National Bureau of Standards

dg

JUL 1 6 1963

^ecknlcai <ylote 192

CALCULATIONS OF THE POTENTIAL AND EFFECTIVE

DIFFUSION CONSTANT IN A POLYELECTROLYTE SOLUTION

SAM R. CORIELL AND JULIUS L. JACKSON

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

CecAmccJ ^ote 192

ISSUED JUNE 28, 1963

CALCULATIONS OF THE POTENTIAL AND EFFECTIVE

DIFFUSION CONSTANT IN A POLYELECTROLYTE SOLUTION

Sam R. Coriell and Julius L. Jackson

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. - Price 25 cents
Contents

Page

1. Introduction 1

2. Model and Calculations 3

3. Discussion 7

II
CALCULATIONS OF THE POTENTIAL AND EFFECTIVE DIFFUSION CONSTANT

IN A POLIELECTROLYTE SOLUTION

Sam R. Coriell and Julius L. Jackson

The results of numerical computations of the electrostatic potential and the effective diffusion constant

of counterions in a polyelectrolyte solution are given.

The potentials for various polyion charge densities

and polyion sizes are presented graphically. The calculated diffusion constants are compared with experimental data on the diffusion of labeled sodium ions in

polyacrylic acid-sodium hydroxide solutions.

1. Introduction

The diffusion of ions in a periodic electric field was considered

as a model for the diffusion of radioactive, counterions in a polyelectrolyte solution by Lifson and Jackson. On the basis of the

theory of first transit times, they obtained an explicit expression for

the observed or effective diffusion constant D' in terms of the hydrodynamic diffusion constant D and the periodic, potential ?(r) for a one dimensional model. Jackson and Coriell (JC) have given a general

theory of the effective diffusion constant of ions in a periodic

electric field in any number of dimensions. Ths calculation of the

diffusion ratio D'/D is reduced to the solution of a particular boundary

value problem over one periodic unit or cell.

The effective diffusion constant D' was shown to be given by

dV

t — I a"*

(1)

dV

Here D is the hydrodynamic diffusion constant and is the reduced

potential, ef/kT. The integrations are over one unit cell. The function

g is a solution of the partial differential equation

6XE> dx

V g - V|

with boundary conditions (for a cube of side 2^ )

(2)

1

S. Lifson and J. L. Jackson, J. Chem. Phys. ^6, 24-10 (1962).

" J. L. Jackson and S. R. Coriell, J. Chem. Phys. 38, 959 (1963).
|(x,^o ,z) = (3)

I

s (x, y, ±t ) =

dz ' J ' o'

For the periodic model of a polyelectrolyte solution, the potential

9(r) is assumed to be given by the solution of the Poisson-Boltzmann

Equat ion

V

2

5(£) = _ ^

{

- p(r) + K exp [ - e*(r)/kT] }, U)

with V? • dS = on the surface. Here e is the absolute value of

the electronic charge, k is the Boltzmann constant, T is the absolute

temperature, and e is the dielectric constant. [ -ep(r)] is the charge

density of the polyelectrolyte and eK exp ( - e'P/kT) is the charge

density of the mobile ions. The constant K is arbitrary, determining

the zero of the potential. In JC, values of the effective diffusion

constant ye re obtained by approximate methods the electrostatic

potential in the polyelectrolyte solution was approximated by adopting

the results of calculations by Wall and Berkowitz for spherical geometry and bounds on the effective diffusion constant were obtained by

using variational principles for D' rather than following the straightforward procedure of solving Eq. 2 and inserting the result in Eq. 1.

In this article, we report on the results of numerical computations

for the potential in a polyelectrolyte solution and the effective

diffusion constant. We present graphs of the potential for various

charge densities and polyion sizes along with the corresponding values

of D ' . The calculated values of the effective diffusion constant are^.

compared with the experimental results of Huizenga, Grieger, and Wall

for the diffusion of labeled sodium ions in polyacrylic acid-sodium

hydroxide solutions. The values of the effective diffusion constant,

obtained numerically, in most cases fall within the bounds which were calculated in JC for an approximate potential.

