NONEQUILIBRIUM THERMODYNAMIC
THEORY OF TRANSPORT IN MEMBRANES

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY ‘
JlNG-SHYONG CHEN.
1971

   
      

  

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Nonequilibrium Thermodynamic Theory of
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presented by

JIng-Shyong Chen

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ABSTRACT

NONEQUILIBRIUM THERMODYNAMIC THEORY OF
TRANSPORT IN MEMBRANES

BY
Jing-Shyong Chen

The steady-state transport theory that accounts
for the osmotic flow of fluids through biological mem-
branes is obtained, subject to certain simplifying
assumptions which may be removed in subsequent work.

The principles of hydrodynamics and nonequilibrium thermo-
dynamics are used to describe the simultaneous transport
of mass and electric current and the osmotic flow rate

in a fluid system undergoing osmosis. In order to under-
stand better the osmotic flow phenomena in biological
systems, the following particular subjects are thoroughly
investigated: (1) the distribution of ions in charged
membranes, including the effect of molar volumes of ions,
(2) the solution of Poisson's equation modified by in-
clusion of the dependence of the dielectric constant of
the salt solution on the salt concentration and the

electric field, and (3) the distribution of pressure

Jing-Shyong Chen

including the effects of ionic concentrations and electric
field. Their contributions to biological systems are dis-
cussed. Moreover, the range of applicability of such
classical monuments as the Boltzmann equation and the
linearized Debye-Hfickel theory is examined.

After these fundamental problems are dealt with,
the transport theory is obtained subject to membrane

-3 C/mz

surface charge density of o z 10 (C = Coulomb;

m = meter), which is apparently reasonable biologically.
A capillary model is used for the membrane separating

two aqueous salt solutions of different concentration at
the same temperature. The theory is valid for total
ionic concentrations between about 0.004 molar and 2
molar, a range which includes systems of biological in-
terest and more concentrated systems. We assume that
there is no hydrostatic pressure difference across the
membrane. The steady-state solutions of the differential
equations derived are obtained for the case of a symmet—
rical, binary electrolyte, yielding the average osmotic
flow rate as a function of initial concentrations. The
result of the theoretical section is then used to explain,
at least qualitatively, the osmotic flow phenomena ob-
served experimentally. In particular, the theoretical
equations predict both the shape of the experimental

curves of flow rate versus concentration and the direction

of flow through charged membranes.

NONEQUILIBRIUM THERMODYNAMIC THEORY OF

TRANSPORT IN MEMBRANES

BY

Jing-Shyong Chen

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Chemistry

1971

To My Parents

ii

ACKN OWLE DGMEN TS

The author wishes to express his appreciation to
Dr. Frederick H. Horne for his guidance throughout the
course of this study.

Appreciation is extended to Mr. James D. Olson
for his computer assistance.

Appreciation is also extended to the Department
of Chemistry, Michigan State University, and the National
Science Foundation for providing financial support.

Thanks are particularly due to his wife, Mei—horng
(Betty) for her assistance and encouragement during his

difficult time.

iii

DEDICATION

TABLE OF CONTENTS

ACKNOWLEDGWNTS O O O O O I O O O O O I O O O O O O 0

LIST OF TABLES O O O O O O O O O I O O O O O O O O 0

LIST OF FI
LIST OF AP

Chapter

GURES O C O O O O O O O O O O O O O C O

PENDICES O O O O O O O O O O O O O O O O I

I 0 INTRODUCTION 0 I O O O O O I O O O O O O O O

A.
B.
C.

II. THE

A.
B.
C.
D.

E.

Membranes and Membrane Phenomena . . . . .
Motivation . . . . . . . . . . . . . . . .
The TheSiS I O O O O O O O O I O O O O O 0

TRANSPORT EQUATIONS . . . . . . . . . . .

Introduction . . . . . . . . . . . . . . .
Chemical Potential . . . . . . . . . . . .
Hydrodynamic Equations . . . . . . . . . .
Principles of Nonequilibrium

Thermodynamics . . . . . . . . . . . . .
The Differential Equations Describing

the Osmotic Flow of Fluids Through

a Charged Membrane . . . . . . . . . . .

III. EQUILIBRIUM DISTRIBUTION OF IONS IN
AN EI‘ECTRIC FIELD I O O O C O O C O O O O O

A.

[310061

Introduction . . . . . . . . . . . . . . .
Derivation of Modified Boltzmann Equation.
Molar Volume and Charge. . . . . . . . . .
Uni-univalent Case . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . .

IV. RADIAL DISTRIBUTION OF ELECTROSTATIC

P

Am
B.

C.

OTENTIAL IN A MODERATELY CHARGED CYLINDER.

IntrOduCtj-On O O C O O O O O O O O O O O O
Poisson's Equation . . . . . . . . . . . .

Simplified Approximations. . . . . . . . .

iv

Page
ii
iii
vi
vii

viii

UTUJH l—’

10
13

18

24

33
33
44

45
47

49

49
51

53

Chapter

6)"!!th

Perturbation Scheme. . . . . . . . . . .
Solutions of the Differential Equations.
Discussion . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . .

V. THE STEADY-STATE DISTRIBUTION OF PRESSURE
IN MODERATELY CHARGED SYSTEMS . . . . . .

A.
B.

Introduction . . . . . . . . . . . . . .
The Macroscopic Osmotic Pressure Between
the Aqueous Phase and the Membrane
Phase in a Moderately Charged System .
The Macroscopic Osmotic Pressure Across
a Moderately Charged Membrane. . . . .
Osmotic Flow of Water in a Charged
Membrane . . . . . . . . . . . . . . .
Conclusion and Discussion. . . . . . . .

VI. THE STEADY-STATE PHENOMENOLOGICAL
THEORY OF OSMOSIS IN CHARGED SYSTEMS. . .

A.

BIBLIOGRAP

APPENDICES

Introduction . . . . . . . . . . . . . .
Transport Equations. . . . . . . . . . .
General Formula for uO . . . . . . . . .
Special Formula for no and its

Consequences . . . . . . . . . . . . .
The Working Equation for

Anomalous Osmosis. . . . . . . . . . .
The Onsager Transport Coefficients . . .
Results and Discussion . . . . . . . . .

HY O O C C O O O O O O O C O O O O O O O

Page
62
65

74
79

81
81

86
91
92
96

98

98
100
104
107
115
120
122
131

135

LIST OF TABLES
Table Page

4.1 Value of 6a and V: of some uni-univalent
electrolytes at 25°C (Sanfeld, 1968) . . . . . 53

4.2 Some calculated values of IO/Il and l/Il . . . 69

4.3 Values of a and b (k = 0,1,...,5) . . . . . 76

k k
4.4 Some calculated values of W, W , and W - W .
-3 -2 O O O
T = 298°K, [cl = 10 c m , a = 4.2 A,

and Ka = 1.692 . . . . . . . . . . . . . . . . 77

5.1 Some calculated values of w and N' at
20 20

various salt concentrations. c (i) =

3 3 0
2.0 x 10 mole/m . a = 17.6 A. o

C/mz. t = 25°C. . . . . . . . . . . . . . . . 94

II
|—'
O

6.1a The Onsager transport coefficients
for Hzo-NaCl at 25°C 0 O I O O O O O O 0 O O O 123

6.1b The Onsager transport coefficients

for H20-KC1 at 25°C. . . . . . . . . . . . . . 124

6.1c The Onsager transport coefficients

for HZO-LiCl at 25°C . . . . . . . . . . . . . 125

vi

LIST OF FIGURES
Figure Page

4.1 The Dielectric constant D0 of water as
a function of field strength [E] (in

volts per cm). . . . . . . . . . . . . . . . . 54

4.2 Plot of the dimensionless parameter

w as a function of Kr. . . . . . . . . . . . . 78

5.1 Plot of NRC (in joules per m3) as a function

of the logarithm of the salt concentration . . 95

6.1 Plot of U0 1 (in cm2 per sec) as a
function of the logarithm of the salt

concentration for negative charge. . . . . . . 127

6.2 Plot of UO 2 (in cm2 per sec) as a
function of the logarithm of the salt

concentration for positive charge. . . . . . . 128

vii

LIST OF APPENDICES

Appendix Page
A A Derivation of the Chapman-Gouy

Equation For a Charged Cylinder. . . . . . 135

B Derivation of Equations (4.87) and (4.88). 139

viii

CHAPTER I

INTRODUCTION

A. Membranes and
Membrane Phenomena

 

 

Transport phenomena across membranes, charged or
uncharged, are encountered in many areas of life and
physical sciences. For instance, chemists and chemical
engineers would like to understand the mechanisms of
membrane transport so that they would be able to fabricate
membranes of any desired property or properties. Biolo-
gists, on the other hand, are interested in understanding
the behavior of complex cell membranes in terms of estab-
lished physicochemical principles.

A membrane, in simple terms, is a phase, inter-
nally heterogeneous or homogeneous, which acts as a bar-
rier to flow of some of the molecules and ions present
in the liquid and/or gases in contact with it. Most
membranes, except the obvious ones such as oil membranes,
are to be considered heterogeneous. Membranes may also
be classified natural or artificial. All biological mem-

branes are natural membranes.

Functionally, a membrane must be more or less
active in the selective transport of some ions when used
as a barrier to separate two solutions or phases, unless
it is too fragile or too porous. Most artificial and
natural membranes have been found to carry ionogenic
groups either fixed to the three dimensional membrane
matrix, as seen in a well-characterized ion exchange
membrane, or adsorbed, as found in some colloidal systems
(Kobatake, 1959). Ionogenic groups and pores (space
occupied by water) in a membrane attribute certain func-
tionality to the membrane. Thus a membrane may be con—
sidered permselective and/or semipermeable depending on
its functionality. The phenomenological transport prop-
erty that controls the former is the transport number or
transference number, whereas the latter is determined by
the so—called reflection coefficient (ratio of the actual
hydrostatic pressure required to give zero net volume flow
to that which is required if the membrane were truly semi—
permeable) introduced by Staverman (1951; 1952).

In the absence of external strains and external
nmgnetic and gravitational forces, the driving forces that
may produce a flow or flux of molecular or ionic species
through a membrane separating two solutions are (l) gra—
dient of concentration, (2) gradient of pressure, (3)

gradient of temperature, (4) gradient of electric

potential, and (5) chemical reactions. These forces may
Operate in various combinations and may generate a number
of transport phenomena. To understand membrane phenomena,
it is necessary to prOpose or design various transport
systems or model systems consisting of membranes with par—

ticular and predetermined specific properties.

B. Motivation

 

The transport of solutes and water through living
systems has been the subject of rather extensive study
since the 19th century. In spite of many experiments,
the fundamental mechanisms involved in the transport
phenomena exhibited by various living systems have re-
mained relatively obscure. There are many reasons for
our failure to advance more rapidly, but a number of the
problems encountered in the study of living systems arise
from the complex nature of living cells when considered
from the point of View of a membrane system. Thus, trans-
port phenomena in various cell membranes are vital to
nmny biological systems.

Among many theoretically-oriented investigators,
Kobatake et a1. (1964) were the first to initiate a study
based on the principles of nonequilibrium, continuum
thermodynamics. They were quite successful in attempting

to explain the experimental findings (Grim, 1957) with

regard to the osmotic flow of fluids through biological
membranes. Since then, various other continuum theories
(sf. Toyoshima, 1967; Fujita and Kobatake, 1968; Gross and
Osterle, 1968) have been develOped. Moreover, nonequilib-
rium thermodynamics is often used to study the discontinu—
ous fluid-membrane systems. The difference in the ap-
proaches between the continuous and the discontinuous
fluid-membrane systems will be discussed in more detail

in the next chapter.

Most of the previous theories, although adequate
in many cases, are, in general, limited in the following
respects:

1. Use of transport equations that are restricted
only to dilute aqueous salt solutions (for detailed analy-
sis, see, for example, Chapman, 1967).

2. Certain transport parameters are treated as
constants.

3. The effect of pressure-induced current on the
overall transport phenomena is neglected, implying that
the pressure is considered independent of salt concentra-
tions and electric field.

4. The dielectric constant of pure water is used
in Poisson's equation, implying that the salt solution is
very dilute and the electric field at the membrane sur-

face is negligibly small.

S. The Debye-Hackel theory is used arbitrarily
and inconsistently; in particular, it is usually assumed
(Kobatake and Fujita, 1964) simultaneously that the Debye
length K-1 is very small and that the ionic strength is
very small.

6. The classical Boltzmann distribution of ions

is used with no attempt at justification.

The main purpose of this work is to obtain a
phenomenological transport theory for osmotic flow in
biological systems that is not subject to the above re—
strictions and that, hopefully, will resolve some of the
discrepancies appearing in the literature (cf. Kobatake
and Fujita, 1964). In addition, we hope to use the theory
to explain experimental observations of osmotic flow
phenomena. In general, it is hOped that our theory can
be used to describe better and to understand further the
membrane systems and the mechanisms of transport processes

encountered in living cells.

C. The Thesis

 

In the following chapters, the steady-state trans-
port theory that accounts for the osmotic flow of fluids
through biological membranes is obtained. We make use of
the principles of hydrodynamics and nonequilibrium thermo-

dynamics in describing the simultaneous transport of mass

and electric current and the osmotic flow rate in a fluid
system undergoing osmosis. In order to understand better
the osmotic flow phenomena in biological systems, we in—
vestigate (1) the distribution of ions in charged mem-
branes including the effect of molar volumes of ions, (2)
the solution of Poisson's equation modified by inclusion
of the dependence of the dielectric constant of the salt
solution on the salt concentration and the electric field,
and (3) the distribution of pressure including the effects
of ionic concentrations and electric field. Their con—
tributions to biological systems are discussed. Moreover,
the range of applicability of such classical monuments

as the Boltzmann equation and the linearized Debye-Hfickel
theory is examined.

The theory is obtained subject to membrane surface
charge density of o z 10-3C/m2 (C = Coulomb; m = meter)
which has been reported (Fair and Osterle, 1971) to be
sensible biologically. A capillary model is used for the
nembrane separating two aqueous salt solutions of differ-
ent concentration at the same temperature. The theory is
valid for total ionic concentrations between about 0.004
molar and 2 molar, a range which includes systems of
biological interest (Woodbury gt_al., 1970) and more con-
centrated systems. We assume that there is no hydro—

static pressure difference across the membrane.

The steady-state solutions of the differential equations
derived are obtained for the case of a symmetrical electro-
lyte, yielding the average osmotic flow rate as a function
of initial concentrations. The result of the theoretical
section is then used to explain the osmotic flow phenomena

observed experimentally.

CHAPTER II
THE TRANSPORT EQUATIONS

A. Introduction

In this chapter the differential equations which
describe macroscopic transport phenomena are presented.
Specialized equations used in the study of osmotic flow
of fluids through charged membranes are deduced, together
with appropriate boundary conditions. We consider only
continuous, isotropic fluids in which no chemical re-
actions occur and which are subject to certain driving
forces but not to a magnetic field. For more detailed analy-
sis of the equations used in the study of various transport
phenomena, refer, for example, to works by Horne (1966),
Kirkwood and Crawford (1952), de Groot and Mazur (1962),
Fitts (1962), Katchalsky and Curran (1967), and Haase
(1969). The more fundamental, rational-mechanical ap-
proach of Truesdell (1969), Bartelt and Horne (1970), and
others is not required here.

Katchalsky and Curran, and Haase, have also con—
sidered membranes. A general description of transport

phenomena in membranes and an extensive list of

references may be found in the recent book by Lakshminar—
ayanaiah (1969). However, nonequilibrium thermodynamics
is frequently applied to the discontinuous fluid-membrane
system which is particularly convenient for experimental
analysis but is not readily subjected to a theoretical
treatment. The system is not regarded as a continuum,
and the state variables are not continuous functions of
space and time. Here the phenomenological or transport
coefficients are functions of the fluid and of the mem-
brane. This approach is a "black box" approach in that
no questions are asked about the interior of the membrane.
Alternatively, if we are able to describe the de-
tails of the inside of the membrane, the fluid-membrane
system could be regarded as a continuum and the state
variables as continuous functions of space and time. In
this approach events in a small volume element are examined
and the macroscopic properties derived by averaging over
the entire volume of the system being studied. Conse-
quently, it will be necessary to construct a dynamic model
and then correlate the model with the macroscopic phenome-
nological description obtained experimentally. The aver-
aging process will require the choice of a proper refer-
ence frame for the flows. Kobatake and coworkers (1964)
and Osterle and coworkers (1968,1970) and Manning (1968)

have also used the continuous approach.

10

B. Chemical Potential

 

The driving forces needed to produce a flux or
flow of molecular and/or ionic species are the gradients
of the chemical potential of all species in the system
(e.g., Haase, 1969). In the absence of a magnetic field,
the total chemical potential “a of Component a may be

written (Horne, 1966)

_ O
“a — “a (T,p) + RT Quad + Ma? + zaFw (2.1)

where am is the activity of Component d,

a = X f p (2.2)

T is absolute temperature, p is pressure, R is the gas
constant, Ma is molecular weight of d, F is the gravita-
tional potential, za is the ionic charge per mole of a,

F is Faraday's constant, and w is the electrostatic poten-
tial, and where u: and the activity coefficient fa are
defined relative to an appropriate standard state in which
there are no effective external fields. Note that fa is

a function of temperature, pressure, and composition.
Since we treat only aqueous solutions here, the appro-
priate standard state is the pure solvent (denoted by the

running index 0),

o _ . _ _ _
“a - 11m [pa RT in Xa Mar zaFw] (2.3)

x *1
o

ll
whence

lim f = 1. (2.4)
a

We are primarily concerned with the gradient of
the chemical potential. Using the chain rule for dif-

ferentiation of “a (T,p,xB,F,¢) we have, in general nota-

tion,
v-2
Vua = uaTVT + papr + 8:0 uaBVxB + uaFVF + pawvw (2.5)
where
= = _’ ' t
uaT (Bud/8T)p,xa,F,w sa (partial molar en ropy)
“up = (Bud/BP)T,XG,F,w = Va (partial molar volume)
“GB = (Bud/3X8)T,p,P,w = RT (Bin xa fa/BXB)T,p.F,w
“a? = (Bud/3F)T,p,xa,w = Md
“aw = (and/aw)T,p,xa,P = zaF (2.6)
Thus,
v-2
= -_ _ + + . 2.7
vua sa VT + van + 830 “a8 VxB MaVF zaFvw ( )

12

The Gibbs-Duhem equation including the external

forces is (Horne, 1966)

v-1
2 xa vua = -E VT + V vp + Mvr + EFvw , (2.8)
d=0
where

_ v-l _ v-l _

s = Z x s , v = Z x v ,
a=0 d d d=0 d d

_ v-l _ v-l v-l

M = Z x M , z = X x z ,0 = Z x u . (2.9)
d=0 a a d=0 d d d=0 a QB

For the case we shall treat, wherein the gravitational
potential is negligible and the temperature is uniform,

(2.8) becomes, upon division by V,

v-1
2 c Vu = Vp + EFVW (2.10)
d=0 a a

where the molarity is defined by
ca = na/V = xa/v (2.11)

and where

2 = z c z = (p/fi)? . (2.12)

Note that E and E vanish for electroneutrality.

