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('4 ‘5 .FH‘JT—V‘»).‘:_u1 — - ...— A
This is to certify that the

dissertation entitled

ON THE ANALYTICAL SOLUTION TO THE
LINEARIZED POISSON-BOLTZMANN EQUATION
IN CYLINDRICAL COORDINATES

presentedby

Richard Eugene Rice

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Chemistry

Major professor
Date July 27 , 19 82

MS U is an Affirmative Action/Equal Opportunity Institution 0-1277 1

 

 

 

 

MSU RETURNING MATERIALS:
Place in book drop to

LJBRARJES remove this checkout from

5—3—- your record. FINES will

 

 

be charged if book is
returned after the date
stamped below.

 

 

 

 

ON THE ANALYTICAL SOLUTION TO THE LINEARIZED

POISSON—BOLTZMANN EQUATION IN CYLINDRICAL COORDINATES

By

Richard Eugene Rice

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1982

ABSTRACT

ON THE ANALYTICAL SOLUTION TO THE LINEARIZED

POISSON-BOLTZMANN EQUATION IN CYLINDRICAL COORDINATES
By
Richard Eugene Rice

The goal of this work is to obtain a complete analytical
solution to the linearized Poisson-Boltzmann equation in cy-
lindrical coordinates:32w/3z2-+(l/r)3(r8w/3r)3r = K2f2(z)¢,
where w is the electric potential, K the inverse Debye
length, and f(z) a particular ionic strength distribution.
We solve it inside a uniformly charged cylindrical pore in
an idealized membrane between two electrolyte solutions at
different ionic strengths. The system is assumed to be con-
tinuous, isothermal,isotropic, incompressible, free of chemical
reactions, and subject only to conservative external forces.

Unable to-separate variables with either an additive or
multiplicative solution, we turn to the solution w(z,r) =
wl(z) + w2(z,r), in which 2 and r are not separable in the
final term. This results in two equations: dzwl/dz2 =
k2f2(z)wl and 82u2/822 + (l/r)3(r3w2/3r)/3r = K2r2(z)w2.

the z-derivative, which is small compared to the r-derivative,

Richard Eugene Rice

is neglected, the second of these equations is the modified
Bessel equation of order zero. The first of them can also
be solved analytically for an explicit f(z).

We choose three expressions for f(z): (l) constant I(z)
(ionic strength), (2) I(z) linear in z, and (3) 2n I(z) lin-
ear in z. The first f(z) yields a solution of hyperbolic
functions, while the latter two yield modified Bessel func-
tions. The boundary conditions on $1 are determined at each
end of the pore in terms of the parameters of the bath-
ing solutions by a mass balance and also by a charge balance.

Next we solve the Poisson-Boltzmann equation for a semi-
infinite cylinder by Laplace transforms; the result has the
form w(z,r) = fl(z) + gl(r) + f2(z)g2(r). Applying such a
solution to the finite cylinder, however, we find it is not
possible to separate variables, and again the solution has
terms in z and terms in z and r.

In the final chapter we discuss our methods and results
in terms of extrascientific considerations involving theory,

metaphor, model, paradigm, language, and explanation.

To the Memory of my Grandparents,
Mr. & Mrs. Bertram E. Weston,
who believed in education and in me

11

But when I consider, how much most of the
qualities of bodies, and consequently their
operations, depend upon the structure of
their minute, and singly invisible parti-
cles . . . I cannot but think the doctrine
of the small pores of bodies of no small
importance to natural philOSOphy.

--Hon. Robert Boyle
The Porosity 9: Animal Bodies, 168A

iii

ACKNOWLEDGMENTS

I wish to thank the Departments of Chemistry and Eng-
lish for financial support in the form of teaching and
research assistantships during my years at Michigan State
University. I also thank the Graduate School for additional
support during several summers.

The International Research and Exchanges Board, the
Soviet Ministry of Higher Education, and the Academy of
Sciences of the Ukrainian SSR made it possible for me to
spend the 1979-80 academic year at the Institute of Colloid
and Water Chemistry in Kiev, where I performed experiments
on capillary osmosis. I thank everyone at that institute, as
well as at the Institute of Physical Chemistry in Moscow
and Leningrad State University where I went on "komandirovka,"
who helped make the year in the Soviet Union a fascinating
and successful one for my wife and me.

Although not a direct part of my education at Michigan
State, a Mass Media Science Fellowship, which I received
from the American Association for the Advancement of
Science during the summer of 1981, enabled me to spend
ten weeks at Omni Productions in New York during the pro-
duction stage of their television science series, "Omni:

The New Frontier."

Each oftflmemembers of my doctoral committee has

iv

 

J

C; A)

(D

n)

contributed to this dissertation. I thank Dr. Christie
G. Enke for sharing his expert knowledge of experimental
electrochemistry with me; Dr. Jack B. Kinsinger for acting
as my Second Reader during his final hectic days at Michigan
State and for selecting me as Chemistry's instructor in the
English Department's experimental writing program for science
students; and Dr. E. Fred Carlisle, who passed up ROTC camp
to attend my oral defense, for teaching me about teaching
and about language as a common basis of both science and
literature, as well as for a critical reading of my final
chapter.

I am especially grateful to my advisor, Dr. Frederick
H. Horne, who is my implicit partner in the "we" of the
first five chapters of this dissertation. He and I have
worked together on the problems involved in solving the
Poisson-Boltzmann equation, and he calculated many of the
residues that appear in Chapter A. An exact and exacting
scientist, Dr. Horne has instilled in me the desire to
be one myself. My wife and I will particularly miss the
Horne family's hospitality (n1 future Thanksgivings.

The other members of my research group have always
been willing to listen and offer their Opinions. I es-
pecially thank Dr. Daniel Bradley, Dr. John Leckey, Mr.
Thomas Troyer, and Mr. Bruce Borey for their helpfulness
and friendliness over the years.

Finally, I thank my family: my parents, Mr. & Mrs.

George A. Rice, who started me on the long road of

education many years ago and never doubted that I would
eventually reach this final destination; and my wife, Dr.
Joanne A. Rice, without whose example, encouragement, and
expert typing and editing skills, I truly would never have

completed this dissertation.

vi

TABLE OF CONTENTS

Chapter Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . ix
LIST OF FIGURES. . . . . . . . . . . . . . . . . . . x

CHAPTER 1 - THE LINEARIZED POISSON-
BOLTZMANN EQUATION . . . . . . . . . . . l

A. Introduction . . . . . . . . . . . . . . . . 1

B. Gouy-Chapman Model of the Double
Layer. . . . . . . . . . . . . . . . . . . 3

C. Debye- Huckel Theory of Strong
Electrolytes . . . . . . . . . . . . 5

D. Assessment of the Debye—HUckel
Theory . . . . . . . . . . . . . . . . . . . 5

E. Motivation for the Present Work. , , , , , , 10

CHAPTER 2 - THE FIRST ATTEMPT: SEPARATION

' OF VARIABLES . . . . . . . . . . . . . . 12
A. Introduction . . . . . . . . . . . . . . . . 12
B. The Method of Separation of
Variables. . . . . . . . . . . . . . . . . . 15
c. Additive Solution. . . . . . . . . . . . . . l6
D. Multiplicative Solution. . . . . . . . . . . 15
CHAPTER 3 - A NEW VERSION OF AN OLD
SOLUTION . . . . . . . . . . . . . . . . 22
A. Introduction . . . . . . . . . . . . . . . . 22

B. Analytical Solutions for wl(z)
and w2(z, r). . . . . . . . . . . . . . . 23

C. An Attempt to Obtain Explicit Boundary
Conditions w1(0) and wl(L) . . . . . . . . . 30

vii

L1.)
(1 I

.‘V
g..~

A»

Va.

“‘99
—

V“

Chapter Page
D. The Bulk Solutions as Boundary
ConditionS. . . . . . . . . . . . . . . . . 37

E. An Alternate Set of Boundary
ConditionS. . . . . . . . . . . . . . . . . 51

F. Final Membrane Potential Formulae . . . . . 56
G. Numerical Calculations. . . . . . . . . . . 59
H. Summary . . . . . . . . . . . . . . . . . . 78
CHAPTER 4 - SOLUTION BY LAPLACE TRANSFORM 80

A. Introduction. . . . . . . . . . . . . . . . 80

B. Perturbation Method . . . . . . . . . . . . 82

C. Zeroth-Order Solution . . . . . . . . . . . 84

D. First-Order Solution. . . . . . . . . . . . 93
CHAPTER 5 - AN ATTEMPTED SOLUTION BASED

ON THE LAPLACE TRANSFORM

RESULTS . . . . . . . . . . . . . . . . 111
Introduction. . . . . . . . . . . . . . . . lll

Zeroth-Order Solution . . . . . . . . . . . 112

Ow>

First-Order Solution. . . . . . . . . . . . 116
D. Critique of the Method. . . . . . . . . . . 129

CHAPTER 6 — MODELS, METAPHORS, AND

MATHEMATICS . . . . . . . . . . . . . 132

A. Introduction. . . . . . . . . . . . . . . . 132

B. Theory. . . . . . . . . . . . . . . . . . . 13“

C. Metaphors and Models. . . . . . . . . . . . 139

D. Paradigms and Language. . . . . . . . . . . 1AA

E. Explanation . . . . . . . . . . . . . . . . 151

F. A Last Look at Electric Potential . . . . . 156
BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . 159

viii

Table

3.1

3.2

3.3

3.A

3.5

3.6

LIST OF TABLES

Values of Parameters Needed in
Calculations.

Calculated Values of wl(z) for
Constant Concentration.
Calculated Values of wl(z) for
I(z) Linear in z. . .
Calculated Values of wl(z) for
in I(z) Linear in z

Calculated Values of <w2(z,r0>r
for I(z) Linear in 2.
Calculated Values 0f’C&(Z:r)>r

for £nI(z) Linear in 2.

ix

Page

60

61

6A

67

72

73

LIST OF FIGURES

Figure 3 Page

2.1 Idealized model of a membrane con-

taining a cylindrical pore . . . . . . . 13
3.1 wl(z) at each end of the pore for

constant ionic strength. . . . . . . . . 69
3.2 wl(z) at each end of the pore for both

ionic strength models. . . . . . . . . . 70
3.3 <w2(z,r)>r along the pore for the two

ionic strength models. . . . . . . . . . 7“
A.1 Semi-infinite cylinder for the Laplace

transform method
6.1 Interrelationships involving theory. . . 135
6.2 Interrelationships involving the

analogue associated with a model . . . . 155

CHAPTER I

THE LINEARIZED POISSON-BOLTZMANN EQUATION

A. Introduction

 

Much of the theoretical work on steady-state membrane
transport rests on some form of the one-dimensional Nernst-
Planck flux equations [20,21,62,8A,88,89,90,93,94,102,10A,
109,110,111,1A5,1A6,159,171],

dfinc V
_ __i _i_d_w_+ _i_e
Ji“'Di°i[ dz + RT dz R dz]+ciu0 ’ (1'1)
where J is the flux of ion i D its diffusion coefficient,

1 ’ i
ci its concentration, 21 its charge, V

i its partial molar
volume, w the electric potential, p the pressure, uO the
velocity of the solvent, F the faraday, R the gas constant,
and T the temperature. In this form the equation applies
to each species in the system except the solvent. Despite
their neglect of interactions between diffusing species
[70], these equations still provide results comparable to
experiment.

Usually, the Nernst-Planck equations are solved, either

analytically or numerically, in conjunction with other

equations. The Navier—Stokes equation [9,11,69], often
in curtailed form such as

nV2u = Vp + F(XCizi)Vw , (1.2)

where n is viscosity, provides the barycentric fluid vel-
ocity u for the convective term (which is usually, but in-
correctly ignored relative to diffusion); Poisson's equa-

tion [3027691173138]:

, (1.3)

provides the required electric potential term in the case
of charged species. Poisson's equation relates the elec—
tric potential w to the volume charge density p, defined

as

p = F2 0121 , (1.“)

and to the dielectric permitivity e of a medium, which is
surrounded by a surface bearing additional charges. This
is exactly the case for the capillary model of a membrane.
A pore, frequently idealized as a cylinder for mathematical
convenience, contains ions in solution as well as a surface
charge, which provides one of the boundary conditions that

the potential solution must satisfy.

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Two well-known solutions of the Nernst-Planck and Pois-
son equations together are those by Planck himself [129]
and Goldman [51]. Generalized in various ways by other
authors [7A,l30], Planck's method, which assumes that the
thickness of the ionic atmosphere is small relative to that
of the diaphragm, gives rise to a potential-difference
equation [99] that has been successfully used for calcu-
lating liquid-Junction potentials, despite the fact that
the method's assumption implies electroneutrality when
that is not the case. Goldman's approximation of a constant
field, which allows immediate integration of the Nernst-
Planck equations, leads to an expression that has proven
popular in working with potential differences of biological
membranes [68,1A7]. Probably the two most famous applica-
tions of Poisson's equation arose in a different context,
that by Gouy [52] and Chapman [27] in their independent
treatments of the double layer, and that by Debye and
Hfickel [3A] for the case of strong electrolytes. It is
instructive to look at both these models, the Debye-Hfickel
theory in particular since many people have written about

its shortcomings and strengths.

B. Gouy-Chapman Model of the Double Layer

 

Gouy and Chapman succeeded in determining the statis-
tical distribution of ions in a double layer by taking into

account both the field from the charged solid surface and

the thermal energy of the ions themselves. Although counter-
ions (ions with the charge opposite to that of the solid
surface) are attracted towards the surface and co-ions

(ions of the same charge) are repulsed, there is no net
movement of ions perpendicular to the wall. By setting

the Nernst-Planck equation with only terms for diffusion

and electromigration equal to zero, Gouy and Chapman ob-

tained a Boltzmann distribution for each ion:

 

m 2 Ft
ci(z) = ciexp[- RT] . (1.5)

The resulting one-dimensional Poisson-Boltzmann equation

for a symmetrical electrolyte,

 

2 2z F 2 Ft
M = + +
dz2 sinh W , (1.6)

has a rather complicated solution [167], which, when plotted
however, decreases more or less exponentially away from
the charged solid wall.

Initially successful in differentiating between thermo-
dynamics and electrokinetic potentials and in systematiz-
ing experimentally observed electrokinetic phenomena [Al],
the Gouy-Chapman model is still important for surface and
colloid chemistry, in both its original form and its sub-
sequent modification by Stern [157]. This model is not

widely known among chemists, however, largely because it

was soon overshadowed by the eminently successful Debye-
Hflckel theory, which Gouy and Chapman actually anticipated

with their work.

C. Debye-Hfickel Theory of Strong Electrolytes

 

Although Michael Faraday [A7] distinguished electrolytes
from nonelectrolytes on the basis of his electrochemical
investigations as early as 1834, the differences in the
behavior of these two classes of substances continued to
bedevil scientists for many years. The first major break-
through in understanding electrolytic solutions was Ar-
hennius' concept of dissociation [58], but considerable
confusion and disagreement lingered until the work of
Debye and Hfickel.

Bockris and Reddy [15] point out that even though
Gouy and Chapman had already devised the general approach,
including the idea of "smooth" charge and the use of the
Poisson equation, Debye and Hfickel's genius lay in choosing
a reference ion whose interactions with the remaining ions
in solution are analogous to those treated earlier between
an ion and a charged surface.

With Poisson's equation in spherical coordinates,

l d 2 d
"z'asl‘ a¥=-%: (1.7)
I’

Debye and Huckel wrote a Boltzmann distribution for each

ion, as in Equation (1.5), but, unlike Gouy and Chapman,
expanded the exponential into a Taylor series and trun-
cated it after the second term. This procedure, which
proved fortuitous but also engendered much of the subse-
quent criticism of the Debye-Hackel theory, results in the

so-called linearized Poisson-Boltzmann equation:

1 d 2 d 2
Fifi-1” ag=KlD o (1.8)

With the boundary conditions that w + O as r + m and

K + O as C1 + O, the solution to this equation is simply

 

w = —— , (109)

which provides the electrostatic potential as a function of

distance r from an arbitrary ion with charge zi.

D. Assessment of the Debye—Hackel Theory

The assumptions underlying the derivation of Equation
(1.9) by Debye and Hfickel are essentially as follows [15,
“8,11A]:

(l) the dissolved electrolyte is completely dis-
sociated;

(2) the ions arising from the dissociated electro-
lyte behave like point charges;

(3) the spherically symmetric atmosphere of excess
charge about the reference ion is a time average
of all possible ionic configurations;

(A) only coulombic interactions occur between

ions, and this allows the potential energy

of mean force in the Boltzmann expression to

be replaced by the mean electrostatic potential;
(5) the ratio ziFw/RT << 1, so that the exponen-

tial in Equation (1.5) can be expanded and

the terms higher than first order neglected;

and

(6) the solvent is a continuum with a constant

' dielectric constant unchanged by the presence
of the electrolyte.

With their model incorporating these sometimes drastic
assumptions, Debye and HUckel provided a simple physical
picture of an ionic solution and derived Equation (1.9),
which led to their famous limiting law for the activity co-
efficient [11“]:

log yi = -A|z+z_|I1/2 , (1.10)

where A is a temperature-dependent constant. This expres-
sion allows comparison with experiment‘and provides the
theory's greatest triumph. As concentration decreases, the
Debye-Hfickel limiting law more accurately predicts experi-
mental behavior.

Yet despite the theory's simplicity and success, or
perhaps because of them, scientists began almost immediately
to tinker with it. The first was Onsager [121], who tacked
on two electrical effects that tend to decrease the mobility

of the reference ion: the asymmetric and electrophoretic

effects [11“]. Bjerrum [l2] and later Fuoss and Krauss

[M9] suggested the formation of transient ion pairs through
electrostatic association to increase the fit between
Debye-Hackel theory and experimental data. Lower dielec-
tric constant and smaller ionic radii favor the formation of
such pairs, which may still be appreciable in a solvent like
water with a high dielectric constant. In 1933 Kirkwood
[77] published an extensive discussion of the substitution
of the mean potential in place of the potential of the mean
force and showed that this approximation, along with some

of the others, resulted in "the neglect of an exclusion-
volume term and a fluctuation term" [12“].

Sixty years after the appearance of Debye-Hackel theory,
trying to improve it is still a popular pastime among
statistical mechanicians [8,lO,92,12A,1A3,156,l69]. Mc-
Quarrie [101] outlines some of these attempts, but he is
forced to concede that statistical mechanics has done little
more than confirm the original theory as the exact limit
for zero concentration. The strongest praise for modern
"improvements" that Levine and Outhwaite can muster is that
the Poisson-Boltzmann equation "has been significantly
modified in a qualitative manner" [92].

Certainly there are many legitimate objections against
Debye—Hfickel theory; some of them are [8,131]:

(1) ions are not point charges, and even the extended

Debye-Hfickel theory, which treats ions as having
finite size, has unrealistic boundary conditions;

(2) the dielectric constant is obviously not constant,
and the solvent's bulk value does not adequately
represent the conditions close to the reference
ion;

(3) many ions do not possess spherically symmetric
electron distributions, and they are therefore
not likely to induce a spherically symmetric ionic
atmosphere;

(A) short-range ion-ion and ion-solvent forces, which
are ignored, may be significant, especially in
more concentrated solutions;

(5) replacing the potential of mean force with the
mean potential is apparently essential since
there seems to be no better approximation, but
the validity of this step is difficult to determine;
and

(6) linearization fails at very small distances from
the reference ion, and this distance increases
with increasing ionic change and decreasing di-
electric constant.

Of these objections, the question of linearizing the
Poisson-Boltzmann equation is perhaps the most interesting.
Since the linearization is an approximation, and apparently
a drastic one, it would seem logical that retaining the
exponential would improve the result. That is not the
case. As Moore, with an uncharacteristic cliché, says,

"To discard the linear approximation, and use instead the
exact Boltzmann equation, is to jump from the frying pan
into the fire" [11”].

