34‘

DIFFUSION OF POLYELECTROLYTES IN
AQUEOUS SOLUTION

Thesis for the Degree of Ph; D -
MICHIGAN STATE UNIVERSITY

Ernest E. KeI'II 7

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This is to certifg that the

thesis entitled

"Diffusion of Polyelectrolytes in Aqueous Solution"

presented by

Ernest E. Kern

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Chemical Engineering

(’Zx/ /////({/ 1

Major professor

 

(‘ '

Date September 26, 1966

 

0-169

ABSTRACT

DIFFUSION OF POLYELEC TROLYTES
IN AQUEOUS SOLUTION

by Ernest E. Kern

The diffusion of polyelectrolytes was studied in order to
investigate the size and shape of the macromolecular ions in aqueous
solution. The translational as well as the rotational diffusion
coefficients were determined. The translational diffusion coefficients
were measured by a very accurate optical method involving the use
of an interferometer, and the rotational diffusion coefficients were
determined by the measurement of the electrical anisotropy effect
of the solutions in a shear field produced by a velocity gradient in
the liquid.

Also of interest was the diffusional driving force as related
to the electrostatic free energy of the counter ions and the polyions.

It was found that the diffusion coefficient for polyelectrolytes
is much greater than the diffusion coefficient of nonionizing polymers,
because of the greater mobility of the counterions as opposed to the
mobility of the polyion.

In the diffusion of polymethacrylic acid it was observed that
the polyion rapidly increased in size as the degree of neutralization
was increased. The increase in the size of the polyion was attributed
to the electrostatic repulsive forces between the charged groups on
the polyion chain. It was also observed that at low concentrations

the increase in size of the polyion occurred in two stages, whereas

ERNEST E. KERN

at higher concentrations the Opening of the polyion ocurred in one
step. At the high concentrations the polyion does not open as far as
it does at the low concentrations.

It was observed that the measured length of the polyion of
polymethacrylic acid at a low concentration and high degree of
neutralization approaches the fully extended length of the polyion

indicating that polyion is nearly completely extended.

DIFFUSION OF POLYELEC TROLYTES

IN AQUEOUS SOLUTION

BY

)‘ "
IL‘
L.
Ernest E. Kern

A THESIS

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

DOC T OR OF PHILOSOPHY

Department of Chemical Engineering

1966

a“.
\

\ I

z
p! ~21

To Christine

 

ACKNOWLEDGEMENT

The author wishes to express his appreciation to Dr. Donald
K. Anderson for his guidance during the course of this work.

The author is indebted to the Division of Engineering Research
of the College of Engineering at Michigan State University and to the
donor of the American Chemical Society Petroleum Research Fund
for providing financial support.

Appreciation is extended to William B. Clippinger for his
assistance in the construction of the laboratory apparatus necessary
for this research.

The understanding and patience of the author's wife, Christine,

is sincerely appreciated.

 

TABLE OF CONTENTS

ABSTRACT

ACKNOWLEDGEMENT ....................
TABLE OF CONTENTS ....................
LIST OF FIGURES ......................
LIST OF TABLES . . . . . ..................
INTRODUCTION ................ . .......
THEORY ............................

Diffusion ........................

Electrical Anisotropy Effect .............

EXPER IMENTAL METHOD ...............

Diffusion Apparatus ..................
Procedure for Experimental Run of Diffusiometer .

Calculation of Diffusion Coefficient ..........
Couette Apparatus ...................
Determination of the Size of the Polyion .......
Synthesis of Potassium Polyphosphate ........
Synthesis of Polymethacrylic Acid . . . . ......

RESULTS AND DISCUSSION. . ............. '. . .
CONCLUSIONS .........................

BIBLIOGRAPHY ........................

APPENDIX I - Sample Calculation of Diffusion Coefficient

APPENDIX II — Tabulation of Diffusion Data .........

APPENDIX III - Tabulation of Electrical Anisotropy

Effect Data . ................

APPENDIX IV - Nomenclature ................

24

24
32
34
39
45
48
51

55
74
76
8O

87

Figure

10

ll

12
l3
14
15

.16

LIST OF FIGURES

Degree of ionization (1' versus degree of
neutralization o .

Osmotic coefficient of counter ions versus
degree of ionization. From Reference 30 . .

