Thus, zones
representative of kinetically fast reactions are more likely to exhibit diffusion-
Iimited plate heights. Conversely, zones representative of kinetically slow
I'eactions are more likely to exhibit mass-transfer-limited plate heights. This
means that larger electric field strengths are necessary to achieve mass-transfer-

- limited plate heights for kinetically fast reactions.

113

FIGURE 3.4A:

Effect of characteristic reaction lifetime on plate height from
theory without diffusion (----), with diffusion (—), and from
simulation (El). Theoretical conditions: V.; = 0.2 cmls; II). =
—1.00 x 10*1 cm2Ns; .13 = —2.00 x 104 cm2Ns; E = 250 Vlcm;
D... =1.00 x 10‘5 cm2/s; K = 1; 0.033 s" (1 =15.15 s) 5 I1... =
kg). 5 10 s" (1 = 0.05 s). Simulation conditions: N = 2000; t =
1.00 x 10'5 s; z = 10 cm; all other conditions same as

theoretical conditions.

114

FIGURE 3.4A

00..

0..

.3 main... zOfiO/mm

..

...0

..0.0

50.0

..000.0

 

 

 

..0000000

 

[I

 

 

..00000.0

5000.0

..000.0

50.0

5.0

...0

(we) iHOlElH Elin

115

 

Simulated and theoretical (with and without diffusion) values of plate
height are shown in FIGURE 3.48 as a function of electrophoretic mobility
difference (1.00 x 10“ 5 Au 5 6.00 x 10" cm2Ns). The data show that plate
height increases with the square of Au, owing to an increase in mass-transfer
contributions. Therefore, zones representative of solutes with larger mobility
differences between species A and B are more likely to exhibit mass-transfer-
limited plate heights. This means that it is important to maximize mobility
difference in order to evaluate kinetic rate constants reliably. For complexation
reactions, this would equate to maximizing the change in size or charge of the
complex.

Simulated and theoretical (with and without diffusion) values of plate
height are shown in FIGURE 3.4C as a function of equilibrium constant (0.01 s K
s 100). The data show that the mass-transfer contribution to plate height
decreases as the equilibrium constant deviates from unity. However, the overall
effect of equilibrium constant on plate height is small compared to the
contributions from rate constant and mobility difference. Whereas mass-transfer-
Iimited plate height increases linearly with t and with the square of Au, it
increases only by an order of magnitude over the entire range of K. This
relatively small contribution from equilibrium constant notwithstanding. more
aCcurate evaluation of kinetic rate constants is achieved when K is near unity.

The simulated values of velocity with varying rate constant, mobility
difference, and equilibrium constant are compared with the theoretical values

calculated by using Equation 13 in FIGURE 3.5A. As shown, the theoretical

116

FIGURE 3.48:

Effect of mobility difference on plate height from theory
without diffusion (----), with diffusion (—), and from simulation

(:1). Theoretical conditions: v3 = 0.2 cmls; —1.00 x 10“

cm2Ns 5 Ap. 5 —6.00 x 104 cm2Ns; E = 250 Vlcm; D... = 1.00

. x 10’5 cm2/s; K = 1; k... = k3,. = 1 s'1 (1: = 0.5 s). Simulation

conditions: N = 2000; t = 1.00 x 10'5 s; z = 10 cm; all other

conditions same as theoretical conditions.

117

FIGURE 3.48

50.0

AmZNEOV mozwmmmnzn. >1_._1__m0_>_

..000.0

 

 

 

58.0

x 50.0

I
-
O.
0

 

 

...0

(we) iHOIEIH EILVTd

118

FIGURE 3.4C:

Effect of equilibrium constant on plate height from theory
without difquion («--), with diffusion (—). and from simulation
(III). Theoretical conditions: v0 = 0.2 cmls; u). = —1.00 x 10"
cm2Ns; .13 = —2.00 x 10*4 cm2Ns; E = 250 Vlcm; D... = 1.00 x
10'5 cm2/s; 0.001 5 K 5 1000; I113 = k3,. = 1 s'1 (1 = 0.5 s).
Simulation conditions: N = 2000; t = 1.00 x 10'5 s; z = 10 cm;

all other conditions same as theoretical conditions.

119

FIGURE 3.4C

000..

.8.

ezfimzoo 2235358

0..

..

—

...0

..0.0

50.0

 

 

 

50000.0

5000.0

..000.0

..00.0

..0.0

 

 

...0

(mo) LHeIaH 21in

120

FIGURE 3.5A:

Comparison Of theoretical velocity with simulation velocity for
simulations with varying equilibrium constant ([1) ‘ and
varying mobility difference (A) on the primary y-axis. The
diagonal solid line represents a linear relationship between
theoretical and simulation velocities. Percent error (X) is
shown on the secondary y-axis. The horizontal solid line
represents zero percent error. Simulation and theoretical

conditions as described in Figures 3.4A and 3.48.

121

FIGURE 3.5A

80883 lNElO8E-Icl

 

 

 

 

 

 

 

 

LO Q Ill)
<1:
I I
co to <1-
‘7- 5 $9. T'. .—o
O 0 0 0 0

(Slum) ALIOO'IEIA OEILVTnWIS

122

0.16 0.17 0.18
THEORETICAL VELOCITY (cm/s)

0.15

0.14

equation provides an excellent description of velocity for the reactive zone. The
error is negligible (< 0.05%) for each velocity because the molecules begin in
equilibrium proportions for each simulation. Therefore, each species contributes
consistently to the zone velocity. I

The simulated values of plate height with varying rate constant, mobility
difference, and equilibrium constant are compared with the theoretical values
calculated by using Equation 24 in FIGURE 3.58. As shown, the theoretical
equation provides an excellent description of the plate height for the reactive
zone. The error is within 10% for nearly all plate heights. However, it should be
noted that two values of plate height Obtained from simulations have greater than
1 0% error. This is because the reactions associated with those data have not
achieved steady state, which is a fundamental assumption Of the plate height
model. Therefore, the accuracy of rate constants calculated via Equation 24 is
limited to solute zones that are representative of reactions at steady state.
3.4.1 Propagation of Random Error

In order to evaluate the reliability of calculated equilibrium constants and
rate constants. it is important to understand the individual contributions to.error.
In order to evaluate the error in equilibrium constant. Equation 13 must first be

rearranged for K

V _
K-A\V (30)
V—VB

From Equation 30, the propagation of random error in K is straight-fonivard

2 2 2
0% 1‘13) o2 +15) o2 4%) o?, (31a)
OVA VA aVB V6 av

123

FIGURE 3.58:

Comparison of theoretical plate height with simulation plate
height for simulations with varying equilibrium constant (El).
varying rate constants (I), and varying mobility difference
(A) on the primary y-axis. The diagonal solid line represents
a linear relationship between theoretical and Simulation plate
heights. Percent error (X) is shown on the secondary y-
axis. The horizontal solid line represents zero percent error.
Simulation and theoretical conditions as described in Figures

3.4A — 3.4C.

124

FIGURE 3.58

0
"1'

, 80883 .LNEIO81-‘Id
O
‘- O

O
N

00

.50 EOE: meson 20:809.:

 

 

 

 

 

...0 ..0.0 50.0 ..000.0
. . .- 38d
1 1 ..00.0
x
x
x x < x
) Ix. x
x x x xx vowx
x
- i . Ed
I
x
m

 

 

 

5

(um) iHOlEIH ELLV‘ld OEILVTnWIS

125

2 2

 

 

02_(1)202+-(_VA_‘LI (.2. V‘VA -1 0%, (31b)
v-vB V11 (v—v3)2 Vs (v-VB)2 v—VB

The individual contributions to the error in K from v, v.., and V.; in Equation
31 are independent and additive when measured in separate experiments. It is
apparent that the errors in K arising from VA and v3 are minimized when v >> v3
and that the error from V.; is further minimized when VA = v. It is also apparent
that the error from v is minimized when v >> v3 and v... = v. FIGURE 3.6A shows
the individual and cumulative contributions to percent error in K arising from VA,
va. and v for Av = 0.025 cmls. When K < 0.1. the percent error from VA and v
increases linearly as K decreases, whereas when K > 10, the percent error from
v3 and v increases linearly with K. Moreover, when K < 0.1, 0.3 is large and
dominated by the error from VA and v, whereas when K > 10, OK2 is large and
dominated by the error from v3 and v. Only when 0.1 < K < 10 is 0.3 minimized
With comparable contributions from each of the sources Of error.
FIGURE 3.68 shows the effects of velocity difference (0.0025 5 AV 5
0-025 cmls) on the percent error in K for a fixed numerical average Of (V). + v3)l2
= 0.1625 cmls. It is apparent that the error in K decreases linearly as Av
increases. This is a result of the ability to distinguish more accurately between
the individual contributions to v from VA and V.; as Av increases. This correlates
Well with the trends observed in FIGURE 3.3, where the linear regime between

Velocity and K increases with Av. Therefore, increasing Av increases both the

rahge and accuracy of discernable equilibrium constants.

126

FIGURE 3.6A: Individual and cumulative contributions to propagated

percent error in equilibrium constant for 0.01 _<_ K 5 100: 0.12
(-—); (aK/av..)2 0.112 (— —); (ck/6113)2 o.,32 (m); (aK/av)2 0.2 (— .—).
Calculation conditions: V.; = 0.2 cmls; E = 250 Vlcm; .111 =
—1.00 x 10“ cm2Ns; .13 = -2.00 x 10“ cm2Ns; o... 6.3, and

0.. =1.00 %.

127

FIGURE 3.6A

00..

._.Z<._1mZOO EDEmESOM

 

 

 

10..

1 00..

 

 

0000 v

80883 _LN3083d

128

FIGURE 3.68:

Effects of mobility difference on propagated percent error in

equilibrium constant for An = 1.00 x 10“ cm2Ns (—); Au =
5.00 x 10'5 cm2Ns (- —); Au = 2.00 x 10'5 cm2Ns (w); and Au
= 1.00 x 10'5 cm2Ns (— - —). Calculation cOnditions: v0 = 0.2
cmls, E = 250 Vlcm, (.1.. + .13)/2 = —1.50 x 104 cm2Ns; o...

Ova, and 0.. = 1.00 %.

129

 

FIGURE 3.68

00..

0..

...Z<..1.wZOO 23_mm_1._30m .

..

...0

n

5.0

 

 

 

10..

1.00..

1 0000..

 

ooooo.

 

80883 .LN3083d

130

In order to evaluate the propagated error in reaction lifetime, it is

necessary to first rearrange Equation 20 for 1

(”‘QFIVWKIZ _ (111.2o...)(1.1<)2

32)
2KAv2 2KAv2 (

 

 

In

From Equation 32, the propagation of random error in 1 is straight-fonrvard, albeit

cumbersome

 

 

 

2 2 2 (33a)
61: 617 at
121-117,) “31115) 09111) “i
2
v(1+K) 02+ H(1+K) + v(1+K) I-I(1+K)2 a
2KAv2 H 2KAv2 2KAv2 2KAv2 HV
( I22 ( )( ) 2 ‘
21+K 2Dm-Hv 1+K
" m 1. * ma “’3“ ‘33”)

 

+ 2(Hv-ZDm)(1+K)_(Hv—2Dm)(1+K)2 202

2KAv2 2K2Av2

Most Of the contributions to the error in 1 are independent and additive.
HOwever, the steady-state velocity and plate height are intrinsically linked and
are measured in the same experiment, thus requiring consideration of their -
C0Variance(o1..,). Covariance is negative when plate height is diffusion limited (H
°‘ 11v) and positive when plate height is mass-transfer limited (H oc v). We have

assumed positive covariance for these calculations. It is apparent that the error

131

in 1 increases with H, v, D..., and K and decreases with Av. It is also apparent
that the error arising from K is minimized when K = 1.

FIGURE 3.7A shows the individual and cumulative contributions to the
percent error in 1 arising from H, v, D..., and Av for K = 1. The errors arising from
v, D..., and Av, as well as the H-v covariance are inversely proportional to 1,
whereas the error from H varies little with 1. Moreover, when 1 < 0.2 s, 0.2 is

dominated by the contributions from Av and v, whereas when 1 > 2 s, 0.2 is

dominated by the contribution from H. This is because of is limited by
separation when 1 is small, but by reaction when 1 is large.
FIGURE 3.78 shows the effects of equilibrium constant (0.1 s K s 10) on
the percent error in 1. When K at 1. 0.2 varies little over the entire range of1,
whereas when K = 1, 0.2 shows a strong dependence for values Of 1 < 2 s. The
trends for 0.2 are mirrored about K = 1, such that errors associated with K < 1 are
slightly larger than their K > 1 counterparts. These trends are a direct result of
the contribution to 0.2 arising from K. When K = 1, the error arising from
equilibrium constant is zero (Equation 33b). However when K at 1, the error
arising from equilibrium constant is large and dominates the combined

contributions from H and Av over nearly the entire range of 1. Therefore, the

accuracy in 1 increases with proximity of K to unity.

3.5 CONCLUSIONS
Generalized derivations are shown for velocity and plate height in reactive

capillary electrophoresis using the non-equilibrium approach described] by

132

FIGURE 3.7A:

Individual and cumulative contributions to propagated

percent error in reaction lifetime for 0.05 5 1 5 500 s: 0.2 (—);
(111/11H)? 01.2 (— —); (111/am)? 0.12 M: (at/av)2 of (— - -);
(.11/60...)2 00.1.2 (— . - —); 2(a1laH)(61/av) 01.10., (——). Calculation
conditions: 11.. = 0.2 cmls; E = 250 Vlcm; .11. = —1.00 x 10‘4
cm2Ns; .13 = -2.00 x 10‘ cm2Ns; K = 1; OH, 00..., o... and 0....

=1.00 %.

133

FIGURE 3.7A

 

 

 

 

 

 

I
, / /
I
r
/
i-
<
I l l l
O O ‘- T“ V- \—
C

80883 .LN3083d

134

100

10

0.1

0.01

REACTION LIFETIME (s)

FIGURE 3.78:

Effects of equilibrium constant on propagated. percent error
in reaction lifetime for 0.05 5 1 5 500 s: K = 10 (- —); K = 2
("-); K = 1 (—); K = 0.5 (_._); K = 0.1 (—--—). Calculation
conditions: v1. = 0.2 cmls; E = 250 Vlcm; .1.. = —1.00 x 10"
cm2Ns; .13 = —2.00 x 10‘4 cm2Ns; o... 09..., 0... and 11..v =

1.00 %.

135

FIGURE 3.78

00..

.o. 0.253.: 20:09.04.
2 F ..o

..0.0

 

 

 

 

..0.0

1 F0

10..

1 00..

 

 

. 82

80883 .LN3083d

136

Giddings. The resulting equations are consistent with their chromatographic
analogs and are validated with an independent stochastic simulation. The
equations for velocity and plate height are logical extensions of chromatographic
theory and thereby illustrate fundamental similarities between chromatographic
and electrophoretic separations. Moreover. these equations exemplify the
shared thermodynamic (K) and kinetic (k) relationships in spite of» different
separation mechanisms. Therefore, the equations for velocity and plate height I
may be used to evaluate equilibrium constants and rate constants in liquid
chromatography, capillary electrophoresis, and capillary electrochromatography.
Velocity for reactive electrophoretic separations is dependent on equilibrium
constant as well as on the mobilities of the relevant species. Velocity can be
used to _detem'iine the equilibrium constants accurately when the individual
mobilities are known. The range and accuracy of accessible equilibrium
constants increases with mobility difference.

On the other hand, plate height is dependent upon equilibrium constant.
kinetic rate constant. and mobility difference. Plate height is shown to pass from
. a diffusion-limited region to a mass-transfer-limited region with increasing electric
field strength. It is shown that the range of accessible rate constants from mass-
transfer-limited plate heights increases with electric field strength and mobility
difference. , It is also shown that, at steady state, mass-transfer-limited plate
heiQ hts are linearly related to kinetic rate constants.