For our purposes here, we will regard the polymer charge distribution

as a given function of r, rather than attempt to write its dependence

on the potential, an approach which introduces a greater order of

complexity.

* (a) F. T. Wall and J. Berkowitz, J. Chem. Phys. 26, 1H (1956); (b)

J. Berkowitz, Ph.D. thesis, University of Illinois, 1955.

5 J. R. Huizenga, P.F. Grieger, and F. T. Wall, J. Am. Chem. Soc. 72,

^228 (1950)
2. Model and Calculations

We assume the following model for a polyelectrolyte solution.

Each poly ion is assigned to an identical cube of volume V, the center

of the polyion coinciding with the center of the cube. Ths center of

the cube is taken as the origin of an (x, y, z) coordinate system. The

distribution of counterions in each cube is proportional to exp (-ef/kT)

.

Two different models have been employed for the polyion charge density,

- ep (r) . In the uniform sphere model the polyion charge density p is

uniformly distributed over a sphere of volume v, and is zero outside of

the sphere. In the Gaussian model a Gaussian distribution of polyion

charge is assumed, i.e., p = a exp (-br ), where a and b are constants

which will be specified subsequently. The potential •£ Is given by the

Poisson-Boltzmann equation (Eq. J+) • Wall and Berkowitz have solved

this equation for the uniform sphere model with spherical symmetry, i.e.,

the boundary conditions are specified on a sphere rather than on a cube.

In terms of the polyion concentration c, and the degree of polymerization s, the volume V per polyion is

Tr - 1000s , ($)

3 where V is in cm. , c in moles of monomer per liter, and N is Avogadro's

number. The charge density p on the polyion is related to per cent

neutralization p by

K

pdv = So > (6 >

where the integral is over the volume of one cube. For the uniform

sphere model, the above equation becomes

Av = A (4/3) nr

Q

3 = ps/100 , (7)

where r is the radius of the sphere or polyion and -eA is the polyion

charge density in the sphere.

The Poisson-Boltzmann equation, Eq. (4) , may be written in a more convenient form by introduction of the reduced potential O = e^/kT and

dimensionless variables (X,.Y, Z) such that X = 4, I = y/t , and Z = z/t , -where 2l is the length of the edge of the periodic cube,

i.e., V = 8v . Eq. (4) then becomes

2 * = da" l

o { ' - K exp ( -*> }

(8)

p p Since K is arbitrary, we can take it equal to (ekT/^ne I ). A different

choice of K simply changes ¥ by a constant amount. For the unifo:

sphere model, the Poisson-Boltzmann Equation can be written as

)rm
V

2 = Cl S(R) - exp (-0) , (9)

2 p 2. —

where R = [X + Y + Y ] 2

, S(R) is a step function eaual„to unity for

R<r A and equal to zero for R > r /I , and c. = 4Ife I A/ekT.

Using Eqs. ($) and (7), c. may be written as

c„. =

3e

2

p(s

2

cN)

L/^ 3

o (10)

; 1 " 3

500 ekT r

o For T = 298°K, and e = 78

C;L = .036^5 pCs

2^

3

(l /r

Q

) 3

. (11)

The boundary conditions on the Pois son-Bolt zmann Equation, Eq. (9), are

f(X =± 1) = 0, g(T=±l) = 0, and

§|(Z =±D - 0. (12)

Because of the symmetry of the potential, i.e., $(X, Y, Z) = $(±X, ±Y, ±Z) , it is sufficient to solve the differential equation in one octant of the cube, e.g., for 0<X<1, < Y < 1, and < Z < 1.