13

For most of our purposes, (2.7) is not the most

useful form for the chemical potential gradient. Instead,
taking the gradient of (2.1) directly, we have
_ o
Vua — Vua (T,p) + RV T in aa + MaVF + zaFvw , (2.13)

which becomes, for the case of negligible gravitational

potential and uniform temperature,
_ —o
Vua - Va Vp + RTVQn ad + zaFvw ,
where

o _ _ . _
(Bud/8p)T’P’w — v — 11m v ,

and where

V x

RTVRn a = Z
d = 8

Note that

---0
Va + RT (Sin fa/Bp)

V
d

T,p,xa

C. Hydrodynamic Equations

 

Equation of Continuity of Mass

+ RT (sen fd/ap)T,p,x

(2.14)

(2.15)

Vp (2.16)
a

(2.17)

In the absence of chemical reactions, for a fluid

containing v components the v independent equations of

continuity of mass are

14
(dp/dt) + pv-E = o , (2.18)

p (dwa/dt) + V-ja = 0 , (2.19)

where p is density, t is time, 6 is the center of mass, or
barycentric velocity, and Wu and 3a are mass fraction and
diffusion flux of Component a, respectively. The barycen—
tric velocity 3 is defined by

v 1

+ _ +
u = Z w u , (2.20)
d d
d=0

where Ed is the velocity of Component a relative to a

laboratory frame of reference. The diffusion flux 3a is
defined by
Era = pa (Ea—E),a= 0,....,\)"l o (2021)

Note that pa = wap. The diffusion fluxes, however, are

not all independent,

v-1
2
a=0

3a = o . (2.22)

Substantial time derivatives d/dt are relative to local

time derivative B/at by
(d/dt) = (a/et) + E-v . (2.23)

The operator "V" is defined by

15

v = I (e/ax) + j (a/ay) + E (e/az) , (2.24)

+
where i, j, and k are the unit vectors of a three dimen-

sional Cartesian coordinate system. For more detailed
analysis in the use of various coordinate systems, see,

for example, Irving and Mullineux (1959).
Navier-Stokes Equation

The Navier-Stokes equation which relates the
velocity of barycentric frame of reference of fluids to

the external forces is (Horne, 1966)

p (dE/dt) + V[(% n-n')(v-E)] 2v-nsymvfi

+
OX " VP I (2.25)

where symVE is the symmetrical part of the tensor V5, and
where n is the coefficient of shear viscosity and n' is
the coefficient of bulk viscosity, both taken as noncon-
stant. In writing (2.25) we have used the equation of

motion,

7* ->

p (dE/dt) — v-E = pX (2.26)

+
+ I
where G is the stress tensor, given approxrmately by the

Newtonian linear phenomenological relation

Q++

2 + 3 +
= - [p + (-3- n-n') (V°u)] I + ZHSYmVu . (2-27)

16
+
and where pX, the net external force, is

pi = - pVF - zpvw (2.28)

Introducing (2.23) into (2.25) and rearranging, we obtain

0 (BE/at) = pi - vp + nvzfi + (% n+n')(7(v.fi)

—> —>
- puV.u , (2.29)

where we take n and n' to be constants.

If the system considered is at steady state, (2.29)

becomes, by setting B/Bt = 0 and rearranging,
+ 2+ 1 + + +
Vp - ox = nV u + (3 n+n') V (V.u) - puV.u . (2.30)

However, most fluids except very dense ones are essentially
incompressible. For an incompressible fluid, the density

p is constant in time and position. Thus according to

(2.18).

v-E = o . (2.31)

The Navier-Stokes equation for an incompressible fluid in
a system at steady state is, then, for constant n,

vp - pi = nvzfi . (2.32)

17

Energy Transport Equation
The general equation of continuity of total energy
is

(apEi/at) + v-EE = o , (2.33)
T

where 3E is the total energy and where ET is the total
T
specific energy,

E = E + % u . (2.34)

where E is the specific internal energy and u2/2 is the
local kinetic energy of the center of mass. Note also

that
++
u = u°u (2.35)

Although it may be preferable (see Bartelt and Horne,
1970; Gyarmati, 1970; and Ingle, 1971) to obtain the
kinetic energy by summing over the kinetic energies of

the components, the difference between the two definitions
is negligible for the present purpose (see Horne, 1966).

The energy transport equation can be expressed as

.+

p (dE/dt) + v-jE = E:VE - pE-§ (2.36)

where 3E is the internal energy flux not due to bulk flow

18

¢ +_ + I + +
3E - ouE + u'0 - (I + puzF) w (2.37)
T

where the electric current I is the sum of the partial

+
currents 1

a
v-l v-1
1 = 2 Id = z pafia (za/MG)F , (2.38)
d=0 d=0
and where
v—l _
pz==paio (Wu/Ma)zd = Z (2.39)

D. Principles of Nonequilibrium
Thermodynamics

 

 

The above treatment of nonequilibrium system is
as far as we can proceed from hydrodynamics alone. In
order to proceed in the discussion of the general theory
of irreversible processes, we could describe the elegant,
rational, fundamental approach of Truesdell (1969),
Mfiller (1968), Bartelt (1968), Bartelt and Horne (1970),
Gyarmati (1970), and Ingle (1971). Alternatively, we
could present the conventional, heuristic approach as
exemplified by de Groot and Mazur (1962), Fitts (1962),
and Haase (1969). For simplicity, we choose the latter
course, and we emphasize that the more fundamental ap-
proach apparently gives the same results for the simple

systems investigated here (Bartelt and Horne, 1970).

19

We need to introduce two fundamental assumptions
regarding the system under consideration. The first
assumption is:

Postulate I: The principle of local state

For a system in which irreversible processes are
taking place, all thermodynamic functions of state exist
for each element of the system. These thermodynamic
quantities for the nonequilibrium system are the same
functions of the local state variables as the correspon-
ding equilibrium thermodynamic quantities.

The second assumption is:

Postulate II: The assumption of locally linear
fluxes

+ . I
The fluxes 3a are linear, homogeneous functions

of the forces Ya. That is,
= 2 L V. (2.40)

The forces are "driving forces" for the fluxes. The
phenomenological coefficients de' are independent of the
forces. The diagonal coefficients Lad relate conjugate

fluxes and forces, while the off-diagonal elements La

a!

(aia') give rise to cross phenomena which are produced
due to interference when twotransport processes take
place simultaneously. As in the case of postulate I,
postulate II is apparently valid when the system is close

to equilibrium. Thus, both postulates apply to systems

20

with small space and time gradients of the local thermo-
dynamic variables. Based on both postulates, we now treat
the linear phenomenological theory for nonequilibrium
systems.

The Gibbsian equation for dE is

dE = Td§ - pdV + z “a dwa+-dF + dew , (2.41)

Applying the chain rule for differentiation of E (T,p,wa,F,w),
we obtain by the principle of local state and various thermo-

dynamic formulas,

dE/dt = ('C'p-pVB) (dT/dt) - (TVB-pVB') (dp/dt)
v-2 — — dr d4
4 ago (Ea-Ev_l)(dwa/dt) + E? + ZF 33 (2.42)

where C? is specific heat capacity at constant pressure

(and external fields),B is thermal expansivity,

_-—-1 -
= V 3V 3T 2.43

B ( / )p.wa,F.w ( )
8' is isothermal compressibility,

B' s - V—l'(8V/3p) (2.44)

T,wa.P.w

and Ea is partial specific internal energy,

Ea==(3E/3ma) (2.45)

T.p.mB,F,w ‘

21

Note also that

WI
II
M

3
t13|

a d (2.46)

Application of the chain rule for differentiation to the

equation of state p = p (T,p,wa) gives a similar relation,

do/dt = - pB(dT/dt) + 08' (dp/dt)

2 v-2 _ _
- 0 ago (Va—Vv_l)(dwa/dt) (2.47)

Substitution of (2.18), (2.19), (2.23), (2.42), and (2.47)

into (2.36) yields, after rearrangement,

+ v-2 _ _,
p6 (dT/dt) - T8(dp/dt) = ¢ -V°q - z j . (H'-H
p l a=0 d

where o1 is the entropy source term for bulk flow,

I Z +
¢l - (o + pI):Vu , (2.49)
3 is the heat flux
-)- f V_l _.
q = 3E - 20 1a Ha . (2.50)
a:

Ha is partial specific enthalpy, and
_' _ _
Ha — Ha-+I‘+-M Fm ,

——1—)F4 . (2.51)

22

It has been shown (Fitts, 1962; Kirkwood and Craw-
ford, 1952; de Groot and Mazur, 1962; Haase, 1969) that

the driving forces conjugate to q and jg are Vin T and

By postulate II, the linear phenomenological fluxes are

v-l

-> .—

-q = QTT Vin T + QEOQGTVTUG

+ V-l _

-ja = QGT Vin T + Z 9&8 VTUB’ d=0,...,v-l (2.53)
B=0

where Q's are the phenomenological, or Onsager coefficients.

These coefficients are not all independent since, by (2.22),

v—1
2 Q = 0, B = 0,...,v-l (2.54)
d=0 dB

Further, due to the requirement of positive definite entropy

production, it has been shown (Bartelt and Horne, 1969) that

2 Q = 0, a = 0,...,v-l (2.55)

23

Thus, for the v independent fluxes 3T = a, 30' 31"°"3v-2’

the linear phenomenological equations are

‘36 = QdT Vin T + 820 ”as VT(“8‘“v-1) '
a = T, o, l,...,v-2 . (2.56)

For most purposes, it is more convenient to use (2.53),
which is permissible because of (2.55). If the fluxes and
forces of (2.56) satisfy Onsager's (1931) condition, which
seems very likely but cannot be shown theoretically (Cole—
man and Truesdell, 1960; Andrews, 1967), then the matrix
of Onsager coefficients is symmetric; i.e., QdB = QBG for
8,8 = T, 0, l,...,v-2. However, since one of the goals

of experimental studies of transport phenomena is the
verification of Onsager's Reciprocal Relations (Miller,
1960), we do not use them.

Substitution of (2.7), and (2.52) into (2.53)

yields
+ v—l _
-Ja = QdT VinT + 320 908 (VB/MB) Vp
v-l v-2
+ Z 0 2 (u /M ) VX
8:0 as y=0 BY 8 Y
v-l
+ Z (ZS/MB) FVw, d=T, 0, l,...,v-l (2.57)

B=0 “as

24

General discussion about the physical implication of various
phenomenological coefficients can be found elsewhere (Fitts,
1962; de Groot and Mazur, 1962; Haase, 1969; Horne, 1966).
E. The Differential Equations

Describing the Osmotic Flow

9f Fluids Through a
Charged Membrane

 

 

The System and the Simplifying Assumptions

We are mainly concerned with the derivation of
the steady-state phenomenological theory for the trans-
port of fluids undergoing osmosis through biological mem-
branes. As has been mentioned in Chapter I, the trans-
port system for the study is composed of a charged,
continuous membrane separating two electrolyte solutions
of different concentration at the same temperature.

Before we can start to write the equations for
our transport system, we must take full account of the
fundamental problems which are pertinent to our present
investigation. As has been recognized for years, a
normally grown living thing is, from thermodynamic point
of view, an Open system in a stationary state (the flows
of all the species are constant in time), inside of which
irreversible processes occur continuously and slowly
(Haase, 1969). If, moreover, the fluxes and the gradients

are small, we may use the principles of nonequilibrium

25

thermodynamics such as those which enable us to use equilib-
rium properties (postulate I) and linear phenomenological
relations for the fluxes (postulate II).

Moreover, for the system in question we consider
(a) an isothermal, incompressible fluid; (b) a capillary
model for the membrane in which uniform cylindrical
capillaries penetrate across the membrane with flows of
all components in the direction of the capillary or
axial axis; (c) absence of chemical reactions; (d)
absence of external magnetic, gravitational, and cen-
trifugal fields; (e) constant viscosities; and (f) slow
motion, i.e., a quasi-steady—state in which local time
derivatives are zero. These requirements can be realized
experimentally, and do not, in principle, introduce error.
It is advantageous to introduce them into the phenome-
nological theory, the net effect being a simplification
of the differential equations.

We do not, however, make use of the previous as-
sumptions such as (i) constant diffusion coefficients and
ionic mobility in the flow equations; (ii) constant
dielectric constant of water in Poisson's equation; (iii)
radial-independent pressure; (iv) the classical Boltzmann
distribution of ions; and (v) the linearized Debye-Hfickel
theory for "all" cases. Moreover, (vi), the assumption

that Ka>>l, where K is the reciprocal of the effective

26

thickness of electric double layer (Debye length) and a
is the radius of the capillary, is found here to be not
always necessary. In the next several chapters we shall
discuss these assumptions in more detail. Our main pur-
pose is to derive, subject only to a minimum number of
assumptions, the steady-state phenomenological theory
that describes for the transport of fluids across bio-

logical membranes.

The Differential Equations and the
Appropriate Boundary Conditions

We now can set up the differential equations
necessary for describing the transport system in question.
Making use of the principles of hydrodynamics and non-
equilibrium thermodynamics and subject to the conditions
considered above, the relevant transport equations, rela-

tive to the local center of mass are

+ v-l _ v-l v-2
-3 — 2 a (v /M ) vp + x Q 2 (u /M ) VX
d B=0 B B B B=O d8 Y=0 BY 8 Y
v-l
+ = —
8:0 0&8 (zB/MB)FVw, a T, 0, l,...,v l (2.58)
+ + _
pu Vwa + V 3a —0 (2.59)
v-2
+ _ _ .+ _ 1 . _'_—l
-TBu-Vp — ¢l V q ago 3a V (Ho Hv-l) . (2.60)

Vp + erw = nvzfi . (2.61)

27

Only (2.58) and (2.61) are considered in what follows; the

other equations deal with effects too small to be observed

in membrane transport in ordinary electrolyte solutions.
The partial electric current relative to the local

center of mass is

e _ e
la — (Zn/Md) ja , (2.62)

and the total current is given by (2.38).
Poisson's equation, which relates the electro-

static potential w to the excess charge sz is
V°DVw = -4nsz = ~4WEF , (2.63)

where D is the dielectric tensor of the electrolyte solu-
tion, taken as nonconstant, and z and 7 are given by (2.41)
and (2.12). The units of w are volts throughout this work.
At this point our description of the system under considera—
tion is, in principle, complete.

In problems of electric conduction, however, it is
useful to relate the diffusion fluxes and electric currents
of all the species to the velocity of the solvent and thus
to adopt.the Hittorf reference system (Haase, 1969). The
solvent-fixed (SF) frame (i.e., (jo)o E 0) is extremely
convenient and transference numbers are usually referred
to it (Miller, 1966). The reference velocity is now

+
denoted by the velocity of solvent molecules, uo.

28

Moreover, since the solutions are relatively dilute, it
is useful to use molarity ca instead of mole fraction xa
to describe composition.

Before we can write explicitly the transport equa-
tions for the SF reference frame, it is essential to re-
quire considerable knowledge about conversion from a
reference velocity Ea to a second velocity 5b. For
simplicity, we follow the treatment of Haase (1969).

We consider, in the general case, the transition

from one diffusion current density

+ —>
ana = ca (um-ma) . (2.64)

to a second diffusion current density

-> +
bja = ca (ua-wb) . (2.65)

ZFor fluid systems, the reference velocity Ea and 3b are

so chosen that the relations

+ V—l —>

ma = :o(wa)aua (2.66)

v-1

2 (w ) = l (2.67)

d a

a:
v-l

+ +

wb = Z (wa)bua (2.68)

29

Z (w )b = l (2.69)
d=0

hold. The (wa)a and (wa)b are the normalized weights,
i.e., "weight factors" for averaging the velocities sub-
ject to the normalization (2.67) and (2.69). For the
barycentric velocity, defined in (2.20), the "weight
factors" are the mass fractions. It can easily be shown

from (2.66), (2.67), and (2.64) that

v-1
2 [(wa)a/Cal
d=0

T

J = 0 . (2.70)

30.

and from (2.68), (2.69), and (2.65) that

v-l
—? —
Z [(wa)b/ca] bja — 0 . (2.71)
d=0
Combination of (2.64) - (2.71) yields the conversion re-

lation between two frames of reference:

v 1

-c 2
a =

B [(wB)b/cB] aje . (2.72)

0

We are, however, primarily concerned with the con-
VeIflSion from the barycentric system to Hittorf's reference

System. Let

= 3 /Ma. baa = (3') (2.73)

30

Note that in Hittorf's frame of reference,

(we)o = l, (Wm)O = 0, or = 1,2,...,v-l , (2.74)
v-l
so that (2.69) is immediately satisfied, i.e., Z (wa)O = 1,
d=0
and
+ _
(30)O — 0 (2.75)

Hence we obtain from (2.74) and (2.72)

(3 ) = 3 /M - c I /c M , (2.76)

yielding the conversion from the barycentric system to
Hittorf's reference system. By definition of jo, the two

reference velocities are related by

+_—>- 7
u — uo (l/Oo)jo . (2.77)

Consequently, if SF reference frame is chosen for
the system, the diffusion flux and electric current of
species a can now be eXpressed, relative to SF reference

frame, as

-r V_l o —o \)-1 o
-(3 ) = 2 9 v Vp + Z 9 RTVin a
d o B=l d8 8 B=l d8 8
v-1 0
+ E 9 z Fvw,cx= 1,2,...,v-l (2.78)
B=l d8 8

31

) (2.79)

where 038 are the phenomenological coefficients in SF
reference frame. The Navier-Stokes equation in SF refer-
ence frame then becomes, according to (2.77),

Vp + EFVw = nVZ E - nV2 ELI;
o

o (2.80)

0
We assume, for the Navier-Stokes equation only, that the

difference between E and Go is negligible and, therefore,

that
_ 2+

Vp + ZFVw = nV uO (2.81)

The appropriate boundary conditions for the problem
are

c := c (0) at x = 0, and c = c (i) at x = i

do do do do

PO(X=i)-po(x=0)=0

0 = 0 (0) at x = 0, and 0 = 0 (i) at x = i , (2.82)

where Coo (0) and Cdo (i) are initial molar concentrations

of Component d at the boundaries, pO (x = i) and p0 (x = 0)
are initial hydrostatic pressures at the boundaries, 0 is
partial electric potential along the axial axis, and 0

(0) and 0 (i) are initial values of 4 at the boundaires.