The propagated errors for K and 1 show that the range and accuracy of

any Calculated equilibrium constants or rate constants will depend most on the

137

 

proximity of K to unity and on the magnitude of Av. For complexation reactions,
this means that the most accurate rate constants will be achieved when the
ligand concentration equals the inverse of K and when the overall size or Charge

of the complex is substantially larger than that of the uncomplexed species.

138

10
11

12

3.6 REFERENCES
Datta, R.; Kotamarthi, V. R., AIChE J. 1990, 36, 916
McEldoon, J. P.; Datta. R., Anal. Chem. 1992, 64, 227
Rathore, A. S.; Horvath, Cs., Electrophoresis 2002, 23. 1211
Knox. J. H., J. Chromatogr. A 1994, 680, 3

Giddings, J. 0., Dynamics of Chromatography, Marcel Dekker, Inc., New
York, NY 1965

Newman, C. I. D.; McGuffin, V. L.. Electrophoresis 2005, 26, 537

McGuffin, V. L.; Krouskop, P. 8; Wu. P. R., J. Chromatogr. A 1998, 828,
37 -

McGuffin, v. L.; Krouskop, P. 8; Hopkins, D. L., Unified Chromatography,
ACS Symposium Series 748, Parcher, J. F ., Chester, T. L. Eds., American
Chemical Society, Washington, DC, 2000

McGuffin, V. L.; Wu, P. R.; Hopkins, D. L., Proceedings of the lntemational
Symposium on Chromatography, Hatano. H., Hanai, T., Eds., World
Scientific Publishing: River Edge, NJ, 1995

Hopkins, D. L.; McGuffin. V. L., Anal. Chem. 1998, 70, 1066

Krouskop, P. 8, Ph.D. Dissertation 2002, Michigan State University

Howerton. s. 8.; McGuffin, v. L., Anal. Chem. 2003, 75. 3539 '

139

CHAPTER 4: EXPERIMENTAL METHODS

4.1 INTRODUCTION

Two instrumental systems were utilized for the studies discussed in
Chapters 5 and 6. In Chapter 5, a commercial capillary electrophoresis (CE)
system was used for diode array absorbance detection at a single location along
the capillary. In Chapter 6.6 novel CE system designed and constructed in-
house was used for laser-induced fluorescence detection at multiple locations
along the capillary. This chapter details the experimental systems and

mathematical functions used to collect and analyze the data in those Chapters.

4.2 1 EXPERIMENTAL SYSTEMS
4.2.1 Capillary Preparation

All separations were performed in W transparent, fused-silica capillaries ,
(Polymicro Technologies). Each capillary was cut to the appropriate length and
detections windows were fabricated by removing the polyimide coating at specific
locations. Following installation, the surface of each capillary was conditioned
with 0.1 M NaOH and distilled water for 10 minutes each, prior to equilibration
with the buffer at room temperature for 24 hours. The capillary surfaces were

further equilibrated with the buffer at each therrnostatted temperature for 3 hours.

140

4.2.2 Capillary Electrophoresis Systems
4.2.2.1 CE System with Diode Array UV Absorbance Detection

The separations discussed in Chapter 5 were achieved with an Agilent
G1600A capillary electrophoresis system operating ChemStation software
(Hewlett Packard). The capillary (id. = 75 um, o.d. = 360 um, L1... = 34 cm, L3... =
26 cm) was thermostatted (8.0 - 60.0 :I: 0.2 °C) with an internal Peltier device.
Sample and buffer vials were thermostatted in situ with a NesLab RTE-10
(T hermo Electron Corporation) Circulating water bath (-25.00 — 150.00 :I: 0.01 °C).
Samples were introduced to the column hydrodynamically (0.5 mbar x 30 s) and
detected via UV absorbance at 210 nm. A schematic diagram of the diode-array
CE system is shown in FIGURE 4.1A.
4.2.2.2 CE System with Laser-Induced Fluorescence Detection

The separations discussed in Chapter 6 were achieved with a capillary
electrophoresis system that was designed, constructed, and validated in-house.
This system contains of a modified incubation oven (LabLine, Model 3500-DT)
capable Of maintaining cOnstant temperature (5.0 - 50.0 :I: 0.2 °C). A plexiglass
insert electrically insulates the oven interior in addition to physically supporting
the individual components Of the separation system with a series of ventilated
. shelves. The capillary (id. = 50 um, CO. = 360 um, L... = 60 cm, L3... = 22 cm
and 38 cm) as well as sample and buffer vials are thermostatted within the oven.

An auto-reversing high-voltage power supply (Bertan High Voltage, Model
2341-A) is used for injection and separation. The power supply can be used to

apply a fixed voltage (0 - 30 W t 0.1%) and monitor the current or to apply a

141

FIGURE 4.1A:

Schematic diagram of the Agilent G1600A capillary
electrophoresis system. Separation system components: Pt
electrodes (E), :l: 0 - 30 kV high voltage power supply (V),
inlet vial (l), fused-silica capillary (0), outlet vial (O), and
ground (G). Detection system components: deuterium light
source (S), focusing lens (L), and UV-vis diode array

detector (D).

142

FIGURE 4.1A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0-30 kV

 

 

 

 

143

 

 

fixed current (0 — 400 LIA :I: 0.1%) and monitOr the voltage. A magnetic switch
(McMaster Carr, Model Schmersal BNS 33) positioned on the incubator door is a
remote safety interlock for the high-voltage power supply and insures that the
power supply is only enabled when the incubator door is closed. A 24 V power
supply (Lambda Power, Model VS758-24) and a 5 V power supply (Lambda
Power, Model VS150-5) are used to power and control the high-voltage power
supply, respectively. These power supplies are housed within a) ventilated
aluminum case which also provides connection between the high-voltage power
supply and a computer via a 15—pin D-connector plate and cable (Black Box.
Model EGM16E—0005-MF).

A computer program written with commercially available software ~
(LabWew. Version 5.1, National Instruments) allows the user to simultaneously
control and monitor the high-voltage power supply. .The program'enables the
user to select the magnitude and duration of either the voltage or current applied
from the power supply. The program is also used to monitor and store data from
the power supply that represents either the current when voltage is controlled or
the voltage when current is controlled.

Samples are introduced to the column electrokinetically (1 W x 15 s) and
detected spectroscopically via laser-induced fluorescence using a 325 nm HeCd
laser (Melles Griot, Model 3074-20M). Solute molecules are excited at each
detector location by transmission of laser light to the capillary via 100 um Optical
fibers. Fluorescence is collected orthogonally to the incident light via 500 um

optical fibers, transmitted through 400 nm long-pass optical filters to

144

monochromators (Instruments SA, Model'H-1061) and finally transmitted to
photomultiplier tubes (Hamamatsu, Model R1463). The resulting photocurrent is
amplified and converted to a voltage (Keithley, Model 480 picoammeter) that is
recorded digitally with a data acquisition board (National Instruments, Model PCI-
MlO-16XE-50) on the computer. Thus, the same LabVIew program is used to
record data from the detectors as well as to cOntrol and monitor the high-voltage
power supply. A schematic diagram of the laser-induced fluorescence CE

system is shown in FIGURE 4.18.

4.3 DATA TREATMENT AND ANALYSIS

After collection, zone profiles are extracted from electropherograms using
previously established conditions for the minimum number of points and
integration limits [1]. Each profile is fit by nonlinear regression with commercially
available software (PeakFit, Version 3.18, SYSTAT Software). The fitted profiles
are then used to calculate the statistical moments of each zone.
4.3.1 Mathematical Functions

Numerous mathematical functions within PeakFit are capable of fitting the
zOne prOfiles. However, the exponentially modified Gaussian (EMG) and
asymmetric double-sigmoidal (ADS) functions almost exclusively provided the
best fit of the experimental data.

The EMG is the result of convoluting a Gaussian funCtion with an

exponential decay [2-4]. This function generally provided the best fit for zones

145

FIGURE 4.18: Schematic diagram of the capillary electrophoresis system
built in-house. Separation system components: Pt
electrodes (E), :I: 0 - 30 kV high voltage power supply (V),
inlet vial (I), fused-silica Capillary (0), outlet vial (O), and
ground (0). Detection system components: HeCd laser (S),
focusing lens (L), excitation optical fibers (EX), emission
optical fibers (EM). optical filters (F), monochromators (M),

and photomultiplier tubes (P).

146

 

. FIGURE 4.18

 

 

PMT P

 

 

 

Mono

 

 

 

 

 

 

 

 

M

 

PMT P

 

 

 

Mono

EM

 

 

 

 

M

F<‘>

 

 

EX EX 1
E G
O

 

 

 

 

 

>0. POGH
, (n
.J

 

 

 

 

 

 

 

 

 

V
0-30kV

 

 

 

 

147

that exhibited tailing profiles. The mathematical form of the EMG is shown in

 

 

Equation 1
A o2 te -t t-tG o )

C(t)=— exp + e - +1 (1)
2" [21:2 t «’20 «I21:

 

 

 

where A is the area, Is is the mean time of the Gaussian component, or is the
standard deviation of the Gaussian component. and 1 is the lifetime of the
exponential component. The parameters of the EMG have previously been
attributed physical significance for the evaluation of chromatographic zone
profiles [511]. However, in this work significance is assigned to the moments of
. the fitted profile rather than to the parameters used to obtain the fit.

The ADS is an empirical function in which each side of the peak is
represented by two distinct sigmoidal functions with different widths [12]. This
function generally provided the best fit for zones that exhibited fronting profiles.
The mathematical form Of the ADS is shown in Equation 2
I-

h 1

(2)
1+ exp{_ t - ghosts} 1+ exp{_ t - 12:11. h

 

 

C(t) =

 

 

 

 

where h is the height of the zone, to is the center, t... is the separation of the two
sigmoidals, w. is the width of the left sigmoidal, and wR the width of the right.
(reverse) sigmoidal. I

Zones that exhibited Gaussian profiles were generally fit equally well by

the EMG and the ADS functions.

148

4.3.2 Statistical Moments

Statistical moments for each zone were calculated at 0.01% of the
maximum height. The first statistical moment (M1) describes the mean of the
zone and the second statistical moment (M2) describes the variance of the zone.
These values are related to the concentration of solute molecules as a function of
time C(t) and may be calculated as

2tC(t)At

‘__2c(1)11 ‘3’

 

_ 2(1411.)2 on)...

2C(t)At (4)

Mz

4.3.1 Figures of Merit
The statistical moments for each zone were used to calculate the zone

velocity (v) and plate height (H)

Z
-_ 5
v M ()
H-214; _ ' - (6)

Mi

where'z is the distance traveled. For data collected at a single detector location,.
2 is the distance from the inlet to the detection window and M1 and M2 are the
moments Of the zone at the detection window. For data collected at multiple
detector locations, 2 is the distance to either detection window or the distance
between detection windows and M1 and M2 are the moments of the zone at or

between detection windows, respectively.

149

Calculation of figures of merit by difference between multiple detectors is
inherently more accurate than calculations using data from a single detector.
Velocities calculated by difference are more accurate because the distance
traveled between detectors is more accurately known than the distance traveled
to any single detector. Plate heights calculated by-difference are more accurate
because, in addition to the advantages mentioned for velocity, the broadening
that occurs between detectors is strictly a result of separation processes of
diffusion and mass transfer. This means that plate heights calculated from
broadening between detectors selectively exclude contributions to zone variance

from injection, which are independent and additive.

4.4 CONCLUSIONS

The use of novel instrumentation and data treatment is necessary in order
to reliably investigate reactive systems with capillary electrophoresis. This
Chapter details the experimental systems and mathematical functions that were

employed fOr data collection and analysis.

150

(”‘10)

10

11

12

4.5 REFERENCES

Howerton. S. 8.; Lee, C.; McGuffin, V. L. Anal. Chim. Acta 2003, 478, 99
Grushka, E. Anal. Chem. 1972, 44, 1733

Grushka, E. J. Phys. Chem. 1972, 76, 2586

Yau, W. W. Anal. Chem. 1977, 49, 395

Foley, J. P.; Dorsey, J. G. J. Chromatogr. Sci. 1984, 22, 40

Jeansonne, M. S.; Foley, J. P. J. Chromatogr. Sci. 1991, 29, 258
Phillips, M. L.; White, R. L. J. Chromatogr. Sci. 1997. 35, 75

'Torres-Lapasio, J. R.; Baeza-Baeza, J. J.; Garcia-Alvarez-Coque. M. C.

Anal. Chem. 1997. 69, 3822

Howerton. S. 8.; McGuffin, V. L. Anal. Chem. 2003, 75, 3539
Howerton. S. B; McGuffin,V. L. J. Chromatogr. A 2004, 1030.3
McGuffin, V. L.; Howerton. S. 8.; Li X. J. Chromatogr. A 2005, 1073, 63

Rundel, R. PeakFit Non-Linear Curve-Fitting Software: Technical Guide,
Jandel Scientific, 1991

151

CHAPTER 5: THEORETICAL PLATE HEIGHT MODEL
FOR THERMODYNAMIC AND KINETIC STUDIES OF

REVERSIBLE, FIRST-ORDER REACTIONS

5.1 INTRODUCTION

The work in this chapter utilizes the theoretical model developed in
- Chapter 3 to calculate equilibrium constants and rate constants for the first-order
isomerization of proline-containing dipeptides from the zone velocity and plate
height, respectively. Whereas in chromatOgraphy, retention factor is proportional f
to equilibrium constant, this relationship is not necessarily maintained in
electrophoretic separations [1.2]. Application of the plate height model to
capillary electrophoresis (CE) retains chromatographic formalism while relating
velocity to equilibrium constant and plate height to rate constant [3,4]. This
relationship is maintained because, under the auspice of‘ separations, zone
velocity is dependent upon the individual velocities of the reactants and products
and the fraction of time spent traveling at each velocity. However, zone
broadening is dependent upon the individual velocities as well as the rates of
reaction [5]. These relationships were validated with the independently
developed stochastic (Monte Carlo) simulation in Chapter 2 [5.6-10].

Unlike other methods for evaluating rate constants in CE that are
restricted to time regimes in which plateau profiles are Observed, the plate height

model is restricted to time regimes in which a single zone is observed. This

152

means that the plate height model facilitates the evaluation of reactions at or
near steady state, whereas other methods require that reactions be far from
steady state. Moreover, only the first and second statistical moments (mean and
variance) of the zone are required for calculating equilibrium constants and rate
constants, respectively. Thus, no assumptions regarding the functional form of
the zone profile are required. It is important to note here that the term steady
state for a dynamic system (eg. separation) refers to the condition in which all
molecules have reacted a sufficient number Of times such that each molecule

has spent an equilibrium distribution Of time traveling as both reacting species.

5.2 THEORY

A generalized development of the theoretical plate height model for
reactive capillary electrophoreSiswas given in Chapter 3. Herein. that model is
applied to the reversible, first-order isomerization of proline. dipeptides. The
isomerization occurs via rotation about the peptide bond between the nitrogen
atom of proline and the carbonyl Of the primary amino acid. This rotation
supports the presence of both trans and cis isomers and can becharacterized by
the equilibrium expression
transJ—tcis (1)

k.

where k. is rate constant for the trans-cis rotation and k. is the rate constant for

the cis-trans rotation. An illustration of this isomerization for Alanine-Proine and

Phenylalanine-Praline is shown in FIGURE 1.

153

FIGURE 1: Illustration of the cis-trans isomerization for Alanine-Praline

(top) and Phenylalanine-Proline (bottom).