The additional boundary conditions used are

:§(X =0) - 0, ||(Y =0) =0, and

§(Z = 0) =0, (13)

which are a result of the symmetry of the potential. It should be

noted that the potential is completely specified by the two parameters

c, and r /t . 1 oo

A similar analysis will yield the relationship between the parameter of the Gaussian distribution of polyion charge and the experimental

quantities. Subsequently, a comparison of the uniform sphere model and

the Gaussian model will be made. We shall wish to relate results for

the two models which correspond to equal total polyion charges whence

J a exp (-br

2

) dV = U/3) nr

Q

3

A. (H)

For the values of b of interest in this investigation, it is an

excellent approximation to write
J a exp (-br

2

) dV = #Ia J r

2

exp (-br

2

) dr = a (n/b) 3^2

. ^

To complete the specification of the relationship of the parameters of

the two models, it is necessary to choose some criterion for the size

of the polyion. It appears reasonable to require that the radius of

gyration of the polyion in both models be equal, i.e.,

J r2

p dV = J r2

a exp (-br

2

) dV. (l6)

V s V

Evaluating the integrals with the same approximation as previously for

the Gaussian model, one obtains

A r

o

5 = (15/8) a (n/b 5

) l/2

. (17)

Combining Eqs. (14) , (15), and (17) yields

b = 5/(2.r

Q

2

) and a = (250/9n) 1/^2

A = 2.9735 A . (18)

The Poisson-Boltzmann equation for the Gaussian model may then be

written as

V

2

<D = 2.9735 c 1 exp[-(5/2)(£

o/ro

) 2

] - exp [-*] . (19)

In order to calculate the effective diffusion constant of counterions in a polyelectrolyte solution, the following steps are necessary:

(l) solution of the appropriate Poisson-Boltzmann equation for the

potential in a polyelectrolyte solution, (2) solution of the differential

equation for g(x,y,z) using the potential obtained in the previous step,

and (3) evaluation of the integrals necessary for the calculation of the

diffusion ratio. We have shown how the parameters in the PoissonBoltzman equation are related to the experimental quantities the per

cent neutralization, degree of polymerization, and the monomer concentration. It is then possible by calculating the diffusion ratio for

different values of these parameters to obtain the diffusion ratio as a function of the per cent neutralization of the polyion.

The solution of the differential equations for and g and the

evaluation of the integrals required for the calculation of the diffusion

ratio have been carried out numerically on an IBM 7090 computer. The

differential equations were solved by the method of succesive block

overrelaxation, '

R. J. Arms, L. D. Gates, and B. Zondek, J. Soc. Indust. Appl. Math.

4, 220 (1956).

This method was suggested and applied to the above differential

equations by R. J. Arms of the National Bureau of Standards. The

Fortran program was written by P. J. Walsh of the National Bureau of

Standards

.
V^ $ dV = Hence > the following relationship

-4>

using a 20 x 20 x 20 grid of points. Iterations were continued until

the maximum change in the function (g or $) in sucessive iterations was less than 10 ^ . Usually this required about 70 iterations. The boundary conditions that the normal derivative vanish on the boundary of the

cube were satisfied by requiring that the value of the function on the

boundary equal its value at the neighboring interior point.

There are additional checks on the solution of the differential

equations. From the. boundary conditions on $ and the divergence theorem,

it is evident that

for the volume integral of e is obtained

J

e^ dV = (411/3) i^l

Q

) 3

Cl . (20)

Since the integral of e is computed numerically in the calculation

of the diffusion ratio, we can compare the numerical value with that

given by Eq. (20). For the 16 calculations made, the average error

in the two values was 2.4% with a maximum error of 6.2%. Usually, the

numerical value was smaller than the value calculated from Eq. (20).

A similar check can be used on the solution of the differential

equation for g. In JC, it was shown that the average current of counterions in the x-direction is proportional to

r+lo ?+lo

I = \_i J_£ e~° (l + m) dy dz (21)

o o

and that hence in the steady state the above integral is independent of

x. Since the calculation of the diffusion ratio requires the evaluation

of this integral, it is possible to check the constancy of this integral

for each of the twenty values of x. If, as a measure of the variation

in I , we take the percentage difference between the maximum and minimum values of I , we find the average of these percentages for the 16

calculations is 3.8%. The largest percentage difference is 10.8%

The results of the numerical calculations of the effective diffusion

constant are summarized in Table 1. The parameters c_ and r /t

appearing in the table are those which occur in the Pois son-Bolt zmann

equation, Eq. (9) for the uniform sphere model and Eq. (19) for the

Gaussian model. In Figures 1 - 16, the potential is plotted as a function of the distance R from the center of the cube along the three

basic directions in the cube, viz. the x axis, the x-y diagonal, and the

x-y-z diagonal. The parame trie equations of the three lines are:

I II III

X = t x = t x = t

y=0 y=t y = t (22)

z=0 z = z = t
3. Discussion

It is apparent from the curves for the potential that except near

the boundaries of the cube the potential is essentially a function of R

alone, and not of the angular coordinates. Also for small values of R,

the Donnan approximation (the assumption of zero net charge) is excellent

for the potential.

For the size ratio of .684., the uniform sphere and Gaussian models

can be compared. The Gaussian model gives considerably larger potential

differences than the uniform sphere model. This is due to the fact that

the polyion charge density at the center of the cube in the Gaussian model

is approximately three times as great as in the unifrom sphere model. However, even though the potential differences are larger for Gaussian model,

the effective diffusion constant is usually greater for the Gaussian model,

i.e., the retarding effect of the field is smaller. As the total charge

is increased, the diffusion constant for the spherical model becomes smaller

than its corresponding value for the Gaussian model. Even though the total

change in potential is not as great for the uniform sphere model, the

potential changes much more rapidly in the spherical model, i.e., the maximum electric field is greater for the uniform sphere model than for the

Gaussian model.

It is interesting to note the effect on the potential of changing

the polyion concentration c, leaving the other parameters, such as the

per cent neutralization and the size and shape of an individual polyion,

unchanged. Then from Eq.ll, as £ is proportional to c~^-, the constant

C]_ is proportional to o~ ^

*

. Thus going to lover values of the polyion

concentration corresponds to higher values of c-^ and smaller ratios of

r /tQ. As the effect of both of these changes is to increase the calculated potential difference, we see that one can expect higher potential

differences at lower concentrations (for fixed polyion sizes). The

numerical calculations S2 and S10 can be interpreted as corresponding to

essentially the same values of p, s, and r

Q

, with S2 corresponding to a

4-. 6 fold dilution from the polyion concentration of S10. The potential

difference in this dilution changes from 3.4-5 at the higher concentration

to $.10 at the lower concentration. Wall and Berkowitz (Figure 9 of

reference 4-a) have also shown this increase in potential upon dilution.

It seems likely that in actual polyelectrolyte solutions, the polyion

will expand upon dilution and that thus the potential will not increase

by such a large amount.

The diffusion constant of radioactive sodium ions in polyacrylic

acid-sodium hydroxide solutions has been measured by Huizenga, Grieger,

and Wall. 5 These experiments were carried out at three different monomer

concentrations and for various percentages of neutralization of the

polyacrylic acid. The degree of polymerization of the polyacrylic acid

was approximately one thousand (1000). In Table 2, the experimental

results are summarized. Using Eq. (ll) the per cent neutralization

corresponding to the numerical parameters c-j_ and rQ/lQ can be calculated

for the three poly ion concent rat ions. These values are given in Table 3.

In Figures 17 and 18, both experimental and theoretical values of the

7
diffusion ratio are plotted as a function of the percentage of neutralization for the polyion concentrations of .0151 and .0378 eq./l., re- spectively. It is apparent from Fig. 17 that the calculations with

r /t = 13/19 = .684. for the uniform sphere model fit the experimental

data extremely well for all percentages of neutralization for the polyion

concentration of .0151 N. For .0378 N polyacrylic acid, the theoretical

points for r /t- = .684- for the uniform sphere model lie slightly below

the experimental results, indicating that a slightly larger value of the

size ratio is required to obtain exact agreement between the experimental

and theoretical diffusion ratios. Although no graph is given for .00378 N

polyacrylic acid (since there is only one experimental measurement), it is

evident from Tables 2 and 3 that the numerical results with r /v = .684. are again in excellent agreement with "the one experimental point Thus, it

is possible with a single value of the parameter, r /t , to obtain good

agreement with the experimental results at three different polyion concentrations and at all percentages of neutralization.