The appropriate boundary conditions to w and 110 are

dw/dr

Sue/8r = 0

dw/dr 4no/Da, and u = 0 at r = a , (2.83)

O

32

where c is the surface charge density fixed on the mem-
brane, Da is the dielectric constant of the solution at
the wall, and uo is the velocity of solvent molecules or
the SF reference velocity along the axial axis.

The experimental transport parameters, for some
binary electrolyte solutions, which are necessary for the
study have been obtained based on the SF frame of refer-

ence and can be found elsewhere (Miller, 1966).

CHAPTER III

EQUILIBRIUM DISTRIBUTION OF IONS

IN AN ELECTRIC FIELD

A. Introduction

 

Before we can obtain the solution of the steady-
state phenomenological theory which describes osmotic flow
phenomena in biological systems, it is necessary to acquire
considerably more knowledge of the local distribution of
ions, charge density, electrostatic potential, and pres-
sure in nonequilibrium system. This information is essen—
tial in describing transport phenomena in membranes and
thus must be resolved and discussed beforehand. In the
present chapter we derive an equation for the local dis-
tribution of ions in an electric field and then obtain
the excess charge density in the system considered. The
principal enabling step is the assumption that the system
is in chemical equilibrium radially, i.e., that the
chemical potential of each species is uniform radially.

The fundamental problem in a study of a membrane
system or any electrochemical system in which the charges

are unequally distributed (i.e., 2 ¢ 0) in the volume or

33

34

on the surface is the knowledge of the macroscopic dis-
tribution of ions at each point of the system since the
presence of a net charge produces a macroscopic electric
field which tends to displace charged species. From
knowledge of this distribution it will be possible to
calculate several electrochemical properties and thus to
describe accurately the behavior of electrolyte solutions
in electric fields.

However, as has been known for years, there are
great difficulties in calculating the distribution of ions
in the field, due partially to the lack of a coherent
theoretical formulation, and partially to the mathematical
difficulty in solving the problem. For these reasons one
often uses the classical Boltzmann distribution equation
without adequate justification (Sf. Kobatake and Fujita,
1964; Fujita and Kobatake, 1968; Gross and Osterle, 1968).

Thus, it is assumed for each species a that

_— __ _ B
xa — Xa eXp ( zaFw/RT) — xa , (3.1)

where Ea is the mole fraction of d in the absence of an
electric field and where, for subsequent use, we denote
the Boltzmann equation mole fraction by x2.

Many assumptions have been made in conjunction
with (3.1); the usual ones are: (a) existence of point

charges, which implies that molar volumes of ions are

35

zero; (b) absence of polarizable components; (c) absence
of activity coefficients; (d) uniformity of pressure,
which implies among other things that the pressure-
induced gradient of the electrochemical potential of
a charged species is negligibly small. Despite the fact
that various authors have attempted to correct the Boltz-
mann equation, often empirically, some by introducing a
finite volume (Sparnaay, 1958), some by introducing
polarization, and others by introducing pressure, none
of the theories is complete. Moreover, many of these
authors treat the parts of the double layer very close
to the interface as special media, very different from
the parts remote from the interface. Further, the micro-
sc0pic nature of these parts of the layer give rise to
difficulties in their description in terms of macroscopic
quantities such as polarization or pressure. The most
complete theory for the distribution of ions in an electric
field so far has been that of Sanfeld (1968) derived by
means of the local thermodynamic method (Prigogine, 1953;
Mazur, 1953; Defay, 1954). Unfortunately, the theory is
limited only to equilibrium systems and is valid only
for very dilute solutions and gas mixtures.

Our purpose now is to calculate this distribution
in the general case and to apply our result for calcula—

tion of the excess charge density in the system.

36

We consider continuous, anisotrOpic fluids in which no
chemical reactions occur and which are subject only to

the external electric field. Unlike many other investiga—
tors, we make full use of the equations of chemical thermo-
dynamics, including particularly all pressure terms. At
present, we only treat systems (particularly, biological
systems) in which the surface charge density fixed on the
surface or on the membrane phase is such that the effects
of polarization are small enough to be neglected. This
requirement can always be realized experimentally without
introducing error, but it could be modified in subsequent
extensions of our theory.

B. Derivation of Modified
Boltzmann Equation

 

Consider a charged, continuous phase (i.e., a
membrane, a barrier, or a cylinder) which is static or
through which there are steady-state flows of matter.

We assume an isothermal electrolyte solution containing
solutes which are dissociated in a neutral solvent,
absence of chemical reactions and external fields except
electric field, and a capillary model for the phase con-
sidered in which uniform cylindrical capillaries penetrate
across the phase. Our results also apply to any single,

non-capillary cylinder.

37

When the only external force is electrical, the
gradient of the molar electrochemical potential “a of

Component a in the absence of polarization may be described,

according to (2.14) by

—O
Vua = Va Vp + RTVin ded + zaFVw, d = 0,1,...,v-l (3.2)

where V: is defined by (2.15). The gradient of “a in the

radial direction is, by (3.2),

_ —o
Spa/8r — Va (Sp/3r) + RT (Bin xafa/ar)

+ zaF (aw/3r) . (3.3)

‘We assume that the chemical potential of each species is

uniform radially,
Qua/3r = 0, d = 0,1,...,v-l . (3.4)

'That is to say, all the radial fluxes are zero everywhere
in the system; chemical equilibrium obtains radially.
lqote, however, that the gradient of “a along the capillary

<Dr cylindrical axis does not necessarily vanish,

Bud/3x ¢ 0 . (3.5)

Tfllus (3.4) by no means requires that the system is in a
5State of total thermodynamic chemical equilibrium (i.e.,

VWJG E 0). Combination of (3.3) and (3.4) yields

38

v: (ap/er) + RT (Bin xafa/ar) + zaF (aw/er) = 0 . (3.6)

The Gibbs-Duhem equation (2.8) yields, for the

radial direction and subject to (3.4),
ap/ar = - 7F (aw/8r) . (3.7)
Substitution of (3.7) into (3.6) yields, for each species,

(1/53) (Bin xa f/Gr)+—(za/V:) (EV/8r) = 2 (aw/er) , (3.8)

where
)1! = Ftp/RT . (3.9)

In all other treatments of electrolyte solutions
in electric fields, electroneutrality is assumed and the
right hand side of (3.8) is set equal to zero. The solu—
tion of (3.8) for electroneutrality is, clearly, (3.1),
the classical Boltzmann equation, as long as fa is con—
stant. We choose to retain the right hand side of (3.8),
and thus to allow, consistently, for non-electroneutrality,
at least near the walls.. Although we could formally take
account of nonconstancy of fa and V: by, say, the perturba-
‘tion scheme of Horne and Anderson (1970), we choose, for
‘the first attempt, to regard fa and V: as constants. The
latter assumption is certainly reasonable since V: varies

<3nly with pressure and since the isothermal compressibility

An»
ad .-

 

A!
a-

bond

u
*

.‘U

I.

39

as defined in (2.44), is of order 10"5 atm_1 for water

and aqueous solutions. Constancy of fa is a stronger
assumption a priori, since it is valid only to the extent
that the composition distribution is not great. We shall
find that this is indeed the case and that, therefore, the
assumption of constant fa is warranted.

With these assumptions, (3.8) becomes, in detail,

(1/63) (Bin xO/ar) = (1/3?) (Bin xl/ar) + (21/38) (EV/8r)

(1/63) (Bin x2/3r) + (22/53) (EV/8r)

(1/58-1) (Bin xv_l/8r) + (20—1/58-1’ (EV/8r)

= E (EV/8r) , (3.10)
or,
—o _ —o —o
(l/Vo) (Bin xO/Br) — (l/va) (Bin xa/ar) + (Zn/Va) (EV/8r)
= 7‘(39/ar) . d = l,...,v-l (3.11)

The first v—l of (3.11) are solved immediately,

x = I (x /i )gOLO exp (—z V)
d d o o d

B

a , 0.: Orl’ooo,\)_l (3012)

— goo
= (xo/xo) x

40

where id is the value of xa for no electric field, where

x: is defined by (3.1), and where

gao = 53/5: . (3.13)
. — gdo
The next task 18 to evaluate (xo/xo) . Note that the

ordinary Boltzmann equation, (3.1), is valid only for

X0 = x0 (or, tr1v1a11y, for gmO = 0).

In order to evaluate x0, we use the facts that

v—l _ __ v-l B B v-1
2 x = l - x , Z x = l - x , Z x = l - x , (3.14)
a=l a 0 6:1 9 0 8:1 9 0
whence, from (3.12),
v-1 9
1 - x = z (x /i ) “0 xB . (3.15)
o d=l o o a
Let
x0 = x0 (1-5) , (3.16)
and then
v-l g
1 - E + x a = 2 (1—6) “0 xB . (3.17)
o o d=l a

By the binomial theorem,

41

._ _ V'1 1 2
l - Xo + on‘;= oil X [l - gdog + 2 goo (goo-l):
l 3
- 6 goo (goo-l) (goo-2)5 + '
v-l v—l
_ _ _ B l2 _
- l X g E Xd gdo + 2 E E Xa gdo (gdo l)
d-l d—l
13)”l
_ E g :1 xa gdo (goo-l) (goo—2) + .. , (3.18)
or,
v-l v-l
_ _ _ _ B 12 B _
xog _ (X x ) E Z Xd goo + 2 E Z Xd gdo (goo l)
d=l d=1
-l
13" B
_ g g :1 xa gao (gaO-l) (gaO-Z) + ... (3.19)

Rather than use the ordinary reversion of series
formula (Abrahamowitz and Stegun, 1964) to solve (3.19)
for g, we observe that the difference (§o-x2) is pre-

sumably very small and so is the value of any particular

x: . This means that the second term on the right hand

side of (3.19) is of second order in smallness since it is

B

a goo (g is of order

a. —_B
the product of 5 ~ (x0 x0) and Z x a0

unity or greater, in general). In order to keep consistent
account of the order of smallness, we multiply both (Vb-x2)
and x: by the bookkeeping parameter A, and we write 5 as a
perturbation-type power series in 1 (see HOrne and Anderson,

1970).

42

“ _ 2 3
E — 151 + A $2 + A 63 + .... (3.20)
where E = 5 when A = l. (3.19) becomes, then,
— 2 3 _ — _ B
xo (A51 + 1 £2 + A 63 + ......) — 1 (x0 x0)
v-1‘
2 B
- K (15 + A E ) Z x g
l 2 d=1 a do
-1
1 2 2 V B
+ 7 A (A g1 ) ail xa goo (gaO-l) + ... (3.21)
Equating powers of l, we find
_ B _
£1 = (xo-xO)/xO (3.22)
v-l B _
£2 = - £1 ail (Xoc/Xo) gao (3.23)
v-l B _
E3 _ — E2 ail (Xe/x0) gdo
v-1
1 2 B —
+ 6 E1 ail (Xe/X0) gdo (goo-l) (3'24)
v-l B _ v-l B _
E4 = - £3 E (Xe/x0) gao + £1 £2 E (Xe/X0) gao (gaO-l)
d—l d—l
v-1
1 3 B —
- 6 51 oil (Xd/Xo) gdo (goo-l) (goo—z)’ (3'25)

and so forth.

43

The modified Boltzmann equation is, then,

_ 9
xa = xa[exp (-zaV)](l-g) do , (3.26)
with gdo = VZ/V: and
e=al+£2+€3+€4+..., (3.27)

the 5i being given by (3.22)—(3.25)

It would be most awkward, however, to have to deal
with all the terms of (3.27) in all cases. As we shall
see in Chapter IV, indeed only the first and the second

terms in (3.27) are important. Then it is quite sufficient

totflm
_ B _ v-l B _
E = [(xo-XO)/XO] [l- ail (Xe/X0) gaO] . (3.28)
xa = xa[exp (-zo“i’)]{1-gOLO [(xo-x:)/§O]

NIH

—B—2
- 90LO (gaO-l) [(xO-XO)/Xo]
_ B _ v—l B _
+ gao [(xO-xo)/xol B:lth/xo) 980} .

a = 0, 1, 2,...,V-l . (3.29)

44

C. Molar Volume and Charge

 

v-l
Since 3 = Z xa za , we have, from (3.29) and
d=l
(3.1),
v-l v-1
'2 = z xB z - z xB z g [(§ -xB)/§ 1
d d d a do 0 o o
a=l d=l
v-1
1 B — B — 2
— 2 oil xd zd gdo (goo-l) [(Xo-xo)/Xo]
v-l v-l
B — B— B—
+ Z x z 9 [(x —x )/x ] Z (x /x ) g
d=l d a do 0 o o B=l B 0 80
+ .... (3.30)
Further,
v-l v-l
v = Z x v = x V0 + Z x V0
a d o o a 0
=0 d=l
- —o v-l B —o
= xovo (1-6) + Z Xa Va (l-gdo E) + .
d=l
v-l
_-—o _ —_B-— B—
.xovo {1 [(x0 xO)/xo] + 6:1 (Xe/X0) gao
v—1

2 (xi/£6) gao (gao-l) + ...} (3.31)

_ B _
- [(xo-xO)/x01 6:1

45

Finally,
v-l
— _ _._ _ — —o -l B
Z — (x/v) - (xovo) { Z xa za
d=1
v—l
_ B B _
- (Xo-xo) I (Xe/x0) za(gdo-l)
d=1
v-l B v-l B _
- ( Z xa za) [ Z (Xe/X0) goo] + ... (3.32)
d=l 8:1
-1 2
where we have used (1 + x) = l — x + — x + ..

D. Uni-univalent Case

 

If all cationic species have charge za = +1 and
all anionic species have charge 2a = -1 (the typical case
of biological interest), then the general formulas found
in the last two sections take on simpler forms. By

electroneutrality of the starting materials,

2 I = z 32 (3.33)
. d . a
cations anions
and by (3.14),
z §=-l-(1-§)= z 32 (3 34)
d 2 o d °

cations

46

Then,
1 - x = Z x exp (-2 V)
d
d=1
= Z xa e-w-+ X x ew
cations anions a
_ l _— -V V
— 7 (1 x0) (e + e )
= (1-ib) coshV , (3.35)
and
i - xB = (1-i ) (c hV-l) (3 36)
o o o 08 .

Further, for the uni-univalent case,

v-1
2 xB z E V e-w - Z ‘V ew
a d

a=1 a

cations

= % (l-SEO) (e‘w -e‘y)

- (1—i6) sinhV . (3.37)

From (3.36), (3.37), and (3.32) we finally obtain

2 = - {(1-V6)/§6Vg}{sinhw + [(1—§0)/§O] sinhV (coshV-l)
+ (1 "‘°) [( hV-l) ( e'w - ew)
/xovo cos v+ v_

- sinhV (v+ e”? + v_ ew)]} , (3.38)

47

where we have used

... -O
v+ = Z x Va
cations
v = 23 V ‘V0
- I a a

anions

E. Discussion

 

For the first time we have a thermodynamic theory
for the distribution of ions in an electric field, which
takes account of the effect of pressure resulting from the
gradient of electrical potential or nonzero local dis-
tribution of charge density, and which simultaneously
takes account of the effect of molar volumes of ions.

The results can be used either to justify the applica-
bility of classical Boltzmann equation or as they stand
when the Boltzmann equation is not applicable. The equa-
tions can be used further to predict the effect of molar
volumes of ions on various electrochemical prOperties and
thus are better in describing the system. Alternatively,
further theoretical and experimental development can now
be improved and interpreted more accurately on the basis
of the equations. In general, use of the equations re-
quires knowledge of molar volumes of ions and the surface
potential or the fixed surface charge density, which we

shall determine in Chapter IV. Further improvement may

48

be made in the equations by taking account of the effects
of variable activity coefficients, polarization, and, in

extreme cases, pressure dependence of partial molar volume.

CHAPTER IV

RADIAL DISTRIBUTION OF ELECTROSTATIC POTENTIAL

IN A MODERATELY CHARGED CYLINDER

A. Introduction

 

As has been shown in Chapter III, knowledge of
the distribution of electrostatic potential in an electro-
chemical system is essential to the understanding of
physical and chemical properties of the electrolytic or
polyelectrolytic solution in question. Poisson's equation
relates the electrostatic potential to the fixed surface
charge dnesity. In the past few years, solutions of
Poisson's equation have been obtained numerically (Kotin,
1962; Gross and Osterle, 1968; Fair and Osterle, 1971)
and analytically (Alexandrowicz, 1963; Philip, 1970)
based on certain geometrical and mathematical models.
Although the results are satisfactory in many respects,
all previous investigators have used the classical Boltz-
mann equation, (3.1), without justifying its use, and all
but one have used the dielectric constant of pure water
rather than the actual dielectric "constant" of the mix-

ture .

49

50

It has been reported (Hasted, 1948; Booth, 1951)
earlier that the dielectric constant of a salt solution
in an electric field decreases monotonically with increas-
ing salt concentration and/or the electric field strength.
Recently, Devillez et_al. (1967) have reported a numerical
method of integration of Poisson's equation in which the
concentration dependence of the dielectric constant is
included. However, no report has yet been made with re-
gard to the solution of such a modified equation by analyti—
cal techniques.

In this chapter our purpose is to specify the in-
fluence of the nature of the electrochemical systems (par-
ticularly, biological systems) and the salt concentrations
of aqueous strong uni-univalent electrolytes on potential.
We combine the macroscopic ion distribution equation de—
rived in Chapter III with Poisson's equation and solve the
resulting equation subject to the requirement that IV] =
(FIwI/RT) : 0.245. The range of salt concentrations used
is 0.004 to 2 mole/i. We solve the differential equation
in cylindrical coordinates since we have employed a capil—
lary model for the problem, and we consider the concentra-
tion dependence of the dielectric constant in Poisson's

equation.