The reason for this is that one of the assumptions of
the Debye—Hackel model is the superposition of potentials;
i.e., the potential at a point is the sum of the potentials

from the nearby individual charges. This is consistent

10

with the linearized Boltzmann equation, but not with the
exact exponential form [15]. Furthermore, the solution of
the linearized equation satisfies requirements of self-
consistency and exact differentials, while that of the
exact equation does not, so that it is the linearization
that causes the Debye-Hackel result to be the exact limit-
ing case [101].

Apparently the best advice is that we "keep in mind the
fact that not only is the linearization procedure unac-
ceptable except for large ions and low concentrations .
but also abandonment of the linearization leads to no im-

provement" [11A].

E. Motivation for the Present Work

For whatever reasons, scientists continue trying to
solve both the linear and nonlinear forms of the Poisson-
Boltzmann equation with both analytical [l,18,30,88,96,
110,126,127,l28,151,152,153] and numerical methods [20,
A6,55,l20,1AA,168]. In membrane-transport studies that re-
quire the electric potential, not only is there no justifica-
tion for not simply choosing the equation in its linearized
form, there are compelling reasons for eschewing any so-
called improvements. Indeed, there is some indication
that the effects of the various corrections are usually
not significant [61,125], and including some corrections

while neglecting others may lead to worse results than those

11

obtained from the linearized Poisson-Boltzmann equation,
warts and all.

Thus solving the linearized Poisson-Boltzmann equation
remains a worthwhile endeavor. The choice between a num-
erical or analytical solution for our purposes should be
equally easy. Although a numerical solution has obvious
advantages, its main disadvantage is that it lumps together
the results so that relative contributions from different
sources may not be obvious. Even in this age of high-speed
computers, solving equations analytically is still an im-
portant activity.

The next four chapters of this dissertation are devoted
to just that: seeking an analytical solution to the lin-
earized Poisson-Boltzmann equation. The resulting w(z,r)
will then enable us to find ci(z,r) from Equation (1.5)
and uz from Equation (1.2), both necessary quantities for
the further application of the Nernst-Planck equations

(1.1) to ion transport across membranes.

C.
a1

9’

CHAPTER 2
THE FIRST ATTEMPT: SEPARATION OF VARIABLES

A. Introduction

Since the ultimate goal of solving the linearized
Poisson-Boltzmann equation is to obtain improved equations
for membrane transport, we choose a model system that is
both mathematically tractable (at least relatively) and
applicable to membranes. We assume an idealized membrane
containing uniformly charged cylindrical pores and posi-
tioned between two infinitely large reservoirs of electro-
lyte solutions, which need not be at the same ionic strength
(Figure 2.1). We consider this system to be continuous,
isothermal, isotropic, incompressible, free of chemical re-
actions, and subject only to conservative external forces.

For the region inside the pore, i.e., 0 i r i a and

0 i z i L, the linearized Poisson—Boltzmann equation is

g%-r 3% = Kgf2(z)w , (2'1)

"SIH

2
Lil...
8Z2

where w is the electric potential at any point z, r; f(z)
is some function describing the ionic strength distribution

along the length of the pore; and K0 is the reciprocal Debye

l2

.mp0d Hmoappcfiamo m maficfimuzoo ocmnnEmE m mo Hopes pmufiflmmcH .H.m onsmfim

13

m
conusaom

 

 

 

 

 

osmanoE

<
soapsaom

 

1”

length defined as

2
Kg = w , (2.2)

eRT
where I(O) is the ionic strength at z = O, and the other
symbols have their usual meaning.

It is not possible to solve the Poisson-Boltzmann
equation completely without specifying f(z) explicitly.
Because neither the individual ion concentrations nor
the ionic strength is available at each point within the
pore, it is necessary to assume a particular axial gradient
of ionic strength. In a recent note [1A0] we discussed
two possible choices: (1) I(z) linear in z and (2) in
I(z) linear in 2.

For the first case, I(z) linear in 2,
1(2) = IA [1 + (82-1)(z/L)l , (2.3)

where

]'l/2 . (2.14)

U)
I“
'—_I
HH
OI?"

v

Thus for this case, Equation (2.1) becomes

2
3 3 3 _ 2 2 2

*SIH

15

For the second case, En I(z) linear in z,

2z/L

I(z) = 1(0)s , (2.6)
and

2

3 l 3 3 2 2Z/L

33"‘55‘51'5‘3‘K03 '“ (2'7)

This form of the linearized Poisson-Boltzmann equation not
only appears simpler mathematically (though it is just an
appearance), but may also be more reasonable physically.
At steady state in the absence of applied pressure and
electric potential gradients, in I(z) is indeed linear, so
that the actual distribution of ionic strength even with
applied gradients may not be significantly different.

For this reason we focus upon the second case and ex-
pend most of our energy trying to solve Equation (2.7).
A third, trivial case, that of f(z) = 1, is also of some

interest for purposes of comparison.

B. The Method of Separation of Variables

 

Separation of variables is frequently the first method
chosen in any attempt to solve a particular partial dif-
ferential equation. Although it is simple and straight—

forward, it seldom works. Nevertheless, Hildebrand claims

16

that despite the method's mathematical restriction "to a
comparatively narrow range of differential equations, for-
tunately this range includes a very large number of those
equations which arise in practice" [66]. This seems suf-
ficiently encouraging to try this method for the linearized

Poisson-Boltzmann equation.

 

C. Additive Solution

We assume a solution of the form

w(z,r) = ¢l(z) + w2(r) . (2.8)

Substituting this into Equation (2.7), we obtain

 

dw
d 2 _ KZSZZ/L(¢1

2+i-l'a'fr’df“ 0 W2). (2.9)

This equation cannot be separated into two ordinary dif-
ferential equations, one of which depends only on 2 and

the other of which depends only on r.

D. Multiplicative Solution

This time we assume a solution of the form

w<z,r) = w1(2)w2(r) . (2.10)

w...

73»

“

17

There is strong precedent for a solution of this type.
In analyzing the interaction between two charged cylinders
(as models of colloid particles), Sparnaay [155] solved a
two-dimensional linearized Poisson-Boltzmann equation with
such a solution. Substitution of Equation (2.10)into

Equation (2.7) leads to

2
LL E—fl + _l— 5L r 333 = 2322/L (2 11)
"1 dz2 mp2 dr dr Ko ’ °

in which the functions of z and r can be separated:

2
d "1 2 2z/L _ 1 d. d"2

l
wl dz2 KOS - '- 5-15-2- 6}- I" ‘3?- . (2.12)

 

Since the left-hand side is independent of r and the
right-hand side independent of z, in order for them to be
equal to each other, they must each be equal to a constant,
which we can designate as 12. This results in the two

ordinary differential equations

 

dzw
l - («2322/L + A2)w = o (2.13)
d 2 o 1
Z
and
dw
d 2 2 2 _

18

The second equation of this pair is immediately recogniz-

able as Bessel's equation, but the first must be trans-

2Z/L before it too can be

formed because of the term 8
solved.

With the change of variable

KOL

z/L
EnS S

. (2.15)

x:

Equation (2.13) becomes

2 dipl 2

 

in S d tn S 2 2 _
2 X a"? X E - (-—'2—- X + A )tlll - 0 . (2.16)
L L
2 2 2 2
If we let A = (n in S)/L , where n need not be an integer,

Equation (2.16) simplifies to

dwl

d 2
x af‘x YEF" (x

+ n2)xp1 = 0 . (2.17)

The solutions to Equations (2.17) and (2.1“) are

z/L z/L
ganlnhs ) + :21 ann(TS ) (2.18)

w1(2)
and

w2(r) = iAnJO(nKOr/T) + iBnYOmKOr/r) , (2.19)

19

where an, b A Bn are constants; I and KH are modified

n’ n’ n

Bessel functions of the first and second kind of order n
respectively; J0 and Y0 are ordinary Bessel functions of
the first and second kind of order zero, respectively;

and

F‘

Ko
T : —fi§ . (2.20)

go

The radial symmetry condition,

39(z,O) = 0

3r , (2.21)

immediately shows that all the Bn must be zero or else the
sum for r = 0 must be zero since Yl(0) is undefined. The

second radial boundary condition,

39(z,a) _

3r , (2.22)

I
mIQ

where o is the surface charge density on the wall of the
pore, is inconsistent with this solution since it re-

quires that

 

_ TO 1
wl(z) - - 8K0 ZnAnJ1(nKOa/T) (2'23)

i.e., that wl(z) be constant. This is incompatible with

the original separation of ¢(z,r).

20

One possible way of salvaging this solution is the
assumption of a z-dependent surface charge density o(z).
Although the derivation of the boundary condition (2.22)
from Gauss' law involves an equipotential surface and thus
a constant surface charge density [138], Gouy-Chapman theory
provides a relationship between surface charge density and
surface potential, and thus between charge density and con-
centration if the surface potential is known [A1,123].

Thus for the boundary condition

 

12%;). = 0(5) , (2.21))

we have that

 

 

- TO(Z) l
wl(z) - - 8K0 XnAnJl(nKOa/TI ’ (2°25)
or
2A J (n r/ )
w(z,r) = - 70(2) n 0 K0 T (2.26)

8K0 EnAnJl(nKOa/T)

There are two major difficulties with this solution.
First, we have no way to determine the constant An. The
natural choice, the orthogonality of the Bessel functions,

would be useful only if we could relate their argument to

21

the zeros of, presumably, J1. Second, there is no inde-
pendent method for obtaining the z-dependence of 0, though
presumably it would be related to the z-dependence of the
ionic strength, and if we simply assume a certain dependence
on 2, we have thwarted our intent of determining the elec-

tric potential w as a function of both 2 and r.

We return to Equation (2.16), and letting 12 = -n2£n28/L2,
we have the solution,
_ z/L z/L
wl(z) - i anIin(rS ) + g anin(rS ) . (2.27)

The solution of Equation (2.14) remains the same. The

only change in the form of the solution for w1(z) is that
the index of the Bessel functions is imaginary. The choice
of 12 = in2 also results only in a change in the index of
the modified Bessel functions of wl(z), in fact a change
for the worse since the index becomes nL/an or inL/an.

None of these choices of 12

provides a complete solu-
tion of u that is consistent with the radial boundary condi—
tion at r = a, nor does the assumption of a z-dependent
boundary condition save the solution. The conclusion is

that the linearized Poisson-Boltzmann equation unfortunately
does not fall within the range of partial differential equa-
tions for which the separation—of—variablas method is useful.

Instead, we turn our attention in the next chapter to a

different kind of proposed solution.

CHAPTER 3

A NEW VERSION OF AN OLD SOLUTION

A. Introduction

 

In this chapter we consider an additive solution ap-
parently originated by Osterle and his colleagues [A6,55,
116]:

w(z,r) = (21(2) + w2(z,r) . (3.1)

This solution has also been used by others as well [78,
88, 1A0,1AA], but the other studies besides our own have
always involved numerical, rather than analytical results.
Lee [88], who did obtain an analytical solution, wrote
the right-hand side of Equation (2.1) as sz, where K im-

plicitly contains the z-dependence of the ionic strength:

2
2 _ 2F I(z)
K - eRT ’ (3'2)

and so his final expressions are dependent on z, but not
explicitly. Moreover, even though he made the above sepa-

ration, he ultimately assumed a linear dependence in z for

22

 

Us

ll!

Vt

2v-
".

23

*1, rather than determining the z-dependence from the re-

sulting equation.

B. Analytical Solutions for wl(z) and w2(z,r)

For the moment we consider the more general Equation

(2.1), which, with the proposed solution (3.1), becomes

2 2
d“’1 3"’2 13 3‘1’2 22
dzz '* 322 'F'F'FF r 3?: = K0f (Z)(Wl+w2) (3-3)

 

 

We now assume that it is legitimate to separate this into
two independent equations; i.e., that there is no mixing of

the two different functions in the above equations:

 

 

2
d "1 2 2
= K f (z)w (3.“)
dzz 0 1
and
32% + 1 3— —8w2 - 2r2< ) (3 5)
322 F3rr3r -Ko th2 . .

In his solution of Equation (3.5), Lee [88] first as-
sumed and later verified that the z-derivative is small
compared to the r-derivative in the case that a/L << 1.

The values of pore radius and pore length of "real" systems

2A

8

are of the order of 1 x 10- m and l x 10-“ m, respectively,
and so this approximation seems like a good one. The omis-
sion of the term a2u2/322 makes the solution of Equation
(3.5) considerably easier. It is in fact the modified

Bessel equation of order zero and may be solved for any

f(z):
w2(z,r) = klIO[KOrf(z)] + k2KO[KOrf(z)] . (3.6)

The radial condition (2.21) demands that k2 be zero
since K0(0) is undefined, and the boundary condition (2.22)

results in the solution

oIOEKOrf(Z)1
w2(z,r) = - (3.7)
6K0f(Z)I1[KOaf(Z)]

 

This result is well known for the special case of constant
ionic strength and is the same as that obtained by Lee
[88], except that his solution's axial dependence is hidden
in the z-dependent parameter K.

Lee did not consider Equation (3.A), but simply assumed
that $1 varied linearly with z. This is obviously not
true since dzipl/dz2 # 0. However, Equation (3.A) cannot
be solved for a general f(z) as can Equation (3.5), and so
we consider the three possible cases of f(z) mentioned

earlier.

n.

.. a
$0

25

For f(z) = 1, the solution is trivial:

¢l(z) = klcoshkoz + k2sinhkoz . (3.8)

For the moment we write the boundary conditions on W1
at z = 0 and z = L only formally as w1(0) and wl(L). This

results in the solution:

L

(3.9)

¢l(Z) = [¢l(0)sinhKO(L-Z) + wl(L)sinhKOz]cschKO

With f2(z) = [1 + (32—1) (z/L)], the solution of Equa-

tion (3.”) is no longer trivial. The change of variable

1/2

c = [l + (82-1)(z/L)] (3.10)
gives rise to the equation
2 dzwi dwl 2 6
C “—37 - C?fi?" 96 C W1 = 0 , (3.11)
dz
where
2KOL
a 3'——75——' (3.12)
3(S -1)

Even though this equation may not be immediately

familiar, it is a special case of a general differential

26

equation discussed by Hildebrand [66]:

2
x2 Q_% + x(a + 2bxq) %% + [c + dx28 - b(1-a-—q)xq
dx

+ b2x2q]y = o , (3.13)

where a, b, q, o, d, s are all constants. The solution of

this differential equation is

q 1/2
y = x(l-a)/2e-bx /qu [d:;_ x2] , (3.1“)
where
2
p E Q;- [($351) - e]l/2 . (3.15)

and
Y (3.16)
for dl/Z/s real, Or

Z=kI+kK (3.17)

1/2

for d /s imaginary. These solutions are required only

 

{D

hi

hi

27

for p either zero or an integer, and if p is not zero or

an integer, then the two solutions may be written

N
ll

k J + k

p 1 p 23- (3.18)

p

and

N
ll

k I + k

p 1 p 21_p (3.19)

respectively.
To specialize Equation (3.13) to our Equation (3.11),

we note that a = -1, b = o, c = o, d = -3a2, s = 3, q = o,

2)1/2

and therefore, since (-a is imaginary,

wl(c) = k1C11/3(ac3) + k2§I_l/3(ac3) , (3.20)
or
w (c) = k CI (o 3) + k K ( 3) (3 21)
1 1 1/3 C 25 1/3 “C ’ °

which are equivalent because of the definition

K = _ % .2____:R (3.22)
sin pa

for p not zero or an integer [66].

Applying the same formal boundary conditions as before

28

to Equation (3.20), we obtain

w1<1)c[I_1/3<as3)11/3<at3>-Il/3<as3)r_l/3<ac3>1

 

_ wl(s>c[I_1/3(a)Il/3(ot3)-I1/3(o)I_l/3(oc3)1

 

3 3 a (3.23)

where g and a are defined in terms of the original variable
and constants by Equations (3.10) and (3.12), respectively.
For the third choice of f2(z) = SZZ/L, the transforma-

tion of Equation (3.A) with the change of variable

a = TSz/L , (3.2u)
where
_ KOL
T = TBS . (3.25)

results in the equation

2
d w dw
2 l l 2
E '-- + E-- - E w = 0 . (3.26)

29

Not surprisingly, this is again a special case of the dif-
ferential equation (3.13), this time simply the modified

Bessel equation of order zero, and thus the solution is

¢1(€) = klIO<g) + k2K0(E) . (3.27)

The solution (3.18) is not an option here since p is zero,
and in (3.27) KO must be retained, since a is never zero
for 0 5 z 5 L.

Application of the same formal boundary conditions

results in the solution

w1(0)[K0(TS)IO(tSZ/L)-IO(IS)KO(tSZ/L)]

 

wl(z) =
IO(T)KO(TS)-IO(TS)KO(T)

¢1(L)[K0(T)IO(TSZ/L);IO(T)K0(tSz/L)1 (3 28)
IO(T)KO(TS)-IO(TS)KO(T) ° °

 

Thus beginning with the linearized Poisson-Boltzmann
equation, we have obtained a complete analytical solution
for different model distributions of ionic strength with
only three additional assumptions:

(1) That the form of the solution is given by Equa-

tion (3.1);

(2) that Equation (3.3) can be separated into Equa-
tions (3.“) and (3.5); and

30

(3) that in Equation (3.5) the z-derivative is small
relative to the r-derivative.

C. An Attempt to Obtain Explicit Boundary Conditions

yl(o) and wl(L)

The major drawback of the solutions for Al in Equations
(3.9), (3.23), and (3.28) is the boundary conditions,
w1(0) and w1(L), which are stated only formally and have
no direct connection with the physical system that we are
trying to model. In order to try to make this connection,
we need to apply boundary conditions that arise from the
electrolyte solutions on either side of the membrane.
First, however, we need to look at the derivation of the
linearized Poisson-Boltzmann equation in more detail, par-
ticularly with regard to the definitions of the inverse
Debye length K and the z-dependent ionic strength I(z).

We begin with the equation describing chemical equilib-
rium for each component i in the radial direction inside

the capillary,

(3.29)

where V1 is the partial molar volume, fi the activity co-
efficient, and x1 the mole fraction, and we assume that
(l) the pressure term is negligible [30,71,88], (2) fi

is constant, and thus aznfi/ar = O, and (3) the total molar

av

V

'I'

e

GU

A)‘ d
{\

9

VA.

#1

41.4

31

volume V is constant so that xi = ci. Ideality is a con-
venient assumption for simplifying the mathematics and
certainly not a bad one for dilute solutions. Equation

(3.29) can then be written as

aznc1(2.r) = 21F 3w(z,r) (3 30)
3:! RT 31" , .

 

the solution of which is

ziF
inci(Z,r) = ‘ fi I!)(Z,I‘) + A(Z) o (3031)

The usual derivation of the linearized Boltzmann equation
includes the assumptions that at infinity electroneutrality
holds and the electric potential diminishes to zero. Al-
though these conditions are certainly met in the bulk solu-
tions outside the capillary, infinity is an unavailable
luxury inside the capillary where the radial coordinate
varies from 0 to a. Even though in practical terms "in-
finity" may be only the distance of a few atomic diameters,
the conditions of electroneutrality and zero electric po-
tential (relative to the charged wall) do not occur any-
where inside the capillary, not even at r = 0 where p and
w(z,r) may be at a minimum, but they presumably are still
not identically zero. Nevertheless, for the time being,

we assume that 31(2) represents the concentration of ion

1 that would occur at 2 (along r = 0) if w(z,0) were zero

32

at that point, and thus these concentrations, even though
they are fictitious, have the advantage of obeying the
condition of electroneutrality.