154

FIGURE 1:

 

 

 

 

 

155

5.2.1 Velocity and Equilibrium Constant

Capillary electrophoresis of these dipeptides results in a zone containing
both isomers, with a velocity that is determined by the velocities of each of the
individual isomers. The velocity (v) for the zone as well as that for the individual

lsomers is calculated by

(2)

z
Vi - ——

M11
where z is the distance to the detector and M... is the first statistical moment or
mean time to the detector for i = cis, trans, zone. In reactive CE. the velocity of

the zone is equal to the weighted average Of the velocities Of the reacting species '

by [4,5]

_vm, ch.s 3
V2” 1+K+1+K (-)

where K = M kr is the equilibrium constant for the reaction in Equation 1.

By measuring equilibrium constants at different temperatures, changes in
the molar enthalpy (AH) and molar entropy (AS) are calculated via standard
Gibbsian thermodynamic equations (Chapter 1, Equations 33 and 34).

5.2.2 Plate Height and Characteristic Reaction Lifetime

Whereas velocity alone provides a reasonable means to estimate
equilibrium constant. it is not sufficient for investigating the rates of reaction.
Zone broadening, on the other hand. is directly related to resistance to mass
transfer and is, therefore, a viable means for evaluating rate constants [4]. Plate

height (H) is Calculated by

156

zllll2
-— 4
M? H

where M2 is the second statistical moment _or variance. Application of the plate

height model to peptide isomerization yields the expression

 

20 217K(vtrans - vols )2
H - +
vzone (1+ K)2 V131...

(5)

where D is the average mobile-phase diffusion coefficient of the dipeptide. ~ The

characteristic reaction lifetime (1 ) is given by

 

(6)

It is noteworthy that evaluation of kinetic rate constants by Equations 5 and 6
requires a priori knowledge of the equilibrium constant and the velocities of the
individual isomers.

By measuring rate constants at different temperatures, the activation

energy (AE‘) is calculated via the Arrhenius equation (Chapter 1, Equation 35).

5.3 SIMULATION METHODS
5.3.2 Simulation Conditions

A stochastic (Monte Carlo) Simulation was used to simulate capillary
electrophoresis separations for the reversible, first-order A e 8 reaction with a
fixed equilibrium constant (K = 1) over a range of rate constants (k1= k. = 0.001
to 10 s"). The input parameters for stochastic simulations are presented in

Chapter 2. These simulation data were used as a known data set to compare

157

the efficacy of the plate height model and a Windows-based mass-balance

simulation method for estimating rate constants from reactive CE zone profiles.

5.4 EXPERIMENTAL METHODS
5.4.1 Chemicals

Alanine-proline (Ala-Pro), phenylalanine-proline (Phe—Pro), and mesityl
oxide were purchased from Sigma. Sodium tetraborate decahydrate (Na28407)
was purchased from MCB Manufacturing Chemists. The electrophoresis buffer
for all separations was 1.0'x 10'2 M Na28407 (PH 9.3) prepared in distilled,
deionized water. Analytical solutions of 1.0 x 10'3 M Ala-Pro and Phe—Pro were
prepared in the buffer solution with 1.0 x 10'3 M mesityl oxide as the
electroosmotic flow marker.
5.4.2 Experimental System

Separations were achieved using the Agilent G1600A capillary
electrophoresis system described in Chapter 4.
5.4.3 Data Analysis
5.4.3.1 Theoretical Plate Height Model

Analysis of the experimental and Stochastic simulation data was
performed .with commercially available software (PeakFit, Version 3.18, SYSTAT
Software) for calculation of the first and second statistical moments (M1 and M2,
respectively) from the fit parameters for the individual zones. These data were
then used to calculate velocity from Equation 2 and plate height from Equation 4.

The velocity data were used to calculate equilibrium constants (K) via Equation 3.

158

 

The plate height data were used to calculate characteristic reaction lifetimes (1)
and rate constants (kg and k.) via Equations 5 and 6, respectively.

5.4.3.2 ChromWin

As a measure Of comparison, thermodynamic and kinetic parameters for
isomerization were also estimated With a Windows-based computer simulation
(ChromWin 2004) developed by Trapp and Schurig [11-14]. The program utilizes
experimental values such as retention time, zone height, plateau height. and
zone width to calculate rate constants for reactive CE separations. Rate
constants can be calculated by matching simulated electropherograms with
experimental data as well as directly via an approximation function by inserting
the datainto a spreadsheet. Prior to use for calculating rate constants from

experimental data. the accuracy of ChromWIn was evaluated with the stochastic

simulation data.

5.5 RESULTS AND DISCUSSION

5.5.1 Simulation Studies

Stochastic simulations of reversible, first-order reactions illustrate the
inflUence of rate constant and extent Of reaction on zone profiles. FIGURE 5.2A
shows the zone profiles as a function 'Of rate constant and detector distance. For

0.1 2 k 2 0.033 s", two poorly resolved zones are observed at early detector
distances and merge into a single, broad zone at later detectors. Whereas only a
single zone is present at each detector for k 2 1 s“, two zones are present for k s

0.01 s". Retention time (from zone apexes). zone height, full width at-half

159

FIGURE 5.2A:

Simulated CE profiles. Simulation conditions: N = 2000
molecules; t = 1 x 10'5 s; L1... = 100 cm; L3... = 2, 4, 6, 8, 10
cm. id. = 100 .1m; v = 25 kV; v... = 0.20 cmls; D = 1 x 10'5
cm2/s; .11 = —1.00 x 10“ cm2Ns; .13 = —2.00 x 10“ cm2Ns; K

=1.

160

 

 

FIGURE 5.2A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

161

maximum, and plateau height were measured at each detector after fast-Fourier
transform (30%) of the zone profiles. These data were used as input parameters
for calculation of rate constant via ChromVlfin. Zone velocity and plate height
were calculated from the zone profiles at each detector. These data, in addition
to the electrophoretic mobilities of species A and 8,.were used for calclation of
rate constant via the plate height model.
5.5.1.1 Accuracy of Theoretical Plate Height Model and ChromWin
FIGURE 5.28 shows the accuracy of rate constants calculated via
ChromWin and Equation 5 for each zone profile. It is apparent that the accuracy
of the calculated rate constants is dependent upon the extent of reaction,
indicated by the Damkohler number (Da = M1/1) in TABLE 5.1. When Da < 0.2,
‘ relatively few molecules have reacted and individual zones are observed for
species A and 8. In this time regime, both ChromWin and the plate height model
fail to calculate rate constants accurately (> 30% error). When 0.2 < Da < 5,
some molecules have spent time traveling as both species A and B and a
- characteristic plateau is observed between the zones representative of those
molecules traveling primarily as species A or 8. In this time regime, ChromVIfin is
able to calculate rate constants quite well (2 — 25% error). whereas the plate
height model is less accurate (2 25% error). When 5 < Da < 100, most
molecules have spent an equilibrium proportion of time traveling as species A
and B and a single zone is observed. In this time regime, ChromVlfin is incapable
of calculating meaningful rate constants because only a single zone is present.

However, the accuracy of the plate height model is actually greatest in this time

162

FIGURE 5.28: Effect of Damkohler number on the accuracy of rate constants

I
calculated by ChromWin (X) and by the plate height mod 6‘ (O) to
the profiles in FIGURE 5.2A.

163

 

FIGURE 5.28

 

 

 

 

E: ”.9 5
O t
00
O )-
OO -
O O
OO
O 00 I
00 :
(D X '
O< r
O X X .
O X X t
0 X X
0 X
0 X :
.8 x X I
O C
O x _
x
O X :
O X Z
O X '
O x -
O O O O ‘- ‘-
O O ‘—
O v-
" 80883 _I.N3083d

164

100

10

0.1

0.01

DAMKOHLER NUMBER

TABLE 5.1

Damkohler Numbers for Simulation Profiles in FIGURE 5.2A.

 

 

 

 

 

 

 

 

 

 

Damkohler number (Da)
k (s"') 2 cm 4 cm 6 cm 8 cm 10 cm

10 2.46 x 102 4.92 x 102 7.38 x 152 9.85 x 102 1.23 x 103
1 2.46 x 101 4.92 x 101 7.38 x 101 9.85 x 10" 1.23 x 102
0.1 2.47 4.93 7.39 9.86 1.23 x 101

0.066 1.63 3.26 4.88 6.51 8.13

0.033 8.15 x 10'1 1.63 2.44 3.26 4.07

' 0.01 2.47 x 10'1 4.94 x 10'1 7.41 x 10'1 9.88 x 10"l 1.24
0.001 2.47 x 10'? 4.94 x 10'2 7.41 x 10’2 9.88 x 10'2 1.24 x 10'T

 

 

 

 

 

 

165

 

regime (0.75 — 25% error). Therefore, the analysis of accurate kinetic
information from zone profiles by ChromWin is limited to the time regime in which
0.2 < Da < 5 and, most importantly, is predicated upon the presence of plateau
profiles. However, the analysis of accurate kinetic information from zone profiles
by the plate height model is limited to the time regime in which 5 < Da < 100 and,
is predicated upon the presence of a single zone.
5.5.2 Experimental Studies
5.5.2.1 Effects of Electric Field Strength

Rotational isomerization about a peptide bond is a reversible. first-order
reaction. Consequently, it is necessary to perform separations under two’time
regimes for each temperature: one in which the individual isOmers are
distinguishable (Da < 1), and one in which only a single zone containing both
isomers is observed (Da > 5). Separations in the short-time regime (Da < 1) are
‘ necessary to determine the electrophoretic mobilities of the individual isomers as
well as for kinetic analysis via ChromWin. These separations are also useful for
evaluating equilibrium constants because (1c... 1133..., and .120... are all available
from the same experiment. Electropherograms obtained in the long-time regime
(Da .> 5) are representative of reactions near steady state and are used to
calculate rate constants with the plate height model [4.5].
5.5.2.1.1 Evolution of Reactive Zone Profiles
I FIGURES 5.3A and 5.38 illustrate the effect of separation time on the
zone profile at constant temperature (10 °C). The electropherogram in FIGURE

5.3A was achieved at 7500 V and Clearly shows both cis-Ala-Pro and

166

FIGURE 5.3:

Effect of separation time on electropherograms of mesitY‘
oxide and Ala-Pro. Separation conditions: 1.0 x 10"2 M
Na28407 buffer (pH 9.3), 10 °C, L... = 343111, L... = 25 cm.
id. = 75 nm. (A) 7500 v, (8) 500 v. UV absorbance

detected at 210 nm.

167

 

FIGURE 5.3A

.55 ms:

.4

L

 

 

r...

can?

osxo Eco:

 

 

 

 

 

168

FIGURE 5.38

00F

.EE. ms:
8. on

 

 

 

 

 

O._n.1m_<

 

 

case _aoos.

 

 

169

trans-Ala-‘Pro in the same zone, connected by a plateau. It is within this time
regime that the electrophoretic mobilities of the individual isomers can be
obtained. The electropherogram in FIGURE 5.38 was achieved at 500 V and
clearly shows only a single Ala-Pro zone. It is within this time regime that the
broadening of the Ala-Pro zone is proportional to the reaction lifetime.
5.5.2.1.2 Identification of lsomers

For both Ala-Pro and Phe—Pro, the faster velocity is attributed to the trans
isomer and the slower velocity to the cis isomer. This assignment presumes that
the cis form exhibits a smaller hydrodynamic radius than the trans form, whereas
'theelectric charge of the isomers is necessarily the same. Therefore, the cis
isomer has the larger charge-to-volume ratio and travels against the
electroosmotic flow with a greater velocity than the trans isomer. This
identification Of elution order is consistent with previous CE inVestigations Of
peptidyl-proline isomerization [1 1.15.16].
5.5.2.1.3 Comparison of Equilibrium Constants and Rate Constants «

TABLE 5. 2 Is a comparison of equilibrium constants and rate constants for
Ala-Pro and Phe-Pro at 10 °C from the plate height model, Chroman, and
reported literature values [11,16]. Errors are reported as the standard deviation
of replicate measurements. It is apparent that the equilibrium constants
determined by these different methods are in good agreement and that the
values calculated via the plate height model and ChromWin are statistically
indistinguishable. The equilibrium constants calculated via 'zone heights for

ChromWin are more precise than those calculated via the mobilities for the plate

170

Comparison of Equilibrium Constants and Rate Constants of Isomerization

TABLE 5.2

 

 

 

 

 

Ala-Pro Phe-Pro Method
1.42 (s 0.27) 7.12 (1 2.50) Plate Height Model
K 1.66 (1: 0.09) 6.00 (1: 0.08) ChromWin
1.20 5.66 [11] e
2.69 [16]
7 5.76 (1 0.30) x 10“ 1.06 (1 0.04) x 10‘3 Plate Height Model
k (8..). 6.24 (11.19) x 10‘4 5.60 (:l: 4.06) x 10'5 Chromwln
' 6.9x 10*4 5.6x 10“ [11]
‘ 1.73x10“ [16]
3.91 (110.20) x 10" 1.43 (1 0.05) x 10“l Plate Height Model
k (8-1) 3.76 (10.65) x 10*4 9.71 (:1 6.62) x 10’5 A ChromVVIn
' 7.4 x 10“ 9.65 x 10'5 [11]
6.42 x 10'5

 

 

 

[16]

 

Forward and reverse rate constants for Ala-Pro and Phe-Pro isomerization at 10

°C calculated by plate height model in the long time regime (Da > 5), ChromWin

in the short time regime (Da < 1), and reported values from [11] and [16].

Buffer conditions in this work: 1.0 x 10'2 M Na28407(pH19.3)

Buffer conditions for [11]: 7.0 x 10'2 M Na28407 (pH 9.5)

Buffer conditions for [16]: 1.0 x 10'1 M Na284Ov (pH 8.4).

171

 

height model. It is also apparent that there is excellent agreement among the
rate constants for Ala-Pro, but less so_for Phe-Pro. In part, this is due to the
smaller magnitude of the rate constants for Phe-Pro and the greater imprecision
in the measurements. The rate constants calculated via the plate height model
are more precise than those calculated via ChromWin under identical buffer and
temperature conditions. This relatively poor precision for ChromVIfin may arise
from uncertainty in determining plateau heights as result of the relatively large
equilibrium constant for Phe-Pro. Slightly different buffer and pH conditions are
used in references 11 and 16, which account for the small discrepancies
observed with the literature values. It is important to recall that the plate height
model was used to calculate rate constants from undistorted zones similar to
those in FIGURE 5.38. Conversely, each of the other methods calculated rate
constants from plateau profiles similar to those in FIGURE 5.3A.
5.5.2.2 Effects of Temperature

In order to evaluate the effect of temperature on equilibrium constant, it is
necessary to first evaluate the effect of temperature on electrophoretic mobility.
FIGURE 5.4 shows electropherograms of Ala-Pro achieved at temperatures of 9
- 25 °C. At lower temperatures there are two zones for Ala-Pro, the resolution of
which decreases with increasing temperature. ‘ This loss of resolution is
attributable primarily to two sources: rate constants generally increase and
equilibrium constants generally decrease with increasing temperature.
5.5.2.2.1 Electrophoretic Mobility V

FIGURES 5.5A and 5.58 show the mean electrophoretic mobilities for

172

FIGURE 5.4: Effect of temperature on electropherograms of mesityl oxide
and Ala-Pro. Separation conditions: 1.0 x 10’2 M Na28407
buffer (pH 9.3), L... = 34 cm, L3... = 26 cm, id. = 75 um, 7500

V. UV absorbance detected at 210 nm.

173

 

 

 

 

 

FIGURE 5.4

 

 

Mesityl oxid -
Ala-Pro
9 °C
, 10 °C w

 

15°C

20 °C

22 °C

 

 

 

 

TIME (min)

174

FIGURE 5.5:

Effect of temperature on (A) 11111111.-» (0). 113-$-11» (>< ). 3111A).
and (B) Wrens-FF (0)1 Hols-FF (X)1 1le (A) separation
conditions same as for FIGURE 5.4. Regression parameters
(slope, intercept, R2): 113333.11: (-3.54 x 106. 8.82 x 10“,
0.9995); .1....» (-3.53 x 10", 6.63 x 10“, 0.9998); .1113 (-3.44
x 10", 8.45 x 10“, 0.9999); .1....13 (-3.20 x 10‘, 7.77 x 10*,
0.9998); .13.... (-307 x 103. 7.29 x 104. 0.9994); .133 (-3.03 x

106, 7.16 x 10“. 0.9996).