51

B. Poisson's Equation

Poisson's equation is, in the general case,
Vobvw = - 4n28 , (4.1)

where D is the dielectric tensor of the salt solution in
question, 0 is the electrostatic potential in volts, F is
Faraday's constant, and 7 is charge per unit volume as

defined in (2.12). We consider in this chapter only the
radial direction and thus solve Poisson's equation for a
circle. Assuming azimuthal symmetry, Poisson's equation

is, in radial coordinates,

D 93 = - 4n2F (4.2)

£1 r
dr dr ‘

HIH

The boundary conditions are

(dw/dr)r=0 = 0, (dw/dr)r=a = 4no/Da = o/ca , (4.3)

there a is the radius, 0 is the charge on the walls, in
C3 nfg (C = coulomb; m = meter), Da is the dielectric con-
stant at the wall, and ea is the electric permittivity at
time wall.

In Chapter III, we obtained an explicit formula
(33.32) for 7 as a function of composition. Composition
is I in turn, related to electrostatic potential (I) by (3.29) .

SiUQDStitution of (3.32) and (3.1) into (4.2) yields a

52

right-hand-side in which the only variable is w. The
result is formidably complicated, however, and, being an
infinite series of various terms of the type exp (—zaV),
could no doubt not be solved. Instead, we use previous
results (e.g., Fair and Osterle, 1971) to estimate various
terms with the goal of restricting the conditions suffi-
ciently that a tractable equation results.

These estimates will also be used in dealing with
the composition and field strength dependence of the
dielectric constant. According to Sanfeld (1968), who
has summarized the results of all previous workers, the
concentration and field strength dependence of the
dielectric constant is quite well represented for binary

uni-univalent mixtures by
D=D+5c+6c+hE, (4.4)

for concentrations c+ and c_ less than about 3 molar and

for field strengths less than about 0.5 x 108 volts per
meter in magnitude. Some values of 6+ and 6_ are tabulated
in Table 4.1. According to Sanfeld, h is found empirically
to be about —3 x 10"16 mZV-Z. D is plotted versus [E] in
Fig. 4.1. For large concentrations the dielectric "constant"
decreases inversely with concentrations (Pottel, 1964), and

(4.4) cannot be used. We assume that (4.4) can be extended

to accommodate any number of ionic species, and we define

53

Table 4.l--Values of gland V8 of some uni—univalent
electrolytes at 25°C (Sanfeld, 1968).

 

 

 

ions V8, cm3/mole 10-3 5“, cm3/mole
Li+ -0.9 -11
Na+ -1.4 -8
K+ 8.8 -8
H+ 0 -17
OH- -5.4 -13
F- -2.2 -5
C1 18 -3
I“ 36.5 -7
6 c = Z 6 c
+ + cations a a
6 c = Z 0 c . (4.5)

C. Simplifying Approximations

The principal goal of this section is to render
(4.2) tractable. In particular, we examine: (i) the
formula, (3.32), for 7, in order to simplify it for use
in (4.2); (ii) the formula, (3.29), for ca, in order to
simplify it for use in (4.4), and thence in (4.2); (iii)

the wall boundary condition for (dw/dr), (4.3), in order

54

 

 

0 A l I 1 I
0 1.0 2.0

IE 107, V/cm +

Fig. 4.l--The dielectric constant I) of water as a
function of field strength IEI (in Volts
per cm). See, for example, Booth (1951).

 

 

55

to estimate IE] and its effect on D as in (4.4); (iv)
(Fw/RT) = V itself, in order to determine under what
circumstances it is permissible to replace sinh? by V.
To these ends, we utilize the classical Chapman-
Gouy relation (see its derivation in Appendix A), valid

only for special circumstances,

2 (2csRT)l/2 sinh (lwal/z) , (4.6)

|9|

only for purposes of estimation. Subsequently we find an
improved version of (4.6), but our estimates here are not
importantly changed. In (4.6), a uni-univalent solution
of large concentration c is assumed. Note that we fully
expect [V] at the wall to be larger than IV] in the solu-
tion, as found numerically by Fair and Osterle (1971),
and we shall therefore estimate the maximum value of IVal
from (4.6).

Let us begin by investigating item (iv) in the

above list; i.e., we require
sinh? = V . (4.7)
Since of course

sinh x = x + gL x3 + E? x5 + ..., (4.8)

we require, in (4.7), that

_1. 2 4

1 >> 3, w + 3% V + ... (4.9)

56

For better than 1% accuracy, we therefore require

0.01 >> 3% 92 + §%-w4 + ..., (4.10)

which yields
IVI < 0.245 (4.11)

So, for Ital < 0.245 (in fact, for [Val < 0.49),

the Chapman Gouy equation becomes

1/2

Iol == (2CERT) IT | , (4.12)

a
and (4.11) yields the following restriction on lol//E ,

2

(|o|//E) < 0.245 (28RT)1/ (4.13)
For T = 298°K, R = 8.314 J mole-l K'l, and e = 7 x 10‘10
C V-1 m-l,

(28RT)l/2 = 1.86 x 10"3 c mole‘l/2 m'l/2 (4.14)

Therefore, for sinh [Val to equal IVaI to better than 1%

accuracy, it is necessary for

3 1/2 m‘l/2 (4.15)

(|o|//E) < 0.45 x 10‘ c mole-

Concentrations of important metal ions in biological
systems (Woodbury et al.. 1970) lie in the range of 0.004 <
c < 0.4, in units of moles per liter, while ion concentra-

tions are often outside this range in other systems of

57
interest. We take c i 2 mole liter"l = 2 X 10-3 mole cm-3 =

2 x 103 mole m-3 since the validity of the simple linear

correction to the dielectric constant, (4.4), would be
questionable for higher concentrations. Then in order for

both (4.15) and (4.16) to hold, Iol may be as large as

3 C m-Z. On the other hand, for c = 0.004 mole

6 mole cm-3 = 4 mole m-3

be larger than 0.9 x 10"3 C m_2. Typically, charge den-

3

20 x 10-

1

liter- = 4 X 10- , Iol must not

sities are of the order of 10— C m_2 (Fair and Osterle,

1971). However, for I0] 2 10-3 C m-Z, we require c 2

4 x 10.3 M. Otherwise, for fixed lo] 2 10.3 C m-Z,

smaller concentrations lead to values of IV] too large

for (4.7) to hold. For such small concentrations (which

do not appear to be of biological interest in any case),

there exists no analytical solution of Poisson's equation.

Fair and Osterle (1971) have obtained numerical solutions
5 -3 2

for c z 10- M and Icl : 10 C m- . We hereafter restrict

consideration to

0.004 M 5 c 5 2 M (4.16)

3 2

and Iol z 10- C m- , the actual upper bound on Io] being
that given by (4.15) for the actual composition in question.
Working backwards, we now accomplish item (iii) by

using (4.3),

IEI z(dV/dr)a = IoI/ea (4.17)

58

For lo] 5 20 x 10'3 c m“2 and e z 7 x 10"10

a
7

c v"1 m‘l,

v m‘l, so that IhlEl2| 5 3 x 10‘16 x 9 x 1014

[El 5 3 x 10
= 0.27, which is less than 1% of the dielectric constant
D0 of water. Since this is the maximum correction to D,

we confidently neglect the contribution of IE]2 to the

dielectric constant and write
D=Do+0c+6c (4.18)

By Table 4.1, the maximum contribution of the concentration
terms to the dielectric constant would be for c+ = c_ =

2 M = 2 X 10-3 mole cm-3; for, say, 2 M LiI, D = 78.3 -

22 - 14 = 42.3, a 46% reduction. The concentration correc-
tion term contributes less than 1% for total salt concen-
trations less than about 0.05 M, depending on the particu-
lar ions involved.

The next task is estimation of xa and ca. For

total salt molarity 5 2 M, Ea is at most 2 M and

— — —0 ~ - —o 3 -l —
x = c v ~ c v = 18 cm mole c ,
O. 0!. 0. C1 0 O.
E 5 0.002 mole cm‘3, Pa 5 0.036, i0 3 0.928 (4.19)

d
For uni-univalent salts, by (3.35),

B _
x0 = l - (1-xo) coshV , (4.20)

and for (1—ib) 3 0.072, [9| 5 0.245,

59

x: 1 1 - 0.072 x 1.03 = 0.926,

F — x3 = (1-I ) (coshW-l) 5 0.00216
0 O O

(4.21)

Thus, correction terms proportional to (xO-xg) in (3.22) -

(3.25) are entirely negligible compared to unity.

Correction terms due to volume terms, however, are

not always negligible. For, say, 2 M KCl, by Table 4.1,

W

(0.036/0.928) e"

0.063 .

R

Thus,for better than 1% accuracy, our equations from

Chapter III become, with

v—l
__ — — _ -o —o
W's—8:1 (xB/xo) [eXp ( zB‘FHWB/Vo) .
the following:
xa = xa exp (-zaV)

(1/6) = (1—V*)/§5 VS

_ B — -o _-* = — _
ca — (Xa/XOVO) (1 v ) ca exp ( zaV)

(8.8/l8) + (0.036/0.928)ew(18/18)

(4.22)

(4.23)

(4.24)

(4.25)

(4.26)

60

0—1
—_ ——O _ - -—
Z — (l/xovo) ( 2 xa za exp ( zaV)) (l v*)
d=l
v-l _
= ail ca za exp (-zaV) (4.27)
In obtaining (4.25) — (4.27), we neglect the difference

between v* and V* compared with unity since

v-l
_ - — —o —o
v-l _ _ —o —o
= v* + 8:1 (xB/xo) (l-exp (-zBV))(vB/VO) (4.28)

and, for small IV] and a uni-univalent mixture,

—* _ * ~ ‘ — “0 ’0
v v ~ ( 8 (XS/X0) (VB/V0)
cations
- z (E /i) (V°/V°)) (4.29)
anions B O B O
For 2 M KC1, e.g., with IT] = 0.245,
7* - v* 2 0.02 X 0.245 = 0.005 (4.30)

Thus, for the prescribed conditions the equations of
Chapter III reduce to the simple Boltzmann equations.
If we further specialize to the uni-univalent case,

we have

61

7 = 2 E e—w - Z '5 ew
cations a anions
= - 2c sinh? = - ZCV , (4.31)
where we have used (4.7) and where
c = z E = 2: E (4.32)

C a O
cations anions

Moreover, the equation for the dielectric constant is

D + 2 6 E e + z 6 E e (4.33)
0 a a! I
cations anions

D

and Poisson's equation is

l. d D dV _ 2 _ 2

EEE r (5;) a? - (8TTCF /DORT) W — K W , (4.34)
where

K = F (8nc/DORT)l/2 (4.35)

is usually called the reciprocal of the Debye length. The

boundary conditions on (4.34) are

(d‘Y/dr)r=0 = 0, (dV/dr)r=a = (4noF/RTDa) , (4.36)
where
—V V
— a — a
Da = D + 2 6a ca e + Z 6 ca e (4.37)
cations anions

62

D. Perturbation Scheme

 

The concentration dependence of the dielectric
constant must be taken into account since it strongly
affects the dielectric constant. However, it complicates
the Poisson equation, (4.34), so much that it still cannot
be solved directly. In order to take account of this last

complication, we solve (4.34) by a perturbation scheme

defined by
6 = DO + a (5+ c+ + 5_ c_) (4.38)
and
@ = W + E? + 62? + O (63) , (4.39)
o l 2
When a = l, D = D and. Q = W. The parameter a is simply

a bookkeeping device which allows us to keep track of
the inclusion of the concentration dependence of D. In
order to obtain an explicit perturbation expression for

c+ and c_, we note that

2
e — l i W0 i 6W1 + O (a )

l 2
+ 7 W0 + ewowl + ... (4.40)

and (4.33), (4.38), (4.40) yield

A _ l 2
D = D + e{ Z 6 c [l—W + — W —'€W (l-W ) + ...]
O cations a O 2 O l O
+ Z 65[1+wo+%wg+ewl(l+4’)+ ]}
anions a a O
= D0 + e{ ( z a E + 2: 5 Ea) (l + % 42)
cations a anions a O
— < 2 5a Ea - 2: 5 E ) we}
cations anions a
+ o (52)
= D + eD + O (82) (4 41)
o 1 ’ °
where
D1 = ( 2 6 E + X 6 E ) cosh?
. a a . d a o
cations anions
- ( Z 6 E' - Z 6 E ) sinh?
. a . a o
cations anions
+ - .
= D cosh? - D SinhW (4.42)
o o
Substitution of (4.41) and (4.39) into (4.34)
yields

(l/r) d/dr {r (1 + eDl/DO + o (52)) (dwO/dr

+ edwl/dr + O (52))}

= K2 (40 + 54 + o (32)) , (4.43)

l

64
with boundary conditions, from (4.36), (4.39), and (4.41),
2
(d‘Po/dr)r=0 + (d‘l’l/dr)r=0 + O (8 ) — O , (4.44)

2
(dwO/dr)r=a + e (dwl/dr)r=a + o (e )

l

= (4noF/RT) (00 + eDl (a) + o (52))" (4.45)
The zeroth order differential equation is, from
(4.43),
(l/r) (dr (dWO/dr)/dr) = K2 we (4.46)
with boundary conditions
(d‘PO/dr)r=0 = 0, (d‘PO/dr)r=a = 4woF/DORT (4.47)

The first order differential equation is, from

(4.43),
(l/r) (dr (dWl/dr)/dr) =
- (l/r)(dr(D1/DO)(dwo/dr)/dr) + K2 41 , (4.48)
and the boundary conditions are
(d‘l’l/dr)r___0 = O, (d‘Pl/dr)r=a = - 4noFDl(a)/D: RT (4.49)

Clearly, higher order equations could be obtained if de-

sired.

65

E. Solutions of the

 

Differential Equations

 

Then

and

Substitution of Variables

First, make the substitution

-1

g = Kr
r = EK
ga = Ka

, d/dr = K (d/dg) r

The zeroth order equations are then

or

(1/5) (d5 (dwO/da)/dg) = w

O

dsz/de2 + (1/5) (dwO/dg) - w = o ,

O

(Mo/dag:O = 0, (dWO/d§)€= = 4woF/KDORT

E

a

The first order equations are

2
d Wl/dg

q1 (E)

2

— (1/5) (da<Dl/DO) (dwO/da)/da)

2 2
- (Di/Do) [d wo/dg + (i/g) (dWO/d€)]

(4.

(4.

(4.

(4

(4.

(4.

(4.

50)

51)

52)

.53)

54)

55)

56)

66

- (dWO/dé) (d(Dl/DO)/d€) ,

- (Di/Do) we - (dwO/da) (d(D1/DO)/d€) , (4.57)

with
_ - 2
(dWl/d€)€=0 — O, (d‘Pl/dg)€=€a — - 4noFDl(a)/DOKRT (4.58)

Zeroth Order Solution
Multiplication of (4.54) by 62 yields

2 n , 2 _
a we + 5W0 - g (0 _ o , (4.59)

which is easily recognized as the modified Bessel Equation
of order zero, whose general solution is (Irving and

Mullineux, 1959)

we = A IO (g) + B KO (5) , (4.60)

where Io and K0 are the modified Bessel functions of order
zero of, respectively, the first and second kinds.
In order to utilize the boundary conditions, we

differentiate (4.60) with respect to E,
I _ _
where we have used the recurrence relations

I") = I , Kg) = K (4.62)

67

As 5+0, Il (€)+O but Kl+w (Abramowitz and Stegun, 1964).
Consequently, since W5 (0) = 0, we must have B = 0 and
therefore
I _
To — A I1 (5) (4.63)

By the second of (4.55),

A = 41TOF/KDORT I1

and the zeroth order solution is

We = A IO (6) = (4noF/KDORT Il (Ka)) IO (Kr)
= (o/VZeocRT) [IO (Kr)/Il (Ka)] , (4.65)

where so = Do/4w . The values of We at two locations are

of particular interest:

W0 (r=0) = To (0) = o/VZeocRT Il (Ka) , (4.66)
To (r=a) = We (a) = o/VZeOcRT) [Io (Ka)/Il (Ka)]
= IO (Ka) W0 (0) (4.67)

Two things are especially noteworthy: (i) We (0) ¢ 0;
(ii) the Chapman—Gouy equation, (4.12), is valid only
for [IC (Ka)/Il (Ka)]+l, which never occurs, but is a
fairly good approximation for Ka >> 1. Recalling that

the motivation for the requirement (4.15) was that we

(Ka) , (4.64)

68

wished to use (4.7), we may easily modify (4.15) in light

of our results here. Formally, all we need is
(|o|//E i 0.45><1o‘3/[10(Ka)/Il(na)] c m"1/2 mole-l/Z (4.68)

where the denominator is tabulated in Table 4.2. For

example, for Ka = 0.5, it is necessary that (Iol//E) :

0.11 X 10"3 C ml/2 mole-l/z; for c = 2 M, the requirement

becomes Iol < 5 C m ; and so forth.