This identifies the integration constant A(z) as in
51(z), and Equation (3.31) becomes

ZiF _
inci(z,r) = - R w(Z.r) + znci(z) , (3.32)

or

 

ci(z,r) = 31(z)e-ZiFw(z’r)/RT (3.33)
The linearized form,
_ ziFw(z,r)
c1(z,r) = c1(z)[l - RT ] , (3.3“)

is accurate to within 5% of the nonlinear expression for
w(z,r) < 0.287 V at 300 K. The net volume charge density,
defined in Equation (1.4), may now be written as

2
Fin(z)zi - Ug—g-El 231(2):); , (3.35)

'0
ll

01"

2—
p = - ELK—1TH w(z,r) . (3.36)

33

since the first sum in Equation (3.35) is zero because of
electroneutrality, and the ionic strength at zero potential,

I(z), is defined as
I(z) = §£ci(z)zi . (3.37)
By substituting Equation (3.36) into Equation (1.3),

we can write the linearized Poisson-Boltzmann equation in

cylindrical coordinates as

2....
V2¢(Z.I‘) = W ¢(Z,I’) 3 (3.38)
or
v2w<z.r) = 22w<z.r> . (3.39)

where E, the inverse Debye length at zero potential, is

defined by

2—
—2 _ 2F I(z)
K - __RT__. . (3.“0)

If we now assume w to be of the form given in Equation (3.1)

and separate the resulting equation as before, we obtain

 

A:

«2.

3A

 

2
d W1(Z) _2
""“"= K w (Z) (3.41)
dz2 1
and
32w2(z,r) l a 3w2(z,r) _2
322 + F a? r "—35-— = K w2(z,r) . (3.112)

These are identical to Equations (3.A) and (3.5) except
that E and the (hypothetical) reference concentration 31(2)
are more clearly defined.
We now return to Equation (3.33) and write it for
r = 0:
-ziFw(z,O)/RT (3.A3)

ci(z,0) = 31(z)e ,

for which we introduce the less cumbersome notation c:(z)

and w°(z) for the quantities ci(z,r) and w(z,r) at r = 0:
c;(z) = 61(z)e‘ziFW°<Z)/RT . (3.uu)
Dividing Equation (3.33) by Equation (3.4“) yields
ziF
ci(z,r) = 03(z)exp{- -§-T—[w(z,r)-w°(z)]} . (3.115)

which has the advantage of involving the actual concentration

(f

(3

(I)

(7.)

' b

or

35

of ion 1 anywhere along the midline of the capillary, but
the apparent disadvantage that 03(2) does not obey the
condition of electroneutrality. In fact, this expression
may also be derived directly from Equation (3.29) with the
same assumptions as before except that the integration
constant is identified as

ziF
A(z) = ifiF'w°(z) + £nci(z) . (3.“6)

The potential difference w(z,r) - w°(z) is equivalent

-.—_-~ ——--— —o

. Q

.to the difference w2(z,r) - w§(z) since ¢l(z), being in-

 

dependent of r, simply cancels out in the subtraction.

The linearized form of Equation (3.“5) is therefore

z F
01(z,r) = 03(Z)(1 - {LT [w2(z,r) - w§(2)1} , (3.147)

from which the net volume charge density is

2
o = rzc;(z)zi - thw2<z,r)-wg<r)12cg<z)z§ , (3.u8)
01"
- ° 2F21°(Z’ E °( )1 ( A9)
0 - F£ci(Z)Zi - __—RT__- ¢2(Z.P)-w2 Z . 3.

where I°(z), defined in terms of 03(2), is the ionic strength

along the capillary midline. With the corresponding definition

.s ‘
«\w

36

of K°, the net volume charge density yields the following

linearized Poisson—Boltzmann equation:

 

 

v2u(z.r) = - g 2cg<z)zi + (K°)2[w2(z,r)-ug(z)1
(3.50)
We again separate this into two equations:
2
d w (.2)
——gl§—— = - £2c3<z>zi - (K°)2ug(z) (3.51)
Z
and
32¢2<z,r) 1 , aw2<z.r) 2
__ = o
322 + r 3r r 3r (K ) u2(z,r) . (3.52)

Comparing these with Equations (3.41) and (3.42), we see
that

Ezwl(z) = - g 2cg(z)zi - (K°)2wg(z) (3.53)
and that E = KO.

It is not immediately obvious that E and K° should be
equal to each other, i.e., that the ionic strength should
be the same in both the presence and absence of an elec-
tric field. Nevertheless, it seems reasonable if we con-

sider ionic strength as an average over the individual

 

 

CE

.«L

.h.»

2.

«\U

37

ions. In the presence of a field, the concentration of
cations shifts away from its zero-field value, but, within
the limits of the linearization of the Boltzmann equation,
the concentration of the anion (for symmetric electrolytes)
simultaneously shifts an equal amount in the opposite direc-
tion so that their average represented in the ionic strength
remains the same as in the absence of a field. This same
argument will reoccur later in a more quantitative manner.
Even if we accept the identity of I(z) and I°(z) (and
thus of E and K°), we must still find an expression for the
sum 2c;(z)zi although w§(z) is already known from the solu-
tion for w2(z,r) at r = 0. Unfortunately, the form of
wl(z) in Equation (3.53) does not satisfy Equation (3.4),
but it must still yield the correct value for a particular
z, and therefore we hope to obtain the boundary conditions
w1(0) and ¢1(L) from it. However, this is still not an
easy task, and we must now consider the bulk solutions at

either end of the capillary.

D. The Bulk Solutions As Boundary Conditions

Since each bulk solution is in equilibrium in the
z-direction, the appropriate expression, with the same as-

sumptions as those leading up to Equation (3.30), is

dznc (z) 2 F
i .. 1 (11(2)
dz " " RT dz ’ (3'5”)

bl.

+9

ha.

3

38

where ci(z) is the z-dependent concentration and ¢(z) the
electric potential in the interphase between the face of
the. membrane and the bulk solution that bathes it, i.e., in
the "region in which there is a continuous transition from
the prOperties of one phase to the properties of the other"
[15]. Not surprisingly, the solution to this equation is

. zip
2n ci(z) = - TEF'¢(Z) + A . (3.55)

Although it is usually legitimate to assume that the
potential is zero at infinity, we will not do that here
because we wish to sandwich together the two mathematical
solutions for electrolyte solutions A and B, and the as-
sumption of ¢ = 0 for each would simply result in A¢ = 0,
hardly an interesting result. Instead, we will assume that

A

for solution A, ¢A(z) = ¢A when ci(z) = ci, i.e., the value

of the potential is the "bulk" potential ¢A when the value

of the concentration of ion i is the concentration c9. In

both cases, the lack of the parenthetical z with each sym-

bol indicates a constant bulk value. And analogously for

B
i.
It seems valid to consider these bulk potentials ¢A

solution B, ¢B(z) = ¢B when cE(z) = c

and ¢B as those obtained from measuring these solutions
individually against a reference electrode, such as that
of the standard hydrogen or saturated calomel electrode.
Presumably either ¢A or ¢B, or at least their difference,

should also be able to include an external potential

39

difference applied across the membrane. This would be
important in certain membrane-transport studies, but we will
not consider any applied potential differences in our study
of the Poisson-Boltzmann equation per se.

Thus for solutions A and B, Equation (3.55) becomes

2 F
62(2) = afiexp{- fir [¢A(z)-¢A1} (3.56)
and
z F
02(z) = c?exp{- 3%F-[¢B(Z)-¢B1}’ , (3.57)

or in their linearized form

2 F
09(2) = ci‘{l — 1%? [¢A(Z)-¢A]} (3-58)
and
2 F
cli(z) = c§{1 — fif— [¢B(z)-¢B]} . (3.59)

The net volume charge density can now be written for

each side of the membrane:

2
9A = cmfizi - i; [¢A(z)—¢A]zc§zi (3.60)

and

no

R)

B 2
DB = cmizi ‘ gT E¢B(Z)-¢B]2C?zi : (3.61)
or
2F2IA
DA = - _?fir—[¢A(Z)_¢A] (3.62)
and
2F2IB
DB = " T E¢B(Z)-¢B] 3 (3063)
since both C? and c? obey the condition of electroneutrality.

The linearized one-dimensional Poisson-Boltzmann equation

can now be written for each interphase:

d2¢A(z) 2F21

_ A

“'57— " “Ent— ”Mm-4’11] (3'6”)
and

d2¢B(z) 2F2IB

-_3;§_— = ‘ERT— [¢B(z)-¢B] , (3.65)
or

d2¢A(Z) 2

7—2—— = KAE¢A(z)-¢A] (3°66)

Z

and

2.

dc

an

41

d2¢B(z) 2
“—E;§—— = KBE¢B(z)-¢B] : (3°67)

where KA and KB are defined as usual.

With the substitutions

C
l

A ‘ ¢A<Z) - ¢A (3.68)

and

C
l

B - ¢B(Z) ' ¢B 3 (3-69)

the solutions to these differential equations can be written

down immediately as

¢A(z) - AleKAZ + A2e-KAZ + ¢A (3.70)

and

¢B(z) BleKBZ + Bze'KBZ + 63 , (3.71)
but since ¢A(z) must remain finite at minus infinity and
¢B(z) at plus infinity, A2 and B1 must be zero. The other

boundary conditions at the membrane itself, i.e.,

d¢ (0) 0
__£___ a 13 (3.72)

dz

FL;

We.

h.

.Qw.

a.

42

and

d¢B(L) OB

Th? , (3.73)
deserve a short digression on the physical meaning -- with-

in the assumed model -- of 0A and GB, the net surface charge
densities on the respective faces of the membrane.

As we will subsequently use them, 0A and GB are a quan-
titative measure of the net charge per unit area on the
"solid" part of the membrane, i.e., the total area of each
face less the area of the pore openings. Because the latter
area is a small fraction of the total surface area (e.g.,
less than 4% in some of the commercially available Nucleo-
pore polycarbonate membranes [119]), the charge on the
"solid" part of the face should be the major contributor to
the electric potential and concentration profiles (relative
to the bulk) in the interphase. Whatever deviations in
potential and concentrations that arise at a pore opening,
relative to those at the "solid" surface of the membrane,
should be small and more or less washed out by the stronger
effects arising from the charge on the rest of the membrane.
A better formulation of 0A (or OB) ought to include the
relative contributions from both the "solid" surface and

the pore openings, i.e., something of the form

0 = Asos + A o , (3.7“)

43

where As and AD are the fractional areas of the "solid"
surface and pore openings respectively, and as and UP
the corresponding charge densities. Presumably, o repre-

p

sents an average charge density at the pore opening since
ci(0,r) (and thus the charge density) varies with r. This
seems a relatively minor consideration and need not impede
us further.

As originally defined, the surface charge densities
0A and GB are equal to each other if the two membrane
faces are identical, although this cannot strictly be
true even then, in light of the discussion of the previous
paragraph since ci(0,r) and ci(L,r) would give rise to dif-
ferent contributions to up at the two faces. However, the
important point is that the two faces need not be identi-
cal, and this is the real reason for maintaining this
distinction between GA and 0B. For example, cellulose
acetate membranes, which are important in reverse osmosis
and hemodialysis, have structurally different faces with
different charge densities.

The boundary conditions presented in Equations (3.72)

and (3.73) lead to the complete solutions
¢A<Z> " ‘— e + 45A (3.75)
and

¢B<z) = -—— e + ¢B . (3.76)

44

The two reciprocal Debye lengths KA and KB are related to

each other through the parameter S, and their definitions,

along with that of S, show that

AB = KAs . (3.77)

Therefore, Equation (3.74) may be rewritten as

 

OB eKAS(L-z)

¢B<z) = + )3 . (3.78)

EKAS

With Equations (3.75) and (3.78), Equations (3.56) and (3.57)

 

become
A A 21,: CA KAz
ci(z) = ciexp - WEE—A- e + ¢A] (3.79)
and
2 F 0' K S(L-z)
c§(z) = cfiexp {_ RéT—[ers e A + ¢B]} (3.80)

The "outside" ion concentrations at the ends of the
capillary, cfi(0) and c§(L), are readily written down, and
a simple mass balance indicates that they should be equal
to the "inside" ion concentrations given by Equation (3.45)

averaged over the radius. Thus

(01(O,r)>r = c§(0) (3.81)

45

and
<ci(L,r)>r = 02(L) , (3-82)

where the symbol < >r indicates the appropriate average

‘ over r, i.e.,

jrarg(r)dr

(goo), = ° . (3.83)

a
}( rdr
O

The average of Equation (3.45) is

 

z F
(c1(z,r)>r = <c‘i(z)exp{- fi-iT-T—[ip2(z,r)-¢§(z)]}>r
(3.84)

Although the relationship
(e-B>r = e-<B>P , (3.85)

where B is some function of r, is not true in general, the
result of applying this relationship is correct as long as
we linearize the final exponentials.

Thus, Equation (3.84) may be rewritten as the simpler

2 F
(ci(z,r)>r = c:(z)exp{- -R=’LT-[<w2(z,r)>r-w‘2’(z)]} .
(3.86)

where, from Equation (3.7),

46

 

 

a
(W2(Z,P)>r = 2 20 -/~ rIOEKOrf(z)]dr
EKOa f(Z)Il[K0af(z)] °
(3.87)
and
w‘2’(z) = 0' . (3.88)
EKOf(Z)Il[KOaf(Z)]

The integral in Equation (3.87) is readily performed [88]

with the result

 

20
w (z,r) = , (3.89)
< 2 >r EKgaf2(Z)

and Equation (3.86) becomes

 

 

- o ZiFo 2 l
(ci(z,r)>r — ciexp — .-——————— -
EKOf(Z)RT Koaf(z) I1[Koaf(z)]

(3.90)

We can now write the matching conditions in Equations
(3.81) and (3.82) explicitly. Equating Equations (3.79)
and (3.90) for z = 0 and Equations (3.80) and (3.90) for

z = L, we obtain

Z F .
ci(0)exp - i o [ 2 - l ]
6K0f(0)RT Koaf(0) IlEKOaf(O)1

 

 

Z F o
= c‘i‘exp {- 4.1.— [3.2% + 8,4} (3.91)

47

 

 

2 F0
ci(L)exp - i [-——i;——— - 1 ] ‘
eKOf(L)RT Koaf(L) Il[K0af(L)]

2 F o
_ B i B
’ ciexp (' ‘RT‘[—EKAS + (Pan - (3'92)

Thus the required linearized expressions for ci(0) and

 

 

 

03(L) are
F0 a 6K ¢ 2K
_ A 2i A A A A
03(0) " Ci " —€KART[-O'_ + O - T—K af2(0)
0
K
+ A (3.93)
KOf(O)I1[KOaf(O)]
and
2 F0 0 8K ¢ 2K
awe? —— -———-—————
EKA 0 ° ' Koaf (L)
K
" A a (309“)

which relate the concentrations ci(0) and ci(L), needed

for the boundary conditions wl(0) and wl(L) from Equation
(3.53), to the corresponding bulk concentrations c? and

B

Ci.

48

However, these expressions also each contain both

KO, defined in terms of I(O), and K defined in terms of

A,
IA“ If we write out the definition for I(O) with Equation

(3.93) we have

 

 

-lo 2__l_A2
I(O) - 2201(0)zi - 2Xcizi
_ Fa 35+EKA¢A 2KA
o 2 2
2€KART O Koaf (O)
K
+ A zcfizi , (3.95)
K0f(0)Il[KOaf(0)]
or
__1_ A2
I(O) - 2 Zcizi (3.96)

for all symmetric electrolytes since chzi = 0. 1(0) is

identically equal to IA within the linearization scheme of
the Boltzmann expression. Thus we also have that K0 and
KA are identically equal to each other, and we can write
Equations (3.93) and (3.94) all in terms of KA, which is

preferable to KO since I is an experimentally determined

A
quantity.

49

 

 

 

 

 

 

Z F0 0 SK ¢
c:(0) = c? l - 1 3% + g A - ____§___
8KART KAaf (0)
+ 1 (3.97)
f(0)Il[KAaf(0)]
and
B 21F0 013 SKA¢B 2
0°(L) = c l - —— + —
i 1 eKART [3" ° KAaf§(L)
+ 1 . (3.98)
f(L)Il[KAaf(L)]

From Equation (3.53) we can now write the boundary condi-

 

 

tions:
w (0) = - F ZcAz + F20 :5 + 35532
1 i i 2 3 o 0
8KA e KART
- ___22___ . 1 zcgzg
KAaf (0) f(0)Il[KAaf(0)]
- ° (3.99)

62Ar(0)11[.Aar<0)1

a

'1‘

1‘...

fi .44
Q.»

Q».
0;

50

and

 

win.) = -

_ ____§———-+ 2c
KAaf (L) f(L)Il[KAaf(L)]

 

 

- ° . (3.100)
EKAf(L)Il[KAaf(L)1

OP

0 8K ¢
6: c? l 3 A - -_—£%—"— (3'101)
A KAaf (0)

l
|
+

 

(11(0)

and

8K ¢
2 B - ———3%——— . (3.102)
KAaf (L)

 

o 0B
¢1(L) - 8K S? +
A
At last, we have obtained explicit expressions for the
boundary conditions that are stated only formally in Equa-
tions (3.9), (3.23), and (3.28). With the appropriate form
of f(z), they can be substituted into each of these three
equations for the complete solution of W1(Z) for the dif-

ferent ionic strength distribution models.

51

E. An Alternate Set of Boundary Conditions

The boundary conditions given in Equations (3.101)
and (3.102) are not an entirely happy formulation since
they contain the bulk potentials ¢A and ¢B which are not
individually accessible. Only their difference can be
determined. We now look at an alternate formulation of
boundary conditions that do not include these potentials.

Previously, we applied a mass balance in order to ob-
tain boundary conditions; this time we apply a charge
balance. In our discussion of the boundary conditions at
the membrane faces, we stated that the net surface charges
GA and GB on the "solid" part of the membrane primarily
determine the potentials ¢A(z) and ¢B(z). If we assume
that Equations (3.75) and (3.77) describe these potentials
near the pore openings as well as near the "solid" membrane
faces, then we can write the following charge balances at

the openings:

d¢A(0) 831(0) at2(o,r)
8?: e —-d—z-—-+ <T (3.103)

and

d¢B(L) dwl<L) 3w2(L,r)

__._.= ——+ <—-—— (3.104)
8 dz 5 dz az r

52

These equations correspond to that given by Levine [91]
in his treatment of the potential distribution across an
interface of two immiscible liquids. Although the physical
nature of our problem is different from his, the two are
analogous in that a charge balance is appropriate at each
interface. Since in our case w2(z,r) varies radially, it
must be averaged over the mouth of the pore.

Thus, taking the z-derivative of w2(z,r) given in

Equation (3.7), we have

3¢2(Z.r) = Of'(Z) PIIEKAPf(Z)]
az af(z) IlEKAaf(Z)]

 

aI [K af(z)]I [K rf(z)]
- 0 A 2 0 A . (3.105)
IlEKAaf(z)]

 

The average, defined by Equation (3.83), yields

<3W2(Z.r) _ 4of'(z) (3.106)
Bz r eKiaf3(z)

and therefore

dw (0) O 1
"‘EE“ 3 g "A + —%£—%Ql- (3.107)
6 GI KAaf (0)

and

53

 

Lilia: = - 2 32 _ “f'(L) (3 108)
(12 E U Kiaf3(L) .

We return first to Equation (3.8). In order to apply
these boundary conditions to this solution, we differentiate
with respect to z:

dwl(2)

= k K

“d§" 1 Asinhl<

z + k2K coshK z . (3.109)

A A A

For the case that f(z) = l, f'(z) = 0, so that the final

result is

_ 1
¢1(Z) - - ——— [oBcoshKAz + o

EKA coshKA(L-z)]cschKAL.