175

 

FIGURE 5.5A

 

 

4:

 

 

N

0.

0°.

‘0.

V. ‘3! ' O.

(\I N \— \— ‘— r- \—
(SNZLUG ..OLX) MITIBOW OIL38OHdO8iO313

176

290 295 300
TEMPERATURE (K)

285

280

FIGURE 5.58

 

CD

 

 

 

or
N
. (

1 I 1 l

0 °C) ‘9. ‘2

“I

 

O.

N ‘- 1- ‘- 1— ‘—
SNzl-UO ..0 IX) AJJ'IIBOIN OLI.380Hd08.LO3'I3

177 '

290 295 300 305
TEMPERATURE (K)

285

280

Ala-Pro and Phe—Pro species. respectively, as a function of temperature. Error
bars are representative of the standard deviation of replicate measurements. In
both figures, it is apparent that the mobilities of the individual cis and trans
isomers (113.3311, (133.3311, (1.,-5-31:, and 113333.311) exhibit similar dependence on
temperature. Moreover, the mobility Of the Ala-Pro zone (.1111) approaches a
median value of (11.15.1111 + nmyz with increasing temperature, i.e. as K
approaches unity. A similar trend is observed for the mobility Of the Phe-Pro
zone (HFPI-

The electrophoretic mobility has a complex temperature dependence
arising from viscosity, dielectric constant, and ionic strength effects [17]. Yet the
dependence of Ala-Pro and Phe-Pro mobilities on temperature within this range
is well characterized by linear regression. Regression parameters are given in
the figure legend, with high correlation coefficients (R2 > 0.999). Consequently,
mobilities obtained at lower temperatures can be used to estimate electrophoretic
mobilities at higher temperatures, where the individual isomers are
indistinguishable.
5.5.2.2.2 Equilibrium Constant

FIGURE 5.6A shows the effect of temperature on equilibrium constant for
Ala-Pro and Phe—Pro isomerization. Error bars are representative of the
propagated error, as described previously in Chapter 3. This propagated error in
K is proportional to (.123... — .1....)2. Thus, the error in K is larger for Phe—Pro than
for Ala-Pro because the difference between (11:11 and (13,-3-11: is smaller than the

difference between 11.1..» and (13,-3.1.: owing to differences in their size. It is

178

FIGURE 5.6A: Effect of temperature on calculated equilibrium constants for
Ala-Pro (A) and Phe-Pro (0). Separation conditions as

described in text.

179

 

 

 

 

 

FIGURE 5.6A

 

 

 

 

 

 

7100

TIII

I

I I'll!!! T l

O ‘-

F

iNVlSNOO Wfll88l'lan3

[IIIITI l

180

 

‘_.
O

3.35 3.40 3.45 3.50 3.55
1 I TEMPERATURE (x10'3 K)

3.30

3.25

apparent that the slopes are positive and that the equilibrium constant
approaches unity for both Ala-Pro and Phe-Pro as temperature increases. It is
also apparent that the slope for Ala-Pro is less than for Phe-Pro. The more
positive slope for Phe-Pro indicates that the isomerization of Phe-Pro has a more
negative Change in molar enthalpy than the isomerization of Ala-Pro. The more
negative intercept for Phe—Pro indicates the isomerization of Phe-Pro has a more
negative change in molar entropy than the isomerization of Ala-Pro.
5.5.2.2.2.1 Molar Enthalpy and Molar Entropy

TABLE 5.3 summarizes the changes in molar enthalpy and molar entropy
for Ala-Pro and Phe-Pro calculated via Equations 33 and 34 (Chapter 1). Errors
represent the standard error from linear regression. It is apparent that trans-cis
isomerization for both Ala-Pro and Phe-Pro is enthalpically favorable and has a
strong entropically unfavorable contribution. This may be a result of a greater
number of polar solvent molecules associated with the cis isomer because of
decreased exposure of the non-polar side-chain at the a-carbon. It is also
apparent that the magnitude of the enthalpic and entropic contributions for Ala-
Pro are about one third of those for Phe—Pro. The entropic excess may be a
result of the bulky aromatic ring of phenylalanine displacing more solvent
molecules during the cis-trans rotation than the methyl group Of alanine. The
enthalpic excess may have contributions from the breaking of hydrogen bonds as
a result of displacing these solvent molecules during the rotation.
5.5.2.2.3 Rate Constants

FIGURE 5.68 shows the effect of temperature on the forward rate

181

TABLE 5.3

Themodynamic and Kinetic Parameters of Isomerization

 

Ala-Pro Phe—Pro
AH (kJ mol") -14.12 (:I: 0.09) -36.34 (:1: 0.63)
AS (J K" mol") -46.64 (:t 0.32) -111.62 (:t 0.21)
AEg1 (kJ mol") 17.20 (:I: 1.84) -23.48 (:t 2.78)
AESt (kJ mol") 31.27 (:I: 1.76) 12.86 (t 2.15)

 

 

 

 

 

 

 

 

 

Thermodynamic changes in molarenthalpy (AH) and molar entropy (AS) as well '
as molar activation energy for the forward and reverse reactions (AE? and AE.*)

for Ala-Pro and Phe-Pro.

182

' FIGURE 5.68: Effect of temperature on calculated forward rate conStants
for Ala-Pro (A) and Phe—Pro (0). Separation conditions as

described in text.

183

 

 

 

)nshnt

tions as

FIGURE 5.68

 

 

 

 

 

 

 

 

 

 

(.-s) J.NV.LSNOO 31V8

184

A
I
|__{ 1 4
l I n 1—
I V I
m
Tllll I l [TTIIII l l [IITIII l I
V'- ‘- T- V-
O O O O
0' 0 0 0
0' 0.: g
0' .
O

3.35 3.40 3.45 3.50 3.55
1 ITEMPERATURE (X10‘3 K)

3.30

3.25

constants for Ala-Pro and Phe-Pro isomerization. Error bars are representative
of the propagated error, as described previously in Chapter 3. This propagated
error in kg has multiple contributions from (13,-s, 1133..., and K. It increases with
(11m — 11...)", decreases as K approaches unity, and is proportional to the error
in K. Consequently, the propagated error in kg for Ala-Pro is substantially less
than that for Phe-Pro. It is also apparent that the slope for Ala-Pro is negative,
whereas the slope for Phe-Pro appears positive. The more negative slope for
Ala-Pro indicates that the trans-cis isomerization Of Ala-Pro has a more positive
activation energy than that of Phe-Pro. The positive slope for Phe-Pro intimates
a negative activation energy. suggesting a departure from Arrhenius behaviOr.

FIGURE 5.60 shows the effect of temperature on the reverse rate
constants for Ala-Pro and Phe—Pro isomerization. Error bars are representative
of the propagated error, as described previously in Chapter 3. Once again, it is
apparent that the error for Ala-Pro is less than that for Phe—Pro. This is for
reasons similar to those discussed above for the propagated error in kg. It is also
apparent that the slopes for both Ala-Pro and Phe-Pro are negative. The more
negative slope for Ala-Pro indicates that the cis-trans isomerization has a more
positive activation energy than that of Phe-Pro.
5.5.2.231 Molar Activation Energy

TABLE 5.3 summarizes the activation energies for the fonlvard reaction
(AE3) calculated via Equation 35 (Chapter 1). Errors represent the standard
error from linearregresSion. It is shown that the activation energy for Ala-Pro is

positive and that the molar activation enthalpy (AH = AE - RT)

185

FIGURE 5.6C: Effect of temperature on calculated reverse rate constants for Ala-
Pro (A) and Phe-Pro (0). Separation conditions as deScribed in

text.

186

 

 

 

FIGURE 5.6C

goose 83.255202.
men one mam 91m _ new one

new

 

 

f\
\l

 

 

 

TIIII

 

 

.50000

..000.0

..00.0

(.-S) .LNVLSNOO 31V8

187

contributes positively to the energy barrier for the trans-cis rotation. It is also
verified. as suggested above, that the activation energy for Phe-Pro is negative.
This type of departure from Arrhenius behavior has previously been attributed to
a mechanistic change in the reaction pathway for protein-folding kinetics [18]. In
the present case, it is unclear whether this trend is mechanistic or indicative Of a
temperature dependence of the pre-exponential factor in Equation 6. Moreover,
this activation energy includes zero at the 95% confidence level (AB = -23.48 :I:
35.48 kJ mol"). Thus, it would be unwise to attribute significant meaning to this
value.

The activation energies for the reverse reaction (AE3) calculated via
Equation 9 are summarized in TABLE 5.3. Errors represent the standard error
from linear regression. It is Shown that the activation energy for Ala-Pro is
positive and that the mOlar activation enthalpy contributes positively to the energy
barrier for the cis-trans rotation. It is also shown that the activation energy for
Phe-Pro is positive, but includes zero at the 95% confidence level (AE.* = 12.86 :t

27.44 kJ mol"). Thus. little meaning should be attributed to this value.

5.6 CONCLUSIONS

This is the first reported application and validation of the plate height
model for thermodynamic and kinetic studies in capillary electrophoresis. This
model has been used to calculate equilibrium constants and rate constants for
isomerization of two proline dipeptides. Equilibrium constants are calculated

directly from velocity, whereas rate constants are calculated directly from plate

188

height. The calculated values for the equilibrium constant and rate constants are
in good agreement with those calculated by ChromWin as well as published
literature values. It is shown that ChromWin and the plate height model are
complimentary methods for extracting rate constants. ChromWin calculates rate
constants more accurately for reactions that are far from steady state and exhibit
plateau profiles. Alternatively, the plate height model calculates rate constants

more accurately for reactions that are at or near steady state.

189

10
11
12
13
14

15

16

17

5.7 REFERENCES

Rathore, A. S.; Horvath, Cs. Electrophoresis 2002, 23, 1211

'Knox. J. H.; J. Chromatogr. A 1994, 680, 3

Giddings, C. J. Dynamics of Chromatography, Marcel Dekker, New York,
1965

Newman, 0. l. D.; McGuffin, V. L. Electrophoresis 2005, 26, 4016

Newman, 0. I. D.; McGuffin, V. L. Electrophoresis 2005, 26, 537

McGuffin, V. L.; Krouskop, P. E.; Wu, P. J. Chromatogr. A 1998, 828, 37
McGuffin. V. L.; Krouskop, P. E.; Hopkins, D. L. In Unified
Chromatography. ACS Symposium Series 748, Parcher, J. F., Chester, T.
L. Eds, American Chemical Society, Washington, DC, 2000

McGuffin, V. L.; Wu. R; Hopkins, D. L. In Proceedings of the lntemational
Symposium on Chromatography, Hatano. H., Hanai. T., Eds, World
Scientific Publishing, River Edge, NJ, 1995

Hopkins,_D. L.; McGuffin, V. L. Anal. Chem. 1998, 70, 1066

Krouskop, P. E. Ph.D. Dissertation 2001, Michigan State University
Schoetz. G.; Trapp, O.-; Schurig, V. Electrophoresis 2001, 22, 2409
Trapp. 0.; Trapp. G.; Schurig, V. Electrophoresis 2004, 25. 318

Trapp. 0.; Schurig, V. J. Chromatogr. A 2001, 911 , 167

Trapp. 0.; Schurig, V. Computers in Chemisry 2001, 25, 187

Ma, S.; Kalman, F.; Kalman, A.; Thunecke, F.; Horvath, Cs J. Chromatogr.
A 1995, 716, 167

Thunecke, F.; Kalman, A.; Kalman, F.; Ma, S.; Rathore, A. S.; Horvath.
Cs. J. Chromatogr. A 1996, 744, 259

Robinson, R. A.; Stokes, R. H. Electrolyte Solutions: The Measurement
and Interpretation of Conductance, Chemical Potential and Diffusion in

190

18

Solutions of Simple Electrolytes, Buttenlvorths Scientific Publications,
London,1995

Oliveberg. M.; Tan, Y. J.; Frescht. A. R. Proc. Natl. Acad. Sci. USA 1995,
92, 8926

191

CHAPTER 6: THEORETICAL PLATE HEIGHT MODEL
FOR THERMODYNAMIC AND KINETIC STUDIES OF I

REVERSIBLE, SECOND-ORDER REACTIONS

6.1 INTRODUCTION

The theoretical plate height model was shown in Chapter 3 to accurately
describe the contributions to zone broadening from diffusion and mass transfer
processes in reactive capillary electrophoresis (CE). The theOreticaI equations
developed in Chapter 3 illustrate that veloCity is independent of kinetic rate
constant and is a direct measure of equilibrium constant. It is also illustrated in
Chapter 3 that the mass-transfer contribution to plate height is inversely
proportional to rate constant. These relationships were validated in Chapter 3
with stochastic (Monte Carlo) simulations as well as in Chapter 5 with well-
characterized [isomerization of proline dipeptides. These successes
notwithstanding, prior application of the plate height model in CE has been
limited to reversible, first-order reactions. I

This chapter describes an attempt to apply the plate height model to a
reversible, second-order reaction. This class of reactions is ubiquitous in
chemical and biochemical systems including DNA intercalation, drug-protein
_ binding, cyclodextrin inclusion, and acid-base equilibria [1-9]. These reactions
are characterized thermodynamically by binding (K3) Or dissociation (K3)

constants and kinetically by association (kg) and dissociation (k3) rate constants.

192

Thermodynamic values are roUtinely calculated in CE from steady-state
measurements of peak heights or electrophoretic mobility. Jia recently reviewed
many of the methods used to perform these measurements, a number of which
are discussed in more detail in Chapter 1 [10]. However, the calculatIOn of rate
constants in CE has historically been limited to reactions that are far from steady
state and exhibit plateau profiles. . While effective, these measurements
inherently require that the plateau profiles be fit by one of a number of
simulations that iteratively estimate kinetic rate constants. These methods are
also described in more detail in Chapter 1. Each of these approaches inherently
assumes that the simulation algorithms accurately describe the kinetic
contributions to peak shape and also limits the range of calculable rate constants
to less than 1 s". Application of the plate height model to reactive CE has the
potential to overcome these limitations by allowing reactions to achieve steady
state, thereby removing any reliance on simulation methods for calculating rate
constants and simultaneously facilitating evaluations of more rapid reactions.
Previous research in our laboratory (unpublished) investigated the selective
Chromatographic retention Of several small drug and pesticide enantiomers by B-
cyclodextrin. The dissociation rate constants for the retained solutes were found
to range between 10 and 50 s"1 at room temperature. These rate constants are
sufficiently rapid to preclude plateau profiles, but still expected to be within the
range of measurable rate constants via the plate height model.

Cyclodextrins are a popular class of compounds known to undergo

selective complexation with a broad range of molecules. Cyclodextrins are

193

torroidally shaped polysaccharides Composed of D-glucose monomers
connected at the 1 and 4 carbon atoms that exhibit a non-polar cavity and a polar
exterior. The a, B. and y-cyclodextrins are composed of 6, 7, and 8 D-glucose
units and are characterized by cavity diameters of 4.7, 6.3. and 8.3 x 10'10 m.
respectively [11]. This combination Of physical and chemical properties enable
cyclodextrins to selectively form inclusion complexes with a wide range of
molecules. The ability to form inclusion complexes that are soluble in water
positions cyclodextrins as ideal vessels to improve the solubility and
bioavailability of pharmaceutical drugs [12].