~

First Order Solution
By (4.42) and (4.7), as supported by (4.68),
D1 = D - D WC (4.69)

Then by (4.57),

<31 (6) = — D31 {[D+ we - D- W2] - 0' W52}
+
= - (AD DO) IO (g)
2 "’ 2 2 4
+ (A D /Do){[IO(€)] + [11(6)] } ( .70)

Now the left-hand-side of (4.56) is the same as
the left-hand—side of (4.54), and we therefore know that

<mne solution of (4.56), if (4.56) were homogeneous, is

“1 = Io (g) (4.71)

69

Table 4.2--Some calculated values of IO/Il and l/Il°

 

 

Ka IO (Ka)/Il (Ka) l/Il (Ka)
0.1 20.02 19.970
0.2 10.05 9.950
0.5 4.12 3.880
0.8 2.70 2.310
1.0 2.24 1.770
1.5 1.68 1.020
1.8 1.51 0.759
2.0 1.43 0.629
3.0 1.23 0.253
4.0 1.16 0.102
5.0 1.12 0.041
10.0 1.05 3.3x 10'4

 

Moreover, KO (6) cannot be a solution because both KO (0)
and K5 (0) are infinite. To obtain the general solution

of (4.56), we write

91 = v (E) ul (6) (4.72)

The boundary conditions are on the derivative,

Vi = V'ul + vui , (4.73)

70

Since 15 (g) = I1 (6) and since I

1
(4.73), (4.71), (4.64), and (4.58) yield

(0) = 0 and 10(0) = 1,

v' (0) = 0 (4.74)

V' (Ka) IO (Ka) + v (Ka) Il (Ka)

2
- 4noFDl (a)/DO KRT

- [D1 (a)/DO] A Il (Ka) (4.75)

Now introduce (4.72) into (4.56). There follows

v'ul + 2v'ui + g-lv'ul + v (ui + 5-1 u' - ul) = q1 (4.76)

But since ul is the homogeneous solution of (4.56), the
expression in parentheses vanishes and the differential

equation determining v becomes

ul + 2v'ui + g-1 v'u = q1 (4.77)

V" l

or

(v')' + (zu'l/ul + 6‘1) v' = ql/ul (4.78)

This equation is of first order in v', with an

integrating factor of the form

exp {1(2ui/ul+ g-l) d5} = guz

1 . (4.79)

71

The solution of (4.78) is therefore

2 . _
gul v — Igulqldg + 81

By (4.74),

and hence,

V'

where

Integration of (4.82) yields, formally,

v = 65(Eui)- uzdg + B

By (4.75) p

32

The only remaining task, in this section, is
evaluation of u2 and its integral, (4.84), in order to

complete the derivation of WI.

the constant is simply

= - (Igu1q1d€)€=0 I

2 -l E
(aul) 3 ulqlda

2 -l
(Eul) 1.12 I

Iggu
O

lqldg

1
2

(4.71), (4.82), and (4.84),

= - Dl (a) A/Do (a) - (1(Eui)-

- u2 (Ka)/Ka Io (Ka) I

(4.71) into (4.83), there follows

uzdg)Ka

Substituting (4.70) and

(4.80)

(4.81)

(4.82)

(4.83)

(4.84)

(4.85)

72

_ + 5 2
u2 - - (AD /Do) 6 £10 d5

5 3 g 2
SIC d: + I $1011 85] (4.86)

+ (AzD-/DO)[I
0 0

It can easily be shown (Appendix B), by integration by

parts, that

g 2 _ 2 _ g 3
0 61011 dg — IO 11/2 0 £10 dg/z (4.87)
g 2 __ 2 2 _ 2
0 EIO dE—E (IO Il)/2 . (4.88)
Then
_ + 2 2 _ 2 - g 3 2
u2 - (A/200){D 6 (11 IO) + D [0 610 dg+-glo IlJL (4.89)
I2
v' = (gui)'l u2 = (A/ZDO) {0+6 (;%) - 1)
E O
- 2 -l 3
+ D [Il + (510) 6 £10 dg] (4.90)
and
v = (A/2DO) [D+ (sl - 52/2)
+ D (ID + 82)] + B2 (4.91)
with
_ 2
S1 (E) - I €(I1/IO) d6 (4.92)
_ 2 -l g 3
s2 (5) - I (610) ( 0 EIO d6) d: (4.93)

Moreover, by (4.85),

w
I

2 _

A
‘ (55—) {2D
0

+

+

+

+__t
KaIO(Ka)Il(Ka)

D-

+

D-

1

[Io (Ka) + S

I1

[Io (Ka)

A

(.....—

2

2

-1- (Ka)2

O

Ka 3
(I) EIOdE

(a) + 0* [s

73

1 (Ka) - %- (Ka)2]

2 (Ka)]

(Ka) I (Ka)

+

E‘a‘iouanluca) 0

O

D (Ka) [TS—TEEY ' TI_TEET]

K3.

1 I El: dil}

 

I (Ka) I (Ka)

+ 1 O
D){D [KaW-Kam+sl

+ 2] + D- [2 IO (Ka) + 82 (Ka)

 

— 2 A 10(Ka)]}

where we have used, by (4.42), (4.7), and (4.65),

where

and

D1 (a) = D+ - A IO (Ka) D-

Finally, put

'9'
II

+
(l/ZDO) {D (s

+

B

2

(ZDO/A)}

1

2 _
- g /2) + D (32+Io)

(Ka)

(4.94)

(4.95)

(4.96)

(4.97)

(4.98)

74

Introducing (4.96) into (4.39) and setting 8 = l, we have

V = W (E) [l + 4

O 1 (6) + ...] (4 99)

Here we include only the first-order perturbation. If
necessary, extra terms may be calculated and added to

(4.99) depending on nature of the system.

F. Discussion

 

It is evident that (4.98) involves the evaluation

of integrals of the form
(5 6f (6) In (6) d6 (4.100)

The evaluation of such integrals in closed form is possible
for only a few functions f (5). Consequently, recourse to
numerical computation is usually necessary (cf. Irving and
Mullineux, 1959). Hence, in evaluating (4.100) we make

use of the following equations (Abramowitz and Stegun, pp.

 

375-377).
(’2 (€/2)2k+n/kz (k+n)!, o i g < 2 (4.101)
k=0
eg 2 2 2 2
I (6) =44 {1 - (4n -l)/8g + (4n -1) (4n -9)/2((8g)
n 1 “205
— (4n2-l) (4n2-9) (4n2-25)/3z(8g)3 + ... , (4.102;

 

a 3 2

75

In general, however, if the actual physico-chemical prOp-
erties of the system are specified, the evaluation of such
integrals can be made tractable without introducing sig-
nificant error.

To clarify this, we consider, as an example, a
charged membrane for which the fixed surface charge den—
sity I0] is 10.3 C/m2 and the radius of the capillary a
is at 4.2%. Consequently, for the case 5a < 2, we can
evaluate the integrals by representing the products of
various modified Bessel functions as a power series in E
with the aid of numerical computation of the various co-
efficients necessary for the validity of such representa-
tion (Watson, 1968).

It can easily be calculated from (4.35) that at

the salt concentration of 1.5 mole/1, the value of 5a = Ka

is 1.69235 at 25°C and hence S1 (6) and 82 (g) can be
represented in practice by
5 2k+4
S = Z a E /(2k+4) + s (g) (4.103)
1 _ k 1k
k—O
and
5 2k+2
S = 2 b E /(2k+2) + e (g) , (4.104)
2 k=0 k 2k

in which it can be shown numerically that the contributions

of elk to S1 and 52k to 82 are immaterial. Note that extra

76

terms, if necessary, can be calculated and added to (4.103)
and (4.104) depending on the nature of the system. The
values of the coefficients ak and bk (k=0,...,5) are
listed in Table 4.3.

Table 4.3--Values of a and b (k=0,...5).

 

 

k k
k ak X 10 bk X 10
0 2500.0 5000.0
1 ~625.00 -625.00
2 143.23 234.38
3 -30.925 -55.882
4 6.4155 9.5554
5 -1.2942 -2.6589

 

Returning to (4.99), the surface potential Iwal
at the wall can be calculated, yielding the following

result

lwal = (RT/F) (0 (6a) [1 + 41 (6a) + ...] (4.105)

where 4

(Ea) is the value of 0 (E) at the wall. From

1 l
the results obtained above, we notice that the solution
represented by (4.99) is valid for the entire domain of

integration, 0 i r i a, since,in particular, no singu-

larity exists at the center of the capillary (r=0).

77

In order to clarify the analytical approximation
obtained above, in this section we carry out a numerical
analysis on 9. As an example, we calculate the values
of W and WC for [cl = 10.3 C/m2 and a = 4.2 X for the
case of 63 = Ka < 2. The salt concentration used in the
calculation is 1.5 mole/R and the temperature is 25°C.
Some values of W, 90 and W - We calculated are listed in
Table 4.4. A comparison of the results of W and WC is

shown in Fig. 4.2.

 

 

Table 4.4--Some calculated values of W, W , and g - yo.
T = 298°K, [0| = 10-3 c m-2 a°= 4.2 A, and
Ka = 1.692.

E W (6) yo (5) V - WC
0.000 0.00855 0.0117 -0.00317
0.051 0.00856 0.0117 -0.00317
0.355 0.00890 0.0121 -0.00320
0.592 0.00954 0.0128 -0.00323
0.728 0.01007 0.0133 -0.00326
0.998 0.01151 0.0148 -0.00332
1.472 0.01558 0.0190 -0.00341

1.675 0.01811 0.0215 -0.00340

78

.mso\o 5-05 0 _o_ .oomm u 0 .a m u m
.mso\0aos muoa x m.H u D .mmw.a u my .55 + 05 n 5 .HH 0>usoo . n 5
.H m>uso .ny mo cowuocsm 0 mm 5 “mumamumm mmmacowmcmfiflp map mo poamllm.¢ .mflm
+.Hv_
om H mvm.o o

1 1 I l. 1
1“ ‘ i d 4 1 1 q u q ‘ I

55.
5

p

_ _ mmm.o
m

mam.H

 

mmma.o
mvmo.o

II ll
(0 .Q

 

79

According to our analysis, we notice that W - To
is nonpositive. This indicates that the inclusion of the
concentration dependence of the dielectric constant in
Poisson's equation actually results in the decrease of the
overall potential function at any point in the system.
However, the decrement W — WC is very small. As a result,
we can anticipate that the solution represented by 90 + 91,

where W is the first-order solution, should give an

1
excellent approximation to V. For larger values of Ka,

more complicated numerical analysis is required, but there
is no conceptual difficulty. For smaller values of c, the

contribution of the concentration dependence of D will of

course be less, and 41 will be less important.

G. Conclusion

 

Including the radial dependence of the dielectric
constant in Poisson's equation, we have obtained analytically,
in the general case, an improved potential distribution equa—
tion for an uni-univalent electrolyte system, subject to

3 to 2 mole/2.

(4.68) and for salt concentrations of ~ 10-
The equation can be used to study various biological sys-
tems. A practical potential distribution equation is also
derived for the case of Ka < 2 where a = 4.2 g. Conse-
quently, the physico-chemical properties of aqueous uni-

univalent electrolyte solutions in biological systems and

other charged systems can now be interpreted more accurately.

80

In the derivation of the equation, we have required
sinh W = W. Recently, MacGillivray gt_§l. (1966) have
demonstrated that the solution of the Poisson-Boltzmann
equation for a charged cylinder can be approximated by the
solution of the Debye—Hfickel equation even when the molarity
of the electrolyte is low, i.e., when T >> 1, provided the
cylinder is "moderately charged," that is, when ZnIOIF
a/DORT < 1. For the example mentioned in the preceding
section, it can be shown that 2NIO|F a/DORT = 0.0118 at
25°C. We have here obtained explicit conditions (4.68)
on the ratio of IOI to /E and have thereby eliminated the
need for restriction on K itself. In fact, we have not

used Debye-Hfickel theory as such at all.

CHAPTER V

THE STEADY-STATE DISTRIBUTION OF PRESSURE

IN MODERATELY CHARGED SYSTEMS

A. Introduction

 

In the study of electrochemical systems, knowledge
of the pressure distribution has been realized to be essen-
tial in the understanding of interfacial diffuse double
layers and thin liquid films (Verwey, 1948; Derjaguin,
1966; Herwitz, 1964; Defay, 1963; Sanfeld, 1966). The
difficulty in obtaining a simple equation for the pressure
distribution is primarily due to the lack of a theoretical
formulation for the problem. As a result, many authors
have either neglected its existence or considered the
pressure to be independent of electric field (9:. Kobatake
and Fujita, 1964) without justification. As we shall see
in the next chapter, the effect of the pressure is impor-
tant and vital to the understanding of osmotic flow
phenomena of fluids across a charged, continuous phase;
i.e., a membrane. Previously, no explicit, practical
pressure distribution equation has been obtained, and
its various important applications have not been dis-

cussed.

81

82

In this chapter, our primary purpose is to make
use of the potential distribution equation derived in
Chapter IV and to derive, in the general case, an explicit
formula for the pressure distribution at any point in a
moderately charged system where the conditions of Chapter
IV are satisfied. We consider a nonequilibrium system at.
steady state, an isotropic fluid, and absence of electric
polarization. A capillary model is adOpted for the mem—
brane. We are primarily concerned with the uni-univalent
case.

One of the fundamental approaches in the study of
flow phenomena in membranes, charged or uncharged, is the
knowledge of macroscopic osmotic pressure in the system.
The differential equation that governs the pressure of
fluid at any point in the system is, in the general case,

by (3.3) r
dp = - (RT/6:) d in X0 , (5.1)

where p is the pressure at any point in the system, x0 is

the mole fraction of water at any point in the system,

w . I
V0 is the molar volume of pure water, R is gas constant,

T is the uniform, absolute temperature, and 9n fO is

assumed constant. From (5.1) we immediately obtain

p = - (RT/52) in X0 + B , (5.2)

83

where B is the constant of integration. This equation can
be used in a variety of ways. For example, the difference
in pressure between any two points a and b in a system is

the osmotic pressure "ha:

“ha = p (b) - p (a) = - (RT/7:) 2n [xo(b)/xo(a)1 (5.3)

By using this equation and the results of Chapters III and

IV, we could calculate n a between any two points in the

b
membrane.

We are, however, primarily concerned with osmotic
flow through the charged membrane. We hereby define “mo
as the steady-state value of the macroscopic osmotic
pressure between the membrane phase m and the aqueous

phase in compartment 0 (the other compartment will be

called compartment L),

“mo = p (m) - pO (0) . (5.4)

where p (m) is the average pressure in the capillary and
po (0) is the average pressure in compartment 0. More-
over, define 020 as the steady-state value of the osmotic

pressure across the charged membrane.
0&0 = p (1) - p (O) , (5.5)

where p (2) and p (0) are the average pressures over the

cross section of the capillary at, respectively, 2 and 0,

84

the positions of the interfaces of the membrane with,
respectively, compartments L and 0. From (5.2), (5.4),

and (5.5), there follows

(rmo = - (RT/R73) 2n [Seem/Xe (0)] (5.6)
020 = — (RT/(7'2) SLn (502/3300) , (5.7)

where xom is the steady-state value of the average water
concentration in the capillary, x0 (0) is the average
value of water concentration in compartment 0, and £02 and
i are the steady—state values of the average water con-

00
centrations over the cross section of the capillary at,
respectively, 2 and O.

In order to obtain the relative distribution of
water, i.e., [Eom/Xo(0)] or (Egg/£00) at steady state, we
could use directly the results of Chapters III and IV to
obtain the desired averages. It is more illuminating,
however, and somewhat simpler to focus attention on the
pressure itself. Hence, our second goal in this chapter
is to make use of the pressure distribution equation and

to derive the practical equations for "mo and n as

lo
functions of the salt concentrations and the electric
field for use in the study of transport phenomena of
water in charged systems.

Consider a charged, continuous phase separating

two aqueous electrolyte solutions of different concentration

85

at the same temperature. We wish to derive, in the
general case, the pressure distribution equation for the
system.

By (3.7) I
8p/8r = - EF (aw/er) . (5.8)

By (4.27) (the conditions which make it valid are there—
fore required here),
v-l _
(Bp/ar) = - (RT) 2 c 2 [exp (—z 9)](89/8r) (5.9)
a a a
a=1
This is easily integrated to yield
v-l _
P = P + RT 2 C [eXp (-z W) - l] , (5.10)
a a
0:1
where po is the value of p for zero electric field.

For a uni-univalent system, (5.10) becomes
p = pO + 2cRT (cosh? - l) , (5.11)

where we have used (4.32). We have now obtained explicitly
the pressure distribution equations which take account of
the concentration dependence of the pressure. Since
osmotic flow is our concern here, we are interested in

the longitudinal, or axial, dependence of the pressure.

Differentiation of (5.11) yields

86
(Sp/8x) = (ape/3x) + 2RT (cosh? - l) (ac/8x)
- RTE (aw/8x) , (5.12)

where we have also used (4.31). If we now assume that the

last term on the right-hand-side is negligible, we have
(Sp/3x) = (apo/ax) + 2RT (coshW - l) (ac/3x) (5.13)

This assumption is reasonable on two counts: (i) we do
not expect the potential difference across the membrane
to be significant, and (ii) the average net charge in the
solution inside the membrane must be small. In any case,
we are interested here in illuminating the contribution
of the concentration gradient to the pressure gradient,
and we therefore use (5.13), wherein W is taken to be
the value of Fw/RT for E, the average salt concentration
in the membrane.
B. The Macroscopic Osmotic

Pressure Between the Mem-

brane Phase and the Aqueous

Phase in a Moderately
Charged System

Making use of the pressure distribution equation
derived above, in the following treatment we derive an
explicit formula to represent the macroscopic osmotic
pressure "mo between the membrane phase m and the aqueous
phase 0 for a moderately charged membrane, separating two

uni-univalent electrolyte solutions.

87
We introduce (5.11) into the following equation

5 (r) = g pdx/(z-O) , (5.14)

I
0
where E (r) is the average axial pressure, and where 2 is

the length of the capillary or the thickness of the mem—

brane. Then, there results

2

'5 (r) = p0 dx/(l—O) + 2RT {cosh‘l’ - l} (:cdx/(i—O) (5.15)

I
0
where we again, and for the same reasons, take W to be
its average in the membrane.