A
(3.110)

Differentiating Equation (3.20) with respect to c, we ob-

tain

Eiliii = 3og3tk I (or3) + k I (6:3)] (3 111)
d; 1 -2/3 2 2/3 ' °

Since c = l at z = 0 and c = S at z = L, the result of apply-
ing the boundary conditions with f(z) = [1+(s2 - 1) (z/L)]l/2
is

54

dwl(l)

d: *CEI2/3(GS: )I1/3(GC3 )- I _2/3(GS3 )I -1/3(:C3 )]

 

¢1(c)=

dul (S)

,Q;, —C[I2/3(G)Il/3(ac3 )- I _2/3(G)I- 1/3(GC3 )]

 

338 3EI2/3(QS3 )I_2/3(a)-I2/3(G)I_2/3(GS3 )]

(3.112)

To obtain the explicit result in terms of the boundary con-

ditions, we need the following relations:

dwl(l)- 2L dw1(0)

d; — 82-1 dz

 

 

and

dwl(S) _ 28L dw1(L)

dc - S2_l dz

 

Thus the complete solution is

0 2(82-1)
w (c) = —9— -5 + ——————- x
1 8"A I: ‘7 KEaL

3 3 3 3

 

C

3 3
I2/3(GS )I_2/3(a)-I2/3(a)I_2/3(GS )

(3.113)

(3.114)

55

 

3 3

 

C
3 3

In the final case we have from Equation (3.27) that

d¢1(€)
——aE—- = klIl(€) - k2Kl(§)

(3.115)

(3.116)

Similarly, E = T at z = O and g = TS at z = L, and the

z/L

result with f(z) = S is

dwl(r)
-———-*[Kl(TS)IO(E)+Il(TS)KO(€)]

 

wlm = (”L
Il(T)K1(TS)-Il(TS)Kl(T)

dw1(TS)
_ -—aE—-—[K1(T)I0(€)+Il(T)KO(€)1

 

I1(T)K1(TS)-I1(TS)K1(T)

(3.117)

Again, the relationships between the corresponding boundary

conditions are

and

and

56

 

(1‘91”) 1 dIPl(0)
T = F; T (3.118)
dwl(rs) 1 dwl(L)

dz ' = KAS dz : (3.119)

the complete solution is

z/L z/L
a GA “Ens K1(TS)IO(TS )+Il(TS)KO(TS )

2
KAaL

 

 

Il(T)Kl(TS)-Il(TS)Kl(T)

+ Kl(r)IO(TSZ/L)+I1(T)KO(TSZ/L)

 

 

0' SE _ LULnS

8K S 0' 2 2

A K aLS
A Il(T)K1(TS)-Il(TS)Kl(T)

(3.120)

Final Membrane Potential Formulae

 

Not only does the charge balance yield dwl(0)/dz and

dwl(L)/dz, Equations (3.107) and(3.108), with much less

effort than was required to obtain w1(0) and wl(L), Equa-

tions (3.99) and (3.100), from a mass balance, but the

lack of ¢A and ¢B in the results of this second formulation

also enables us to write an equation for the total membrane

57

potential EM. With our symbols in the expression given by

Morf [115], we have
EM = ¢B - ¢A = E¢A(O) - ¢AJ ‘ E¢B(L) ’ ¢BJ

+ [wlm - wlmn + Edgar», - <w2<o,r)>r] ,

(3.121)

where the first two differences in brackets represent the
total boundary potential difference, and the last two dif-
ferences in brackets represent the diffusion potential.

We can now write down the total membrane potential for
each of the three ionic strength distribution models.
Substituting Equations (3.75), (3.76), (3.89), and either
(3.110), (3.115), or (3.120) in the above, we obtain for
f(z) = l:

U - U U - C
E = -fl————9 + —£————§ (coshK L-l)cschK L , (3.122)
M A EKA A A

which is identically zero for GA = GB; for f(z) =

[1 + (82-1)(z/L)]l/2:

a o [35 , jg] _ 20 [82-1]
EKA 0 So EKATa S2

 

58

3 3 3 3
I2/3(as )[SIl/3(GS )-Il/3(a)]-I_2/3(a3 )[SI_1/3(GS )’I_l/3(a)]

 

3 3
12/3(GS )I_2/3(a)-I2/3(G)I_2/3(GS )

 

+ U SE _ 2(82-1) X
EKAS2 O KiaLS
3 3
I2/3(G)ESIl/3(as )‘Il/3(a)]-I_2/3(a)[SI_l/3(a3 )_I-l/3(a)]

3 3
I2/3(GS )I_2/3(a)-I2/3(a)I_2/3(a3 )
(3.123)

and for f(z) = sZ/L:

 

 

K1(TS)[I0(TS)‘IO(T)]+I1(TS)[KO(TS)'K0(T)1

 

Il(t)Kl(IS)-11(TS)K1(T)

° 2aLS2

+_Q_S_B_-_fln_s_ x
KA

Kl(T)[I0(TS)-Io(r)]+Il(T)[KO(TS)-KO(T)J (3.12”)

 

I1(T)K1(TS)-Il(TS)Kl(T)

59

G. Numerical Calculations

We now calculate values for wl(z), <w2(z,r)>r, and EM

with the equations that we have derived. We choose two

sets of conditions: (1) IA = 0.01 M with 32

= 10, and
(2) IA = 0.5 M with 82 = A. There are also a number of
parameters in these equations for which we must obtain tabu-
lated values or estimates. Probably the most problematic
parameter is surface charge density. Since we are ultimately
interested in models that simulate biological systems, we
choose a value of a that corresponds to the electrokinetic
charge of the red blood cell. Its value was calculated to
be in the range from 3.6 x 10'3 C/m2 to 12.3 x 10"3 C/m2
[31], and therefore we choose 8.0 x 10.3 C/m2 as a typical
value. For the purpose of these calculations, we assume
that GA and GB are also equal to this same value of a.

From an analysis of the osmotic data of Grim and Sell-
ner [SH], Lee [88] estimated the pore radius of their oxy-
hemoglobin collodion membranes to be of the order of

'8 M m.

10 m, along with a thickness of 1 x 10’ Lee also

used the value of the absolute permittivity of water at

25°C, 7 x 10‘10

C/Vm, as an approximate value for electro-
lyte solutions. We do likewise.

The inverse Debye lengths for the two different concen-
trations chosen are calculated from Equation (2.2), a from

Equation (3.12), and 1 from Equation (3.25). The values of

these and all other necessary parameters are tabulated in

60

Table 3.1. Values of Parameters Needed in Calculations.

 

 

o = 8.0 x 10'3 C/m2 e = 7 x 10'10 C/V m
a = l x 10.8 m L = l x 10'“ m

R = 8.31M J/K mole F = 96,500 C/mole

T = 298 K

IA = 0.01 M: IA = 0.50 M:

52 = 10 32 = u

KA = 3.277 x 108 m“1 KA = 2.317 x 109 m‘1
a = 2.u27 x 103 a = 5.1u9 x 10"

r = 2.8u6 x 10" r = 3.3u3 x 105

 

 

We look first at the equation derived for w1(z) in the
case of no concentration gradient. Replacing the hyper-
bolic functions of Equation (3.110) with exponentials, we
obtain the approximate form

-x (L-Z)
e A ] . (3.125)

-K 2
_ .[A
w1(z) - - EKA e +

The values calculated from this equation for the two dif-
ferent concentrations are tabulated in Table 3.2. For each

concentration, wl(z) is tabulated in two ways, both as the

6l

 

 

 

 

 

 

00.0 H0H10H x m.ml 00.0 mHI0H x mw.ml H00.0 >I0H x H
00.0 HmI0H x 00.:I 00.0 010H x 50.»: m000.0 0I0H x m
oo.o HH-OH x em.mu mm.H . , memo.ou Hooo.o muOH x H
00.0 0I0H x Hm.ml 0>.0 I mzmH.0I m0000.0 0I0H x m
0:.01 0000.0! MH.mm| 00m>.0l H0000.0 mloH x H
mm.HI 0:Hm.0| H0.mml 00:0.0I m00000.0 0HI0H x m
Hm.mu mmmm.0l 05.mm: 0500.01 H00000.0 0H10H x H
mm.:1 0000.0! Hm.:ml mmwm.01 m000000.0 HHI0H x m
mm.:l Humm.0l 0w.:m| 5000.0: H000000.0 HHI0H x H
mm.zl 0000.HI 00.:ml 0000.HI 0 0
Amos 2 so nee.\ev Amos 2 so A<.o\sv o\N less
3:. E:
,----...--.mm.m.uafl fizz.-- - -- .. s. ammo-”4H-
.COHpmppcoocoo pcmumcoo pom ANVHa mo mosz> omumHsono .m.m oHnme

62

 

 

 

 

 

 

 

 

 

m0.zl 0000.H: 00.:m: 0000.H: H :IOH N H
00.3: H550.0: 05.:m: 5000.0: 0000000.0 :I0H x 0000000.0
0m.:: 0000.0: Hm.:m: 0000.0: m000000.0 :I0H x m000000.0
H0.m: 0005.0: 05.mm: 0500.0: 000000.0 :IoH x 000000J0
mm.H: 0:H0.0: H0.0m: 00:0.0: m00000.0 :I0H x m00000.0
03.0: 0000.0: MH.mN: 00m5.0: 00000.0 :IOH x 00000.0
00.0 0:0H x Hm.0: 05.0: mz0H.0: 00000.0 ::0H x 00000.0
00.0 HHI0H x 50.0: Nm.HI 0500.0: 0000.0 :IOH x 0000.0
00.0 HmIOH N 00.0: 00.0 0:0H x 50.5: 0000.0 :IOH x 0000.0
oo.o HOH-OH x m.m- 00.0 mH-OH x am.m- aam.o :-OH x aam.o
:3 1:: 2033 $3 .2: 1.33 on is:
E: E;
z om.o u eH z Ho.o u <H
.ooscfipsoo .m.m oHooe

63

fraction of U/EKA (which has the units of volts) and as
the numerical value in V x 103 (i.e., as millivolts). Be-
cause wl(z) rapidly decreases inside the cylinder, only
the values very close to the ends are tabulated.

For the model with I(z) linear in z, the modified Bes-
sel functions of negative order in Equation (2.115) are re-
written in terms of ones of positive order from Equation
(3.22), and these are approximated by the asymptotic ex-
pressions for large argument [2]. The resulting equation

is

1
E e ‘T7E‘I7E

-o:(z;3-l) + l e-a(S3-I;3)]
S C

w1(c) = - EEZ-[
(3.126)
and the calculations based on it are tabulated in Table
3.3.
Finally for the model with in I(z) linear in z, Equa-
tion (3.120) is also simplified with the asymptotic ap-

proximations for the Bessel functions, the result being

 

z/L
w1<2) = ° [ 1 ”7(8 ‘1)

— e
EKA Sz/2L

+ (3.127)

 

1 -r(s-sZ/L)]
S1/252/2L

 

6i:

 

 

 

 

 

oo.o HOHIOH x o.m: 5smwmzaoo.fi oo.o mauoa x mz.m: ommmw::oo.a Hoo.o
oo.o Hm:OH x Hm.:: masmzsooo.a oo.o mnoa x om.5: :5:5:NNOO.H mooo.o
00.0 HHIOH x No.0: mmmm:aooo.a Hm.H : 55mo.o: mommzaooo.a Hooo.o
oo.o 0:0H x mm.m: 5mm=5oooo.a 55.0 : m:ma.o- msmzmmooo.a moooo.o
m:.o: mmmo.o: ooomaoooo.fl ma.mm: mom~.o: mmm::oooo.a Hoooo.o
mm.H- mmam.o: oompooooo.a Hm.mm- 00:0.0: oommmoooo.H mooooo.o
Hm.m: mmm5.o: oomaooooo.fi @5.mm- w5mm.o: oomaooooo.H Hooooo.o
mm.:: oomm.o: omsoooooo.d Hm.:m: 5mwm.o- ommmooooo.a moooooo.o
mm.z- H55m.o: omaoooooo.a 05.:m- 50mm.o: om:oooooo.a Hoooooo.o
mm.:: oooo.H: H mm.:mu oooo.H- H o
Amoa x >0 Aaymxov U Amoa x >0 A<gu\o0 u qu
E: E;
z om.o u aH z Ho.o n «H

 

 

.N Ga pmocfiq ANVH pom Anvaa Mo mmzam> oopwHSono .m.m mamas

65

 

 

 

 

 

 

 

 

5:..0: 0000.0: 0 00.HH: 0000.0: 00055000H.0 H
mm.m- msg:.o- mmmmmmmmm.a mm.OH- Omam.o- mflmafimmofl.m mommmmm.o
om.a- mmmm.o- mmmmammmm.H 0:.QH- moom.o- msmsammefi.m mammmmm.o
mm.H- mzflm.o- ommmmmmmm.a :m.m- Hmmm.o- ummspmmmfi.m mmmamm.o
:m.o- mmzo.o- ommmmmamm.H 0m.w- nmwfi.o- m:moammoa.m mammmm.o
00.0: 0:0H x 00.3: 000000000.H H00: 00HH.0: 00:00000H.0 00000.0
00.0 HHI0H x 005: 000000000.H 00.0 0:0H x 05.H: 00000000H.0 00000.0
00.0 H0IOH x 35.0: 000:00000.H 00.0 0:0H x 00.0: :mmmma00a.m 0000.0
00.0 H0.m:0._u x min: 000:00000.H 00.0 00:0H x 00.0”: 000000HOH.0 0000.0
00.0 000:0H x 0.0: 0000:0000.H 00.0 0::0H x 00.0: 0H0:0000H.0 000.0
“meg x >0 A<ym\ov u AMOH x >0 A<¥w\ov u a\N
ANVHe ANVHQ
z om.o u aH z Ho.o u <H
.emscfipcoo .m.m magma

66

The calculated values from this equation are presented in
Table 3.“.

There are three particularly noteworthy things about
these three sets of calculated values for ¢l(z). That
part of w(z,r) that depends only on z is essentially zero
through 99.99% of the capillary. The precipitous change
in wl(z) at each end of the cylinder is obvious in Figure
3.1, which graphically presents the data from Table 3.2
for IA = 0.01, and in Figure 3.2, which presents the ana-
logous data from Tables 3.3 and 3.u. At z/L = 0.0001, i.e.,
at a distance from the end equal to the radius of the tube,
the value of wl(z) diminishes to less than h% of that at
z/L = 0.

Secondly, the change in wl(z) near the end at z = 0
is virtually identical for all three concentration models.
Small differences begin to appear at z/L = 0.0005, but by
then the value of 01(2) is indistinguishable from zero.

In other words, tl(z) diminishes to zero before differences
in the concentration models have any effects on it.

And finally, the values for the constant-concentration
model are symmetric at the two ends of the capillary, while
those for the other two models are not. Although the con-
centration gradient does not affect wl(z) inside the tube,
it shows up in the assymmetry of the values at the two ends
(Figure 3.2).

Unlike wl(z), w2(z,r) is nonzero inside the capillary,

 

67

 

 

 

 

 

oo.o HOHIOH x «.m- oo.o mauoa x m5.m: Hoo.o
oo.o Hmuoa x m5.=u oo.o muoa x No.5: mooo.o
oo.o Hausa x mm.wu Hm.H : 55mo.o: Hooo.o
oo.o ouoa x mm.m- 05.0 : mamH.o: moooo.o
m:.o: mmm.ou ma.mm- 0005.0- Hoooo.o
mm.au mmam.on Ho.mm: mm:m.o: mooooo.o
Hm.m: mmm5.o: o5.mm- m5mm.o: Hooooo.o
00.0: 0000.0: H0.:0: 5000.0: 0000000.0
mm.z: m55m.o: 05.:m: 5omm.o: Hoooooo.o
mm.:: oooo.au mm.=m: oooo.a: o
AmOH x >0 A<ym\ov AmOH x >0 Aayu\ov 45N
E: E;
: om.o u <H z Ho.o a «H

 

 

.N CH pmocfiq ANVH :a non vaaa mo mmSHm> umpmasoawo .=.m manme

68

 

 

 

 

 

 

5:.m: ooom.on mo.HH: mmHm.o: H
00.0: 0550.0: 00.0H: 00H0.0: 0000000.0
00.H: 0000.0: 5:.0H: 0000.0: 0000000.0
00.0: 0:H0.0: 300.0: H000.0: 000000.0
30.0: 0030.0: 50.0: :00H.0: 000000.0
No.0: mIOH x 00.3: Hm.m- mmHH.o: mmmmm.o
00.0: HH:OH x mm.:: 00.0: m-OH x m5.H- mmmmm.o
00.0 HN-OH x =5.m- 00.0 0.3 x aa.m- mamm.o
oo.o HOH:OH x m.H: 00.0 mNIOH x Ho.H: mmmm.o
oo.o mom:OH x :.m: 00.0 o=-OH x mm.m: mmm.o
AHOH x >0 A<gw5ov AHOH x >0 A<¥w500 4\N
HNVHa ANVHe
z om.o u <H z Ho.o u <H
.emscHucoo .=.m mHnme

 

 

 

69

.zpwcmpum OHCOH ucmumcoo pom whoa on» no 0cm seam um Amvaa .H.0 onswfim

I.

_ .. 00m00¢ i .0000 . 0...
:_\N

 

 

 

 

 

 

 

7o_

 

 

.mHmUoE numcoppm OHCOH 2909 you whoa on» no new comm um Auvas .0.0 mpzmfim

a ammmo _ .r .0000 _ . o_-

4\~

0.0..

 

 

 

71

and its profile differs for the two models involving con-
centration gradients. We calculate ¢2(z,r) from Equation
(3.89), which provides values averaged over r.

For the constant—concentration model,

<w2(z’r)>r = 2 : (3.128)
8K a
A
there is obviously no concentration dependence in the z-
direction, and the calculated values are 17.4 x 10"3 V
, for IA = 0.01 M and 2.u7 x 10‘3 v for IA = 0.50 M.
For the models with I(z) linear in z and in I(z) linear

in z, the expressions are

20
w (2.r) = (3.129)
< 2 >P eKi[1+(S2-l)(z/L)]

 

and

<w2(z’r)>r = 2202 L (3.130)
EKAS

respectively. The values calculated from Equation (3.129)
are presented in Table 3.5; those calculated from Equation
(3.130) are presented in Table 3.6. The values for both
models are presented graphically in Figure 3.3.

Of course neither 01(2) nor w2(z,r) is experimentally
available, and so we look at expressions and values for

E the measurable potential difference. Substituting

M)

72

 

 

 

 

 

 

 

 

 

 

00H.0 00H00.0 00.0 0H.0 00H00.0 00.0H H
0HH.0 00000.0 05.0 00.0 00500.0 0H.0 00.0
00H.0 00000.0 00.0 00.0 00050.0 00.0 00.0
HOH.0 00000.0 00.0 05.0 05050.0 05.5 05.0
50H.0 00500.0 0H.0 00.0 00000.0 00.5 05.0
00H.0 00000.0 00.0 00.0 50000.0 00.0 00.0
05H.0 00000.0 00.0 50.0 0HHH.0 00.0 00.0
00H.0 00000.0 00.0 00.0 500H.0 00.0 00.0
000.0 00000.0 00.H 05.0 000H.0 05.0 00.0
000.0 00000.0 05.H 00.0 050H.0 00.0 00.0
000.0 00000.0 00.H 00.5 00H0.0 00.0 00.0
000.0 00000.0 00.H H0.HH 0H00.0 00.H 0H.0
000.0 00000.0 H 00.H0 00H0.0 H 0
0 <
AmoH x >0 A gw\o0 mu AMOH x >0 A gwxov mu 0\N
2:...st 225$
z om.o u 0H z Ho.o u 0H
.N :H pmmch ANVH pom LAAL.NV0aV mo mmsHm> cmumHSono .0.0 mHnme

73

 

 

 

 

 

 

 

 

 

00H.0 00H00.0 000.0 mH.0 00H00.0 000.0H H
00H.0 05000.0 000.0 00.0 00050.0 000.5 0.0
00H.0 00000.0 H00.0 50.0 05000.0 0Hm.0 0.0
H0H.0 00000.0 000.0 05.0 000H.0 000.0 05.0
H0H.0 H5000.0 000.0 00.0 0H0H.0 0H0.0 5.0
00H.0 50500.0 500.0 00.0 mm0H.0 H00.0 0.0
0H000 0Hm00.0 000.0 05.0 000H.0 00H.m 0.0
000.0 00000.0 H05.H 00.0 0000.0 0H0.0 0.0
H00.0 00000.0 0H0.H 50.0H 0000.0 000.H 0.0
H00.0 00H00.0 0H0.H 50.HH 0000.0 055.H 00.0
000.0 00000.0 000.H 00.0H H00m.0 000.H 0.0
H5m.0 0H050.0 00H.H H0.0H 0000.0 000.H H.0
000.0 00000.0 H 00.H0 00H0.0 H 0
000H 0 >0 0000\00 0x000 000H 0 >0 0000\00 0\000 0\0
0A1“..0000 “Aig..0000
2 00.0 n 0H z H0.0 u 0H
.0 SH LamCHa ANVH :0 pom LAAL.NV0SV mo mmsHm> UmpmHSono .0.0 mHnt

7H

.mHmcoE :pwcmmvm 00:00 0:» map you whoa mnp wCOHm 0AAp.0000v .0.0 093000

0.0 0 .