Vector diagram in gap of Couette apparatus

Relative anisotropy of conductivity in X-,
Y-, and Z-directions as a function of the
velocity gradient q, for rod-shaped particles

Position of the major axis of the particle,
coordinates y and <13 . . .....

Position of the particle relative to the
directional measurement of conductivity,
coordinates 9 and [3 . . . . . . . .

Schematic diagram of interferometer showing
position of mirrors. . . .

Photograph of Mach— Zehnder interferometer
showing components . . . . . . .

Set of photographs taken during diffusion
run. (Experimental Run No. A- 6)

Diagram of diffusion cell . . . . . . . .....

Photograph of diffusion cell for measurements
of diffusion coefficients

Diffusion cell coordinates . ......
Fringe pattern .

Couette apparatus (outline). .

Couette apparatus without stator . . . .....

Stator for Couette apparatus . . . .....

The normalized anisotropy K((r) for the size
determination of rod-shaped particles based
on calculations by Schwarz Z and Gotz

Page

10
15

20

20

25

26

28

29

30
34
36
40
42

43

47

l8

19

20

21

22

23

24

25

26

27

28

LIST OF FIGURES (continued)

Plot of ns C versus concentration for
potassium olyphosphate in 10% solution
of tetramethylammonium bromide . . . . . . . .

Plot of reduced viscosity versus concentration
for polymethacrylic acid in 2N sodium hydroxide

Plot of intrinsic viscosity versus degree of
polymerization of polymethacrylic acid in 2N
sodium hydroxide. From Reference 25 . . . . .

Plot of diffusion coefficient versus degree of
neutralization for PMA at C = 0. 0320
monomoles/liter, moleculaigvvgeight =
470,000............... ..... ..

Plot of diffusion coefficient versus degree of
neutralization for PMA at C 2 0. 0320
monomoles/liter, moleculaiglezgéight =

127, 000 ............ . . . .......

Plot of diffusion coefficient versus degree of
neutralization for PMA at C = 0. 0213
monomoles/liter, moleculaig‘flgéight =
860,000............. .........

Plot of diffusion coefficient versus degree of
neutralization for PMA at C = 0. 0213
monomoles/liter, moleculaiglfzge'ight =
357,000 ........ .....

Plot of the relative change in conductivity versus
R. P.M. for PMA at 0. 0058 monomoles/liter,
parameter a = degree of neutralization . . .

Plot of the relative change in conductivity versus
R. P.M. for PMA at 0.0116 monomoles/liter,
parameter 0. = degree of neutralization .....

Plot of the relative change of conductivity versus
R. P.M. for PMA at 0. 0213 monomoles/liter,
parameter a 2 degree of neutralization. . . . .

Plot of the relative change of conductivity versus

R. P.M. for PMA at 0. 0291 monomoles/liter,
parameter a : degree of neutralization. . .

vi

50

53

54

56

57

58

59

61

62

63

64

29

30

31

32

33

34

35

LIST OF FIGURES (continued)

Plot of the relative change in conductivity at 400
R. P.M. versus degree of neutralization. Curve 1,
C = 0. 0058 monomoles/liter. Curve 2, C =

0. 0116 monomoles/liter ............ . .

Plot of the relative change in conductivity at 400

R. P.M. versus degree of neutralization. Curve

3, C = 0. 0213 monomoles/liter. Curve 4, C =

0. 0291 monomoles/liter ..............

Plot of apparent viscosity versus degree of
neutralization for PMA solutions, molecular
weight: 375,000 . . . . . ...... . .

Plot of apparent viscosity versus degree of
neutralization for PMA solutions, molecular
weight = 860, 000 ...............

Plot of apparent viscosity versus degree of
neutralization for PMA solutions, molecular
weight = 470, 000 O O O O O O O O O O O O O O 0

Plot of apparent viscosity versus degree of
neutralization for PMA solutions, molecular

weight = 127,000 ..................

2
1 1 .
Plot of ((Xj + xk)/(A + B) )avg. versus t1me

vii

65

66

68

69

70

71

86

Table

III

IV

VI

VII

VIII

IX

XI

XII

XIII

XIV

XV

LIST OF TABLES

Polyion size of polymethacrylic acid and
rotational diffusion constants of polymethacrylic

acid solutions.... ..... . . . . . .........

Diffusion data for potassium polyphosphate

s oluti ons .......................

Diffusion data fOr polymethacrylic acid solutions
at an average concentration 2 0. 0320 monomoles/

liter, molecular weight = 127, 000 ...... . . .