Clearly, the evaluation of integrals in (5.15) re-
quires knowledge of the explicit expressions of p0 and c
as functions of x. We assume, as a first approximation,

that po (x) and c (x) are linear functions of x. That is,

we may write
p0 = (APO/2) X + pO (0) (5.16)
c = (Ac/2) X + c (0) , (5.17)
where ApO and Ac are defined as

Apo = po (2) - po (0) (5.18)

Ac = c (12,) - c (0) (5.19)

Combination of (5.16), (5.17) and (5.15) yields the ex-

plicit expression for the radial pressure in the system,

88
E (r) = {pO (2) + pO (0)}/2 + RT (coshw-1}{c(2)+c(o)} (5.20)
We rewrite (5.26) as
O

'5 (r) - p (0) = Ape/2 + RT {cosh‘P-l}{c()2.)+c(0)} (5.21)

The macroscopic osmotic pressure “mo between the
membrane phase m and the aqueous phase 0 can be deduced

from (5.21), yielding the following equation

"mo = ApO/Z + RT{c (2)
a a
+ c(O)}[I 28r{coshW-l}dr/ I 20rdr] , (5.22)
0 0

where Wmo is defined by (5.4) in which p (m) is given as

a _ a
I anp (r) dr/ 6 andr . (5.23)

p (m) 0

In the absence of initial hydrostatic pressure difference,

i.e., when ApO = 0, then,(5.23) becomes

"mo = RT {c(£)+c(0)} 6a an {coshy—l} dr/Tra2 . (5.24)

For experimental convenience, in general, the salt con-
centration at one side of the membrane is fixed while that
at the other side of the membrane is varied. Thus, in

practice we may rewrite (5.24) as

"mo = c(£)RT(l+q) 6a 2r (cosh‘i’-l)dr/a2 (5.25)

where q is given as

q = c(0)/c(2) . (5.26)

89

The value of q can then be varied during the experiment.
To evaluate the integral in (5.26), we make use

of the conditions of Chapter IV. There follows, then,

a a 2
I 2r (coshW-l) dr 5 0 r? dr (5.27)
0
where we have used
coshw = 1 + 92/2 + 5% 44 + ... ~ 1 + % 92 (5.28)

Introducing (4.50) and (4.52) into (5.27), we write

5a
6a :92 dr = (1/K2) I £72 66 , (5.29)
0

where K is the reciprocal of Debye length defined by (4.35),

and hence (5.25) becomes

5

«an 5 {c (0) RT (1 + q)/€:} (1)3 642 as; . (5.30)

Making use of the substitution: W = To + 91,
where WC and 91 are defined by (4.65) and (4.96) respec-

tively, we have

Ea 2 ga 2 5a
I 69 6: = I 59 as + 2 I 69 w 66
O O o 5 O l
Ea 2 5 31)
+ 6 5W1 d6 . ( .
For simplicity, denote
Ea 2
L1 = I SW 66 (5.32)
0 O

90

ga

L2 = a 5W0 W1 d5 (5.33)
Ea 2

L3 = a £91 dE (5.34)

The evaluation of (5.32) can easily be carried out. We
introduce (4.65) into (5.32) and make use of (4.88).

There follows

2 2

O (Ea) - Il (Ea)}/2 . (5.35)
However, from (4.96) and (4.98) we notice that there is
great difficulty in evaluating (5.33) and (5.34) in closed

forms, and hence the evaluation of such integrals requires

knowledge of the physical situation. Hence, (5.30) be-

come S
“mo 5 {c (0) RT (1 + q)/€:} [A2 a: (I: (6a)
2
- 11 (6a)}/2 + 2L2 + L3] . (5.36)

The equation can, moreover, be used to determine
the relative distribution of water between the membrane

phase m and the aqueous phase 0, since by (5.6),

i /x0 (0) = exp (- 8 62/81) (5.37)

om mo

91

C. The Macroscopic Osmotic
Pressure Across a Moder-
ately Charged Membrane

 

 

 

In this section, we derive a formula to represent

the macroscopic osmotic pressure 0 across a moderately

20
charged membrane. Returning to (5.13), we integrate the
equation over the length of the capillary, i.e., 0 i x i 8.

There results
Ap (r) = ApO + 2RT {cosh W-l} Ac (5.38)

where Apo and Ac are defined by (5.18) and (5.19), re-
spectively.
The macroscopic osmotic pressure across a charged

membrane can, then, be deduced from (5.38), yielding

_ a 2
020 — Apo + 2RTAc 6 2r {coshW l} dr/a (5.39)

In the absence of initial hydrostatic pressure difference,

i.e., when Apo = 0, (5.39) becomes

a

6,0 = 2RTAc I 2r (coshW—l) dr/a2 (5.40)
0

From (5.27), (5.29), (5.31), (5.35), and (5.26), there

follows

«,0 = 2 {c(4) RT (l-q)/€:} (A2 6: {IO (€)2

— I (ga)2}/2 + 2L

1 + L3] . (5.41)

2

 

92

This equation can be also used to determine the relative
distribution of water at the interfaces with the two

aqueous phases, by (5.7):

xoi/xoo = exp (- n

—O
lovo/RT) (5.42)

D. Osmotic Flow of Water in
a Charged Membrane

 

 

As has been known for years, knowledge of the
macrosc0pic osmotic pressure is useful in the understand-
ing of the mechanism of transport process of solutes and
water across a charged membrane. In the following treat-
ment we make use of (5.41) to study the osmotic flow
phenomena of water across a charged membrane. As an
example, we consider a charged membrane for which the
fixed surface charge density is 10"3 C/m2 and the radius
of the capillary is 17.6 X, The temperature is 25°C.

The concentration of one solution is fixed at 2.0 M
whereas the concentration of the salt solution varied is
in the range of 0.02 to 2.0 M, where 5a > 5. Hence, the
conditions of Chapter IV are valid throughout.

Subject to the external conditions considered
above, we now compute the macroscopic osmotic pressure

n across the charged membrane against the concentration

20
varied. It is apparent that at very large 5a the effect

of the concentration dependence of the dielectric

93

constant on potential becomes negligibly small and, hence,

in computing 0 , in general, we may neglect the contribu-

£0
tions due to L2 and L3 in the equation. To clarify this,
we perform numerical computation on ”(o and Rio, where
fig and n' are defined as
o 80
~ 2 2 2 2
4,0 = 2 {c (0) RT (l-q)/€a} [A Ea {IO (Ea)
- I (a )21/2 + 2L 1 (5 43)
l a 2 '
and
4' = c (2) RT (1 - q) A2 {I (a )2 - I (a )2} (5 44)
lo 0 a l a ‘

Note also that 0i
0

zero—order solution in W and that in writing (5.43) the

is the contribution due only to the

contribution due to L3 is excluded since it is easily
realizable to be immaterial. Some calculated results of

n and Rio are listed in Table 5.1. It can be shown

£0
.. | I
from the results that the effect of “£0 020 020

in general, about 1% at very large Ea. Hence, at very

on is,

large 5a, equations (5.36) and (5.41) tend to be

c (2) RT (1 + q) A2 {Io (Ea)

2 2
-11 (6a) }/2 (5.45)

:1
ll

mo

2c (2) RT (1 - q) A2 {10 (5a)2

2
1r - Il (Ea) } (5.46)

20

without introducing significant error. These equations

are useful in studying and interpreting osmotic flow of

94

Table 5.l-—Some calculated values of Rio and ”'0 at

 

 

vargous salt concentratigns. c (£) = 2.0
10‘ mole/m3. a = 17.6 A. I0] = 10’3 C/mz.
t = 25°C.
° ‘3212763-3' .. “(cg/:03 3837'
0.02 5.819 0.291 0.2928
0 05 5.862 0.280 0.2818
0.80 6.851 0.105 0.1059
0.95 7.032 0.085 0.0854
1.55 7.714 0.027 0.0274
1.97 8.158 0.002 0.0015

 

water across various loose or porous charged membranes,
when there is no initial hydrostatic pressure difference
across the membrane. Making use of (5.46), 010 is plotted
against the logarithm of the salt concentration varied in
Fig. 5.1.

On the basis of our calculation, it is found that

0 required decreases monot0nically with increasing con—

£0
centration varied, and that7&0 is positive when 0 (0)/c (£)
< 1. Hence, our results are in good agreement with the
experimentally observed phenomena that the flow of water

Inoves toward the more dilute solution. This is contrary

'to the flow of solutes which tends to move toward the more

II...“

4:“:

.oomm 0 .ms\0aos moa x o. m n 151 0 s\0 3 n _6% .101 o .qoaumuuaqoqoo
pawn mcu MO San. Homoa mcu mo coauoc m m m AME no mmasofl Gav Can we uoamula.m .on

+ 2: 0 ca

hom.b mm.m . m
. . . 4 . . . . q . . . . . . . 4 . . .mN.OI

95

mmm.o H n
mm~.o

ll
(6

.hvh.m

 

96

concentrated solution in charged membranes. The latter
phenomena is referred to as "anomalous osmosis," which

we shall disucss in more detail in the next chapter.

E. Conclusion and Discussion

 

Considering the field dependence of the pressure,
i.e., Bp/Bw # 0, for the first time we have obtained, for
a v-component system, and an uni-univalent electrolyte
system, the macrosc0pic pressure distribution equations
which simultaneously take account of the concentration
dependence of the pressure. As a result, the macroscopic

formula for the osmotic pressure 0 across a moderately

£0
charged membrane has been obtained for an uni-univalent
electrolyte system and, accordingly, the relative distri—
bution of water in two aqueous phases has been calculated.
We have also deriVed, for an uni-univalent electrolyte
system, the macroscopic formula for the osmotic pressure
”mo between the membrane phase m and the aqueous phase 0
and hence calculated the relative distribution of water
between the phases. These equations are useful in the
treatment of osmotic flow phenomena of water in a porous,
moderately charged membrane at steady state, subject to
various salt concentrations and geometrical conditions of

the membrane in question. Consequently, the fluid-

membrane system dealing with an uni-univalent electrolyte

97

solution can now be described more adequately, and experi-
ments in the study of transport phenomena of water across
a porous, moderately charged membrane can now be inter-

preted more accurately.

CHAPTER VI

THE STEADY-STATE PHENOMENOLOGICAL THEORY

OF OSMOSIS IN CHARGED SYSTEMS

A. Introduction

 

Osmotic flow phenomena of fluids across a charged
continuous phase separating two aqueous solutions of dif-
ferent concentrations have been known for years to be
important and vital in the understanding of various
mechanisms of transport process encountered in many
areas of physical and life sciences. It has been
realized experimentally (Grim, 1957) that the flow in
a charged continuous phase, in contrast to the normal
experience obtained with uncharged phases or with non-
electrolyte solutions, occurs toward the more concentrated
solution (anomalous positive osmosis), and its rate is
roughly proportional to the concentration difference.
Moreover, when the concentration of one solution is fixed
and that of the other is varied, plots of the flow rate
against the logarithm of the varied concentration often
give an N-shaped curve.

Since the early findings of Dutrochet (1835),

various transport theories have been developed (gf.

98

99

Kobatake and Fujita, 1964; Gross and Osterle, 1967;
Toyoshima, 1967; Fujita and Kobatake, 1968), in order

to interpret the mechanism of osmotic flow in a charged
membrane. Although most of the theories are satisfactory
in many respects, they are all inadequate in one way or
another—-e.g., the theory of Kobatake and coworkers con-
tains many contradictory assumptions, while the theory

of Osterle and coworkers is restricted to extremely dilute

 

solutions. Moreover, the conditions of numerous experi-

V'. H . .. _

ments so far reported have usually not been well-defined.
As has been known for years, membranes--artificial
or natural--vary so widely in structure and function that
it is impossible to say that anything approaching a general
membrane transport theory has been established. In this
chapter, we make use of the principles of nonequilibrium
thermodynamics and the equations of hydrodynamics without
recourse to most of the restrictive simplifications re-
quired by previous workers, and we derive a steady-state
phenomenological theory that accounts for the osmotic
flow of fluids across a charged continuous phase. It is
hoped that the theory can be used better to describe and
understand transport phenomena in charged membrane systems.
In the following treatment, we consider the system

which is composed of a moderately charged membrane for

3 2

which the fixed surface charge density is about 10- C/m

100

and which separates two aqueous uni—univalent electrolyte
solutions of different concentrations at the same tempera-
ture without the presence of initial hydrostatic pressure
difference. A capillary model is used for the membrane.

We make use of the macroscopic distribution equations of
ions, pressure, and electrostatic potential in the trans-
port equations derived in the previous chapters. The re—
sulting steady-state differential equations are then solved
to yield the average osmotic flow rate as a function of
initial salt concentrations. Finally, the theory is com-

pared with experimental observations.

B. Transport Eguations

 

As mentioned in Chapter II, the transport equations
dealing with electrochemical systems are conveniently
written with respect to the solvent-fixed (SF) frame of
reference. In the following treatment we consider the
fluid-membrane system in which the conditions of Chapter
IV are all satisfied. Moreover, we hereafter confine the
discussion to the system which separates two aqueous
solutions containing the solvent molecules and a single
uni-univalent electrolyte of the same kind. Positive
ions and negative ions are denoted by l and 2, respec-
tively. It is of course quite important for biological

and other purposes to consider also solutions of many

 

101

components. In that context, our purpose in this chapter,
and in this thesis, is to lay the foundation upon which a
general theory can be built. Transport equations for
multi-component systems are similar in form to those for
a binary system, but they are so long and complicated
that they could obscure our simple, central purpose.

The diffusional fluxes of species 1 and 2 in the
direction of the capillary axis relative to the SF refer-

ence frame are, according to (2.78),

(jl)O = - (02152 + QEZV§)(3p/8x) - 911 RT (alncl/ax)
- QEZRT (alncz/ax) - (011218 + 0122 2F)(80/ax)
(j2)o = - (021Vl + 922V 2)(Bp/ax) - Q; lRT(81ncl/8x)
- 032RT(81nc2/8x) - (021218 + 0222 2F)(80/8x) , (6.1)

where we also make the important assumption that the total
electrostatic potential 0 is separable into a part 9 which

depends only on x and a part E which depends only on r:
I) (x,r) = 9 (x) + E (r) . (6.2)

The second term, W (r), on the right-hand-side of (6.2)

is what we have considered exclusively in Chapters II--V.
A deeper analysis may reveal that the separation of (6.2)
is not adequate [e.g., it may well be that w (xr) = 0 (x)

+ W (x,r)] to satisfy all the governing equations.

wamutr.mrmi .

102

It should certainly be an excellent approximation physi-
cally, however, since it is clearly experimentally pos-
sible to vary independently the trans-membrane potential
difference A¢ = ¢ (£) - ¢ (0) and the charge.

Also in (6.1), we have taken in fa to be constant,
and Cd is the concentration of species a in mole m-3, za
(a = 1,2) is the charge valency of species a per mole,

R is the gas constant, T is the uniform temperature, F is
the Faraday constant, and the 9:8 (d,8 = 1,2) are the
phenomenological coefficients in the SF reference frame.
We see that for the problem at hand the solute ions are
subject to three kinds of forces, namely, osmotic forces
corresponding to concentration gradients, mechanical
forces corresponding to pressure gradients and electrical
forces corresponding to potential gradients. The solvent
molecules, however, are subject to only osmotic forces
and mechanical forces according to (5.13) in Chapter V.
These "forces" are of course not independent--there are
only two independent "forces," the chemical potential
gradients of l and 2.

The partial electric current and the diffusion
flux of the salt along the capillary axis relative to the
SF reference frame are, by (2.78) and (2.79),

2
(i ) = Z 2a F (jd)o (6.3)

d=l

103

(6.4)

By (2.65), the electric current and the diffusion flux of
the salt along the capillary axis both relative to the

capillary wall are, then,

c u (6.5)

2
Z zd'F(ja)o +

2 F c u (6.6)
d=l a=1 a a

d=l a

where 110 is the axial component of the reference velocity
in SF frame of reference which, under the assumptions
which give (2.81), satisfies the axial component of the

Navier-Stokes equation in the form
(ap/ax) + ‘z'F (dCD/dx) = nr-l[(d/dr)r(duO/dr)] , (6.7)

where we have used (6.2) and where we assume that uO is
independent of x. In (6.7), n is the isothermal shear
viscosity of the fluid, taken as a constant, and E is de-
fined by (2.12).

Poisson's equation which relates the electrostatic
potential to the excess surface charge density is, by

(4.1) and (6.2),

{1 [(d/dr) (rD(d$/dr))] = — “2p , (6-8)

 

104

where D is the dielectric tensor of the electrolyte solu-
tion, taken as nonconstant. This equation was solved in
Chapter IV, under the conditions listed there. For uni-

univalent electrolytes, we have, by (4.99),

I = (RT/F) A10 (5) [1 + ¢l (£)] , (6.9)

where E Kr, A = 4noF/KDORTI (Ka), K = (817F2 E/DORT)l/2,

l
and $1, defined by (4.98), results from inclusion of the

concentration dependence of the dielectric constant. We

‘v‘mnma

have also, by (5.13),
(Sp/8x) = (dpo/dx) + 2RT(coshW—l)(dc/dx) , (6.10)

where W = FW/RT, where po (assumed a function of x only)
is the pressure when $ = 0, and where c is assumed a
function of x only-~thus, c is effectively, the average
over any cross section of the capillary. Finally, for

any uni-univalent electrolyte, by (4.31),
‘z' = - 2c sinh‘i’ , (6.11)

C. General Formula for 110

 

In the following treatment, we combine all the
differential equations, solve the resulting equations sub-
ject to the appropriate boundary conditions and derive a

steady-state phenomenological theory of osmosis in a

105

charged continuous phase for an uni-univalent electrolyte
system. It has been realized experimentally (Sollner,
1945) that anomalous osmosis of fluid occur only if the
continuous phase is in a charged state and is porous to
some degree. It is evident that anomalous osmosis does
not occur for semipermeable phases. Thus, we take a and
Ka very large compared to unity.

We have demonstrated in Chapter V through numerical
computation that at very large Ea = Ka, (6.9) can, in prac-
tice, be represented by the zeroth-order solution of the

form
E = (RT/F) AIO (£) (6.12)

since, for large Ka, A is very small.
Making use of (6.10) and (6.11), we rewrite (6.7)

as

nr‘l [(d/dr) (rduO/dr)] = F (x,r) (6.13)

where
F (x,r) = de/dx + 2RT (coshW-l) (dc/dx)
- 2chinhW (d¢/dx) . (6.14)
we solve (6.13) subject to (2.83), yielding the expression,

a _ r
nuo = I r l I F (x,r) rdrdr (6.15)
r O

 

._ -‘TI'IS_ATTA f'i r W

 

106
Introduction of (6.14) into (6.15)

u0 = + {(aZ-r2)/4n}(dpo/dx)

+ (RT/n) Ia r-1 Irwzrdrdr(dc/dx)
. r

0

a -l r'
— (2cF/n) I r 0 Wrdrdr (d¢/dx)
r

where we have used
cosh? = 1 + wZ/z , sinh) = w
Substitution of (6.12) into (6.1) yields

u0 = + {(E: - 52)/4K2n}(dpo/dx)

C
+ (AZRT/2K2n) 1 ag (Ii-Ii) d§(dc/dx)

E

2
- (2AcF/K n) {10(Ea)-IO(€)} (d¢/dx)

where we have made use of the substitutions

and where we have used the recurrence formulas

d (aIl)/da= 5:0 . d IO/da = II .

a
I 51: d6 = £2 (Ii—Ii)/2
0

(6.16)

(6.17)

(6.18)

(6.19)

(6.20)

 

107

D. Special Formula for 1.10

 

and its Consequences

 

Although the general formula (6.18) could be used
for any experimental situation which fulfills the require-
ments previously listed in Chapter IV and in this chapter,
the occurrence of the integral term on the right-hand-side
would require numerical integration. In order to obtain
formulas to which we may easily attach physical signifi-
cance, we now specialize to the case in which the second
term on the right-hand-side of (6.18) is negligible com-
pared to the third term. The requirement we find, (6.22),
is so easily met that we may claim with confidence that
our "special" formulas are in fact those which apply to
the great majority of experimental situations.