 

 

 

0 A q — fl 0 a A O
.32.: 03 H :_\N A<¥w\.ov
.—
. N
.62.: 00: 5 At 3 9v

 

0.0

 

 

75

exponentials into Equation (3.122) and retaining only
significant terms (KAL mlOS), we obtain
2(OA-OB)

EKA

 

As we remarked earlier in connection with Equation

(3.122), E is identically zero when there is no concentra-

M
tion gradient, as long as 0A = UB' However, there can be
a potential difference generated across the membrane of a
constant-concentration system if 0A # 0B. Some values of
EM are tabulated in Table 3.7 for several differences in
charge density between the two membrane faces.

Applying the asymptotic approximations for the Bessel

functions [2] to Equations (3.123) and (3.12”), we obtain

['11
ll

1 2(s2 _____1__S) 2(32-1) 1 .
M {ix—2% [1- g] K2aL [libs—5]}

A (3.132)

and

 

__ __g__ _ 2_(___s 2-1)+ ans _1_
EM - SKA{2[1-S-]- :aL [1 + 33]} , (3.133)

where the final term in parentheses in each expression is

small, so that the two equations for the membrane potential

76

Table 3.7. Membrane Potentials of Constant-Concentration

 

 

 

 

 

Systems.
IA = 0.01 M IA = 0.50 M
UA"°B EM EM
(C/m2 x 103) (v x 103) (v x 103)
0.5 n.36 0.617
1.0 8.72 1.23
2.5 21.8 3.08
5.0 “3.6 6.17

 

 

 

Table 3.8. Membrane Potentials of Systems with Concentra-
tion Gradients, 0A = CE = o.

 

 

 

 

 

IA = 0.01 M IA = 0.50 M
EM EM
Ionic Strength 3 3
Model (V x 10 ) (v x 10 )
I(z) linear 28.5 “.61

in I(z) linear 28.5 “.61

 

 

77

are essentially equal to each other. Calculations from
these expressions are presented in Table 3.8 (page 76).

In order to determine whether these are reasonable
values, we look at some equilibrium potentials calculated
for concentration cells. The EMF of such a cell without
transference is given by the equation [11a]

E = 5,?- in 23-, (3.1311)

1
where al and a2 are the ion activities on either side of
the cell partition. Replacing the activities with concen-
trations, which are equal to the ionic strengths, we find
the values for IA = 0.01 M and IA = 0.50 M are 59.1 x 10"3
V and 35.6 x 10"3 V respectively.
However, for a concentration cell with transference,

which is more similar to the system that we are consider-

ing, the EMF is given by

F _ , (3.135)

where t+ is the transference number of the cation, in the
case that the electrodes are reversible to the anion.
Even though such a cell is not reversible, this expression
is still a good approximation [87,113,15N].

For the salt KCl, for example, t+ is O.u9 over a wide

range of concentrations, and thus the values calculated

78

from Equation (3.135) are 29.0 x 10‘3 v and 17.u x 10'3 v

for I = 0.01 M and I = 0.50 M, respectively. For the

A A
lower ionic strength, this is surprisingly close to the
value in Table 3.8, and the larger difference between the
values at higher ionic strength is not unexpected since

replacing activity by concentration is not as good as for

the more dilute system.

H. Summary

In this chapter we have derived expressions for 01(2)
and w2(z,r), whose sum gives the total potential w(z,r)
inside a membrane capillary between two outside solutions.
Besides the assumptions implicit in the original linearized
Poisson-Boltzmann equation itself, the only other assump-
tions needed to solve this equation was that the second 2-
derivative of 02(z,r) is small compared to its second r-
derivative. By also solving a one-dimensional Poisson-
Boltzmann equation for each interphase region between
the face of the membrane and the bulk solution, we were
able to obtain boundary conditions on 01(2) both from a
mass balance and from a charge balance at each end of the
capillary.

Furthermore, the boundary conditions obtained from the
charge balance enabled us to derive expressions for the
membrane potential, i.e., the potential difference across

the entire cell consisting of a membrane separating two

79

electrolyte solutions of different ionic strengths. The
values calculated from these expressions are consistent
with those from equilibrium thermodynamic theory. Al-
though the agreement is not as good for the values cal-
culated for the higher ionic strength, they are still of
the same order of magnitude.

It is now possible to use the expressions that we
have derived for w(z,r) within the capillary in conjunc-
tion with the Navier-Stokes (1.2) and Nernst-Planck (1.1)
equations in order to obtain flow equations for steady-
state membrane transport. However, since that is another
dissertation project in itself, we take a different look

at the Poisson-Boltzmann equation in the next chapter.

 

CHAPTER u

SOLUTION BY LAPLACE TRANSFORM

A. Introduction

 

The Laplace transform [66,105,135] of a function f(z),

symbolized by f(p),
L{-f(z)} = ?(p) =_/;°°.e‘pzr<z)dz , (u.1)

is sometimes a powerful mathematical tool for solving par-
tial differential equations. Unfortunately, the boundary
conditions on the Poisson-Boltzmann equation that we are
considering do not lend themselves to a solution by this
method. Nevertheless, a purely mathematical treatment de—
void of any physical meaning may give us information about
the form of the solution.
Thus we consider a semi-infinite cylinder (Figure

U.1) of a radius a, for which we wish to solve the equation

3
'51-; 31" = A q; . ((4.2)

"'5“—J

2
1%..
32

Although we retain the same symbols as in previous chapters,

80

81’

.605005 Epommcwpp moM0aw0 on» now meC00mo mp0S0MC0I0Emm

O

 

 

 

 

.0.: opsm0m

82

they are intended here to be mathematical quantities with-
out necessarily any physical significance.

The term S2z/L, which is equivalent to exp[(22/L)2n 8)],
introduces additional problems to the Laplace transform

method because
L{ebzf(Z)} = f(p-b) , (u.3>

in which f(p-b) represents a linear translation of f(p)
by b units in the positive p direction. Thus we write

the problematic term as a series,

2z/L _ 2z 1 22 2
S - 1 + T7 ins + 2T (T7 AnS)

+ g? c%; £nS)3 + . . . , (u.u>
and attempt a solution with a perturbation scheme.

B. Perturbation Method

Assuming a solution of the form
w(z,r) = wo(z,r) + wl(z,r) + w2(z,r) + . . . , (N.5)

we rewrite Equation (u.2) as

83

2
0 l 2
it? (5 wo + 8 $1 + 8 02 + . . .)

l 3 3 O l 2
+ F 5? r 3; (5 $0 + 8 wl + 8 02 + . . .)

_ 2 0 122 2 1 22 2
-K‘A[(5 +5 T£DS+O '21- TRI’IS) +. . .] X
0 l 2
(8 00 + 8 01 + 8 02 + . . .) , (u.6)

where 6 is merely a bookkeeping symbol. This can be written

as separate equations:

 

 

 

a w 30
0 1 a 0_ 2
322 + F 3? 1" "‘ar ‘ l<A“’0 ’ (”'7)
2
a w 30
1 l 3 l _ 2 2z
az2 +ffir—5?"<A (”’1+‘1’0T2n5) , (11.8)
2
3 w 80
2 l 3 2 g 2 fig
322 ”'Ffir Tr "A [‘92+“’1 L “‘3
+0... 8022] ’ (“-9)

etc.

84

C. Zeroth-Order Solution

 

The Laplace transform is more easily applied to the
z-coordinate than the r-coordinate because of the forms of
their respective derivatives. Under this transformation,

Equation (N.7) becomes

 

322 0 3r 3r
= w e-sz2w dz (u 10)
o A O . .

The parameter p is simply a constant in this integration,
as is the coordinate r, and thus the derivative and integral
in the second term on the left-hand side can be inter-

changed:

 

n: ]; e—pzwodz . (14.11)

The first integral on the left-hand side can be done
by parts, or the result can simply be found in a table of

transforms [66]. With the notation that

85

L{w0(z,r)} = Eo(p,r) , (n.12)

Equation (4.11) becomes

3w (0 r) d
2— 0 ’ 1d 0 2—
p wO-pwo(0,r) ‘ az +F3‘r7r'd—r ’ KA I"0

 

(“.13)

As long as w0(0,r) and awO(O,r)/3z are both constants --

it may be necessary to average these functions over the
radius of the cylinder or define them at a particular value
of r in order to remove their r-dependence -- this equation
is now a linear second-order differential equation in only

one variable. Rearranging, we get

 

d aw (0 r)
l d O 2— _ O ’
g a; r 757 + 8 $0 - pwo(0,r) + 32 , (”.1“)
where
B2 = p2 - K: . (H.15)

If we consider the homogeneous part of this equation,

i.e., with the left-hand side set equal to zero,

dy -
d H 2 _
a; r'EET + B yH ' 0 ’ (”'16)

"SIH

86

we see that this is once again Bessel's equation of order

zero, and the homogeneous solution yH is
ya = liO(er) + k2Y0(Br) , (n.17)

where J0 and Y0 are the ordinary Bessel functions of order
zero of the first and second kind respectively. By inspec-
tion we see that a particular solution of Equation (H.1h)
is

8w (0,r)
_ .3; 0
yp - :% 00(o,r) + s2 32 , (H.18)

 

and thus the complete solution is

_ 3w (0,r)
w0(p,r) = yH + yP = :22— ‘p0(0’r) + 81? 032

 

+ liO(Br) + k2YO(Br) . (8.19)

This solution already contains the two boundary condi-

tions in z; the two in r are

l—L—a (z 0) = 0 (£1.20)

Br

and

87

352(223.) = Q; ('4
e ’ °

Br

21)

where o/e is merely some constant. Before applying these

conditions, we transform them as we did Equation (“.7):

- 0 (“
O
and
f e-pz Ml dz =f e-pz 9. dz , (u.
.0 3r 0 E
or, in the notation of Equation (“.12),
—‘1’—P-I-—d_( 0) = 0 (0
dr
and
d- a) = 5L. (u
dr ep °

We will force $0 to satisfy these conditions, and thus

both of these derivatives will be zero for all other

wi.
Differentiating Equation (“.19) with respect to r,

.22)

23)

.2“)

.25)

88

we have

d
0 -
a? " “k18J1(Br) - k2BYl(Br) : (14.26)

and it is immediately obvious from the boundary condition
in Equation (“.2“) that k2 must be zero since Y1(O) is un-
defined. The remaining unknown constant k2 can be deter-
mined from the other boundary condition given in Equation

(“.25):

 

= _ 0
k1 epBJl(Ba7 (”'27)
Equation (“.19) may now be written as
aw (0,r) J (Br)
- = 12. __9___ .. 2 0
WO(P,1") B2 ¢O(O,I’) + '8‘: 32 . 8 p—ml Ba .
(“.28)

This is the expression that we need to transform back to
the original coordinate z in order to obtain the solution
wo(z,r) The inverse transforms of the first two terms on

the right-hand side may be found in tables [“5]:

L'l{l%} = cosh KAZ (“-29)

and

89

-l l l
L {g5} = szsinh KAZ . (“.30)

The inverse of the final term on the right-hand side
is not so readily found, and we must turn to the methods

of complex integration, since by definition,

c+im

L‘1{F(p)} = 5%Iw/.j. f(p)epzdp . (“.31)
c—co

This integral can be solved by direct contour integration

[105] or indirectly as the sum of residues [66], i.e.,
drf(mepzdp = 2wiZRes(pi) , (“.32)

where f(p) has a pole or order m at each point pi and

m-l
ReS(pi) = TrrT-ITYT{EC:-rfi-_l' [(p-pi)m?(p)epz]}p=pi (“.33)

In our case then,

 

J (Br) c+im J (Br)epzdp
szlma) 7' C-ioo PB 1 1

(“.3“)

90

It is necessary to find all the singularities of the above
function and determine the residue of each. The simple

pole at p = O is obvious from inspection, and thus

 

pz
JO(Br)e }
p=0

Res(0) ={831(Ba) (“.35)

Evaluating this function at p = 0 results in imaginary
arguments for the Bessel functions and thus turns them

into modified Bessel functions:

JO(1KAr) 10(KAr)

Res(0) = iKAJl(iKAa) = - KAI1(KAa)

(“.36)

 

 

The residues at the other poles at p = :KA are not so
easy to evaluate since J1 also has a singularity at each
of these values of p. In fact, because of the oscillatory
behavior of the ordinary Bessel functions, Jl has singu-

larities at each value of p that satisfies the relation
a = 31 8. (14.37)

In order to determine the order of these poles, we

first look at the series.

W-gl—m(2r-a)+..o o (03)

91

Thus we see immediately that at p = iKA there are also
simple poles. The sum of the residues at these two points

is
Res(KA) + RGS(-KA) = Egg cosh KAZ . (“.39)
A

Still not obvious, however, is the order of each pole
corresponding to a 31,3 for s‘: 1. For this it is possible
to show (with much differentiation) that the functions-

[P i (Ji,s + Ki32)1/2(1/a)]/J1[(p2 - Ki)1/2a] are analytic
at the points p = 1 (Ji’s + K§a2)1/2(l/a) respectively;
i.e., both the function and its derivative are finite at
the particular point. Thus the poles at these points are
also simple ones. Applying Equation (“.33) to the com-
plete function in Equation (“.3“) and then l'Hospital's
rule to determine the limit, we find that the sum of the

residues for these two poles is

ReS[(J§’S + K§a2)l/2(l/a)] + ResL(J§,S + K2A32)1/2(1/a)]

2a cosh[(Ji’s+ria2)1/2(z/a)]JO(Jllsr/a)

(3%,s*Kia2)J0(31,s)

(“.“O)

 

for s 3 1.
Since the residues at p = iKA can be written as Equa-

tion (“.“0) for s = O, we combine them all together into

92

one summation. Thus the inverse transform in Equation

(“.3“) is

 

 

_1 JO(BP) IO(KAr)
L PBJ1Z835 = - KAI1(KAa)
+ 2 i coshUJi s+|<!2\_a2)1/2(z/a)] JOUI Sr/a)
a 2 2 2 ’
s=0 J1,s + "A3 J0(J1,s)

(“.“l)
The ultimate goal, the solution ¢0(z,r) is finally then
3w0(0,r) 1

32 FA- sinh KAZ

 

w0(z,r) = w0(0,r)coshKAz +

+ a 10(KAr)
EKA I1(KAa)

 

co
‘2KAa Z 2 2

cosh[(Ji,s+xia2)l/2(z/a)] JO(J1,Sr/a)}
2 O
s=0 Jl,s + KAa J0(Jl,s)

(“.“2)
This is the complete solution of Equation (“.2) for the
special case that the right-hand side is equal to Kiw,
i.e., without the z-dependent term 82Z/L. Since we did not
assume that the z-derivative is small relative to the r-

derivative, this is the exact solution for the equation

with no additional assumptions.

93

D. First-Order Solution

We apply the Laplace transform to Equation (“.8) and

obtain

 

a) 2 co
3 w aw
-pz 1 —pz 1 a 1
I e 322 dz +1: e r 8r r 3r dz

2 ° -pz 2K§£nS °° -pz
KA o e wldz + _—TT__ 0 e zwodz . (“.“3)

The only additional transform that we need here is [66]

L-1{zf(z)}=- Qggfil , <"-““>

and thus Equation (“.“3) becomes

 

_ 301(0,r) ]_ d7
p2"’1 ‘ pl”1(0’1') ‘ T + F 8% r "a'fl'
2_ 2K§2ns 880
= KAwl ‘ L dp (£1.05)

From Equation (“.28), we find that

 

dEO p2+nfi 2 300(o,r)
7&7“ -‘—;3r‘¢o(°’r) ‘ g%' 32

 

 

-——————— + +
82J1(8a) 628J1(0r) 02J§(Ba)

le

[rJl(Br) JO(Br) aJ0(8a)JO(Br)]
(“.“6)

9“

We assume that wo has satisfied all the boundary condi-
tions so that w1(0,r) = 3wl(0,r)/az = O, and therefore,
Equation (“.“5) is

 

 

 

 

 

— 2 2 2
11. __.2- (o )
r dr r I‘ l - I. B“ 11’0 ,r
+ 3%3w0(0’r) - g rJ1(8r)
B 32 e B2J1(Ba)
- g JO(Br) - g aJO(Ba)JO(Br) (“ “7)
‘5 p28J1<sa) e §J§<ea)

Again, the homogeneous solution simply consists of Bessel
functions and is identical to the homogeneous solution for
$0 in Equation (“.17). It is not so simple as before, how-
ever, to find the particular solution, and we resort to a
method for reducing the order of an inhomogeneous linear
differential equation if one homogeneous solution is known
[66].

Writing the solution in the form
W1 3 myH (“.“8)

where w is also a function only of r, and substituting

this into Equation (“.“7), we have

wh

q,

95

 

.d. 9.0 1 A “(H . .1. s10
dr dr yH dr r dr
dzyn * 1 dyH 2 w h
+ + — ——— + B y —— = —— (“.“9)
dr2 r dr H yH yH ’

where h represents the entire right-hand side of Equation
(“.“7). We see that the expression in parentheses is
identically zero since it is Just the homogeneous equation,

and thus we have

dw+[2 dyH l]dm=h

d
21?"; gar—”Farr? y; : (“-50)

which is first order in dw/dr with an integrating factor,

 

q,
_ l . _
q - exp[/% dr] - r . (“.51)
Equation (“.50) then becomes
k
22 = 1 Jffhy dr + -01- , (“.52)
dr 2 H 2
ryH ryH
or
2K22nS p2+K2
g-“l=-——A —-;4—Aw(0r)
3
dr Lryfi B O

96

8w (0,r)
+ 2% 032 J./;[liO(Br)+k2YO(8r)]dr

 

B

 

- ° rJ (8r)[k J (Br)+k Y (Br)]dr
€82J1(83)Jf 1 1 0 2 0

1 l aJ0(Ba)

k
+ k2Y0(Br)] +-—3% . (“.53)
ryH

We look at each of these integrals in turn. The first

is simply

./r[k1JO(Br) + k2YO(Br)Jdr
= % rElil(Br) + k2Y1(8r)] , (u 5“)

and even though the other two are not trivial, they are
still relatively simple and may be performed with known

relations among the Bessel functions [165]:

.lr2Jl(Br)[liO(8r) + k2Y0(Br)]dr

k k
= —— r2J§<8r) + 1:3 P3EJ1(BP)YO(BP)

k
- JO(Br)Yl(Br)] + 5% r2Jl(Br)Yl(Br) , (“.55)

am

The

and

1.0

~12:

+

97

(8r)[liO(Br) + k2YO(Br)Jdr

r2 [Jg(8r) + Ji(Br)]

K‘

7% r2 [Jo(3r)Yo(Br) + J1(BP)Y1(BP)] . (“.56)

The expression for dw/dr is thus

0..