Diffusion data for polymethacrylic acid solutions
at an average concentration = 0. 0213 monomoles/
liter, molecular weight = 357, 000 ........

Diffusion data for polymethacrylic acid solutions
at an average concentration : 0. 0213 monomoles/

liter, molecular weight = 860, 000 . ...... . .

Diffusion data for polymethacrylic acid solutions
at an average concentration = 0. 0320 monomoles/

liter, molecular weight = 470, 000 .........

Apparent viscosities of polymethacrylic acid
solutions, molecular weight = 127, 000 . . .

Apparent viscosities of polymethacrylic acid

solutions, molecular weight = 357,000 . . . . . . .

Apparent viscosities of polymethacrylic acid

solutions, molecular weight = 860,000 . . . . . . .

Apparent viscosities of polymethacrylic acid
solutions, molecular weight = 470, 000

Conductivity data from Couette apparatus,
concentration = 0. 0291 monomoles PMA/liter

Conductivity data from Couette apparatus,
concentration = 0. 0116 monomoles PMA/liter

Conductivity data from Couette apparatus,
concentration : 0. 0213 monomoles PMA/liter

Conductivity data. from Couette apparauts,
concentration = 0. 0058 monomoles PMA/liter

Relative change in conductivity of polymethac rylic

acid solutions in Couette apparatus at 400 R. P. M. ,

Page

73

87

87

88

89

90

91

91

92

93

94

99

104

109

115

INTRODUCTION

In a macroscopic state of equilibrium in the absence of external
forces, the concentration of any molecular species, at constant temperature
and pressure, is uniform throughout any single phase. If the distribution
of molecules is not uniform, they will tend to move from regions of higher
to those of lower concentration. Such a flow is a necessary condition of
the second law of thermodynamics, since the entropy of the system is at
a maximum when the molecules are distributed uniformly throughout.
This process of flow, or diffusion results from the thermal energy of
the molecules which gives rise to their Brownian movement. The speed
with which the molecule diffuses is indicated by its diffusion constant
which is in part a function of the size and shape of the molecule.

The Brownian movement of molecules involves both translation
and rotation. The study of translational and rotational diffusion provides
a good set of physical methods for investigating the size and shape of
large molecules in solution.

In the determination of the translational diffusion constant, free
diffusion is permitted to take place; while in rotational diffusion, the
Brownian movement of the molecules is opposed by an external con-
straining force, which tends to orient them. This orienting force may
be a shearing force produced by a velocity gradient in the liquid. If
the molecules are electric dipoles, the external force may be an

electric field.

In order to better understand the rotational diffusion, let us
consider a system of ellipsoidal molecules in solution, whose
orientation may be described in terms of the angles between their
principal axes and some fixed direction in the surrounding medium.
If the molecules are oriented by an external force, so that their
principal axes are all parallel, and this orienting force is then
suddenly removed, the parallel orientation will gradually disappear.
In the final state of equilibrium, the distribution of orientations among
the molecules will be completely random. The rotational diffusion
constant is a measure of the speed with which this equilibrium is
approached.

In this study both the translational diffusion and rotational
diffusion constants of polyelectrolytes are investigated in order to
obtain the size and shape of the polyion of polymethacrylic acid. The
translational diffusion constants were measured with the use of a
Mach-Zehnder diffusiometer and the rotational diffusion coefficient
was determined by investigating the electrical anisotropy effect of the
polyelectrolyte solution.

Viscosity measurements were also made to verify the results

of the diffusion measurements.

 

THEOR Y

Diffusion

In 1855, Fick11 suggested that the process of diffusion could
be treated as analogous to that of the conduction of heat. Fick's
law of diffusion also holds true for electrolyte solutions. This

law of diffusion is represented by the equation
J = - D(dc/dx)

for one-dimensional diffusion. J is the flux or rate of material
transfer across a plane of unit area and dc/dx is the-concentration
gradient. D is the proportionality constant called the diffusion
coefficient.

The change of concentration with time at any point along the

diffusion cell in the x-direction is given by the relation
dc/dt = D ch/dxz (1)

which is Fick's second law.

These equations imply (that D is independent of the
concentration. In general this is not true, but the analysis of
diffusion experiments is usually made on the assumption that D
can be treated as a constant over a narrow range of concentrations.