Rather than deal directly with (6.18), we find a

stronger condition by requiring, in (6.14),

|2RT (coshW-l) (dc/dx)l : 0.01 |2chinhW(d¢/dx)| (6.21)
By (6.17) and the definition of W (E FE/RT), (6.21) becomes

|(d 2n c/dx)/(d¢/dx)| 3 0.02|'()7|‘l (6.22)

In Chapter IV, we required [WI 3 0.245. For 298°K,

F/RT = 38.93 V"1 and we therefore have

[El < 0.0063 v ,

IEW—l 3_158.9 v‘1 (6.23)

-lll I'll

 

108
So for the largest allowed value of |$|, we get
|(d 2n c/dx)/(d®/dx)| 1.3-2 (6.24)

Ordinarily, I$l is much smaller than 0.0063 V because it
decreases strongly with the square root of average com-
position and with increasing pore size. Noting that it

is fin c which appears in the denominator of (6.22) and
(6.24), it is clear that (6.22) will ordinarily be easily
satisfied. Typical values of the trans-membrane potential
A¢ are 60 to 100 mV (Woodbury, et_al., 1970) and therefore
(6.24) would be satisfied for IA£n<3Ii 0.3. For smaller,
more typical value of (0|, IAln cl may be even larger but
still negligible.

We therefore reduce (6.18) to

2 2
no = {(5a - a )/4K20} (dpo/dX)

2
- (ZACF/K 0) {IO (Ea)-Io(€)} (d¢/dx) (6.25)

Equation (6.25) is generally valid for loose membranes
with very large pores. Moreover, it is mathematically
more tractable.

The average values of the reference velocity,
electric current density, and the diffusional flux of the
electrolyte component over the cross-section of the

capillary are

[Mn fa‘a‘auil Hw'F‘E'JT

109
U - 1a 2 ( ) d a 2 d
o - 0 nr uo r/(I) TTI‘ r
_ a a
I = I an (I) dr/I 20rdr
0 0
_ a a
J = 0 an (J) dr/I 20rdr
0

Introducing (6.25) into (6.26), there follows

U0 = A01 (de/dx) + A02 (d¢/dx)

where the coefficients A and A are
01 02

2 2
A01 _ ga/8K n

A

02 - (4AcF/€:Kzn) {6: 10(Ea)/2-€aIl(€a)}

(6.26)

(6.27)

(6.28)

(6.29)

(6.30)

where we have used (6.20). It is interesting to note that

in the coefficient A02 of (d0/dx), usually referred to as

the electroosmotic coefficient, is a function of the salt

concentration. Previous existing theories for the

anomalous osmosis (Schlogl, 1955; Kedom, 1961; Kobatake,

1958) have not taken this fact into account. As we shall

see later, this effect is essential for the observed

phenomenological behavior of the osmotic flow in charged

membranes.

We substitute (6.1) into (6.5) and (6.6) and hence

have, for the general case,

Wm~m.mzn:n:r 3mm)

110
2 o —o o —o 2 o o
J = - E (levl + Qa2v2)(8p/3x) - E (901 + 0&2)
d—l a—l
2
o 0
RT (dincaO/dx) ail (90121 + 9&222)F(d¢/dx)
2
+ ail Cao exp (-zaW) uO (6.31)
2 o —o o *0
=.. F
I 6:1 (Qalvl + QaZVZ) 2a (Sp/BX)
2 o o
— ail (gal + Qaz) z£7RT (dincaO/dx)
2 o o 2
- ail (Qalzl + 0&222) zaF (d¢/dx)
2
+ ail Cao exp (-zaW) zaFuO (6.32)

where we have made use of the substitution,
dflnca/dx = dincaO/dx , (6.33)

since (6.33) is assumed independent of radical coordinate.

For an uni-univalent electrolyte system, there result

2
J = — 2 (0:16? + 03263) (ap/ax)
a=l
2 o o 2 o o
- 2 (gal + QGZ)RT(d£nc/dx) - Z (Gal—0a2)F(d¢/dx)
d-l a=1

+ 2c cosh? 110 (6.34)

‘mmrmmfiqugwmmzr

111

2
_ _ o —o o -o _ 0+1
I — ail (Qalvl + Qazvz) ( 1) E’(3p/3x)
2 o 0 +1
- Z (0 1 + 0 2) (-l)a F RT (anC/dx)
d=l a
2 o o d+1 2
- 2c F sinh? 110 (6.35)
The practical equation for the axial pressure gradient
(Sp/3x) can be deduced from (6.10), yielding
ap/ax g de/dx + RT )2 (dc/dx) (6.36)

where we have

come

used (6.17). Hence, (6.34) and (6.35) be-

2
: (0: 6° + 0° 6°) (de/dx)

a l 1 1 a2 2

2 o —o o —o 2

:1(Qalvl + Qazvz) RT W (dc/dx)

2 o o 2 o o

ail (gal + Qa2)RT(d2nc/dx) - a:l(Qal-Qa2)(d¢/dX)

26 (1 + 22/2) uO (6.37)

W¥-fiflmvnmsrrmrz-ml ‘.

112
1 — - i (0° 6° + 0° ’°) (-1)°‘+l F (d /d )
‘ = 61 1 62V2 po X

2
- z (0: 6° + 03263) (-1)""'1 F RT (2 (dc/dx)

2
- 2 (031 + 032) (-1)°+1 F RT (dlnc/dx)
d=l
- (0:1 0:2) (-1)O‘+1 F2 (d¢/dx)
=1
+ 2cFWub , (6.38)

where we have used (6.17).
Introduction of (6.37) and (6.38) into (6.27) and

(6.28) and use of (6.12) and (6.19) yield

+
o —o + Q0 —0 a l

61V1 a2v2) (’1’ F (dpo/dX)

o —o 0+1 2 2
01 1 + Qa2v2)(-1) {(ZA FRT/ga)

a 2
I 51 d5} (dc/dx)
0 o

2
- 2 (0° + no 0+1

a1 02) F RT (dinc/dx)

{-1)

d+1 2

° ° -1) F (d6/dx)

2
- g (9.1 - 9.2) <

2 5a
+ (4AcF/ga) I gIOuOdg (6.39)

0

Wfi“fi:flflr“?‘fiw1 .

113

2
— _ _ o —o —o
J - ail (levl + QaZVZ) (de/dx)
2 o —o —o 2 2 5a 2
— ail (levl + 002v2)(2A RT/Ea) 0 EIO d5 (dc/dx)
2 o o 2 o o
_ 0:1 (901 + QaZ)RT(d9.nc/dx) - a:1(Q°1-Q°2)F(d¢/dX) E
F; E‘
+ 2 a 2 2 h
(4c/ga) I 5(1 + A 10/2) uO dg . (6.40) E
O .
E
The integrals in (6.39) are readily evaluated, yielding E
4 5
a _ 2 2 _ 2
0 SI d5 — 5a {IO(€a) Il (5a) }/2 (6.41)
and
E

where we have used (6.

2
(l/ZKzn) 4a I2 (Ea)(dpo/dX)

(2AcF/K20) [gale

(Ea)Il (ea)

52

a

2

(a ) - 11 (5a)2}/21 (d4/dx) (6.42)

{10 a

20) and the recurrence formula

3
a

2
d: 4 Il (Ea) - 26a 12 (Ea) (6.43)

Nevertheless, it is apparent that at very large Ea, the

integral in (6.40)
E
a
I
0

2

g{1 + A IO

(6)2/2} no 65 (6.44)

114

tends to be

5
1a gu dg (6.45)
0 0

because A + O as Ka + w. This can be evaluated easily in

closed form, yielding the following result,

(aa)/2

+ (Si/IGKZU) (de/dx) - (2AcF/Kzn){§: IO

,7 1‘1er gain.
I

- Ea I1 (53)} (d0/dx) (6.46)

-..l‘l‘

Introduction of (6.41), (6.42) and (6.46) into (6.39) and

 

(6"
‘1

(6.40) finally yields, in the general case,

I = BOl (de/dx) + B02 (d¢/dx) + BO3 (dc/dx)
+ BO4 (dlnc/dx) (6.47)
and
J = ROl (de/dx) + R02 (d¢/dx) + Ro3 (dc/dx)
+ RO4 (anC/dx) , (6.48)
where the coeff1c1ents BOi and Roi (1 = l,....4) are
B — - g (0° -0 + 0° ‘0)(-1)OL+1 F + (2AcF/K20) I (E )
Ol ‘ 6:1 61V1 azvz 2 a
B _ _ g (0° _ Qo )(_l)d+1 F2
02 0:1 dl 02
2 2 2 2 2
- F
(8A 0 /€aK n) [Ea Io (Ea) 11 (Ea)
2 2

2
- 4a {10 (a ) - 11 (Ea) 1/21

a

115
2 o —o o —0 0+1 2 2 2
B03 = - 0:1 (Qalvl + Qazvz) (-l) (A FRT){IO(§a) -Il(€a) }
2 o 0 0+1
B04 = - 0,21 (901 + {202) (-1) FRT
2 o -o o —o 2 2
R01 = - 0:1 (Qalvl + Q02v2) + Ea C/4K n
2 o o 2 2 2 2
R02 = - 0:1 (901 - 9&2) - (8Ac F/K 05a){€a10(€a)/2-§a11(§a)}
2 o —o o —o 2 2 2
R03 = ' 6:1 (Qalvl + Q62V2) (A RT) {Io (ga) —Il(€a) }
2 o 0
R04 = - 0:1 (901 + 0&2) RT (6.49)

Although the (dc/dx) term and the (d in c/dx) term could
obviously be combined in (6.47) and (6.48), and elsewhere,
no useful simplification is thereby obtained. Formally,
the coefficient of(d.£n c/dx)ixx(6.47) would be (5 B03 +
304) and the coefficient of(d.£n C/dX)le(6.48) would be
(5 R034-Ro4), where E is some average value of c.

E. The Working Equation for
Anomalous Osmosis

We have so far obtained explicitly, making use of
some eXplicit, justifiable simplifying assumptions, the
equations for the averages of reference velocity, electric

current density and diffusional flux of the electrolyte

WHIP-immanmmrma ,

116

component over the cross section of the capillary. From
these equations we now derive a working equation which can
be used to interpret the characteristic behavior of osmotic
flow through a charged membrane. As mentioned before, we
consider that initially there is no hydrostatic pressure
difference between the two sides of the membrane, and
assume that the system has attained a steady state. The
steady-state assumption is justifiable since it is real-

izable in most biological systems and physicalchemical

Wmm1nnmr1‘m‘fi’)

systems.
Returning to (6.29), (6.47) and (6.48), at steady
state, from the conservations of mass and electric current

density and from the incompressibility of the fluid there

follow
d UO/dx = 0, df/dx = 0, dE/dx = 0 . (6.50)

Furthermore, in the present system there is no applied

(electric field across the membrane and, hence, from (6.60)

we have

Uo = constant, T = 0, 3 = constant. (6.51)

11ccording to the assumptions made above the apprOpriate

boundary conditions for the problem are, then,

c = c (0) at x = 0 and c = c (£) at x = 2

117

<I>=<I>(O)atx=0and4>=<l>(£)atx=IL (6.52)

Making use of the conditions given by (6.51) we
now integrate (6.29), (6.47), and (6.48) over the length
of the capillary, i.e., 0 i x i 2, subject to (6.52).
However, by examination of (6.30) and (6.49) we imme-
diately notice that the coeff1c1ents A02, B01, B02, R01

and R02 are concentration-dependent. Hence, we may write

. ‘_ ‘Tg'i-mflfifpv

 

_ R _ a;

U0 — AOl (Ape/1) + 6 A02 (d¢/dx) dx/Q — constant (6.53)
I = I2 E (dp /dx) dx/E + I2 B (d¢/dx)dx/£

0 ol 0 o 02

+ BO3 (Ac/2) + Bo4 (Ainc/E) = 0 (6.54)
3 — )2 R (dp /dx)dx/£ + 1" R (dCI>/dx)dx/IL

01 o o 02
+ RO3 (Ac/R) + RO4 (Alnc/l) = constant (6.55)

\nhere Apo, Ac, and Alnc are defined as

ApO = po (x = i) - po (x = 0)

Ac=C(£)-C(0)

A£nc = inc (2) - inc (0) in {c (2)/c (0)} (6.56)

FHthhermore, we have required in Chapter V that

118

P0 = (APO/R) X + pO (x = 0)

0
II

(Ac/IL) x + C (0)

0
II

(04/2) x + 4 (0) , (6.57)
whence
dpo/dx = ApO/£

dc/dx

Ac/l

‘f’merww‘m 1.1:).

d0/dx A0/2 (6.58)

With the aid of the above equations, the evaluation of
the integrals in equations (6.53)-(6.55) then can easily

be performed. The resulting expressions are

2 _ 2 2 2
3 A02 (d¢/dx)dX/£ — - (ZAF/EaK n) c (£)(l+q){€aIO(€a)/2
- Ea Il (5a)} (04/2) (6.59)
It B (d /d )d /2 " (A /2) [- g (00 VG + 52° VON-DOW]?
0 01 p0 X X _ po 0:1 01 l 02 2
- (AF/K20) 12 (6a)c(4)(1+q)1 (6.60)
1% B (66/6 )d /2 — (44/2) [- 2 (0° -0° )(-1)°‘+1F2
0 02 X X ‘ “=1 01 02

-(8A2F2/E:K2n){{c (9.)2 (1 + q + q2)/3}

2 2 2
[Ea IO (6a) 11 (Ea) - 6a {10(Ea) -Il(€a) }/2]}](6.61)

119
2

2 o -o o —o
0 R01 (dpo/dx)dx/£ = (Ape/2) [- 0:1 (0alvl + 0a2v2)

- (ii/BKZU) c (2) (1 + q)] (6.62)
1% R (d¢/dx)dx/£ = (0¢/£) [- g (00 - 0O ) F
0 02 OF]. a]- 012

- (8AF/K2€:n){{c (IL)2 (1+q+q2)/3}

2
{Ea IO (Ea)/2 - 5a I1 (Ea)}}] (6.63)

“ mac-0731::ms‘ .

We introduce equations (6.59)-(6.63) into equations

(6.53)-(6.55). From (6.52), there follow

“F

Uo = A02 (0¢/£) = constant (6.64)

0 (6.65)

I (Afinc/fi)

B (00/0) + BO

02 (Ac/2) + BO

3 4

J = R02 (00/2) + RO

(Ac/2) + Ro4 (Alnc/i) constant (6.66)

3

where A02, 302’ and R02 are given by

__ 22 2 _
A02 - - (ZAP/EaK n){c (£)(l+q)}[€a 10(Ea)/2 Ea11(Ea)] (6.67)
2
— o 0 0+1 2
B02 = - E (901 - Q02) (-1) F
0—1
- (SAZFz/EiKZn){{C (1)2 (1+q+q2)/3}
[ I ( ) I ( )- 2(1 ( )2-1 ( )2}/2]}(6 68)
5a o E;a l E(a ga o 8‘:a 1 ga '
— 2 o o 2 2 2 2
R02 = - ail (0&1 - 0&2)F- (8AF/gaK n)HC(£) (1+q+q )/3}

[5: 10 (ga)/2 - 5a I1 (ga)]} (6.69)

120

Our chief goal, however, is to obtain explicitly
the expressions for U0 and 3 as functions of salt concen-
trations. Hence, we combine (6.64) through (6.66) appro-
priately and rearrange the resulting equations. There

follow, then,

UO 2 = (AOZBo3/B02) Ac - (AozBo4/B02)A2nc (6.70) g
6
a“
J 2 = {R03 - (R02303/Boz)} Ac f

+ {Ro4 — (ROZBO4/B02)} A2nc (6.71)

which give (a) 2 and 3 2 as functions of salt concentra—
tions. As we shall see later, these equations can be used
to study and interpret osmotic flow phenomena of fluids in
loose, moderately charged membranes.

F. The Onsager Trans—
port Coefficients

 

 

Before we can proceed further to investigate our
results numerically, it is essential to acquire consider-
ably more knowledge of the physical and chemical proper-
'ties of the linear phenomenological coefficients, or
Onsager Coefficients, 9:8. For simplicity, we follow

the treatment of Miller (1966) since it is probably the

best reference which deals with the determination of

121

I I 0 I O O
ionic transport coeff1c1ents QaB’ for isothermal vector

transport process in binary electrolyte systems based on
the SF reference frame.
Consider an isothermal system consisting of a

neutral solvent, e.g., water, and a binary electrolyte

j I

which ionizes completely into the solution giving two
ions. The independent flows for the system are those of
the two ions since (30)o E 0 for the SF reference frame.

Hence, the transport equations are

IMmew-Pm‘wmnr

+
i + (2° :2 (6.73)

where the 2a (a = 1,2) are the conjugate forces, as given
explicitly in Chapter II. These equations completely
describe the isothermal vector transport properties in a
binary electrolyte solution provided the 9:8 are known as
functions of the temperature T, the pressure p, and com-
position.

The 9°

a8
because they arise from a fundamental thermodynamic theory

are the fundamental transport coefficients

as outlined in Chapter II. Any isothermal transport
process in a binary system is completely characterized
by equations (6.72) and (6.73) together with a knowledge

of the Q: as functions of c, T, p. Moreover, the same

8

122

9:8, apply when the phenomena is two or three dimensional,

e.g., where an electric field is perpendicular to the

diffusion direction. Consequently, even though the con-
centration dependence of the 9:8, is determined experi-
mentally from the one-dimensional special cases, the re-
sulting numbers can be applied to any process no matter

how complex. The experimental Onsager transport co-

m: wm‘r'“?! (

efficients for some uni-univalent electrolytes are

listed in Table 6.1.

“_lin‘fl
' I " . - ‘ I

1:!