The thermodynamics of cyclodextrin complexation have been studied
extensively via numerous methods. In 1998 Rekharsky and lnoue published a
lengthy compilation of cyclodextrin complexation thermodynamic data published
or in press as of October 1997 [13]. Their results showed that structural and
electronic attributes of the guest molecule such as hydrophobicity, shape,
hydrogen bonding ability. aromatic substitution. and flexibility strongly affect the '
complexation thermodynamics. It was also concluded that only the portion of the
guest molecule that “senses” a change in environment upon inclusion contributes
appreciably to the overall thermodynamics of complexation. Although a small
fraction of the reported data were collected by Chromatographic methods, and
most were collected spectroscopically or by calorimetry. none were collected -
electrophoretically.

However, the use of CE to investigate thermodynamics of cyclodextrin

complexation has been the topic of numerous research articles. Lemesle—

194

Lamache et al. used CE to investigate the complexation of native and derivatized
B-cyclodextrin with a bronchodilation stimulant [14]. Penn et al. investigated the
complexation of native and derivatized a— and B-cyclodextrins with a series of
nitrophenolates [15]. Davis and co-workers investigated the complexation of
native B-cyclodextrin with neutral and charged aliphatic dansyl amino acids,
benzoate, and 4-nitrophenolate [9]. Additionally, applications of cyclodextrins for
chiral CE separations have been discussed in several reviews [16-19]. This
wealth of thermodynamic data notwithstanding, there is a noticeable lack of
literature regarding the kinetics of cyclodextrin complexation. It is the intent of
the work discussed in this chapter to employ CE to investigate therm‘odynamics
and kinetics of cyclodextrin complexation via application of the theoretical plate

height model.

6.2 THEORY
Reversible, second-order complexation reactions can be expressed by the

general chemical equation

ka
(1)

 

where A and 8 are reactants and AB is the complex of A and 8, k. is the rate
constant for association, and k3 is the rate constant for dissociatiOn. The reaction

in Equation 1 may be characterized by the equilibrium expression

k. [AB]

“:Tnli ‘2’

195

where K3 is the equilibrium constant for association. The approximation arises
from the assumption that activity coefficients for A, B, and A8 are e 1. The rates

for association (ra) and dissociation (rd) may be expressed as

ra - ka [A] [6] (3a)

rd .. k... [AB] (3b)
Thus, the rate of association is second-order and is dependent upon [A] and [8],
whereas the rate of dissociation is first-order and independent of [A] or [8].

Moreover, if [8] (or [A]) is sufficiently large or buffered such that it may be

considered constant, then K3 and k. may be approximated as a pseudo-first-

Order quantities
K'b- K11 [B] (4)
k'a-ka [B] (5)

Therefore, controlling [8] facilitates simultaneous control of k’.. as well as the

equilibrium proportions of A (GA) and A8 (011(3)

GA - A] - 1 (6a)
A + AB 1+Kb|BI

a _ [AB] - Kb [B] (6b)

AB |A|+|A8| 1+Kb|8l

The studies inclusion of dansyl phenylalanine by B—cyclodextrin discussed in this

chapter demonstrate reactions in which pseudo-first-order conditions are

imposed by maintaining cyclodextrin concentrations in exceSs of dansyl

phenylalanine. ' Investigations of the equilibrium constants and rate constants for

196

this system were performed as a detailed suite of experiments evaluating the

effects of electric field strength, concentration, and temperature.

6.2.1 Velocity and Equilibrium Constant

In capillary electrophoresis, the Observed velocity (V333) of the zone
representative of species A interconverting between the free (A) and complexed
states (AB) is equivalent to the weighed average of the individual velocities of A
(VA) and AB (v.13) by
Volos - 0LAVA + GABVAB (7)

Substitution of Equations 6a and 6b into Equation 7 yields

VA + VAB Kb [3]

1+Kb[8] 7 (8)

 

Vobs "

from which K3 may be calculated by evaluating v33. as a functiOn of [8], provided
VA and v1.3 are known.
6.2.2 Plate Height and Rate Constant

Whereas velocity alone provides a reasonable description of equilibrium
constant, it is not sufficient for describing the rates of reaction. Zone broadening
on the other hand is directly related to resistance to mass transfer and is
therefore a viable means for evaluating rate constants [20.21]. Application of the

plate height model deVeloped in Chapter 3 to the equilibria in Equation 1 yields

2

_ 2K3 [8](VA — 11.3)2
Vobs

kd (1+ Kb [8])3 va8

DA , +DABKDIB:
1+K..[B] 1+K.,[B]

 

H (9)

 

 

 

 

 

197

where, DA is the mobile-phase diffusion coefficient of A, D13 is the mobile-phase
diffusion coefficient of A8, and the quantity in brackets represents the weighted
average of the diffusional contributions to plate height. Thus, accurate evaluation
of kinetic rate constants may be evaluated from the plate height. but require a

priori knowledge of K3, DA, DAB, VA, and v.13.

6.3 EXPERIMENTAL METHODS
6.3.1 Chemicals
Dansyl amide (DnsA), dansyl phenylalanine (DnsF) and B—cyclodextrin (80D)
were purchased from Sigma. FIGURE 6.1 shows the molecular structures of
DnsA, DnsF, and BCD. Sodium tetraborate decahydrate I (Na28407) was
purchased from MCB Manufacturing Chemists. Electrophoresis buffer solutions
containing 0 - 1.5 x 10'2 M 8CD were prepared with 1.0 x 10'2 M Na28407 (pH
9.3) in distilled, deionized water. Analytical solutions of 5.0 x 10‘5 M DnsF were
prepared in each buffer solution with 4.0 x 10'5 M DnsA as the electroosmotic
flow marker. These experiments were performed at 283, 285, 288, and 293 K.
6.3.2 Experimental System

Separations were achieved using the capillary electrophoresis system with
laser-induced fluorescence detection described in Chapter 4 and depicted in
Figure 4.18. Fluorescence excitation was achieved at 1.3.31. = 325 nm and

emission was monitored at A33... = 570 nm.

198

FIGURE 6.1: Molecular structures of dansyl amide. dansyl phenylalanine.

and B-cyclodextrin.

199

FIGURE 6.1

\N/ \N/
NH2 . HN /C
Dansyl amide ' \TH \OH
CH;

Dansyl phenylalanine

 

o
’16
19 9°
0
go
0
0 o
0
B—cyclodextrin

200

6.3.3 Data Analysis

Zone profiles were extracted from electropherograms with commercially
available software (Origin, version 7.5, OriginLab) and iteratively fit for calculation
of statistical moments using the procedures described in the Data Treatment and
Analysis in Chapter 4 (Section 3). The presence of multiple detector locations for
the laser-induced fluorescence system enables more accurate calculation of

velocity and plate height by difference between detectors

H_(Zz'z1)[MzI2"Mz)1] . (11)
[~01 -M.).i

 

where 21 and 22 are the distances todetector locations 1 (22.73 cm) and 2 (38.05
cm), M1). and M1)2 are the zone means at z. and 22. respectively, and M2). and
M2)2 are the zone variances at 21 and z2, respectively.) By calculating velocity
and plate height between detectors, extra-column contributions (Chapter 1.

Section 1.2.4) to mean and variance are isolated and do not contribute to the

calculated equilibrium constants or rate constants.

6.4 RESULTS AND DISCUSSION
6.4.1 Velocity and Equilibrium Constant
6.4.1.1 Effects of Electric Field Strength and Concentration on Velocity
In order to accurately determine solute electrophoretic mobility and
equilibrium constants for [30D inclusion, separations were performed at electric

field strengths (E) Of 418, 368, and 334 Vlcm (25, 22, and 20 kV, respectively).

201

 

FIGURE 6.2 shows representative electropherograms of DnsA and DnsF with 0 -
1.0 x 10'2 M 6CD obtained at detector 2 with 416 Vlcm at 265 K. For each
separation samples are introduced at the anode, electroosmotic migration is
towards the cathode, and the electrophoretic migration of DnsF is towards the
anode. As expected, the velocity of DnsF increases with 8CD concentration
whereas the velocity of DnsA varies little with 6CD concentration.

Addition of 80D to the electrophoresis buffer is known to effect buffer
viscosity directly as well as indirectly (via buffer conductivity), thereby resulting in
uncontrollable influences on both electroosmotic and electrophoretic velocities
[10.14.15]. When unaccounted for, these additional contributions to velocity
have been shown ‘to result in underestimations of K3 [15]. It has also been
shown that the ratio of buffer viscosities with and without BCD (n and 110,
respectively) is linear with the ratio of separation current with and without BCD (i
[and in, respectively) [14]. Thus, the complicated effects of BCD on buffer
viscosity and, hence, electrophoretic velocities (v... = v333-v33) may be corrected
via [14.15]

V' 311' vep(-,i—) (12)

I0
where v33 is the electroosmoticvelocity and v’ep is the corrected electrophoretic
velocity.

FIGURE 6.3 shows the effect of electric field strength and BCD
concentration on v”... at 285 K. It is apparent that v’3p is approaches a minimum

when [80D] < 1 mM, experiences the greatest increase between 1 < [8CD] < 4

202

FIGURE 6.2:

Effect of [ICU concentration on electropherograms of DnsA
and DnsF. Separation conditions: L... = 59.85 cm, L3... =
36.05 cm. 265 K10 x 10'3 M Na28407 buffer (pH 9.3), v =

25 W, [660] = o - 1.0 x 10'2 M. Fluorescence detection

conditiOns: 1.3.3.. = 325 nm, A3"... = 570 nm.

203

 

ns of W

Ii 9.3))“

e detect?“

FIGURE 6.2

 

 

 

_<_—g
_. J

' 2'

E

'42

O

 

 

1mM
2mM

 

4mM

 

10mM

 

 

204

TIME (min)

FIGURE 6.3:

Effect of 8CD concentration on electrophoretic velocity at
334 (O), 368 (A), and 416 (El) Vlcm. Separation conditions:
L... = 59.65 cm. L... = 38.05 cm. 285 K, 1.0 x 10'3 M Na28407

buffer (pH 9.3), [BCD] = o - 1.0 x 10'2 M. Fluorescence

detection conditions: 1.3.31. = 325 nm. 1.3.31. = 570 nm.

205

FIGURE 6.3

5.0

3.95 .00...
See

_._..P.

5000

5000.0

 

 

...0-

1 00.01
1 00.01
1 N001
1 00.01
1 00.01
1 .4001
1 00.01

1 No.01

 

 

..0.01

(S/UJO) A.L|3013/\ OIJ.38OHdO8.LO313

206

mM, and increases less when [BCD] > 4 mM. This sigmoidal dependence of v’33
on 8CD concentration can be generally described by three regions. The region
in which v’.... is a minimum ([800] < 1 mM) corresponds to DnsF traveling
predominantly as free DnsF. The region in which v’.p increases most with BCD
(1 < [8CD] < 4 mM) corresponds to increasing proportions of DnsF traveling as
~ both the free and BCD inclusion complex. The region in which v’33 begins to
approach a maximum ([BCD] > 4 mM) corresponds to DnsF traveling
predominantly as the BCD inclusion complex. The inflection of the curve
corresponds to DnsF traveling equal prOportion's of time as both free DnsF and
the BCD inclusion complex. Therefore, the concentration of [ICU at the inflection
is 1/K3, such that the pseudo-first-order equilibriuchnstant is unity (Equation 4).
6.4.1.2 Effects of Temperature and Electric Field Strength
on Equilibrium Constant

Corrected electrophoretic velocities were used to calculate equilibrium
constants (K3) fOr inclusion "via non-linear regression to Equation 8. TABLE 6.1
summarizes the equilibrium constants Obtained at 283, 285, 288, and 293 K for
each applied electric field strength as well as the average equilibrium Constant
(ng) and standard deviation for each temperature. These values are in good
agreement with previously reported binding constants for BCD with dansyl amide
and dansyl amino acids (1 — 4 x 102 M") [9.11.13]. It is apparent that the data
show excellent agreement between equilibrium constants calculated at the same
temperature for each electric field Strength. Although the data shows minor

increases in equilibrium constant with electric field strength, evaluation of the F

207

statistic for each temperature indicates that this trend is not statistically
significant. It is also apparent that the data also Show a general trend of
decreasing equilibrium constant with increasing temperature.
6.4.1.2.1 Molar Enthalpy and Molar Entropy

The effects of temperature on K3... were used to calculate the molar
enthalpy (AH) and molar entropy (AS) for inclusion via the standard Gibbsian
thermodynamic relationships described in Chapter 1 (Section 2.1). FIGURE 6.4
is a plot of In K3... vs 1fl'. It is apparent that the plot is linear with a positive sIOpe
and a positive intercept. The molar enthalpy for inclusion is calCulated from the
slope as -12.66 kJ/mol (:I: 0.87 lemol); the molar entropy for inclusion is
calculated from the intercept as -2.19 J/K-mol (i 3.06 JlK-mol). These data show
that BCD inclusion Of DnsF is enthalpically driven and has a competing entropic
contribution, in good agreement with previously reported trends [11 .13].
6.4.2 Plate Height and Rate Constants

I In order to differentiate the contributions of diffusion and mass

transfer to plate height it is necessary to estimate the diffusion coefficients DA
and DAB in Equation 9. Electrophoretic mobility can be used to estimate diffusion
coefficients via the Einstein relation (D°. = (1°.RTIzF). However, this equation is
only accurate at infinite dilution and zero ionic strength and, thus, does not fully
account for the dependence of diffusion coefficient and electrophoretic mobility
on temperature and ionic strength. As discussed in Chapter 1, Li et al. have
adapted the detailed conductivity theory of Pitts to more accurately describe the

dependence of electrophoretic mobility on temperature and solution electrical

208

TABLE 6.1

Calculated Equilibrium Constants for BCD Inclusion of DnsF.

 

 

 

 

 

 

 

 

 

 

 

 

334 Vlcm 358 Vlcm 4.18 Vlcm
283 K 278 277 280 278 (d: 1)
285 K 268 273 277 ‘ 272 (:t 4)
288 K 259 261 263 261 (:I: 2)
293 K 228 232 237 233 (:l: 4)

Equilibrium constants for BCD inclusion of DnsF calculated from separations as a

function of electric field strength and temperature.

209

FIGURE 6.4: Effect of temperature on calculated equilibrium constants for
8CD inclusion of DnsF. Experimental conditions same as

given for FIGURE 6.3.

210

FIGURE 6.4

V0.0

googmmazmmosmt. _
em mew .. can 33

h n — n

. Ned

v.0

 

 

 

II<

 

000 F

 

00F .

.LNVLSNOO INI'II88I'IIIIO3

211

properties [9,22-24]. From their description of electrophoretic mobility, 11°. can be
calculated iteratively via the self-consistent algorithm developed by Survey et al.
[9.25], which may then be used to calculate D01. Diffusion coefficients may then
be corrected for temperature and ionic strength via Onsager’s treatment for
diffusion of ions at near-infinite dilution discussed in Chapter 1 (Equations 7, 8,
12, and 13) [9.26].

TABLE 6.2 lists the diffusion coefficients of free and bound DnsF with (Do.
and D13) and without (DA and D’AB) Proper corrections for temperature and ionic
strength. It is apparent that direct application of the Einstein relation to
electrophoretic mobility tends to overestimate DA and D113 by about 1%.
However, this relatively small discrepancy translates into > 5% error in calculated ,
rate constants when the mass transfer contribution to plate height is < 20 times
the contribution from diffusion.
6.4.2.1 Effects of Electric Field Strength and Concentration on

Plate Height

FIGURE 6.5A shows the effects of electric field strength and [30D
concentration on H at 285 K. It is apparent that plate height changes little when
[BCD] < 1 mM, is generally slightly greater for intermediate values of [8CD] (2 - 5
mM), and decreases with [3CD] > 5 mM. However, the error associated with
these values is quite large. obscuring any Significance of these trends. It is also
apparent that plate heights obtained at 334 Vlcm are generally greatest, whereas
those obtained at 418 Vlcm are generally least. This means that plate height is

larger at slower velocities than at faster velocities, intimating that diffusion is a

212

TABLE 6.2

Calculated Diffusion Coefficients of Free and Bound DnsF.