For the Gaussian model of the charge density, a single value of the

size ratio will not fit the experimental data at all percentages of

neutralization. To obtain agreement with the diffusion data, it is

necessary to assume that the size ratio decreases slightly as the percentage of neutralization is increased.

Before proceeding with the discussion of these results, it is

desirable to point out that in the theory of the effective diffusion

constant which has been presented, it has been assumed that none of the

counterions are localized or site-bound to the polyion. If some of the

counterions are not free, then this would change the values obtained for

the effective diffusion constant in the calculation. The theory given in

this report applies to the "free" counterions in the polyelectrolyte

solution. Of course, the "free" counterions may be somewhat bound to the

polyion by the electrostatic forces, but are not localized to a specific

site on the polyion. If the fraction of free counterions is denoted by f

and if the dependence of the effective diffusion constant on the percentage

of neutralization is denoted by D' (p), then the observed diffusion constant will be given by

D

Qbs

= fD'(pf). (23)

Thus, if f was known as a function of p, then the above equation could be

used to obtain the observed diffusion constant as a function of the percentage of neutralization.

Assuming for the present that there are no localized ions, the

numerical calculations indicate that the experimental diffusion data can

be explained by taking a single size ratio, viz., r

Q/l = 0.684-. For a polymer of degree of polymerization of 1000, the monomer concentrations of

.00378, .0151, and .0378 moles/liter correspond to values of l of 380,

240, and 176 A, respectively. The radii of the equivalent spheres needed

to fit the diffusion data are then 260, 164, and 120 A for the three con- centrations. The radii of gyration of these spheres are 201, 127, and 93

A. For a random walk model of polyacrylic acid with a statistical element

8
of four monomers, the radius of gyration of an "uncharged polymer of degree

of polymerization of lCr is 66 A. Experimental measurements by Oth and

Doty^ on polymethacrylic acid indicate that this is a reasonable estimate

of the size of the uncharged polymer.

The result that the size ratio necessary to fit the experimental

diffusion data is essentially independent of the polymer concentration

implies that the polyion size decreases as the polymer concentration

increases. This conclusion is in qualitative agreement with viscosity

data on polyelectrolytes. However, the result that the experimental

diffusion data can be explained by a single value of the size ratio which

is independent of the percentage of neutralization is in strong disagreement both with theory and with experimental measurements of the size of

polyions as a function of percentage of neutralization. Since most

methods of determining polymer size involve extrapolation to zero polymer

concentration, no data are readily available on the size of polyions as

a function of percentage neutralization at finite concentrations* However, at zero polymer concentration, the results of Oth and Doty for

polymethacrylic acid indicate that the radius of the polyion changes by

a factor of five or six in going from zero to one hundred per cent

neutralization. Most of this change occurs at low per cent neutralization;

the root mean square end to end distance increased from 210 to 94-0 A. from

zero to 25 per cent neutralization, and at 70 per cent neutralization

reached a value of 1180 A. Although the change in size is probably not as large at finite concentrations, there is undoubtedly an increase in

size as the charge on the polyion is increased. The smallest percentage

of neutralization for which the diffusion ratio was measured was 9.6

per cent. To explain the results obtained in the present model, it is

necessary to assume that the polyion expands at low percentages of

neutralization (below 9.6%), and that the size then remains fairly constant. It should also be pointed out that at low percentages of neu- tralization a relatively large change in the size ratio has only a small

effect on the diffusion ratio.

Another possible explanation of the experimental results may be

given in terms of site binding. Thus, at high per cent neutralizations,

the polyion might expand to a size greater than those for which we have

done our calculations leading to a higher value of the effective

diffusion constant, and this increase could be compensated for by the

existence of bound ions.

The authors wish to thank Dr. R. J. Arms and P. J. Walsh of the

National Bureau of Standards for their assistance with the numerical

computations. We also wish to thank Dr. Shneior Lifson of the Weizmann

Institute of Science for his continued interest and for many helpful

suggestions.