 

 

 

2 380(0,r) l
B 32 8 r [lil(Br) + k2Yl(Br)]
k k
0 1 1 2 2 2 3
— -— r J (8r) + r [J (8r)Y (Br)
8 B2J1(Ba){ e 1 1f 1 1

k
JO(Br)Y1(Br)] + 5%r2J1(8r)Y1(Br)}

o 1 1 aJo(Ba) k1 2 2 2
2: m [7 + 0311857 2 r ”0‘8” " Jim”

k2 2 3
.2— r [JO(8r)YO(Br) + J1(er)yl<sr)1 + —

98

To obtain m, we simply integrate again,

 

2 2 2
2Ka2nS [p +KA

(L) = —_‘D (Car)
L 35 0

 

 

s ' 32 [li0(sr) + k2Y0(Br)]2

0'

e :2 1 dr
Jl(8a)

 

1.l‘i'JlHSr)[lil(8r)+k211(8r)J
{83' [k1J0(Br)+k2YO(Br)]2

 

”2 .
+ 2% jfr [J1(Br)YO(8r)—JO(Br)Y1(Br)J dr}

[liO(Br)+k2YO(Br)]2

 

o l 1 aJ0(Ba) rJO(Br)dr
' E 2§bitea5 E5 + 031(0a) liO(Br)+k2YO(Br)

+ rJ1(8r)[k1J1(Br)+k2Yl(Br)] }
2 dr
[li0(Br)+k2YO(Br)J

 

 

+ k3] dr 2 + kn . (14-58)
r[li0(Br)+k2Y0(Br)]

As formidable as these integrals appear, they may be per-
formed individually or in pairs with relative ease. Again,
‘we go through these six integrals, designated II to I6.

The first may be solved by a simple substitution since

99

dyH = -3[k1J1(8r) + k2Yl(Br)]dr; . (“.59)
thus
dy
1 H l

The sixth integral may be performed by combining formulae
6.539.l and 6.539.2 from Gradshteyn and Ryzhik [53]:

fl k2J0(Br)+le0(Br)

(“.61)
2(kg-ki)

I6 = -
yH

The second and third integrals and the fourth and fifth
integrals may each be solved in pairs. The fourth and
fifth are simpler, and their solution is also required
for the other pair. It is important to note that I2 and
I5 are the same integral, but we continue to distinguish

them for the time being for convenience. We integrate 15

 

by parts:
k J (8r)+k Y (Br)
dv = 1 l 22 1 dr (“.62)
yH
and

u = rJl(Br) . (“-63)

100

From the result of I1, we know that

_ .1.
v - ByH , (“.6“)
and by direct differentiation, we have
du = BrJO(8r)dr . (“-55)
Thus
rJl(Br) rJO(BP)
I5 = ——————— - ————-——— dr , (“.66)

but the integral on the right-hand side is simply I“’ so
that

rJ1(8r)
In + I = ——————— . (“.67)

5 ByH
The integration of 12 and I3 is a but more complicated.

We first rewrite I with the relationship [66]

3
- 2 .
J1(BP)YO(BP) - JO(Br)Yl(Br) - 58? . (“.68)
k2 I 3 3:; rdr
7r 3 2w8 2
yH

2
- J. r JO(Br) ]_ rJO(Br)

101

We see that the integral on the right-hand side is I”

again, and thus

2
k r J (Br)
1 2 l O
‘2"B’Iu‘TI33F8'T' (“'70)
Substituting from Equation (“.68) and I2 for I5, we ob-

tain

k rJ (Br) r2J (Br)
1 2 l 0
72—3- I2 + T I3 = "'—_2 "' _—""—' 3 (”'71)
2B yH “ByH
which is Just the relationship required.

All of these integrals, i.e., Equations (“.60), (“.61),
(0.67), and (“.71), may be substituted back into the ex-
pression for w, Equation (“.58), which is simply multiplied
by yH to obtain W1

2 2 2
_ 2KA2ns p +KA 300(0,r)

2
1p1 "' L 86 70m,” + :8 az‘

 

1
283J1(Ba)

IQ

 

finger) - % r2J0(sr)]

N

 

I
0H0

- 37Eé%;§7[k2Jo(Br) + leO(Br)] + ku[liO(Br)+k2Y0(Br)]-
2 1 (u.72)

102

Since the radial boundary conditions, Equations (“.20)

and (“.21), were applied to 80, here we require that

 

072’ (2.0)
and
d$1(z,a)
_—dr = O (“.7“)
Differentiation of Equation (“.72) leads to
E:l = 2KilnS - g r2Jl(Br)
l" L E “82J1(Ba)
0 1 1 aJo(Ba) '
- z W 172 + W “0‘3“
k3"8 E ) < )1
+ ——-——- R J (Br + R Y 81"
2(kg'ki) 2 1 1 1
- kuB[k1J1(Br) + k2Yl(Br)] . (“.75)

Even though there are only two boundary conditions for
this expression with four unknowns, the last two terms can
t>€3 combined to yield only two undetermined constants. We

C351r1 achieve this by setting k3 = k2 = O to remove Yl(0);

103

this leaves only one constant, klk“’ to find from the re-

maining condition at r = a:

 

 

2
_ _ g a. l
k1k“ - e ”B3 Jl(Ba)
aJ (Ba) J (8a)
0 l. O a 0
6 [p2 BleBaS] 282 Ji<8a)

Thus the complete solution for $1 is

 

 

 

 

 

 

 

2 2 2
_ 2K Ans p +K 3w (0,r)
Wl(p,l‘) = AL { 86A 1P0(0,I’) + ‘3? Oaz
+ g (r2-a2)JO(Br) _ g rJ1(er)
8 “B3J1(Ba) e 20"J1(8a)
- g rJl(Br) _ g aJO(Ba)JO(BP)
8 2p282J1(Ba) S 2p282J§<Ba)
2 2
O aJO(Ba)rJ1(Br) a a Jo(8a)J0(Br)} (u 77)
E 283J§(8a) 8 283J§(8a) . .

Again, this must be transformed back to the original co-
(DIPGinate z in order to obtain the solution wl(z,r).

Each of the first two terms can be expanded by the
method of partial fractions and their inverse transforms

12(314nd in tables [“5]:

10“

L_1{-—z—p2+KA} = 22 sinh K z - z cosh K 2 + -—l— sinh K z
B HEX A 2;: A “K2 A
(“.78)
and
-1 Z2 2 .
2L {1%} = ——§ cosh KAZ - ——§ sinh KAZ . (“.79)
8 "KA ”KA

.The remaining terms must again be transformed through
complex integration. The method is the same as in the pre-
vious section, although the actual computation of the resi-
dues is much more difficult since all the terms to be
transformed contain a number of second- and even third-
order poles. The results of these inverse transformations,

in which we let
2 2
= + a “.80
Jl,s KA ( )

for convenience, are as follows:

 

-1 JO(Br) 2r2-a2 z
L 3 = - —————- sinh KAZ + —§— cosh KAZ
B Jl(8a) “KAa KAa

2azsinh(ySz/a) JO(Jl r/a)

 

 

l
- Kg; sinh KAZ + J2
A l,sYs

105

 

 

 

J (Br) 2_ 2
L l { ,1 } P<r a ) sinh KAZ
B Jl(8a) SKAa
zr
+ cosh K 2 - sinh K 2
2K§a A 2K2a A

 

 

1 (“.82)

3
J1,sYs

 

1‘1 { Jl(Br) } = _ 2% 11(2Ar)
p282J1(8a) KA I1ZKAa5

2a3sinh(ySz/a) J1(Jl38r/a)

 

 

 

 

 

r
+ sinh K 2 +
3 A 3
2KAa Jl,sY5 JO(Jl,s)
(0.83)
L'1 {JO(Ba)JO(Br)} = 2L IO(KAa)IO(KAr)
P2B2J§(Ba) Ki I§(KA3)
r2 2Z 6 h
- ——§—§ sinh KAZ + _“_§ cosh KAZ - —§—§ sin KAZ
2KAa KAa KAa

22azcosh(ySz/a) J0(Jl sr/a)

2
S JO(Jl,S)

+

 

Y

106

2a2rsinh(ySz/a) J1(Jl Sr/a)

3
J11,313 (JO(Jl,s)

 

6a3sinh(ysz/a) J0(Jl’Sr/a)

 

 

 

 

 

 

 

 

 

 

 

 

- 5 (“.8“)
Y3 J0(Jl,S)
J (Ba)J (Br) 3
L-1 O 3 l } = - I. 2 sinh KAZ
B J1(Ba) ZKAa
+ 2221; cosh KAZ - 2r sinh KAZ
KAa KAa
+ azcosh(ysz/a) Jl(Jl’Sr/a)
31,3Y: J0(31,s)
+ 2arsinh(YSz/a) JO(Jl,sr/a)
J1,sYs JO(Jl,s)
“azsinh(YSz/a) J1(Jl’Sr/a)
-'—— 3 J0(31,s)
Jl,sYs
23281Hh(YSZ/a) J1(31,sr/a) . (u 85)
31,3Y2 JO(Jl,S)

107

 

 

L-l {J3(Ba)JO(Br)} = ru+2a2r2-au

 

 

sinh K z
3 A
B3Ji(Ba) 8KAa
2(2r2+a2) 222+2r2+a2 i h
— cosh K 2 + s n KAZ
2Ki33 A 2Kia3

Z

3
- —“_— cosh K 2 +-————§ sinh K z
KAa3 A 2Kza A

(r2-a2)sinh(ysz/a) JO(Jl,sr/a)

 

 

 

 

 

J1,sYs JO(Jl,s)
z2sinh(YSz/a) J0(,jl Sr/a)
+ 2
Y3 J0(31,s)
2arsinh(ySz/a) Jl(jl Sr/a)
+ 2
JLs)“: J0(31,s)
2arsinh(ySz/a) Jl(Jl sr/a)
+ L
J1,sYs JO(Jl,s)
3a2sinh(ySz/a) JO(Jl Sr/a)
+ L

 

5
Y8 J0(31,s)

108

22rcosh(ySz/a) J1(Jl Sr/a)

Jl,sY§ J0(Jl,s)

 

3zacosh(y z/a) J (J r/a)
- 1; S O 1’s . (“.86)
YS J0(Jl,s)

 

Combining these results, we obtain the complete first-
order solution wl(z,r), i.e., the inverse transform of

Equation (“.77):

 

KAZnS Z2
W1(Z,I') = L 1110(0,I') ‘21:; sinh KAZ

 

z 1
— —§ cosh KAZ +-§;§ sinh KAZ]
"A A
3w0(0,r) Z2 2
+ 32 2 cosh KAZ - ——§ sinh KAZ
2KA ZKA

“ 2

2
0 r r z 9
8 [ KAa K28 KZa ZKEa] A

2
0 Z]? Z

- E [T + T 00811 KAZ
KAa KAa

109

 

 

 

 

 

 

 

+ 9; E5: 11(KAr) - Eé'. IO(KAa)IO(KAP)
8 K2 I1(KAa) Ki I2( )
1 "Aa
°° 2 2 )4
_ 2 Z Z a + 28
8 2 2 2 1/2 2 2 2 2 1/2
3:1 (3193+KAa ) Jl,S(':I:L«,S+KI-Xa' )
“ J (J r/a)
' 2 332 2 5/2 Sinh [(31,S*K§a2>1/2(z/a)1 0 1,3
(3123+KA8 ) J0<Jl,s)
°° 3
+ 9’ Z za cosh [(J2 +K2a2)l/2(z/a)1 X
8 3:1 (35’s.... ia2)l/2 1,3 A
J (J r/a)
0 l,s } (“.87)
Jo<31,s>

Although the solution for 11 appears more complicated
than that for $0, Equation (“.“l), its form is similar.
It contains functions of z and functions of z multiplied by
functions of r, and there is no reason to believe that the
higher-order solutions will yield any different result.
Since 00 also contains a function of r in addition to
the ones mentioned above, the general form of the result

for w from the Laplace transform method is

w(z,r) = fl(z) + gl(r) +T2: fé(z)g2(r) . (“.88)

110

This differs from the form of the solution that we

found in Chapter 3, i.e.,

w(z,r) = f(z) + g(z,r) , (“.89)

in which 2 and r are not separable in the final term. In
the next chapter, therefore, we attempt to find a solution
of the form of Equation (“.88) for our model of a cylinder

of finite length.

CHAPTER 5

AN ATTEMPTED SOLUTION BASED ON THE

LAPLACE TRANSFORM RESULTS

A. Introduction

 

In this chapter we again consider a cylinder of finite
length (0 g z 5 L). With a proposed solution of the form
obtained in the previous chapter, we impose z-dependent
boundary conditions at z = O and z = L. The particular

Z/L, and

model we choose to solve is again for f(z) = S
the complete equation is given in Equation (“.2). Writing
SZ/L as a series as in Equation (“.“) and (z,r) as a series
as in Equation (“.6), we obtain the same differential equa-
tions as those given in Equations (“.7) through (“.9).

For each 00,01, etc., we choose a solution of the

form
wi(z,r) = fl(z) + s1(r) + f2(z)g2(r) . (5.1)

We apply the boundary conditions in Equations (3.107) and
(3.108) and in Equations (2.21) and (2.22) to fl(z) and
gl(r) of we respectively and require that these derivatives

be zero for every other function f and g in each wi'

111

112

B. Zeroth-Order Solution

Assuming Equation (5.1) for we and introducing it into

Equation (“.7), we separate the result into three equations:

 

 

d2fl 2

— K f = 0 (5.2)
dz2 A 1 ’

d8
1 d. 1 2 _
its??? “mgr” ’ (5'3)
and
d2f2 1 <1 dg2 2

s2 d 2 + f2 FEE? 1” d'r— " KN252 = 0 ° (5'1”

Z

The final equation can be separated into z-dependent and

r-dependent parts, and writing the separation constant

as 12, we obtain

 

d2f2 2 2
2 - (A + KA)f2 = 0 (5.5)
dz
and
d8
1 d. 2 2 _
FEPF+Ag2-O o (5-6)

The solutions to Equations (5.2), (5.3), (5.5), and (5.6)

are

113

f1 = klcosh KAZ + k2 sinh KAZ , (5.7)
gl = k3IO(KAr) + kUKO(KAr) , (5.8)
2
f2 = kscosh(A2 + KA)l/2Z + k6s1nh(x2 + K§)l/2Z ,
(5.9)
g2 = k7JO(Ar) + k8YO(Xr) . (5.10)
The z-dependent boundary conditions,
df (0) o
l
‘32— : -€_A (5.11)
df (L) o
___§z = _ _EB. , (5.12)

result in the solution

1

f(z):...
l EKASinhKAL

[oBcosh KAZ + oAcosh KA(L-z)] ,
(5.13)

which is the same result as Equation (3.110), that obtained
for wl(z) in the absence of an ionic strength gradient.

The r-dependent boundary conditions applied to g1(r) yield

010(KAr)

EKAT1(KAa) ’ (5'1“)

 

gl(r) =

11a

the same result as w2(z,r) in Equation (3.7) with no ionic
strength gradient.

We now apply the conditions

dg2(0) dg2(a)
f2-———-—— = f2-—-—- = O (5.15)
dr dr
and
df2(0) df2(L)
dz dz
to the product f2(z)g2(r):
ds
2 _ 2 2 1/2
f2 757 - —A[k5cosh(k +KA) a

+ k6 sinh(12+xi)l/2z][k7Jl(1r)+k8Y1(1r)]
(5.17)

The requirement that this expression be zero at r = O forces
k8 = 0 sinceYlO. r) is infinite at this point. At r = a,

we have

-A[k5cosh(l2+Ki)l/22 + k6sinh(>.2 + Ki)l/ZZ] X

Jl(1a) = o , (5.18)

115

where k7 has been subsumed into R5 and k6. This condition

is satisfied for all

 

is = a_ . (5.19)

where Jl s is the s-th zero of J1.
3

The second set of conditions given by Equation (5.16)

applies to the expression

df m
2 .2 2 1 2 .2
g2 TE? = i :25 (J1 S+Ki a )‘/ {k5,sSinh[(Jl,s+KAa 2)l/2(z/a)]

2)l/2

+ k6’scosh[(ji’s 'tKAa (z/a)]} JO(Jl,sr/a) 2

where the constants k5,s and k6,s may now depend on 3.
However, the function JO(jl’Sr/a) is orthogonal in the
interval 0 i r g a because its derivative vanishes at both
r = 0 and r = a [66], and thus we can exploit this property
to determine these constants.

At z = O,

00

K2 a2 1/2
=0 (31, 3+ KaA )

l - _
‘8': k6,SJO(Jl,Sr/a) - O . (5.21)

S

Multiplying by rJO(jl tr/a), where t is a particular value
)
of s, and integrating over r from O to a, we obtain

.2
1

(J 2)”2

nflm

+K a

2 2 . _
,t A k6,t J001,0 - 0 , (5.22)

116

and thus k6 s = O for every 3. Similarly at z = L, apply-
’

ing the condition of orthogonality to the expression
1 °° 2 2 2 1/2 2 2 2 1/2
a s§0 (Jl,S+KAa ) k5,ssinh[(Jl,S+KAa ) (L/a)] x

o<Jl,s g) = o (5.23)

results in the requirement that

2 2)l/2

%(J§,t+KAa sinh[(J§,t+K2a2)l/2(L/a)1Jg(31’t)=0-

k5,t A
(5.2M)
This is true only for RS 3 = 0 for every 3.
2

Thus our complete zeroth-order solution is simply

 

 

_ 1
w0(z,r) - - EKASinhKAL [oBcosh KAZ
oIO(KAr)
+ oAcosh KA(L-Z)] + EKAIl(KAa) (3.25)

C. First-Order Solution

We now turn to the first-order solution, for which we

must solve the differential equation

117

 

 

2
3 w 3W 2K zlnS
l .1; i l = 2 _ A
8.22 + I." a]? r 3r KAlpl ELSinhKAL [GBCOShKAZ
20KAzan

 

+ o COShKA(L-Z)] +

A 10(KAr) . (5.26)

€LI1(KA3)

Again, we assume a solution of the form given in Equation

(5.1). The separation of the resulting expression,

 

 

‘2 2
“1. 12.12.,dg1.f12_,dg_2
2 g2 2 rdr dr 2rdr dr
dz dz
2K zlnS
_ 2 2 2 A
‘ KA f1 + KA g1 + KAf2g2 ‘ eLsinhKAL [OBCOShKAZ
20KAz£nS
+ oAcoshKA(L-z) + SLIl(KAa) IO(KAP) , (5.27)

is problematic. Although we can readily separate out the

z-dependent equation,
d2f 2K zlnS
l A

____.- 2f = -
dz2 |<A 1 eLsinhK

 

[oBcoshKAz + oAcoshKA(L-z)] ,
(5.28)

AL

The remaining z-dependent and r-dependent terms cannot
be separated from each other.
We choose to place the remaining inhomogeneous term

with gl since the homogeneous solution to the equation

118

for gl is also modified Bessel functions. Thus we have

the two equations

d dgl 2 20KAzlnS

 

 

1 ..
and
(1212 1 d dg2 2
g2d2”f'2E-"cT:E-'r'E-"""Af2‘52=O' (5°30)
Z

Equation (5.30) is the same as Equation (5.4) in the solu—
tion of mo. Since the boundary conditions are the same
as before, we see that f2g2 = O, and we need only solve
Equations (5.28) and (5.29), treating 2 as a constant in
the latter.