 

283 K

285 K

288 K

293 K

 

*—=——-F
10 04cm Is)

2.35 (+ 0.01)

2.43 (+ 0.01)

2.69 (+ 0.01)

3.15 (+ 0.02)

 

1 OWD’A (cm2/s)

2.38 (+ 0.01)

2.46 (+ 0.01)

2.72 (+ 0.01)

3.19 (+ 0.02)

 

%Error in D’).

1.06

1.11

1.19

1.37

 

10'7 013(cm’rs)

7.20 (+ 0.26)

7.84 (+ 0.20)

8.45 (+ 0.21)

10.1 (+ 0.33)

 

10'7 0’13 (cm2/s)

7264+ 0.26)

7.93 (+ 0.20)

8.55 (+ 0.22)

10.2 (+ 0.34)

 

 

%Error in D’AB

 

1.06

 

1.11

 

1.19

 

1.37

 

Diffusion coefficients of free and bound DnsF with (D1. and DA...) and without (DA

and D’AB) corrections for temperature and ionic strength at 283, 285, 288, and

293 K.

213

 

FIGURE 6.5A:

Effect of BOD concentration on plate height at electric field
strengths Of 334 (O), 368 (A), and 416 (El) Vlcm.
Separation conditions: L... = 59.85 cm, L... = 38.05 cm, 285
K, 1.0 x 10'3 M Na28407 buffer (pH 9.3), [BCD] = 0 - 1.0 x
10'2 M. Fluorescence detection conditions: A3,... = 325 nm,

Aemit = 570 nm.

214

FIGURE 6.5A

 

3.68 .00...
..o 8.0 .86 88.0 88.0
.F.p. EF-._. 3 H ...br. . . L .....hb _ r 00000.0
- «888
d
.. 880.0 m
....—
. mm
- 088 o m
.. H
.. .. In... L
14.31.. .. .Ii -“.. . - 88cc )
ffieii ._ ...
. . 289°

 

 

 

N..000.0

215

. more dominant contribution to broadening than mass transfer from reaction.

FIGURE 6.53 shows the effects of electric field strength and pCD
concentration on H corrected for diffusion at 285 K. It is shown in Equation 4 that
[300 concentration is proportional to K’b, thus FIGURE 6.58 also shows the
effects of K’b on H corrected for diffusion. It is apparent that plate height is least
when [BCD] < 1 mM (K’b < 0.27) and greatest when [BCD] > 1 mM (K’b > 0.27). It
is also apparent that plate height changes little within these regimes. This is in
contrast to the trends shown in FIGURE 3.4C (Chapter 3) where the mass
transfer contributions to plate height are greatest at K’b = 1 (here, [fiCD] = 3.7
mM) and decreases monotonically as K deviates from unity. Thus intimating that
the dominant contribution to broadening is from neither diffusion nor mass
transfer.
6.4.2.2 Effects of Temperature and Electric Field Strength on

Rate Constants

Ignoring the obvious lack of statistical significance or dependence of plate
height on [500] for these data, one can attempt to use plate heights corrected for
diffusion to calculate rate constants for dissociation (kd) via Equation 9 and
association (kg) via Equation 2. TABLE 6.3 summarizes the average rate
constants (km,g and Ram) and standard deviations obtained at 283, 285, and
288 K. It is apparent from these data that kd are all roughly 20 -~ 30 s", increase
marginally with E, and remain essentially constant with temperature. This trend
is in contrast to the “rule-of-thumb” expectation that rate constants typically

double for every increase of 10 K. Moreover, it was discovered sometime after

216

FIGURE 6.58:

Effect of 500 concentration on plate height corrected for
diffusion at electric field strengths of 334 (O), 368 (A), and
416 (El) Vlcm. Separation conditions: Ltd a 59.85 cm, Ld.‘ =
38.05 cm, 285 K, 1.0 x 10'3 M NazB407 buffer (pH 9.3),
[BCD] = O - 1.0 x 10'2 M. Fluorescence detection conditions:

km“ = 325 nm, Item“ = 570 nm.

217

FIGURE 6.58

_..o

l-P-lb - p

3:25 303
:3 god 58o

...-L... - in.-.. . —.b_.-.. .

5000.0

 

 

..u\; ..
.1 .... \t
.W...‘.:
1:3 l \
51‘ . “V
A n. “my: nn’...
, H! II I.
aux
as

o

1 525.0

- Nooood

r mooood

.. Vooood

- mooood

I 000006

- hooood

 

 

mooood

(we) .LHSIEIH 5mm!

218

TABLE 6.3

Calculated Rate Constants for BCD Inclusion of DnsF.

 

kwg (M'iI S") kd.a_vg (8'1)

283 K 6.42 (+ 0.44) x 103 2.31 (4- 0.16) x 101
285 K 6.41 (+ 0.59) x 103 2.36 (+ 0.26) x 101
288 K 5.77 (+ 0.49) x 103 2.21 (+ 0.20) x 10'r

 

 

 

 

 

 

 

 

Average rate constants for association (kam) and dissociation (Ram) of fiCD
inclusion of DnsF calculated from separations at 334, 368, and 418 Vlcm as a

function of temperature.

219

completion of these studies that theserate constants are ."103 times slower than
those reported for the dissociation of [3CD with cobaltocene and ferrocene [27-
29]. It is also apparent that all k, are > 5.0 x 103 M'1 s“, increase marginally with
E, and decrease as temperature increases. This trend arises in the data strictly
as a consequence of calculating ka via Equation 2. Moreover, these rate
constants are 104 times slower than those reported for the dissociation of BCD
with cobaltocene and ferrocene and 103 times slower than those expected

assuming kd ~ 10‘ 5".

Thus, although the calculated equilibrium constants
compare reasonably with literature values, the calculated rate constants do not,
because mass transfer is not a significant contribution to broadening.

Using the calculated values of k, and kd, the characteristic reaction
lifetimes near K’b~ 1 are on the order of 10 — 20 ms, resulting in Damkohler
numbers > 1000. As shown in Chapter 5 (FIGURE 5.13) the plate height model
cannot be used to accurately determine rate constants from Damkohler numbers
> 100 without correcting for diffusion. _However, correcting for diffusion only
extends the viability of the plate height model by an order of magnitude to
Damkohler numbers 5 1000. Therefore, the calculated values for ka and kd are

not only unreasonable in contrast to published values for. BCD inclusion but also

unreliable with respect to the experimental limitations of the plate height model.
6.5, CONCLUSIONS

This chapter details the first attempted application of the plate height

model to the reversible, second-order inclusion of dansyl phenylalanine by B-

220

cyclodextrin. Separations were performed with a dual, on-column laser-
fluorescence detection CE system with varying concentrations of BCD over a
range of temperatures and electric field strengths. The plate height model was
used to calculate thermodynamic equilibrium constants and kinetic rate constants
for inclusion from 283 - 293 K.

Equilibrium constants are calculated by evaluating the effects of [SOD
concentration on velocity. It is shown that the electrophoretic velocity of DnsF
increases sigmoidally with BCD concentration _in agreement with the observed
dependence of velocity on equilibrium constant shown in Chapter 3 (FIGURE
3.3). This trend was used to elucidate equilibrium constants via non-linear
regression to Equation 8. The calculated equilibrium constants, molar activation
enthalpy, and molar activation entropy {are shown to be in agreement with
previously reported values for BCD inclusion of dansyl amino acids.

Rate constants are calculated by evaluating the effects of BCD
concentration on plate height. It is shown that plate heights are largely
unaffected by BCD concentration. It is also shown that plate heights generally
increase as electric field strength decreases, intimating that diffusion is a more
dominant source of broadening than mass transfer from reaction. Plate heights
were corrected for diffusion to increase the accuracy of calculated rate constants;
however, the rate constants determined via the PHM were ~103 times slower
than the only other reported rate constants for BCD inclusion in aqueous buffers.

It is presumed that the calculated rate constants for BCD inclusion of DnsF

are grossly underestimated by the plate height model because of both inherent

221

physical limitations of the experimental system as well as inherent physical
. limitations in the mathematical model. It was shown in Chapter 3 that the range
of calculable rate constants by the plate height model under ideal chemical
conditions (K’b = 1) is limited by: 1) the amount of time molecules have had to
react prior to detection; and 2) the difference between the velocities of products
and reactants. For the 3CD inclusion of DnsF, velocity increased with Kb’;
however, the differences between free and bound DnsF are quite small (< 0.05
cmls). This means that even using the maximum achievable electric field
strength (501 Vlcm), the fastest rate constants calculable for the DnsF-[3CD
inclusion system are ~20 — 30 s". Thus, calculation of rate constants in excess
of this limit requires faster separations (smaller Damkohler numbers) via shorter
detection distances, larger electric field strengths, or both.

The most likely method to meet these requirements is to perform the
separations on chip. The primary advantages of chip CE over conventional CE
are shorter separation channels (3 - 25 cm) and a larger range of electric field
strengths (~1 kV/cm). Chip CE also facilitates detection at shorter detection
distances than conventional CE systems, thereby decreasing diffusional
contributions to broadening. However, unreasonably large electric field strengths
(> 5 kV/cm) are necessary in order to access rate constants near 104 s”. Thus,
even given optimal instrumentation, this reaction is likely still too fast to calculate
rate constants from the plate height model with any degree of reliability.
Moreover, it is difficult to predict successful application of the plate height model

for a particular reaction without some a priori knowledge of electrophoretic

222

mobilities and diffusion coefficients in addition to a reasonable estimation of at
least one of the rate constants. The effects of these limiting factors on the

maximum calculable rate constants are evaluated in detail in Chapter 7.

223

10
11

12

13

14

15

16

17

6.7 REFERENCES

Maiti, S.; Chaudhury, N. K.; Chowdhury, S. Biochem. Biophys. Res.
Comm. 2003, 310, 505

Jia, 2.; Ramstad, T.; Zhong, M. Electrophoresis 2001, 22, 1112
Jia, Z.; Ramstad, T.; Zhong, M. J. Pharm. Biomed. Anal. 2002, 30, 405

Ostergarard, J.; Schou, G; Larsen, C.; Heegaard, N. H. H.
Electrophoresis 2002, 23, 2842

Ohnishi, T.; Mohamed, N. A. L.; Shibukawa, A.; Kuroda, Y.; Nakagawa, T.;
Gizawy, S. E.; Askal, H. F.; Kommos, M. E. E. J. Phann. Biomed. Anal.
2002, 27, 607

Baumy, P.; Morin, P.; Dreux, M.; Viaud, M. C.; Boye, S.; Guillaumet, G. J.
Chromatogr. A. 1995, 707, 311

Larsen, K. L.;Zimmem1ann, w. J. Chromatogr. A. 1999, 836, 3
Guillaume, Y. C.; Peyrin, E. Anal. Chem. 1999, 71, 2046'

Seals, T. H.; Sheng, C.; Davis, J. M. Electrophoresis 2001, 22, 1957
Jia, 2. Curr. Phann. Anal. 2005, 1, 41

Catena, G. C.; Bright, F. V. Anal. Chem. 1989, 61 , 905

Stella, V. J.; Rao, V. M.; Zannou, E. A.; Zia, V. Adv. Drug Deliv. Rev.
1999, 36, 3

Rekharsky, M. V.; lnoue, Y. Chem. Rev. 1998, 98, 1875

: Lemesle-Lamache, V.; Taverna, M.; Wouessidjewe, D.; Duchene, D.;

Ferrier, D. J. Chromatogr. A 1996, 735, 321

Penn, S. G.; Bergstrom, E. T.; Knights, l.; Liu, G.; Ruddick, A.; Goodall, D.
M. J. Phys. Chem. 1995, 99, 3875

Chu, Y.-H.; Cheng, C. C. Cell. Mol. Life Sci. 1998, 54, 663

Britz-McKibbin, P.; Chen, D. D. Y. Electrophoresis 2002, 23, 880

224

18

19
20
21
22
23
24
25

26

.27.

28
29

Kapnissi-Christodoulou, C. P.; Zhu, X.; Warner, I. M. Electrophoresis
2003, 24, 3917

Gubitz, G.; Schmid, M. G. Electrophoresis 2004, 23, 3981
Newman, C. I. D., McGuflin, V. L. Electrophoresis 2005, 26, 573
Newman, C. l. D.‘, McGuffin, V. L. Electrophoresis 2005, 26,4016
Li, D.; Fu, 8.; Lucy, C. A. Anal. Chem. 1999, 71,687

Pitts, E. Proc. R. Soc. A 1953, 217,43

Pitts, E.; Tabor, B. E.; Daly, J. Trans. Faraday Soc. 1970, 66, 693

Survay, M. A.; Goodall, D. M.; Wren, S. A. C.; Rowe, R. C. J. Chromatogr.
A 1996, 741, 99

Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, The Measurement
and Interpretation of Conductance, Chemical Potential and Diffusion in
Solutions of Simple Electrolytes, Butterworths Scientific Publications:
London,1955

lsnin, R.; Salam, C.; Kaifer, A. E. J. Org. Chem. 1991, 56, 35

Wang, Y.; Mendoza, S.; Kaifer, A. E. Inorg. Chem. 1998, 37, 317

Guyard, L.; Hapiot, P.; Jouini, M.; Lacroix, J.-C.; Lagrost, C.; Neta, P. J.
Phys. Chem. 1999, 103, 4009

225

CHAPTER 7: THEORETICAL LIMITATIONS OF THE

PLATE HEIGHT MODEL

7.1 INTRODUCTION

The theoretical development in Chapter 3 showed that mass-transfer
limited plate heights are inversely proportional to rate constant. It was also
shown in Chapter 3 that plate height is mass-transfer limited when the
contributions from diffusion are 5 20% the total plate height. As demonstrated
previously, the diffusional contribution to plate height is dependent upon physical
parameters of the solutes such as diffusion coefficient (Chapter 3, Equation 2)
and electrophoretic mobility (Chapter 1, Equation 7 and Chapter 3, Equations 1
and 22) in addition to physical parameters of the experimental system such as
temperature (Chapter 1, Equation 7) and ionic strength (Chapter 1, Equation 12)
as well as electroosmotic mobility and electric field strength (Chapter3, Equations
1 and 22). It was also shown in Chapter 3 (Equations 20 and 22) that the mass—
transfer contribution to plate height is dependent upon physical parameters such
as electric field strength; electroosmotic mobility and electrophoretic mobility as
well as chemical parameters such as equilibrium constant and rate constant.
Therefore, the maximUm rate constants calculable via the plate height model will
vary with both the physical and chemical attributes of the individual system

investigated.

226

7.2 THEORY
In order to evaluate the individual physical and chemical contributions to
the maximum calculable rate constant, it is helpful to consider their effects on a

normalized plate height (h)

hJ. (1)

Hb
where H is the total plate height from both longitudinal diffusion (H.,) and mass-
transfer (H.,).

For the reversible, first-order reaction

e A B (2)

 

kAB and kBA are the rate constants for transfer from species A to B and from
species B to A, respectively. From Chapter 3, the contribution to plate height

from longitudinal diffusion is described by

 

(3)

H -—--—
b .v v1+K+1+K

2D 2( D A K03)
where v is the observed velocity of the reactive zone, D is the observed diffusion
coefficient of the reactive zone, DA is the diffusion coefficient of species A, D3 is

the diffusion coefficient of species B, and K is the equilibrium constant for the

reaction in Equation 2.

227

The contribution to plate height from mass transfer is described in Chapter
3 by
21(sz

H a
c kBAv(1+K)3

 

(4)

where Av is the velocity difference between species A and B (VA and v3,
respectively). Therefore, h approaches unity when H is diffusion limited and
increases with increasing contributions from mass transfer.