A. Oth and P. Doty, J. Phys. Chem. j>6, A3 (1952).
Table 1

SUMMARY OF NUMERICAL CALCULATIONS UNIFORM SPHERE MODEL

No. C l

r

oA D'/D

SI 100.0 .474 .72

S2 500.0 .474 .27

S3 847.0 .474 .18

S4 33.18 .684 .88

S5 99.00 .684 .64

S6 165.9 .684 -51

S7 281.0 .684 .39

S8 382.0 .684 .33

S9 108.0 .789 .66

SIO 182.9 .789 .55

GAUSSIAN MODEL

No. c lr A D'/D

Gl 49.17 .600 .81

G2 245.9 .600 .45

G3 416.7 .600 .36

G4 33.19 .684 .86

G5 165.9 .684 .55

G6 281.0 .684 .46

10
Table 2

EXPERIMENTAL DIFFUSION RATIOS*

Poly ion Concentration Per Cent Neutralization D'/D

.00378 N 61.7 .63

9.6 .92

24.0 .79

.0151 n g;3

.62

.51

81.6 .39

97.9 .38

9.6 .88

24.0 .71

41.3 • 59

.0378 N g;2 .49

• 41

97.9 .38

* Da-t;a of Huizenga, Grieger, and Wall.

Table 3

RELATION OF NUMERICAL CALCULATIONS TO EXPERIMENT

Radius Per Cent Neutrali:zation

Numerical Ratio Polyion Concentration

No. D'/D r A .00378 N

o o .0151 N .0378 N

SI .72 .474 18.8 11.8 .8.7

S2 .27 .474 93.8 59.1 43.5

S3 .18 .474 159. 100. 73.8

S4 .88 .684 18.7 11.8 8.7

S5 .64 .684 55.8 35.2 25.9

S6 .51 .684 93.5 58.9 43.4

S7 .39 .684 158. 99.8 73.5

S8 .33 .684 215. 136. 100.

S9 .66 .789 93.4 58.9 43.4

S10 .55 .789 158. 99.7 73.5

Gl .81 .600 18.7 11.8 8.7

G2 .45 .600 93.5 58.9 43.4

G3 .36 .600 159. 99.9 73.6

G4 .86 .684 18.7 11.8 8.7

G5 .55 .684 93.5 58.9 43.4

G6 .46 .684 158. 99.8 73.5

11
-<D

l.V 1 1 1 1 1 1

SI

1 1

6.0 C, = I00 r /H = 474

D7D = .72

5.0

' -

4.0 -

3.0 \ -

2.0 ^- I

-

1.0 - ^

N

"***aaas=:^ TT TTT

ii

JJLL

1 1 1 1 1 1

1 1 8 6 20

I9l7£

24 • 28 32 36

Figure 1. Ths potential as a function of relative distance from the

center of the polyion.

12
7.0 i r

6.0

5.0

4.0

*

3.0

2.0

1.0 -

S2

V

D'/D = .27

C, =500 Xo /i

Q

=474

in

j L 4 8 12 16 20 24 28 32 36

I9r/i,

Figure 2. The potential as a function of relative distance from the center of the polyion.

13
7.0

6.0

5.0

4.0

-<j>

3.0

S3

C,=847 r /£ =.474

D7D=.I8

n

in

J I I L 4 8 12 6 20

I9r/i

24 28 32

Figure 3- The potential as a function of relative distance from the center of the polyion.

14
7.0

6.0-

-<&

8 16 20

I9I7JL

24 28 32

Figure 4.. The potential as a function of relative distance from the center of the polyion.

15
$

1 ,\J

6.0

1 1 1 1 1 1

S5

C, = 99. r /* =684 1 D7D = .64

1 i

-

5.0 ' —

4.0 - \ -

3.0 \ -

2.0

\>— I

TT

-

n

TTT ill

1.0 —

i i i i i i i 1 4 8 16 20

I9I7JL

24 28 32 36

Figure 5. The potential as a function of relative distance from the

center of the polyion.

16
7.0

6.0

S6

C, = I65.9 r /ft =.684

D7D=.5I

*

5.0

4.0

3.0

2.0

m

OL _L _L 16 20

l9r/JU

24 28 30 36

Figure 6. The potential as a function of relative distance from- the center of the polyion.