We use the same method here for finding the complete
solution when the homogeneous solution is known as we
did in the previous chapter. The homogeneous solution

of Equation (5.28) is

yH = klcoshKAz + k2sinhKAz . (5.31)

Assuming a complete solution of the form

f1(z) = w(Z)yH(z) , (5.32)

we find from Equation (5.28) that

119

 

 

.2. 22. . .2. 1H 22 .
dz dz yH dz dz
2KAz£nS
- eLstinhKAL [oBcoshKAz + OACOShKA(L-Z)] .
(5.33)
The integrating factor is 1 so that
2K ins
%% = - 2A (klcoshKAz
eLstinhKAL
k3
+ k2sinhKAz)[oBzcoshKAz+oAzcoshKA(L-z)]dz + —5—
yH
(5.3”)

Integration yields

2K ins

dw A [l 2
___=_ kg z
dz eLygsinhKAL { 1 B E

 

1 1 2
+ 2E_ ZCOShKAZSinhKAZ --——§ (COSh KAZ
A 8KA

2 l 2
+ sinh KAZ)] + RZOB [HEX z (cosh KAZ

+ sinh2KAz) - —;§ COShK

AzsinhKAz
“KA

120

l 2
+ kloA [E-z COShKAL

2

1 2
- FE; z[sinhKAL(cosh KAZ + sinh KAZ)

2COShKALCOShKAZSinhKAZ]

1 2 2
- 8K2 [coshKAL(cosh KAZ + sinh KAZ)
A

-2$inhKALcoshKAzsinhKAZJ]

+

l 2

2

1 2
+ Hz; 2 [coshKAL(cosh KAZ + sinh KAZ)

ZSinhKALCOShKAZSinhKAZ]

+ —l§ [sinhKAL(cosh2K z + shinZKAz)

KA

A

k
2COShKALCOShKAZSinhKAZJ]} + —§i-. (5.35)
y
H

The second integration to obtain m itself presents a

121

number of integrals that appear difficult. However, by
rewriting Equation (5.35), we can obtain the results of
several integrals collectively with relative ease, as
we did in seeking the general solution for the first-
order solution in terms of Bessel functions in the pre-
vious chapter. We combine terms in Equation (5.35) and

integrate as follows:

2K ins k 2
= - A o l 5— dz

“ eLsinhK L B 1F 2
A yH

 

 

 

1 j{zcoshrAz(klsinhKAz+k2coshKAz)

 

 

+ HKA 2 dz
yH
+ 1. j[ZSinhKAz dz _ l COShKAZ dz
11KA yH 82E yH
l sinhKAz(klsinhKAz+k2coshKAz) d
‘ “2 2 2
8K y
A H
kl z2
+ oAcoshKAL 1r —5 dz
yH
dz

 

2

+ l j[ZCOShKAZ(klsinhKAZ+k2COShKAZ)
HK
A yH

 

l zsinhKAz l COShKAZ
+I‘_l(_ dZ-—§ ———dZ
A yH 82A yH

dz

 

8K2 2

- 1 ijinhKAz(leinhKAZ+kZCOShKAZ)
A yH

122

 

k 2 ZCOShK z
+ O’AS1nhKAL ‘l‘gfz—z' dz - —L —$’-—A-— dZ
- l ZSinhKAZ(k181nhKAZ+k2COShKAZ) dz
4K 2
A yH

1 COShKAZ(k181nhKAZ+k2COShKAZ

2 2
8KA yH

’sinhK z '
+ —l§ J[—————5— dz + k3 33 dz + kn
8KA yH yH

 

dz

(5.36)

Each group of integrals within square brackets can be

solved as a unit. We demonstrate with the first set

ignated I through I

1 5'

, des-

The only other integral needed is

the final one in Equation (5.36), which can be found with

relative ease:

 

 

 

]_ d - sinhKAz - COShKAZ
—2‘ Z‘W‘"m‘
yH 1 A H 2 A H
Using this result to do the other integrals by parts
find:
2
z sinhKAz 2
I = — I ,
1 leAyH leA 3
ZCOShK z
KAyH KA

(5.37)

, we

(5.38)

(5.39)

123

SinhKAZ
I5 = --——————— + I“ . (5.AO)

KAyH

Thus, the expression in the first square brackets

of Equation (5.36) is

 

 

 

 

k
1 1 l 1 1
T11+Fr12+w13"‘21u-_2'I5
A A 8K 8K
A A
= H‘l“ z sinhKAz - 2l‘ I3
KAyH KA
1 1 1
- “K2 ZCOShKAZ + “K2 IA + 3:; I3
AyH A
1 1 1
+ FE; I3 - 525.1” + 8K3 sinhKAz
A AyH
1
' 8:2 In
A
_ 1 2 1
_ EEX§E z sinhKAz - 2 ZCOShKAZ
AKAyH
+ 8 3 sinhKAz . (5.“1)
K y
A H

Proceeding similarly for the other groups of integrals,we

eventually find that

12A

Ans

 

_ 2 2
cu - - L {UBE2KAZ sinhKAz

2
AeKALstinhKA

2 2
2KAZCOShKAZ + SinhKAZJ-OAC2KAZ sinhKA(L-z)

k
3
ZCOShKA(L-Z)+SinhKA(L-Z)J} + k__—_—

+ 2K
lKAyH

A sinhKAz+ku ,

 

 

(5.“2)
and thus that
f (z) = - ins o [2K2Z2SinhK z
1 2 B A A
48K LsinhK L
A A
- 2K zcoshr z + sinhK z]-o [2K2Z2SinhK (L-z)
A A A A A A
+ 2KAZCOShKA(L-Z) + sinhKA(L-z)]}
+ klcoshKAz + kzsinhKAz , (5.“3)

where we have lumped the final terms with undetermined co-

efficients together.
In order to determine kl and k2, we apply the condi-

tions

125

dfl(0) _ dfl(L)

dz - dz = 0 ° (5.AA)

The z-derivative of Equation (5.43) is

df
15% = - _ 2£nS {OBE2K222008hKAZ
HeKALsinhKAL

 

+ 2K§ZSinhKAZ - KACOShKAZ]
+ o [2K322coshK (L-z) - 2K2ZSinhK (L-z)
A A A A A

- KACOShKA(L-Z)]} + leAsinhKAz + k2KACOShKAZ

 

 

(5.45)
At 2 = 0, we find that
k2 = - 2£nS (GB + oAcoshKAL) , (5.“6)
Aer LsinhK L
A A
and at z = L,
k1 = 2 fins 2 [2oB(KiL2coshKAL
AEKALsinh KAL
+ K LsinhK L) + o (Sinh2K L + 222L2)] (5 47)
A A A A A ' '

126

Thus the complete solution for fl(z) is

fl(z) = - 2£nS {oB[2K§zzsinhKAz
AEKALsinhKAL

- 2KAZCOShKAZ + sinhKAz] - GAE2K§2281nhKA(L-Z)

+ 2K ZCOShKA(L-Z) + sinhKA(L-Z)1}

A

+ 2 ins 2 {GBEZKEchosthLcoshKAz
AEKALsinh KAL

 

+ 2KALSinhKALCOShKAZ - sinhKALsinhKAz]

+ OAE2KiLZCOShKAZ + sinhKALsinhKA(L-z)]} . (5.“8)

We now need to find the solution to Equation (5.29) in

the same way. The homogeneous solution is

yH = klIO(KAr) + k2KO(KAr) , (5.39)
and thus we assume a complete solution of

gl(r) = w(r)yH(r) . (5.50)

Substituting this solution into Equation (5.29) yields

127

1g + [2 dyH + 1] dw 2oxAz£nS
dr _

a; ‘ eLyHIl(KAa) IO(KAr)

:1
dr
(5.51)

With an integrating factor of r, this equation yields the

solution

dw 2OKA22DS

__ = r1 (K r)[k I (K r)
dr eLrySIl(KAa) Jf O A 1 0 A

 

<<:|PT
:Emoo

+ k2KO(KAr)]dr + (5.52)

These integrals are straightforward [165], and the result
is

 

20K z£nS k
eLryHIl(KAa)
k2 2
+ 77 r [10(KAr)KO(KAP) + 11(KAr)K1(KAr)]
k
+ _% (5.53)
yH

The third integral can be performed analogously to
tnutt involving ordinary Bessel functions as in Equation

(“.61), the result being

128

I l k2IO(KAr)-leO(KAr)

 

3 = 2 2
k2+kl yH

The first two integrals are found to be

rIl(KAr)

 

-I =
l 2 yH

so that the final result is

ozfinS

= eLyfif;(KAa) rIl(KAr)

*- k3 kZIO(KAr)-leO(KAr)

2 y
k +kl H

 

2 + kn
2

Thus the complete solution to Equation (5.29) is

ozlnS

31(r) ‘ eLIl KAa rI1(KAI')

+ klIO(KAr) + k2K0(KAr) ,

(5.55)

(5.56)

(5.57)

(5.58)

where we have clumped the four constants into a new k

k2. The boundary conditions

d82(0) d82(a)
dr g dr

finally result in the solution

13

(5.59)

129

rIl(KAr) aIO(KAa)IO(KAr)
I1(KAa) -

 

 

2 . (5.60)
I1(KAa)
Thus the complete first-order solution is the sum of
Equations (5.48) and (5.60). Higher-order solutions may
be found similarly, except for the expenditure of more

energy.

D. Critique of the Method

 

The difficulty with determining w(z,r) in this way
is the impossibility of separating the z-dependent terms
in the equations for wl(and for higher—order solutions
as well). The separation that we have chosen is equivalent

to

wl(Z,P) = f(Z) + g(z,r) , (5.61)

where we neglect 62g/322

as being small compared to
(l/r)3(r3g/3r)3r.

The method here is also equivalent to our earlier
attempt in Chapter 2 with the additive solution of Equa-
tion (2.8). There we did not proceed because it was not
possible to separate the partial differential equation with
this solution into two ordinary differential equations.

If we had separated Equation (2.9) as we have here, we

would have obtained

130

2
d f (2)
dz
and
1 <1 d31(r) 2 2z/L

Treating the variable 2 as a constant in Equation
(5.63) leads to the solution in Equation (3.7), where the
ionic strength distribution model f(z) is simply SZ/L.
The perturbation scheme presented in the previous sec-
tions thus leads only to an infinite series of hyperbolic
functions and modified Bessel functions that are presumably
equivalent to the solutions for wl(z) and w2(z,r) with

Sz/L

in Chapter 3.

The form of the solution prOposed in Equation (5.1)
is apparently due to the imposition of both z-dependent
boundary conditions at Z‘= 0. An examination of the
zeroth-order solution in Equation (u.u3) shows why this
is so. For the condition w0(0,r), the hyperbolic cosine
is equal to l, and since it is possible to show that the

Fourier-Bessel series for IO(KAF) is [165]

m l JO(31Lsr/a)

I (K r) = 2K aI (K a) Z:
0 A A 1 A s=0 3% 5+K§32 J0(Jl,s) ,
,

 

(5.6M)

131

those terms cancel, and wo(z,r) obeys the boundary con—
dition. Similarly, for the boundary condition 3w0(0,r)/
82, the term IO(KAP)/I1(KAa) is zero, as is the hyperbolic
sine, and again these two terms vanish so that w0(z,r)
obeys the boundary condition.

This relationship between the modified Bessel functions
and its Fourier-Bessel series does not work for boundary
conditions at values of z other than zero, and thus the
term f2(z)g2(r) is zero for each solution $1 as we try
to impose a solution of the same form on a finite cylinder.
Although the method of the Laplace transform has not ob-
tained any new solution for us, it has added further sup-
port to the method presented in Chapter 3, i.e., the sepa-
ration of w(z,r) as in (3.1) and the subsequent neglect

of the second z-derivative.

CHAPTER 6

MODELS, METAPHORS, AND MATHEMATICS

A. Introduction

On November 10, 1619, the Angel of Truth appeared to
Rene Descartes, assuring him that mathematics is indeed
the key to unlock the physical universe [22]. Thus Des-
cartes had more tangible evidence for this metaphysical
belief than do most of us, who believe in it because others
do and because it seems to work. It works in the sense
that mathematical predictions about the universe can be
tested experimentally; for example, by following the cal-
culated trajectory from earth, Apollo 8 succeeded in orbit-
ing the moon. There can be dramatic corroborations of
Newtonian mechanics even in this age of quantum theory.

Mathematical prediction, however, is only one part of
the "systematic coherent formulation" that David Hawkins
[60] cites, along with empirical verification, as the dis-
cipline of science. Science must also provide explanation,
distinct from prediction, within this framework, although
Erwin Chargaff warns us that eXplanation is not understand-
ing [29] and chides us that "even the most exact of our

texact sciences float above axiomatic abysses that cannot

132

133

be explored" [28].
No less a scientist than Pierre Duhem states unequi-
vocally that physical theory is not an explanation. The

aim of theory ”is to summarize and classify logically a

 

 

group of experimental laws without claiming to explain
these laws" [40]. Duhem rejects any explanatory role for
theory because such a role, in his view, subordinates
physics to metaphysics. How he could avoid such ultimate
beliefs is not clear; even his choice to be a physicist
rather than a biologist, for example, reflects certain at-
titudes towards the world. Such beliefs and attitudes are
bound up in the research traditions and language of any
scientific field, and therefore it seems better to be aware
of them than to deny their existence.

For this reason, we use this chapter as an opportunity
to step out of the narrow niche of our specialized re-
search topic [139] and take a somewhat broader look at the
connections between the object of our study (the distribu—
tion of electric potential across a charged membrane) and
the tools that we study it with (the Poisson-Boltzmann
equation and ancillary mathematics and the membrane model
of cylindrical pores). Thomas Kuhn [81] notes that scientists
generally have little need for such speculation except during
times of crisis that precede scientific revolutions, but we
believe occasional reflection on these "extrascientific"

questions can save us from the plight of Tolstoy's

13H

plasterers

who are set to plastering one side of a church
wall and who, taking advantage of the absence
of the work's chief supervisor, in a burst of
enthusiasm plaster over the windows, icons,
woodwork, and the still unbuttressed wall, and
rejoice that from their point of view as plas-
terers, everything is now smooth and even [162].

Science should never appear smooth and even.

B. Theory

Practicing scientists have had theory bred into their
bones, but almost always in an informal or tacit way, so
that they are usually uneasy or inept in trying to discuss
it as a concept. On the other hand, philosophers of science,
whose business it is to define such things, usually lack
the scientist's intuitive feel for theory and often portray
it as something quite unrecognizable to the scientist. In
fact, the current consensus among philosophers of science
is that the ideas presented in this section are inadequate
for analyzing scientific theory [158], but since philoso-
phers are not able to agree among themselves on a single
acceptable alternative and since these ideas provide us
with a way of looking at theory and lead us to other topics,
we present them anyway, generally as they have been dis-
cussed by fluent scientists [23,HO,95,118,137].

A theory (T) consists of at least two sets of state-

Inents. The first is termed the "calculus" (H)--Campbell

135

calls this the "hypothesis"--and, depending on the scientist
doing the defining, is either just made up or deduced from

a large number of empirical laws. However it is proposed,
the calculus by itself is abstract and may appear quite
arbitrary.

The calculus becomes valuable when it is joined with
the second set of "correspondence rules" (D), which Camp-
bell refers to as the "dictionary." The correspondence
rules relate or connect the ideas of the calculus to con-
cepts (C), ideas that have meaning within the context of
physical laws (L) known to be valid empirically.

It must always be possible to deduce additional state-
ments or propositions (P) from the calculus, and if these
additional statements, interpreted through the correspon-
dence rules, imply or are consistent with known physical

laws, then the proposed theory is deemed valid (Figure 6.1).

HvC

1 D t

Figure 6.1. Interrelationships involving theory.

If P turns out to be inconsistent with L, then there are

usually renewed attempts to find another P that is consistent.

136

If this is not possible, T itself may be called into ques-
tion.

In the present study of the Poisson-Boltzmann equation,
for example, T is the theory of electrostatics, from which
may be deduced P, including Poisson's equation, the par-
ticular solutions for wl(z) and w2(z,r) in Equations
(3.120) and (3.7) respectively, and finally the expression

for E in Equation (3.121).

M

From D we know the correspondences between H and C.

For example, w(z,r) in H corresponds to electric potential
in C, p in H, through the definition in Equation (1.“),

is related to the Faraday constant, ionic charge, and con-
centration in C, etc.

We did not perform all the previous mathematical deri-
vations in order to determine whether electrostatics is
valid or not; we implicitly believe that it is. Our intent
was obviously much more modest. By beginning with a theory
that we accept, we are able to test the various (and num-
erous) assumptions that we made in moving from T to P.
Indeed, Equations (3.123) and (3.12“) are the main pre-
dictive results of our endeavor in that they can be tested
against experimental results. If our equations agree with
experiment, then we can also feel confident about our
results for $1, Equations (3.115) and (3.120), and $2,
Equation (3.7), even though these are not experimentally
accessible quantities.

If, however, our equations are not in accord with

137

experiment, then our first impulse will be to question the
experiment's results and/or suitability. Did the experi-
menter make any errors in his measurements? Is there a
systematic error in the method that could account for the
differences? Once we are satisfied that there is no ex-
perimental error, we will ask whether the experiments per-
formed actually are a test of our equations. Do the ex-
perimental conditions conform to those that we invoked at
the beginning of the derivations?
If the experiments seem both correct and applicable,
we will set about deriving a new P to test against experie
ment. Another P incompatible with experimental result will
still be unlikely to cast doubt on the theory of electro-
statics. Instead, we would probably question the validity
of the Poisson-Boltzmann equation, a macroscopic statement,
to the system we wish to describe. We may then find a
lower limit in size for which the equation is applicable.
In this sense the failure of P to agree with experiment
may provide us with more definitive information than would
agreement, which can tell us nothing about the uniqueness
of the derived P.
We assume that we have a T and, we hope, a P consistent

with L, but Campbell claims that this is not enough:

Any fool can invent a logically satisfactory

theory to explain any law.. . . If nothing but

this were required we should never lack theories

to explain our laws; a schoolboy in a day's work
could solve the problems at which generations have

138

laboured in vain by the most trivial process of
trial and error. . . . It is never difficult to
find a theory which will explain the laws logically;
what is difficult is to find one which will explain
them logically and at the same time display the
requisite analogy [23].
The key word in this quote is the final one: analogy. Camp-
bell insists that besides being logical, a successful theory
must also be analogous to known laws, although Duhem, in
acknowledging the heuristic importance of analogy, allows
the analogy to exist with other theories.

Modern scientists from Sir Humphrey Davy [33] and James
Clerk Maxwell [107] to the present [35,86,182] have recog-
nized the role of analogy in the progress of science, and
most writers on this topic equate analogue (the thing
analogous to T) with model. In trying to minimize the dif-
ferences between Duhem and Campbell, who are invariably
cited as the starting point for all discussions about scien-
tific models, D. H. Mellor [112] argues that both physicists
sharply distinguish between analogues and models, i.e.,
mechanical models, and more or less agree on their relative
importance, although Campbell regards the former as essen-
tial to theory, while Duhem does not. Mellor then proceeds
to distinguish models as consisting of observable entities-—
in the sense of being part of a successful theory's obser-
vation language--referred to by the set of laws that make

up the analogue.

This is very close to the idea of model that we wish

139
to use. In order to make this idea more clear, we back
up and approach it from a different perspective, that of

the interaction theory of metaphor proposed by Max Black.

C. Metaphors and Models

 

The modern study of metaphor begins with I. A. Richards,
who first named the two parts of metaphor. The terms
"tenor" and "vehicle," the "thing-meant" and "thing-said"
respectively [AH], refer to the "two thoughts of different
things active together and supported by a single word,
or phrase, whose meaning is a resultant of their inter-
action" [1&1].

Max Black, however, is usually credited with the de-
velopment of the interaction view of metaphor some twenty-
five years after the work of Richards [13,1N]. He improves
Richards' terminology, pointing out that a metaphor con-
sists of some words that function literally, the "frame"
or "principal subject," and some that function metaphor-
ically, the "focus" or "subsidiary subject." In the ex-
ample, "Metaphors are pretty little maidens," the words
"pretty little maidens" are the subsidiary subject, which
in some nonliteral way describes the principal subject
"metaphors."

After commenting on the shortcomings of the substi-
tution and comparative views of metaphor, Black presents

the interaction view. Richards says that the mind

1&0

"connects" the metaphor's two ideas [1A1]; Black suggests
how. A person reading the metaphor mentioned in the pre-
vious paragraph understands its intent by transferring his
generally held conceptions about pretty little maidens to
metaphors. Dictionary or literal meanings of the sub-
sidiary subject are not so important as Black's "system of
associated commonplaces." Pretty little maidens are just
that: pretty and little, soft and cuddly, pleasant to
look at, but insubstantial. So are metaphors. Such common-
places obviously need not even be true for an effective
metaphor; they need only be readily evoked.