In electrophoretic separations, velocity is dependent upon the applied
electric field strength (E), the electroosmotic mobility (Pea). and the

electrophoretic mobility of the individual species (HA and us) such that
VA -E(p.eo+p.A) (5)
v3 -E(ueo+u3) (6)

Analogously to diffusion coefficient (Equation 3), the observed velocity of the

reactive zone is the weighted average of VA and v3

-VE+KVB- - 119+ng 7
V 1+K E(”e°+ 1+K H

In order for the reaction in Equation 2 to be evaluated by the plate height
model one or both species must exist in the mobile phase. If both A and B exist
in the mobile phase, then VA and v3 are non-zero and the reaction is‘evaluated
with the plate height model via capillary electrophoresis (CE). However, if one
species is in the mobile phase (A) and the other is immobilized (B), then only VA
is non-zero and the reaction is evaluated with the plate height model via capillary

electrochromatography (CEC). Consequently, the mass-transfer contributions to

228

plate height in CE and CEC influence the maximum calculable rate constants
(kmax) differently. Therefore, the theoretical limitations for CE and CEC are not

necessarily identical and should thus be evaluated separately.

7.3 CAPILLARY ELECTROPHORESIS

As stated previously, theoretical limitations to evaluating reactions via the
plate height model in CE are dictated by the physical parameters of the
instrumentation and solutes as well as by the chemical parameters of reaction.
Each of these factors influences H, and He differently and is therefore considered
individually. Values of km, are identified by h = 5, the minimum value at which
plate height is mass-transfer limited and inversely proportional to rate constant.
7.3.1 Effect of Electrophoretic Mobility Difference on km in Reactive CE

FIGURE 7.1 is a graph of normalized plate height as a function of rate
constant (10’2 s k s 10‘ s") and electrophoretic mobility difference (0.1 s Ap. s 2.0
x 10“ cm2Ns). It is apparent that the range of rate constants over which h 2 5
increases with Au. This means that smaller values of Au result in diffusion-
limited plate heights _at slower rate constants, whereas larger values of Au permit
calculation of faster km”. It is also apparent that increasing Au by 10 increases
kmax by 100 and that the maximum rate constant calculable for these conditions is
~30 s". It is also apparent that for more realistic values of Au (5 1.0 x 10“
cm2Ns), the maximum rate constant calculable is s 10 3". Thus, to whatever

extent possible, Au should be maximized to increase kmax.

229

FIGURE 7.1: Effect of electrophoretic mobility difference on maximum
calculable rate constants via capillary electrophoresis.
Calculation conditions: E = 500 Vlcm, (robs = 6.0 x 10"
cm2Ns, (“Amp/2 = -1.5 x 10" cm2Ns, o = 1.0 x 10‘5 cm2/s,
K =1,10'2 < k <104 s", and All. =1.o x 10*" cmst (D), 2.0
x 10'5 cm2Ns (A), 5.0 x 10'5 cm2Ns (O), '1.0 x 104 cmst
(I), 2.0 x 10*4 cm2Ns (A), 5.0 x 10* cm2Ns (c). h = 5 (- -). *

230

FIGURE 7.1

 

be kzfimzoo ME». ,
82:. 82 2: 2 F to . 5o

 

 

.. cor

ooov

 

oooor

 

 

oooooe

iHOIElH 2|le CEIZI'IVINHON

231

7.3.2 Effect of Electric Field Strength on kmax in Reactive CE

FIGURE 7.2 is a graph of normalized plate height as a function of rate
constant (10'2 s k s 10‘ s") and electric field strength (100 s E s 1500 Vlcm).
This range of electric field strengths encompasses the reasonable operating
range for conventional scale (capillary) and chip-based electrophoresis. It is
apparent that the range of rate Constants over which h 2 5 increases with E. This
means that smaller values of E result in diffusion-limited plate heights at smaller
rate constants, whereas larger values of E permit calculation of larger km”. It is
also apparent that increasing E by a factor 3 increases kmax by 10 and that at
even the largest values of E k...ax is only ~ 100 s". Thus, increasing electric field
strength by employing shorter columns orlchips has a greater influence on kmax
than Au. 1 -

7.3.3 Effect of Equilibrium Constant on k...." in Reactive CE _

FIGURE 7.3 is a graph of normalized plate height as a function of rate
constant (10'2 s k s 104 s") and equilibrium constant (10'? s K s 102). It is I
apparent that the range of rate constants over which h 2 5 increases as K
approachesunity and is mirrored symmetrically as K deviates from unity. This
means that kmax is greatest when K ~ 1 and, to whatever extent possible, efforts
should be made to impose this condition. Although little can be done to adjust K
to unity for true first-order reactions, it is possible to manipulate K to unity for
* pseudo-first-order reactions by judicious control of concentration.

7.3.4 Effect of Diffusion Coefficient on km“ in ReactiveCE

FIGURE 7.4 is a graph of normalized plate height as a function of rate

232

FIGURE 7.2: Effect of electric field strength on maximum calculable rate
constants via capillary electrophoresis. Calculation
conditions: it,” = 8.0 x 10“ cm2Ns, [LA = -1.0 x 10'5 cm2Ns,
(is = .2.0 x 10'5 cm2Ns, 0 =1.0 x 10‘5 cm2/s, K =1,10'2 < k
< 104 s", and E = 100 Vlcm (c), 200 Vlcm (A), 500 Vlcm

(I), 1000 Vlcm (O), 1200 Vlcm (A), 1500 Vlcm (Cl). h = 5

(- -)-

233

FIGURE 7.2

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Ase wpzfiwzoo mEE
ooe or e

prP - —:-pr - p

...o

—.:-n p b -

 

5.0
_ F

o_.

oo_.

coo _.

 

ooooo _.

.LHOIEIH SlV'Id GEIZI'IVINHON

234

FIGURE 7.3:

Effect of equilibrium constant on maximum calculable rate
constants via capillary electrophoresis. Calculation
conditions: E = 500 Vlcm, n... = 8.0 x 10“ cm2Ns, in = -1.0 x
10'5 cm2Ns, (is = -2.0 x 10'5 cm2Ns, o = 1.0 x 10“5 cm2/s,
10'2 < k < 10‘ s", and'K = 0.01 (o), 0.1 (A), 1 (I), 10(0),

100 (A). h = 5 (- -).

235

 

 

FIGURE 7.3

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82: 82 8F 2 V e to 8.0
4!... . _ l1. . C... . . .....ee. . 5:... . F
m
woe
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m oooe.
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236

lHOIEIH HIV—Id GEIZI'IVINEION

FIGURE 7.4:

Effect of diffusion coefficient on maximum calculable rate
constants via capillary electrophoresis. Calculation
conditions: E = 500/ Vlcm n... = 8.0 x 10* cm2Ns, m = -1.0 x
10'5 cm2Ns, (is = -2.0 x 10'5 cm2Ns, K = 1, 10'2 < k < 10‘ s",
and o = 1.0 x 10“5 cm2/s (c), 2.0 x 10‘ cm2/s (A), 4.0 x 1045
cm2/s (I), 6.0 x 10*5 cm2/s (O), 8.0 x 10" cm2/s (A), 1.0 x

10'5 cm2/s (El). n = 5 (- -).

737

 

FIGURE 7.4

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Do? or F

 

 

 

 

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238

constant (10‘2 s k s 10‘ s") and the observed diffusion coefficient of the reactive
zone (0.1 s D s 1.0 x 10‘5 cm2/s). It is apparent that the range of rate constants
oirer which h 2 5 increases as D decreases. This means that larger values of D
result in diffusion-limited plate heights at smaller rate constants, whereas smaller
values of D permit calculation of larger km. It is also apparent that decreasing D
by a factor of 10‘.increases km proportionately and that the maximum rate
constants calculable for these conditions is ~ 100 s". Thus, to whatever extent
possible, D should be minimized to increase kmax.

The preceding evaluation for CE illustrates the effects of Au, E, K and D
on kw. It is shown that km, is influenced most by physical parameters (E, An,
and D) and least by chemical parameters of the reaction (K). It is also shown
that the largest values of kmax are only attainable when reactions are
characterized by first-order or pseudo-first-order equilibrium constants of unity.
Of these, the greatest values of kmax are achieved under high electric field
strengths for species that exhibit large mobility differences and small diffusion
coefficients. Of these influences, there is greater experimental control over E
and K (when pseudo-first-order); however, application of large E should be

chosen judiciously to avoid excessive Joule heating and buffer decomposition.

7.4 CAPILLARY ELECTROCHROMATOGRAPHY
In this section, the individual contributions to Hi, and H.,- are again
considered separately, but for the case where Equation 2 represents the

transition of a molecule existing in solution (A) and at a surface (B). In CEC, this

239

transition is manifest as species A traveling in solution with VA > 0, and species B
remaining on the surface with v3 = 0. As shown below, this distribution of
velocities results in kmam values that are considerably larger than those obtainable
via CE. 1
7.4.1 Effect of Electrophoretic Mobility Difference on k.m in CEC

FIGURE 7.5 is a graph of normalized plate height as a function of rate
constant (10'2 S k s 104 s") and electrophoretic mobility difference (-1.0 5 Apt 5
1.0 x 10" cm2Ns) where Ap. = [1A because its - 0. For this evaluation, it is
assumed that surface diffusion is negligible (D9 = 0). It is apparent that the range .
of rate constants over which h 2 5 has little dependence on Air. This is in
contrast to the large dependence of km on An in CE. It is also apparent that
kmax for CEC (~ 1000 s") is much greater than kmax for CE (~ 10 s") with the
same values of Au and (1.0. This is because in CEC bulk flow contributes to Av
(Equation 4) and this contribution dominates kmax. The minimal effect of Au
notwithstanding, it can be seen that large, positive values of Au permit calculation
of slightly larger km, than large, negative values of Au. Thus, to whatever extent
possible, Av should be maximizedvia 090110 increase kmax.
7.4.2 Effect of Electric Field Strength on km, in CEC

FIGURE 7.6 is a graph of normalized plate height as a function of rate
constant (10'2 s k s 104 s“) and electric field strength (100 s E s 1500 Vlcm). For
this evaluation, it is assumed that surface diffusion is negligible (D3 = 0).- It is
apparent that the range of rate constants over which h 2 5 increases with E. This

means that smaller values of E result in diffusion-limited plate heights at smaller

240

FIGURE 7.5:

Effect of electrophoretic mobility difference on maximum
calculable rate constants Via capillary
electrochromatography. Calculation conditions: E = 500
Vlcm, it... = 8.0 x 104 cm2Ns, D). = 1.0 x 10'5 cm2ls, K = 1,
10'2 < k < 10" s", and a. = 1.0 x 104 cm2Ns (1:1), 5.0 x 10‘5
cm2Ns (A), 0.0 x 10'5 cm2Ns (O), -5.0 x 10'5 cm2Ns (A),

.10 x 104 cm2Ns (I). h = 5 (- -).

241

FIGURE 7.5

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-:-. - u -

 

 

Ase Ezfiwzoo ME».

009. oo_. o_.. _. ...0 5.0

—.-..-p. b run... . n :-.-.P. - bE-pb-p. - -:-PF- -

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ll

 

 

_.

o _.

oo _.
ooo _.
oooo _.
ooooo F

000000 _.

.LHOIEIH arena CEIZI‘IVINEION

242

FIGURE 7.6:

Effect of electric field strength on maximum calculable rate
constants via capillary electrochromatography. Calculation
conditions: pea = 8.0 x 10“ cm2Ns, px = -1.0 x 10'5 cm2Ns,
oA =10 x 10'5 cm2/s, K =1,10'2 < k <1o‘ s", and E = 100
Vlcm (O), 200 Vlcm (A), 500 Vlcm (I), 1000 Vlcm (O),

1200 Vlcm (A), 1500 Vlcm (1:1). h = 5 (- -).

243

FIGURE 7.6

 

 

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1000000 2:
1 00000
1 0000
1 000
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1 0

lHOIEH BLV'Id CIEIZI'IVINHON

244

 

I IIIU

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j I I rIIIIl

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10
RATE CONSTANT (s“)

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rate constants, whereas larger values of E permit calculation of larger kmax. It is
also apparent that increasing E by a factor 3 increases km by 10, similar to the
trend shown for CE. However, the largest values of E yield km...x ~ 100 s‘1 for CE
whereas the same values of E yield kmax ~ 10,000 s'1 for CEC. The reason for
this discrepancy is because bulk flow contributes linearly to Av in CEC whereas
bulk flow has no contribution to Av in CE. Thus, increasing bulk flow via the
electric field strength by employing shorter columns or chips has a greater
influence on kmax than Au.
7.4.3 Effect of Equilibrium Constant on km, in CEC

FIGURE 7.7 is a graph of normalized plate height as a funCtion of
rate constant (10'2 s k s 10‘ s") and equilibrium constant (10'2 s K s 102). For
this evaluation, it is assumed that surface diffusion is negligible (D3 = 0). It is
apparent that the range of rate constants over which h 2 5 increases with K. It is
also apparent that the effect of increasing K on km. decreases as K increases
such that km. ~ 2000 s'1 is achieved by K = 100. These trends are the
cumulative manifestation of the effects of K on velocity (Equation 7) and diffusion
coefficient (Equation 3). In CEC, larger K values necessarily result in faster
observed velocities for solutes with negative electrophoretic mobilities and slower
observed velocities for solutes with positive electrophoretic mobilities. Moreover,
larger K values also necessarily result in smaller D in CEC because the diffusion
coefficient of a molecule on a surface (D3) is significantly less than that for a
molecule in solution (DA). While some surfaces may permit molecules to diffuse

within the phase, those diffusion coefficients are still quite small (D3 ~ 10’7

245

FIGURE 7.7:

Effect of equilibrium constant on maximum calculable rate
constants via capillary electrochromatography. Calculation
conditions: E = 500 Vlcm, pea = 8.0 x 10" cm2Ns, HA = -1.0 x
10'5 cm2Ns, o. = 1.0 x 10'5 cmz/s, 10'2 < k < 104 s", and K =

0.01 (c), 0.1 (A), 1 (I), 10(0), 100 (A). h = 5 (- -).

246

FIGURE 7.7

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247

cm2/s). However, the dependence of km, on K for 0 < D3 < DA can be expected
to have a similar effect as for CE, but with the inflection at K > 1 and km ~ 1000
— 2000 s". This means that surface phases that minimize surface diffusion will
permit calculation of larger km. .
7.4.4 Effect of Diffusion Coefficient on km, in CEC

FIGURE 7.8 is a graph of normalized plate height as a function of rate
constant (10'2 s k s 104 s") and the diffusion coefficient. of species A (0.1 5 DA 5
1.0 x 10'5 cm2/s). It is apparent that the range of rate constants over which h 2 5
increases as DA decreases. This means that larger values of DA result in
diffusion-limited plate heights at smaller rate constants, whereas smaller values
of DA permit calculation of larger km”. It is also apparent that decreasing Diby a
factor of 10 increases kmax proportionately, similar to the trends shown for CE.
However, in CEC km. is extended to 10,000 s‘1 for the smallest values of DA

whereas in CE km is extended only to ~ 100 s".

7.5 CONCLUSIONS

This chapter discusses the physical and chemical contributions to the
plate height model in terms of their individual effects on the maximum calculable
rate constants in CE and CEC. It is shown for both CE and CEC, that the
physical contributions from electric field strength, electrophoretic mobility
difference, and diffusion coefficient have greater influence on km. than the

chemical contributions from equilibrium constant. The greatest influence on km.

248

FIGURE 7.8:

Effect of diffusion coefficient on maximum calculable rate
constants via capillary electrochromatography. Calculation
conditions: E = 500 Vlcm, 1130 = 8.0 x 10‘ cm2Ns, m = -1.0 x
10'5 cm2Ns, K =1, 10'2 < k <104 s", and 0. =1.0 x 10‘6
cm2/s (c), 2.0 x 10’5 cm2/s (A), 4.0 x 10" cm2/s (:1), 5.0 x
10‘6 cm2/s (e), 8.0 x 10‘ cm2/s (A), 1.0 x 10‘ cm2/s (I). h

=5(--).