17
-*

l.\J

1 1 I 1 1 1 1

S7

i

6.0 C |=28l r /C =.684

D7D = .39

-

5.0 A -

4.0 \

-

3.0

\ — I

-

2.0

^

H

TTT

-

ill

1.0 - -

i i i i i i i 1 8 16 20

I9I7JU

24 28 32

Figure 7. Ths potential as a function of relative distance from the center of the polyion.

18
7.0 i r

6.0

S8

C, = 382 rn /Je n =.684

5.0

4.0

3.0

2.0

m

i.o

01 1

4 8 6 20

I9r/je rt

24 28 32

Figure 8. Ths potential as a function of relative distance from the center of the polyion.

19
-*

2.0 -

Figure 9. lentefofth.

1 & ° &

. function of "^tive distance from the center ol the polyion.

20
7.0

6.0

SIO

C, = 182.9 r /fl = .789

D'/D = .55

<S>

5.0

4.0

3.0

2.0 m

i.o

Ol L4 8 12 16 20

1917*

24 28 32 36

Figure 10. The potential as a function of relative distance from the center of the polyion.

21
7.0

"i r

6.0 C| = 49.I7 r /* =.600

DVD =.81

5.0

_<j>

2.0

-m

0L

4 8

J L 6 20

l9r/£„

J i 24 28 32 36

Figure 11. The potential as a function of relative distance from the center of the polyion.

22
7.0 i r i r

*

C,=245.9 r /£ =.600

DVD =.45

-m

1.0

j i

l _L

4 8 12 16 20

I9r/j?„

24 28 32 36

Figure 12. The potential as a function of relative distance from the center of the poly ion.

23
-$

7.0

6.0

5.0

4.0

3.0-

2.0-

C, =416.7 r /£ =.600

D7D = .36

m

1.0-

J L J L 4 8 12 16 20

I9I7JL,

24 28 32 3

Figure 13. The potential as a function of relative distance from the

center of the polyion.

24
7.0 i r

6.0

G 4

C

. = D'/D=.86

C =33.19 r /je = .684

5.0

4.0

*

2.0

4 8 12 16 20

I9|7JL

24 28 32 36

Figure 14. The potential as a function of relative distance from- the center of the polyion.

25
7.0

6.0 =

4.0 - -$

3.0

2.0

G5

C, = 165.9 r /{ =.684

DVD =.55

m

.0

J I 4 8 6 20

I9r/JL

J I 24 28 32 36

Figure 15. The potential as a function of relative distance from the

center of the polyion.

26
7.0

-*

4.0

3.0

2.0

C, =281 r /2 =.684

D'/D =46

m

1.0

J I 4 8 12 16 20

19 r/ft

24 28 32 36

Figure 16. The potential as a function of relative distance from the center of the polyion.

27
I 1 EXPERIMENTAL DIFFUSION RESULTS

NUMERICAL CALCULATIONS

SPHERE GAUSSIAN

.474 O .600

O .6 84 .684

A .789

D'D

40 60

PERCENT NEUTRALIZATION

100

Figure 17. The diffusion ratio D'/D as a function of per cent neutralization.

28
1.0

.2 -

EXPERIMENTAL DIFFUSION RESULTS

NUMERICAL CALCULATIONS

SPHERE GAUSSIAN

.474 O .600

<> .684 .684

A 789

.0378 N. POLYACRYLIC ACID

20 40 60

PERCENT NEUTRALIZATION

80 100

Figure 18. The diffusion ratio D '/D as a function of per cent neutralization.

29

IV U. S. GOVERNMENT PRINTING OFFICE : 1963 O - C91350

U. S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A.'V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards at its major laboratories in Washington, D.C.,and

Boulder, Colorado, is suggested in the following listing of the divisions and sections engaged in technical work.

In general, each section carries out specialized research, development, and engineeringin the field indicated by

its title. A brief description of the activities, and of the resultant publications, appears on the inside of the

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