While the subsidiary subject bears on the principal
subject, the latter turns back on the former--Polanyi
[132] likens this to the effect of the thing symbolized on
the symbol--and the two "interact" to form a whole. Those
traits of the subsidiary subject that make sense in terms
of the principal subject are brought to the foreground,
while those that do not are relegated to the background.
The metaphor acts as a filter; it "suppresses some details,

emphasizes others--in short, organizes our View" of meta-

 

phors in terms of pretty little maidens [1”]. It may even
"generate new knowledge and insight by changing relation-
ships between the things designated" [13].

William Empson makes somewhat the same point when he

says that

141

a metaphor may be a matter of "insight"; it

may be used to survey a whole complicated mat-

ter as if from a height; it is a device for

letting you handle the proportions of the mat-

ter intuitively, instead of fiddling about with

first one part and then another [AM].
The similarity between the functions of metaphor and theory
in this respect is apparent from Duhem's comments in speak-

ing of the economy of thought in theories:
To bring directly before the visual imagination
a very large number of objects so that they
may be grasped simultaneously in their complex
functioning and not taken one by one, arbitrarily
separated from the whole to which they are in
reality attached-—this is for most men an im-

possible or, at least, a very painful opera-
tion [MO].

With metaphor well in hand, Black proceeds to models.
He distinguishes four types, of which "theoretical models"
interest us here. Unlike most other kinds of models,
theoretical models cannot be physically constructed; they
consist of descriptions and assumptions: "the heart of
the method consists in talking a certain way" [1“]. Peter
Achinstein outlines the characteristics of a theoretical
model as follows [A]:

l) a set of assumptions about some object or
system;

2) a description of what might be called inner
structure, composition, or mechanism;

3) a simplified approximation;

1&2

A) a part of a broader framework of some more
basic theory or theories; and frequently

5) a formulation on the basis of an analogy.

In our work we applied the machinery of electrostatics,
and the Poisson-Boltzmann equation in particular, to the
distribution of electric potential across a charged mem-
brane. To attempt this with an actual membrane raises
insurmountable mathematical problems: the membrane's
geometry is irregular, and necessary physical characteris-
tics, including boundary conditions, are unknown. Instead
we chose an idealized membrane with straight, rigid, uni-
form cylindrical pores in place of the presumably tortuous
routes or channels that somehow open and close to allow
ions across a real membrane. Our theoretical model provides
us with a way of talking and applying the mathematics of
electrostatics. In Stephen Toulmin's words, our model
puts "flesh on the mathematical skeleton" [16“].

Richard Braithwaite states that a theory and an ap-
propriate model share the same calculus [17]. In terms
of the idea of model that we wish to develop here, perhaps
we should modify this to say that there exists a one-to-

one correspondence between the statements of the theory

and those of the analogue with which the model is associated.

In the view of Carl Hempel, a theoretical model has the
character of a theory with a more limited scope of applica-

tion [63], and in fact in our study, the analogue to the

143

theory is a specific part of the theory, so that this ac-
counts for the two having the same calculus.

Black discusses the problem of scientists' attitudes
towards theoretical models. He quotes Maxwell, who ini-
tially talks about the ether as "merely a collection of
imaginary properties" [106], but later discusses it as some-
thing physically real that fills all of space [108]. There
is a significant difference between treating a model as a
heuristic fiction and as a physical reality, the difference,
as Black points out, between thinking of space éfi if it
were filled with an actual medium and as bging filled
with it.

In connection with this, Romer says

If the model is no more than an intellectual

tool, then there is little point in discussing,

for example, wave-particle duality . . . It is

only as we believe that lightreally behaves

like waves and like particles that the implied

contradiction begins to interest us [1&2].
Perhaps in our case, there is less chance of identifying
our model with the inner structure of a real membrane be-
cause the differences are quote obvious, yet there are com-
mercially available artificial membranes with straight,
uniform cylindrical pores produced from neutron bombard-
ment [119]. Here at least the geometrical distinctions
between our model and a real physical system are not so

great, and we may feel, like Maxwell, that the results

from our model must say something about the actual

1&4

situation in such a membrane under the conditions we
impose.

However we-view the ultimate nature of our model,
we must admit that it resembles a metaphor in terms of its
"analogical transfer of a vocabulary" [1“]. Black talks
of the commonplace implications in metaphor, while he
admits that specific scientific knowledge is required
for model construction. Although Black himself says little

about this topic, Thomas Kuhn has a good deal to say.

I:

D. Paradigms and Language

 

Probably no philosopher of science has created so

much furor in recent years as Thomas Kuhn, whose Structure

 

of Scientific Revolutions [81] presents a theory of science

 

that emphasizes the sociological and revolutionary, rather
than cumulative nature of scientific progress. His critics
are legion [26,83,85], and Kuhn has responded to them on
numerous occasions [79,80,82]. Regardless of the short-
comings of his theory for the philosophy of science in
general, his idea of paradigms fits very nicely with our
specific analysis of scientific models.

Basically, Kuhn portrays the development of a science
as proceeding from a period of "pre-science," which con-
sists of more or less random fact gathering, to "normal
science," which he characterizes as "research firmly based

upon one or more past scientific achievements, achievements

1H5

that some particular scientific community acknowledges for
a time as supplying the foundation for its further practice"
[81]. The sociological nature of his ideas is already
obvious. That these achievements are capable of spawning
such normal science, which Kuhn frequently refers to as
puzzle solving or even mopping up, arises from two condi-
tions:
(1) these achievements were sufficiently un-
precedented to attract an enduring group
of adherents away from competing modes of
scientific activity; and
(2) they were sufficiently open-ended to leave
all sorts of problems for the redefined
group of practitioners to solve [81].
Kuhn terms achievements that fulfill these two condi-
tions "paradigms":
By choosing it, I mean to suggest that some ac-
cepted examples of actual scientific practice--
examples which include law, theory, application,
and instrumentation together--provide models
from which spring particular coherent traditions
of scientific research [81].
Much of the criticism directed against Kuhn concerns the
imprecise, and in fact multiple meanings of paradigm, and
in much of his response he tries to define it more exactly,
even renaming it "disciplinary matrix," yet as he responds
to each critic, the notion of paradigm becomes less and
less clear.

The object of normal science is to work out the implica-

tions of the paradigm. Naturally, there are always problems

146

that cannot be solved in terms of the paradigm; although
they challenge the foundations of the paradigm, they are
usually ignored. It is the build-up of such problems,
termed anomalies by Kuhn, to a point that they cannot be
ignored that brings on a period of crisis, a time when
scientists uncharacteristically question the ultimate
basis of their paradigm. The crisis may eventually be
solved with the methods of the current paradigm; the anom-
alies may be set aside; the paradigm may be rejected in
favor of a new one, which usually arises in some extra-
scientific way. This latter case is termed a scientific
revolution, and scientists, at least those who can shift
their allegiance, set about working out the "normal-science"
implications of the new paradigm.

It is the idea of paradigms as bodies of research tra-
dition that interests us here. Whether science does or does
not develop in such a manner, it seems to us that the exist—
ence and importance of particular research traditions can-
not be denied.

Students are initiated into particular research com-
munities through a long apprenticeship that begins with
introductory science courses and frequently culminates in
a doctoral dissertation. During that time the student has
absorbedtnuaparadigm of his chosen research specialty and
has mastered problems that progressively "become more com-
plex and less completely precedented" [81]. Kuhn argues

that scientists never learn concepts, laws, theories, etc.,

1U7

in the abstract, but "with and through their applications,"
which "are not there as embroidery or even as documentation.
On the contrary, the process of learning a theory depends
upon the study of applications, including practice problem-
solving both with a pencil and paper and with instruments
in the laboratory" [81].

Thus an undergraduate majoring in chemistry learns to
view the world in terms of the paradigm held by the com-
munity of scientists who consider themselves chemists. The
problems available for study, the theoretical and experimen-
tal tools available for attacking those problems, the ac-
ceptable forms of answers, these are all determined by the
paradigm. As a graduate student who further specializes in
physical chemistry, the trainee comes to regard the world
in terms of a more specialized paradigm. In equipping a
scientist to do research in a particular discipline, a
paradigm implicitly imparts to that individual a particular
view of the world that he is to study. He expects the
world to be a certain way, and as long as it is, the para-
digm is successful. It is impossible, in these terms, to
imagine how Duhem, trained as a nineteenth-century physicist,
could escape having the ultimate beliefs of nineteenth-
century physics ingrained in him.

Despite the importance of language for paradigms--
since it is the means by which we learn science, dissemi-
naterunvfindings, and ultimately hold our world view--

Kuhn spends remarkably little time on it, but what he

1H8

does say is pertinent. The language used by scientists
"embodies a host of expectations about nature and fails to
function the moment these expectations are violated." He
adds:
This is not to suggest that pendulums, for
example, are the only things a scientist could
possibly see when looking at a swinging stone
. But it is to suggest that the scientist
who looks at a swinging stone can have no ex-
perience that is in principle more elementary
than seeing a pendulum. The alternative is not
some hypothetical "fixed" vision, but vision
through another paradigm, one which makes the
swinging stone something else [81].

E. Fred Carlisle, in his discussion of discourse as
it relates to both literature and science [2h], points out
similar attitudes in the writings of other philoSOphers of
science [59,67,85,l33] and makes a beginning at developing
some of these ideas. Within the framework of James M.
Edie's so-called appresentational View of language [”3],
Carlisle applies the four levels of reference, or layers
of meaning in discourse, to the way that words bear meaning
in scientific, as well as literary language.

"Words designate and signify. They point to experi-
enced things and they carry meaning. These two primary
functions permit us to organize and communicate experience"
[24]. This is the level of the "given," that which is
directly and intentionally presented. The three other

levels of reference refer to the "given-with," that which

is appresented, only implied or expressed indirectly.

149

Discourse "directs attention toward our experience
itself . . . toward the ways we organize and interpret
experience and differentiate regions of experience" [24].
According to Carlisle, areas of discourse develop mainly
through shared interest, attention ("the way one attends
to reality"), and intention ("how one takes reality").

Discourse, as a form of intersubjective communica-
tion, also refers

to the expressive-communicative context out
of which it arises . . . through expression and
communication, speakers and listeners . . . build
up a store of sedimented meanings and functional
codes that define regions of discourse and con-
stitute the world view of a linguistic community
or the scientific and artistic vision of more
specialized communities [24].
Although this level obviously overlaps the previous one,
the emphasis here is on the speaker and the listener them-
selves, the people involved in the diScourse, while the pre-
vious level emphasizes the experience that provides the
basis for the discourse.

Finally, discourse contains "the 'store of sedimented
meanings,'" which include "the circumstances, institutions,
and traditions within which a discourse occurs and out of
which it arises, and it also involves the original, prior
meaning of words or a discourse" [24].

The three appresentational levels of reference in par-

ticular indicate that the adherents to a particular para-

digm speak their own language [149]. In a paper on the

150

philosophy of chemistry, D. W. Theobald [160] suggests

that there are many answers to the question "What is an
electron?" A solid-state physicist has a different con-
ception of an electron from that of an organic chemist,

and indeed their answers may even be incommensurable with
each other, yet each is a valid answer within the framework
of its appropriate paradigm. This indicates why scientists
from different disciplines frequently have difficulty dis-
cussing topics of supposedly mutual interest with each other.
They may be using the same words, but with different ap-
presentational levels of reference.

Thus we see in paradigms the complex network of implica-
tions that can function for models as Black's commonplaces
function for metaphors. Just as a layman can evoke from
his general background a list of commonly held ideas about
pretty little maidens in order to complete the metaphor, a
physical chemist can evoke from his paradigmatic background
a comparable list of commonly held ideas about charge and
ions and electric potential in order to evaluate the model
we have chosen to study membranes. If a listener from a
culture where there are no pretty little maidens or where
ideas about them are vastly different from those of the
speaker, the listener may either not understand the meta-
phor or ascribe to it a different meaning from that in-
tended. Similarly, an adherent to a paradigm in which
charge, ion, and electric potential have no meaning or

different meanings from those in physical chemistry may

151

construe our model differently than we do.
At last, we have reached a position to consider the
final topic of this chapter, the ways that models, para-

digms, and language shape scientific explanation.

E. Explanation

 

Margaret Boden strikes close at least to the dictionary
definition of "explain" when she says that it is to make a
thing "somehow more intelligible" [16]. Ian T. Ramsey
speaks of "insight" that allows us to "articulate" [136].
For W. H. Leatherdale, it is the retention of this insight
that is essential to understanding and enables other people
as well to understand the discovery [86]. Karl Popper
relates this to "testability" [134]. There are as many
views of explanation as there are explainers [3,6,19,64,
72,163,170], but the one idea that constantly recurs is
that of the analysis of complex phenomena into the "natural"
or "familiar" that requires no further explanation.

If the listener can somehow decompose the phenomenon
being explained into terms that are familiar to him, he
feels that he has grasped its meaning; he can make the
phenomenon compatible with his own framework of knowledge.
In terms of scientific explanation then, it is the frame-
work of knowledge developed within a paradigm that enables
its adherents to "understand," both as students trying to

learn the paradigm and as practicing scientists trying to

152

work out its implications.
This agrees with the view of D. W. Theobald, who dis-
cusses explanation in terms of emphasis and style:
Explanations have to be rational . . . Rationality
means no more than that explanations have to be

commonly reasonable in the circumstances, and this
alone guarantees our understanding [160].

 

Although a paradigm cannot be said to guarantee understand-
ing, it certainly provides the "natural" and "familiar,"
the language for articulation, the means for insight, the
criteria of testability, all the necessities for rationality
within the circumstances. Indeed, each paradigm is its own
justification for the explanations that it sanctions.

This makes it clear why the solid-state physicist and
the organic chemist cannot accept each other's explana-
tions about the electron. Committed to a particular para-
digm and the understanding that it affords, a scientist
can understand only in terms of that paradigm; he has
neither the means nor the inclination to understand in
terms of someone else's. He knows what constitutes an
explanation--his paradigm tells him--and therefore it is
not what the adherent to another paradigm says it is. The
unwary scientist may become a victim of his paradigm in
the sense that Colin Turbayne refers to as being used by
metaphor:

the victim of a metaphor accepts one way of sort-
ing or bundling or allocating the facts as the

153

only way to sort, bundle, or allocate them. The
victim not only has a special view of the world
but regards it as the only view, or rather, he
confuses a special view of the world with the
world. He is thus, unknowingly, a meta-
physician [166].

So far we have discussed only the circumstances that
allow explanation. In order to try to look at explanation
itself, we return to theory. As one of the characteristics
of theory, Romer lists explanation, which, he says, "im-
plies that we have identified the true cause of it" [142].

We have three objections to this. First, in terms of
paradigms, there can be no true causes in the absolute
senSe, only appropriate or rational ones. Science is not
some asymptotic approach to Truth, but a succession of in-
creasingly successful visions of various truths. Secondly,
the theory, at least its calculus, can often be applied to
quite different kinds of phenomena. For example, even a
quick look at chapters on heat flow in cylinders [25] and
chemical diffusion in cylinders [32] reveals the mathematical
similarities to our equations of potential distribution in
a cylinder. There is nothing in the theory itself to explain
the particular set or sets of phenomena that it applies to.

Finally, and most compellingly, mathematics, which
is the basis of theory, is a descriptive language; it pro-
vides no explanations.

Mathematics can be expected to do no more than

draw consequences from the original empirical
assumptions. If the functions and equations

154

have a familiar form, there may be a background
of pure mathematical research readily applicable
to the illustration at hand. We may say, if we
like, that the pure mathematics provides the
form of an explanation, by showing what kinds
of function would approximately fit the known
data. But causal explanations must be sought
elsewhere [1 .
The continual appearance of Bessel functions, both regular
and modified, in our attempts at a solution for wl(z)
results from the mathematical analysis of the problem and
provides no basis for explaining that particular form of
the electric potential.

If theory does not explain, do theoretical models? A
number of philosophers of science think so, but how these
models provide an explanation is not always clear [5,73,
100,161]. Mary B. Hesse discusses explanation as a meta-
phoric redescription of the phenomenon to be explained in
terms of the model [65]. The model's terminology is trans-
ferred to the principal subject, whose own observation
language is both altered and extended. Peter Gardenfors
suggests that such redescription must be evaluated within
a "knowledge situation" [50], i.e., in our terms, within
a particular paradigm.

With our idea of model associated with an analogue
A analogous to H, we can diagram a model (Figure 6.2)
as we did theory earlier. In this case, however, the

concepts CA and laws LA are valid for the model rather than

any physical system. The replacement of the theory

155

A

\ /CA

Figure 6.2. Interrelationships involving the analogue
associated with a model.

 

diagrammed in Figure 6.1 by the model diagrammed above
constitutes this metaphorical redescription and provides
the basis for explaining the phenomenon. Concepts C and
L are replaced by concepts CA and laws LA respectively,
and this tacit correspondence imparts the physical reality
to our model that provides it with its explanatory power.
As Herbert Butterfield says,
this modern law of inertia is calculated to
present itself more easily to the mind when a
transposition has taken place--when we see, not
real bodies, moving under the restrictions of
the real world and clogged by the atmosphere, but
geometrical bodies sailing away in empty Euclidean
space [22a].
Michael Scriven charges that all laws of nature are
known to be in error [148]. This seems natural if we

consider these laws refer directly to a model and only

indirectly to a genuine physical system. This is implicit

156

in a scientist's discussion of a particular law and the
reasons why it does not describe nature exactly. There
are always conditions or restrictions on the physical
system that must be met, restrictions that in essence
define the model for which the law is exact. Thus models
are substitutes for real systems, and this provides them

with explanatory power.

F. A Last Look at Electric Potential.

As we remarked earlier, there are numerical solutions
of the Poisson-Boltzmann equation for a system similar to
ours [46,55,144]. We have claimed that they offer numbers
with little accompanying explanation. What of our method?

We separate the total potential inside the capillary
into wl(z) and w2(z,r), in the latter of which, we claim,

z and r cannot be further separated. In the first papers
that we can find that make this separation, the authors
consider w1(z) to result macroscopically from the imposi-
tion of a concentration gradient across the interface
separating the two solutions [46,116]. This is essentially
the diffusion potential across the interface. Once a
capillary structure is assumed for the membrane, ¢2(z,r)
represents the distortion of the original potential; i.e.,

it is a measure of the effects of the departure from electro-
neutrality.

The separation of the total potential into these two

157

parts is very closely related to the model explaining
capillary osmosis, which is the volume flow in the presence
of a concentration difference, but in the absence of applied
pressure and electric potential gradients [37]. B. V. Derya-
gin and his colleagues propose that in the presence of a
tangential concentration gradient, the interaction between
the ions and a charged solid surface produces microgradi-
ents of both pressure and electric potential [36,38,39,4l].
These induced gradients then produce a volume flow analogous
to that of reverse osmosis and electroosmosis respectively,
so that adherents to paradigms in which these two phenomena
are familiar have received a satisfactory explanation of
anomalous osmosis.

Electroosmosis, for example, may be familiar to certain
scientists in terms of ionic rearrangement under the in-
fluence of an electric field and the resulting flows of
ions along with their associated hydration shells [7].

This might then be viewed in the context of the Kedem-
Katchalsky formulation of nonequilibrium thermodynamics

[75] or a more rigorous Onsager formulation [57], but in
any case, the imposition of a model occurs within a para-
digmatic context with which the scientist feels comfortable.
Thus if our model and treatment of the electric potential
within a charged capillary membrane make sense in terms of
the scientific commonplaces that they evoke, then we may

feel satisfied that we have explained this particular

158

phenomenon by presenting it in terms that need no further

clarification.

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