G. Results and Discussion

 

In the following treatment we make use of equation
(6.70) to study anomalous osmosis. We consider a porous,
moderately charged membrane for which the fixed surface
charge density '0' is 10.3 C/m2 and capillary radius 150 g,
whence Ka >> 1 in the range of the salt concentrations used

for this study. In particular, we assume a negatively

charged membrane with 0 = - 10-3 C/mz, and choose the

following experimental conditions: the salt used is

NaCl, the temperature is 25°C, the isothermal shear

viscosity of water, n at 25°C is 8.903 X 10-10 J sec

cm-B, the electric permittivity of water, i.e., Do/4W,

10 1 -l

at 25°C is 6.94 x 10. C V- m , and the partial molar

volumes of Na+ and Cl- ions at infinite dilution are -l.4

and 18 cm3 mole-l, respectively (see also Table 4.1,

123

Table 6.1a--The Onsager transport coefficients for
H O-NaCl at 25°C.

 

 

2

'6', 1012 x (221/6, 1012 x (232/5, 1012 x (232/6,
mole/2 mole cm2/J sec mole cmz/J sec mole cm2/J sec
0.0000 5.381 0.000 8.201
0.0010 5.363 0.026 8.177
0.0005 5.341 0.058 8.146
0.0010 5.325 0.081 8.123
0.0050 5.263 0.170 8.036
0.0100 5.219 0.233 7.974
0.0500 5.065 0.440 7.742
0.1000 4.971 0.554 7.601
0.2000 4.851 0.682 7.435
0.5000 4.613 0.840 7.121
0.7000 4.484 0.882 6.950
1.0000 4.311 0.911 6.772
1.5000 4.053 0.923 6.370
2.0000 3.812 0.911 6.035
2.5000 3.581 0.884 5.708
3.0000 3.366 0.858 5.393

~ ..f1;_m1"
'r l
.b

 

. |

124

Table 6.1b--The Onsager transport coefficients for

 

 

E, 1012 x 021/5, 1012 x 0:2/5, 1012 x 052/6,
mole/2 mole cmz/J sec mole cmZ/J sec mole cmZ/J sec -
0.0000 7.892 0.000 8.198 E
0.0001 7.872 0.026 8.176 §
0.0005 7.844 0.059 8.148 E
0.0010 7.826 0.086 8.129 1
0.0050 7.746 0.187 8.045
0.0100 7.694 0.256 7.991
0.0500 7.520 0.503 7.810
0.1000 7.430 0.647 7.715
0.2000 7.331 0.809 7.613
0.5000 7.193 1.038 7.478
0.7000 7.140 1.124 7.425
1.0000 7.077 1.214 7.365
1.5000 6.972 1.304 7.265
2.0000 6.866 1.362 7.160
2.5000 6.754 1.404 7.050
3.0000 6.634 1.440 6.929

 

 

125

Table 6.1c—-The Onsager transport coefficients for

H20-LiCl at 25°C.

 

 

1

 

E, 1012 x 0‘1’1/‘5, 1012 x 0‘132/6, 1012 x (232/6,
mole/2 mole cmz/J sec mole cmZ/J sec mole cmz/J sec
0.0000 4.153 0.000 8.197
0.0001 4.137 0.021 8.166
0.0005 4.115 0.051 8.136
0.0010 4.103 0.073 8.112
0.0050 4.046 0.162 8.020
0.0100 4.011 0.223 7.956
0.0500 3.870 0.417 7.718
0.1000 3.774 0.513 7.547
0.2000 3.624 0.614 7.290
0.5000 3.316 0.700 6.827
0.7000 3.159 0.714 6.593
1.0000 2.942 0.699 6.286
1.5000 2.662 0.677 5.877
2.0000 2.406 0.622 5.475
2.5000 2.156 0.548 5.073
3.0000 1.954 0.500 4.718

 

1W Wfi‘tfinmmmsm) .

126

Chapter IV). The experimental Onsager transport co-
efficients for the NaCl-HZO system at 25°C are taken from
Table 6.1. The ratio of the concentrations of two solu-
tions is fixed. Plots of U0 (2) calculated from (6.80)
at q = 0.5 and 0.25 against the logarithm of the salt
concentration varied are shown in Fig. 6.1. It can be

seen from Fig. 6.1 that bell-shaped curves are obtained

.. WWfi‘rr—mr'

for both cases, and the maximum flow rate at q = 0.5

occurs approximately at the salt concentration of 0.06667

 

mole 2-1 while that at q = 0.25 occurs approximately at
the salt concentration of 0.08 mole 2-1.

Moreover, to see that our equation is also useful
for positively charged membranes, we plot Uo 2 calculated

3 C/m2 against

from (6.70) at q = 0.5, taking 0 = + 10‘
the logarithm of the salt concentration varied. The re-
sult is shown in Fig. 6.2. It is found that the values
of U0 2 are all negative. This indicates that the
direction of the osmotic flow of fluid is changed and
the flow moves toward the more dilute solution (anomalous
negative osmosis) as can be seen from Fig. 6.2.

As a result, it is apparent that the direction of
the osmostic flow of fluid in charged membranes changes
as the sign of 0 changes. This fact has not been ex-

plained by previous theories (sf. Kobatake, 1964). The

failure of previous theories to account for this fact

cm sec

(1),

10’9

70+

60‘

50‘

404

30‘

20‘

10‘

 

127

II

1 1 1 1

—ll -10 -9 -8 -7 -6
2n c (0) +
Fig. 6.1-—Plot of U0 2 (in cm2 per sec) as a function of the
logarithm of the salt concentration, c (0). Curve I,
q = 0.25. ‘Curve II, q = 0.5. The glectrolyte used
is NaCl. 0 = -10-3 C/mz. a = 150 A. t = 25°C.

 

U (2), cm sec

10

-1O

-20

-30

-4O

 

-12

Fig. 6.2-—Plot of U 2 (in cm2 per sec) as a function of the

logarithmoof
0.5. 0 = 10
a = 150 X.

128

4

2n c (0) +

-7

She salt concentration, c(0).

C/mz. The electrolyte used is NaCl.

t = 25°C.

q:

VMH“na.wr:—m:m

129

apparently arises from the use of restrictive simplifica-
tions in their treatment of the theories, which we have
discussed before. These restrictions have been excluded
in our theoretical development. Hence, we are confident
that our equation is much more complete and better in
describing various membrane systems. Consequently, the
experiments in study of osmotic flow phenomena in charged
membranes can now be interpreted more accurately, in

principle. Unfortunately, so little experimental infor-

mwgwnr-tm'mflmw’

mation is available regarding the actual values of mem-
brane charge and pore size that we cannot make a full
comparison with eXperiment. Perhaps our equation can

be used to determine charge and pore size in conjunction
with fully—characterized experiment. The shapes of our
curves are qualitatively the same as those found experi-
mentally (Kobatake and Fujita, 1964; Tasaka, Kondo and
Nagasawa, 1969) and we certainly predict the proper
direction of flow.

In the theoretical development of this chapter,
we have made use of the assumptions that Ka >> 1, and
that the gradient of 2n c is small enough compared to
the trans-membrane potential that (6.22) is satisfied,
and have adopted only the zeroth order solution for the
potential. Although we have clearly made only a start

at obtaining a complete theory of transport through

130

membranes, it is at last, a well-formulated, self—
consistent start. Hopefully, we have laid the founda—
tion sufficiently firmly that future investigators,
including ourselves, will not need to reexamine the

foundation, but may instead build upon it.

I

"’E

_fl‘n
I
{‘5 .

,‘

t"

1, fi:W-T"‘r
C

 

a?

BIBLIOGRAPHY

I"
.'

.'I‘-

.
I“
-
,___
‘I " 7

 

BIBLIOGRAPHY

Abramowitz, M., and I. A. Stegun, Handbook of Mathe-
matical Functions, National Bureau of Standards,
Applied MathematiCs Series, 55, U.S. Government

 

, Printing Office, Washington, D.C., 1964. E:
Alexandrowicz, A., and A. Katchalsky, J. Polymer Sci. Al, i
3231 (1963). E
Andrews, F. C., Ind. Eng. Chem. Fundamentals 6, 48 (1967). %
Bartelt,J;Ih, Ph.D. TheSis,Michigan State University, ;4

1968.

, and F. H. Horne, J. Chem. Phys. 51 (l), 210
(1969).

, and F. H. Horne, Pure and Applied Chemistry, 22,
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Bellman, R., Perturbation Techniques in Mathematics,

Physics and Engineering (Holt, Rinehart, and
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Booth, F., J. Chem. Phys. 19 (4), 391 (1951).
Chapman, T. W., The Transport Properties of Concentrated
Electrolyte Solutions, Ph.D. Thesis, University

of California, Berkeley (University Microfilms,
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Coleman, B. D., and C. Truesdell, J. Chem. Phys. 33, 28
(1960).

de Groot, S. R., and P. Mazur, Nonequilibrium Thermo-
dynamics (North-Holland, Amsterdam, 1962).

Defay, R., and P. Mazur, Bull. Soc. Chim. Belges, 63, 562
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, and A. Sanfeld, J. Chim. Phys. 60, 634 (1963).

131

132

Derjaguin, B. V., Research in Surface Forces (Consultants
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Devillez, C., Sanfeld, A., and A. Steinchen, J. Coll.
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Dutrochet, M., Ann. Chim. Phys. 69, 337 (1835).

Fair, J. C., and J. F. Osterle, J. Chem. Phys. 53 (8),
3307 (1971).

Fitts, D. D., Nonequilibrium Thermodynamics (McGraw-Hill,
New York, N.Y., 1962.

 

Fujita, H., and Y. Kobatake, J. Coll. Interf. Sci. 21,
609 (1968).

Grim, E., and K. Sollner, J. Gen. Physiol. 40, 887 (1957).

Gross, R. J., and J. F. Osterle, J. Chem. Phys. 49 (1),
228 (1968).

Gyarmati, I., Nonequilibrium Thermodynamics (Springer—
Verlag, New York. N.Y., 1770).

Haase, R., Thermodynamics of Irreversible Processes
(Addison-Wesley Publishing Co., Reading, Massa-
chusetts, 1969).

Hasted, J. B., Ritson, D. M., and C. H. Collie, J. Chem.
Phys. 16 (l), 1 (1948). '

Horne, F. H., J. Chem. Phys. 45, 3069 (1966).

, and T. G. Anderson, J. Chem. Phys. _3 (6),
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Hurwitz, H. D., Sanfeld, A., and A. Steinchen, Electrochim.
Acta, 9, 929 (1964).

Ingle, S. E., Ph.D. Thesis, Michigan State University,
1971.

Irving, J., and N. Mullineux, Mathematics in Physics and
Engineering (Academic Press, New York, N.Y.,71959).

 

Katchalsky, A., and P. F. Curran, Nonequilibrium Thermo-
dynamics in Biophysics (Harvard’Univ. Press,
Cambridge, Massachusetts, 1967).

 

133
Kedem, 0., and A. Katchalsky, J. Gen. Physiol. 45, 143
(1961). _—

Kirkwood, J., and B. Crawford, J. Phys. Chem. 56, 1048
(1952) . ‘—

Kobatake, Y., and H. Fujita, Kolloid Z. 999, 58 (1964).
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, Bull. Tokyo Inst. Technol. Ser. B, p. 97 (1959).
, J. Chem. Phys. 99, 442 (1958).

Kotin, L., and M. Nagasawa, J. Chem. Phys. 99, 873 (1962).

Lakshminarayanaiah, N., Transport Phenomena in Membranes
(Academic Press, New York, N.Y., 1969).

 

MacGillivray, A. D., and J. J. Winkleman, J. Chem. Phys.
99 (6), 2184 (1966).

Manning, G. S., J. Chem. Phys. 99 (6), 2668 (1968).

Mazur, P., and I. Prigogine, Mem. Acad. Roy. Belg. 99, l
(1953).

Miller, D. G., Chem. Rev. 99, 15 (1960).

', J. Phys. Chem. 19 (8), 2639 (1966).
Mfiller, I., Arch. Rational Mech. Anal. 99, l (1968).
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Philip, J. R., and R. A. Wooding, J. Chem. Phys. 99 (2),
953 (1970).

Pottel, R., Chemical Physics of Ionic Solutions, ed. B. E.
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Prigogine, I., Mazur, P., and R. Defay, J. Chem. Phys.
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Sanfeld, A., and R. Defay, J. Chim. Phys. 63, 577 (1966).
, Introduction to the Thermodynamics of Charged

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134

Schlogl, R., Z. Physik. Chem. N. F. 9, 73 (1955).
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Sparnaay, M. J., Rec. Trav. Chim. 77, 872 (1958); 81, 395
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Verwey, E. J. W., and J. Th. G. Overbeek, Theory of the
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Watson, G. N., A Treatise on the Theory of Bessel Functions,
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A. i
.. _. . r
.31... NJ...»

. «”3 a.
Per-L. aft-...EHE a

APPENDICES

APPENDIX A

A DERIVATION OF THE CHAPMAN-GOUY EQUATION

FOR A CHARGED CYLINDER

Consider a charged cylinder containing an aqueous,
binary, uni—univalent electrolyte solution. The Chapman-
Gouy equation can be derived from the usual Poisson—

Boltzmann equation in the form
B B
(Do/r) (dr (dw/dr)/dr) = - 40F (c+ - c_) (A.l)

where r is the radial coordinate, D0 is the isothermal
dielectric constant of water, 0 is the electrostatic
potential, F is the Faraday constant, and where c: (a = +,-)

is the Boltzmann equation molar concentration of species a,
B _
Ca = c exp (-zaF0/RT) , (A.2)

where E is the average salt concentration, 2a is the
valency of a, R is gas constant, and T is the absolute

temperature. The boundary conditions are
dw/dr = O at r = 0, and dw/dr = 4no/DO at r = a , (A.3)

where o is the surface charge density fixed on the cylinder

wall, and a is the radius of the cylinder.

135

 

136

We introduce (A.2) into (A.1) and make use of the

following substitutions:
Fw/RT = w
r = E/K

where K—1 is the Debye length

 

K = /80F26/DORT
There results, then,
(l/E) (d€(dY/d€)/d€) = sinh? .

where we have made use of the transformation

We rewrite (A.7) as
d2 Y/dgz + (1/5) (dY/dg) = sinh?
Equation (A.9) reduces to
d2 W/dgz = sinh? ,

for (1/5) (dY/dg) << d2 Y/dgz, in particular, for Ka >>

(A.4)

(A.5)

(A.6)

(A.7)

(A.8)

(A.9)

(A.10)

1

and (dW/dg)o = 0. It is also assumed in this derivation

that W vanishes at E = 0. The boundary conditions on W

then become,

WWI” WW .1

dW/dg 0 and W = 0 at E = 0 (A.11)

dW/dg

4noF/DORTK at S = E , (A.12)

a

where Ea is the value of g at the wall, i.e., Ea = Ka, and
where we have used (A.3) and (A.4).
We now solve (A.10) subject to (A.11) and (A.12).

Making use of the transformation

dW/dg = p , (A.13)
whence

(129/dz:2 = p (dp/dY) , (A.14)
there follows, from (A.10),

p (dp/dW) = sinh? . (A.15)
The solution to (A.15) is immediately obtained, yielding

2

p /2 = cosh? + b , (A.16)

where b is the constant of integration which is readily

evaluated, subject to (A.11), to give
b = -1 . (A.17)
Hence, (A.16) becomes

p2/2 = coshY - 1 (A.18)

 

.. Vfifi “m Tfi‘.
4 (

 

“14.1.71-
‘ 1'}!

138

The Chapman-Gouy equation relates the surface
potential to the surface charge density fixed on the wall.

Hence, we have, from (A.12), (A.13), and (A.18),

l 2 _ _
f (4noF/DORTK) — cosh‘i’a 1 , (A.19)

where Ya is the value of W at the wall. By rearrangement,

(A.19) becomes

0 4 (zeRTE) (coshYa - 1)/2 (A.20)

 

where e is the electric permittivity

e = Do/40 . (A.21)
From (A.20), there follows, then,

0 = 2 (268TE)1/2 sinh (Ya/2) , (A.22)

where we have used

 

sinh (Ya/2) = /(cosh‘i’a - 1)/2 . (A.23)

Equation (A.22) is the Chapman-Gouy equation.

. 1.:
-D :' . v ~ I ‘3 Q . 7" ". r'
J {r- . ' '. ... r ' i ‘ -.. ml

0
1 '1‘ ' ‘ ..1 "" 1!
m . y.)
4 I 4 : 1‘ \
‘ ...
( « . .1
- 1"“ I
. , . . ) c)
. ‘( . ... . l
A.
I. H! V 4 ~( 1
I I ‘ . "
. ’ ‘A v " ' I
. , . . ._.
r . . H,

 

 

APPENDIX B
DERIVATION OF EQUATIONS (4.87) AND (4.88)

In the following we derive equations (4.87) and

(4.88). From the recurrence formula,
d EIl/dg = 510 , (B.l)

we have, in (4.87),

E

I 3 = 5 2 = E 2
o 610 dg 0 IO (:10) d6 3 Io dg Il . (8.2)
By integration by parts, there results

3 _ 2 _ g
0 d6 — 51110 2 g gIlIOdIo (8.3)

15 61
0

Note, however, that

dIO/dg = Il , (8.4)
whence

:5 61 I d1 = 15 61 1 (dI /dg)dg = I5 6121 dg (B 5)

0 1 o o 0 1 o o 0 1 o '

Substitution of (B.5) into (B.3) yields, by rearrangement,

equation (4.87),

E 2 _ 2 _ g 3
0 gIOIldg — 61011/2 0 gIodg/z (8.6)

139

 

140
To obtain equation (4.88), we write

)5 Elida = 15 I:d(€2/2) (B.7)
0 0

By integration by parts, there results

a 2 _22 _62
3 gIodg - a 10/2 I g IodIO (8.8)

From (3.4), we have

{'1'}. I.,!“ = . El .I '” lllllim.

E 2 _ E 2 _ E
a a 10910 - 3 6 IO (dIO/da) d6 — 5 (EIO)(€Il)d€
_ E _ 2 2

0

where we have used (B.l). Substitution of (B.9) into (B.8)

yields equation (4.88),

2
o

(g 41 d: = 62 (I:

_ Ii)” (8.10)
0

 

(mm)

462

WM”

80