249

FIGURE 7.8

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250

comes from E. This is fortuitous because it the parameter over which there is the
greatest degree of experimental control. Whereas the relative effect of E on k...am
is identical for CE and CEC, larger km,m can be achieved via CEC because of
additional contributions from bulk flow to Av. The effect of Av on km is
evaluated via Ap. for CE and via ILA for CEC. It is shown that the effect of Au in
CE has a greater influence on kmax than the effect of m in CEC. However, CEC
permits calculation of larger kmax because of the additional contribution to Av from
bulk flow. Evaluation of the effect of K on kmax shows marked differences
between CE and CEC. The largest kmax for CE is achieved at K = 1 and all other
values of km. are distributed symmetrically as K deviates from unity. In contrast,
the largest km for CEC is achieved at K > 1 and the value of K at which kmax is
greatest approaches unity as D3 approaches DA. It is also important to note that
CE permits investigations of second-order reactions via imposition of pseudo-
first-order conditions by controlling concentration. In CEC, pseudo-first-order
conditions are assumed because the number of surface sites is considered to be
in excess. However, manipulation of pseudo-first-order equilibrium constants is
more difficult for CEC than for CE. This is because it is more difficult to
quantitatively vary of the number of surface sites in CEC than it is to
quantitatively vary the concentration in CE. The effect of the observed “diffusion
coefficient is evaluated via D for CE and via DA for CEC. It is shown that the
relative effect of D on k"...x is identical for CE and CEC. However, larger km. can
be achieved via CEC because only species A is the dominant contributor to

diffusion. Thus, the plate height model may be used to investigate rate constants ~

251

in CE and CEC. Smaller rate constants can be calculated via CE for first-order
and pseudo-first-order reactions, whereas larger rate constants can be calculated

via CEC. However, CE permits greater control over pseudo-first-order

equilibrium constants than CEC.

252

CHAPTER 8: CONCLUSIONS AND

FUTURE DIRECTIONS

8.1 INTRODUCTION

In the 25 years since Jorgenson and Lukacs demonstrated the potential
efficiency of capillary electrophoresis (CE) [1], it has become a staple for rapid
separations of charged analytes. However, applications of CE for investigating
reactions have only recently come into popularity. This delay is, in large part, the
result of deficiencies in theoretical development more so than deficiencies in
‘ instrumentation. Whereas mathematical models for evaluating reactions in CE
have been developed, much of this development has been limited in scope to
evaluating thermodynamic quantities [2-4]. Rather, it is the lack of theoretical
development for evaluating kinetic quantities that has been the true limitation.
Moreover, the methods for evaluating kinetics in CE have historically relied on
the presence of plateau profiles and the computational fitting of those profiles.
Additionally, these methods and have little to no relationship with the models for
evaluating thermodynamic quantities [5-8]. .

The work presented in this dissertation attempts to bridge the existing gap
between models for evaluating thermodynamics and models for evaluating
kinetics. In doing so, physical and theoretical descriptions were presented of the
thermodynamic and kinetic contributions to separation and broadening that are
consistent with chromatographic theory. In Chapter 2, stochastic (Monte Carlo)

simulations were used to illustrate the physical (electrophoretic mobility) and

253

 

chemical (equilibrium constant and rate constants) contributions to zone profiles,
velocity, plate height, and skew. In Chapter 3, theoretical development of the
plate height model was extended to reactive CE. It was through this theoretical
development that zone velocity was related to equilibrium constant and zone
plate height was related to rate constants. In Chapter 5, the plate height model
was applied to the reversible, first-order isomerization of peptidyl-proline
dipeptides. The equilibrium and rate constants calculated via the plate height
model were compared to those calculated with ChromWin as well as to literature
values and shown to be in good agreement. In Chapter 6, the plate height model
was applied to the reversible, second-order reaction of dansyl phenylalanine
inclusion by B—cyclodextrin. Equilibrium constants were calculated from velocity
and shown to be in good agreement with literature values. In contrast, rate
constants were calculated from plate height and are shown to be grossly
underestimated. This unsuccessful application of the plate height model was

attributed to physical limitations of the experimental system that are not expected
. to be overcome under realistic CE operating conditions. In Chapter 7, these
limitations of the plate height model were evaluated for capillary electrophoresis
and capillary electrochromatography. This chapter discusses avenues of further
theoretical and experimental development for investigating reactions via CE and

potential difficulties of each.

8.2 EVOLUTION OF REACTIVE ZONE PROFILES

The work in Chapter 1 provided a detailed physical description of how

254

zone profiles evolve during reaction. However, what is lacking is an accurate
mathematical description of the process of separation and broadening prior to
steady state. In order . to achieve this, it is first necessary to develop a
mathematical description of the evolution of the profiles representative of the
individual species. This latter task requires a description of how both the mean
and variance change for the individual solute zones as a function of time. The
crux of the matter will lie in developing a consistent mathematical treatment for
the effects of kinetics on the mean and variance for the individual solute zone.
Once this is achieved, it should be a relatively simple matter to apply the
descriptions of the individual solute zones to the net zone. The end result of this
is a complete and continuous mathematical model of how reaction influences the

evolution of zone profiles.

8.3 THEORETICAL DEVELOPMENT

In addition to further development of the physical and mathematical
descriptions of how zone profiles evolve as a result of reaction, work is needed to
extend and validate the plate height model to consider multiple equilibria.
Although the theoretical development .in Chapter 2 advances the range of
accessible rate constants over previous models, it too is limited in scope. This
development of the plate height model is limited to systems at or near steady
state that can be described by a single equilibrium expression. Steady state is a
fundamental assumption of the plate height model; however, there are no

fundamental assumptions regarding the number of equilibria. In order to be truly

255

 

general and more applicable to a broader range of chemical systems, the model
should be extended to consider multiple equilibria. It is not a difficult task to
describe zone velocity as a function of multiple equilibria. The crux will lie in
accurately describing plate height from multiple equilibria and extracting kinetic
rate constants from the experimental data. Experimentally, this should be
possible by treating each equilibrium expression independently; however,
application of such an approach will be predicated upon large electrophoretic
mobility differences among species as well as large differences (2 10‘) between
the individual equilibrium constants. The stochastic simulation described in
Chapter 2 is the ideal candidate to investigate the effects of multiple equilibria on
zone broadening and to validate the proposed extension of the plate height
model. i

8.4 INSTRUMENT DESIGN

The mathematical model developed in Chapter 3 illustrates that increasing
the electric field strength increases the range of calculable rate constants. Thus,
calculation of rate constants in excess of ~ 20 s'1 requires shorter capillaries,
larger applied voltages, or both. This optimization of instrumental parameters is
best realized with chip-based CE systems. Chip-based systems facilitate
separations over distances a small as a few centimeters with electric field
strengths up to ~ 1 kV/cm [9]. This combination of large electric field strengths
over extremely short distances will extend the range of calculable rate constants

to ~ 103 s". The crux with performing these measurements will lie in the

256

appropriate choice of chip substrate to ensure electroosmotic flow and effective
heat dissipation, fabrication of uniform separation channels, and positioning of

detection elements within extremely close proximity (~ 1 cm).

8.5 EXPERIMENTAL DESIGN

The unsuccessful application of the plate height model in Chapter 6
illustrates an additional physical limitation that cannot be overcome with
instrumentation alone. The upper limits of calculable rate constants are dictated
not only by electric field strength and detector locations, but also by the inherent
mobilities of solutes in the free and bound states. In order to control and
maximize the effect of mobility on plate height it will be necessary to affix one of
the reactants to a stationary surface and perform the experiments as capillary
electrochromatography (CEC). The most ideal approach to these experiments
would be to affix one reactant to the walls of open channels in chip CE, thereby
extending the range of rate constants to ~ 104 s“. Chip CEC has been
performed with packed channels and monoliths [10]; however, these stationary
phase supports introduce additional contributions to broadening. These
experiments will share the same potential difficulties associated with choice of
chip substrate and positioning of detection elements discussed in the previous
section. Moreover, an additional challenge in these experiments will arise from
coating the channels with the stationary phase [11] and determining the

mobilities of any non-neutral solutes under identical conditions a priori.

257

10

11

8.6 REFERENCES

Jorgenson, J. W.; Lukacs, K. 0., Anal. Chem. 1981, 53, 1298

Busch, M. H. A.; Carels, L. B.; Boelens, H. F. M.; Kraak, J. C.; Poppe, H.,
J. Chromatogr. A 1997, 777, 311

Busch, H, M. A.; Kraak, J. C.; Poppe, H., J. Chromatogr. A 1997, 777, 329
Jia, 2., Curr. Pharm. Anal. 2005, 1, 41

Avilla, L. 2.; Chu, Y.-H.; Blossey, E. C.; Whitesides, G. M., J. Med. Chem.
1993, 36, 126

Okhonin, V.; Krylova, S. M.; Krylov, S. N., Anal. Chem. 2004, 76, 1507

Patterson, D. H.; Harmon, B. J.; Regnier, F. E., J. Chromatogr. A 1996,
732, 119 .

Schoetz. Trapp. 0.; Schurig, V., Electorphoresis 2001, 22, 2409
Dolnik, V.; Liu, S., J. Sep. Sci. 2005, 28, 1994

Stachowiak, T. B.; Svec, F.; Frechet, J. M. J., J. Chromatogr. A 2004,
1044, 97

Dolnik, v., Electrophoresis 2004, 25, 3589

258

 

APPENDIX

259

APPENDIX: VALIDATION OF INSTRUMENT CONTROL
PROGRAMS AND CAPILLARY ELECTROPHORESIS

POWER SUPPLY

This appendix includes descriptions of the function and validation of the
programs written to operate the capillary electrophoresis with laser-induced
fluorescence detection (CE-LIF) system described in Chapter 4. These
programs were written with cOmmercially available software (Laerew, Version
5.1, Nation Instruments) and executed on computers running the Microsoft
Windows 98 operating system. Additionally, this appendix includes a description
of the validation of the high voltage power supply (HVPS) used to achieve
separations with this system.

FIGURES A.1A and A.1B show a typical screen view and program
schematic of the capillary electrophoresis power supply control program (step-
logic2). The program allows the user to specify the time, duration, polarity and
magnitude of applied voltage or current. The program also allows the user to
update the specified values in real-time as well as monitor the resultant current or
vouage.

FIGURES A.2A and A.2B ’show a typical screen view and program
schematic of the Program for Unified Computer Control (PUnCC) used to control
and monitor the separation progress. Signal from the detectors is monitored in

real-time and written to a file with the PUnCC. Additionally, the PUnCC is used

260

FIGURE A.1A:

Typical screen view of the-step-logic2 program. The toggle
switch allows the user to choose either voltage or current
control of the HVPS. The digital controls labeled “T1” - “T6”
allow the user to specify the elapsed time at which voltage or
current changes occur. The buttons labeled “set T4” - “set
T6” allow the user to choose additional (> 3) voltage steps.
The digital controls labeled “kV1” - “kV6” and “uA1” - “uA6'
allow the user to specify the magnitude of the applied
voltage or current. The digital control labeled “T max" allows
the user to specify the maximum time for voltage or current
application. The digital indicator labeled “Time” shows the
elapsed experiment time. The digital indicator labeled
“Output" shows the voltage output from the computer to the

HVPS.

261

{if step

FIGURE A.1A

 

262

FIGURE A.1 B: Schematic of the step-logic2 program. The schematic
shows the data flow as a logical cascade from T1 through T6

to the output channels that interface with the HVPS.

263

FIGURE-A.1B

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264

FIGURE A.2A:

Typical screen view of the PUnCC program. The dialog box
labeled “Input Comments” allows the user to add comments
regarding each experiment to the data file. The toggle
switch allows the user to monitor either the current or voltage
from the HVPS. The digital controls labeled “Buffer size”,
“Scan rate”, and “Scans to read at time” allow the user to
specify the number of scans stored in the data buffer, the
number of scans recorded each second, and the number of
scans written to a file, respectively. The button labeled “New
file” allows the user to choose to store the data as a new file
or to append the data to an existing file. The digital control
labeled “Input limits” allows the user to specify the board
gain. The digital indicators labeled “Scan backlog“ and
“Time Elapsed” show the number of scans in the buffer and
the elapsed time, respectively. The button labeled “STOP”
allows the user to stop the experiment. The waveform
charts and digital indicators show the real-time response
from the detectors and the monitored current or voltage from

the HVPS.

265

p Pllnl‘l‘. v- '

FIGURE A.2A

 

266

FIGURE A.28:

Schematic of the PUnCC program. The schematic shows
the data flow of specified inputs from the screen view to the
data acquisition loop and the data file. The schematic also
shows the interface to the step-logic2 program as a sub-
routine (labeled “V or I Step”) within the data acquisition

loop.

267

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268

to control and monitor the HVPS via the program described above as an
embedded sub-routine. The PUnCC allows the user to specify whether voltage
or current is applied for the separation, the rate of data acquisition, the rate at‘
which data are written to the display and to a file, as well as the gain of the data
acquisition board prior to each experiment. The PUnCC also allows the user to
add pertinent comments to each data file.

Output from the HVPS and its control program was validated with a
1000:1 high-voltage probe (Fluke) and a 495 (:1: 2%) Mn precision resistor.

FIGURE A3 is a graph of the programmed voltage applied versus the
output voltage measured by the high-voltage probe, together with the percent
error. It is apparent that there is a linear relationship between the programmed
voltage and output voltage with a slope of unity. It is also apparent that the
output voltages are within the calculated upper and lower output limits specified
for the power supply and that the output voltage error is greatest (> 2%) below 1
kV and least (< 2%) above 1 kV. ‘

FIGURE AA is a graph of the programmed voltage applied across the 495 -
MD resistor versus the measured current, together with the percent error. It is
apparent that there is a linear relationship between the programmed voltage and
the measured current. The resistance calculated from the slope is 508 MO,
within 2.1% of the expected value of 495 MO. It is also apparent that the
' measured current error is greatest at lower programmed voltages and decreases
monotonically with increasing voltage.

FIGURE A5 is a graph of the programmed current applied through the

269

FIGURE A.3:

Programmed voltage versus the output voltage (—) and
percent error of the output voltage (X). Dotted lines

represent the specified output accuracy for the power

supply.

270

 

 

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FIGURE A.3

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PROGRAMMED VOLTAGE (kV)

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FIGURE A.4:

Programmed voltage versus the measured current (El)
across the 495 MO precision resistor and the percent error
(X) of the measured current calculated assuming a

resistance of 495 MO.

272

 

FIGURE A.4

 

 

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FIGURE A.5:

Programmed current versus the measured voltage ([3)
across the 495 MO precision resistor the and percent error
(X) between the measured voltage and that calculated.

assuming a resistance of 495 M0.

 

274

 

FIGURE A.5

 

 

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275

495 MO resistor versus the measured voltage, together with the percent error. It
is apparent that there is a linear relationship between the programmed current
and the measured voltage. The resistance calculated from the slope is 508 M0,
~within 2.1% of the expected value of 495 MO. It is also apparent that the
measured voltage error is greatest when the programmed current is less than 5
[LA and remains relatively constant at < 5% when the programmed current is > 5
3A.

The ability of the PUnCC to accurately record data (0 - 10 V input) was
validated with a Dial-A-Volt voltage source (Model DAV-450, General Resistance
Inc.) and a Fluke Digital Multimeter 77 (DMM).

FIGURE A6 is a graph of the specified voltage versus the voltage
measured by the PUnCC and the DMM, together with the percent error of the
voltage measured by the PUnCC. It is apparent that there is negligible difference
between the voltages measured by the PUnCC and the DMM and that the

percent error of the voltage measured by the PUnCC is 1% or less.

276

FIGURE A.6: Input voltage versus the voltage measured by PUnCC (Cl),
DMM (A), and the percent error (X) of the voltage

measured by PUnCC.

 

277

 

 

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FIGURE A.6

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INPUT VOLTAGE (V)