The diffusion of polyelectrolytes in anionizing medium is unlike
the diffusion of other polymeric substances because of the presence
of ionized groups. The counterions of the polyelectrolyte have a much
greater mobility than the polyion. In order to maintain electrical

neutrality, the counterions exert a strong drag effect on the polyion,

thereby substantially increasing the diffusion rate above that of
nonionizing polymers.

Nernst36 showed that the differences in mobility of the
positive and negative ions would tend to separate them during
diffusion, which in turn would establish an electrostatic potential
gradient within the diffusion cell. This gradient will have such a
sign that it retards the faster ion and accelerates the diffusion of
the slower.

Nagasawa and Fujita3'7 have studied the concentration
dependence of the diffusion coefficient of a polyelectrolyte in aqueous
solution and have shown that the diffusion coefficient remains relatively
constant except at very low concentrations. The diffusion coefficient
of polystrenesulfonic acid was shown to drop Off abruptly at infinite
dilution.

Most diffusion studies of polyelectrolytes have been made
with an added electrolyte. The added electrolyte eliminates the
Nernst potential by suppressing the ionization of the counterions.
Malmgren35 determined the diffusion coefficient for polyrneta-
phosphates in 0. 4 molar sodiumthiocyanate and obtained a value of
l. 5 x 10"7 cmZ/sec. However Katchman and Smith'28 carried out
the diffusion of polymetaphosphates in the absence of an added salt
and observed the diffusion coefficient to be larger by a factor of
20 or more. Thus showing the effect of the counterions upon the
Nernst potential and in turn the effect upon the diffusion coefficient.

One of the most interesting properties of polyelectrolyte

solutions is the chain configuration as dependent on the degree of

ionization. The degree of ionization of a polyacid, such as poly-A
methacrylic acid, can be controlled by the addition of a strong base.
The degree of ionization is related to the degree of neutralization
c1 , as shown by Wall. 19
Katchalsky24 observed that the viscosity of polymethacrylic
acid solutions increase 100 times as the degree of neutralization
increased from 0 to 50%. This increase in viscosity is caused by
the expansion of the polyion. The unionized macromolecules are
hypercoiled by intramolecular hydrogen bonding between the acid
groups and therefore are relatively small. Upon addition of sodium
hydroxide to the aqueous polyelectrolyte solution, the ionic dissociation
leads to large repulsive forces among the charged groups in the chain.
These repulsive forces give rise to an expanded configuration of the
polyion.
The diffusion of a solute in solution can be described by the
mutual diffusion coefficient, D, which is given by

D : Nip 11%) p’T (2)

where n is the moles of solute per milliliter, p is the hydrodynamic
resistance of a single molecule, N is Avogadro's number, and [J is
the chemical potential of the solute. This equation also holds for

solutions of simple electrolytes. In this case:
9 = ”1 p1 + ”2 p2

in which 1/1, V2 are the numbers of ions per molecule and p 1, p2

are their hydrodynamic resistances.

The values of the hydrodynamic resistances are given by

Stokes'8 law
p : 6Trnor (3)

for a sphere of radius r, and

67m a
— ° (4)
p ‘ 1n 2a7b
for a rod with length of 2a and thickness of b, where no is the
viscosity of the medium.
From equations (2), (3) and (4) it can be seen that the diffusion
coefficient for a polyelectrolyte varies inversely with the size of the

24’ 26’ 38 have shown that the size of

polyion. Several investigators
the polyion increases with increasing degree of ionization. The
degree of ionization, a ' , for polymethacrylic acid is a function of
the degree of neutralization, a . This relationship was studied by
Huizenga, Grieger and Wall19 who developed a diffusion method for
determining the amount of counterion binding. They assumed that
the mobility Of the free ions (in their case sodium, Na+) is the same
as in simple electrolyte solutions and that the bound ions have no

mobility, and then calculated the amount of ion binding, f, from

the lowering of the apparent self-diffusion constant.

 

Thus,
D
Na
f = ,5 (5)
Na
with D DZ, the self-diffusion constants of sodium ion in a simple

Na’ Na

salt solution and the polyelectrolyte solution, respectively.

The diffusion constant is easily measured by use Of Nazz.

In one compartment of a diffusion cell, Which has a glass frit at the
center, is placed a solution having no isotope, and in the other
compartment is placed a solution having the same concentration
but containing Nazz. The apparent diffusion constant can be
calculated from the amount of Na22 transported across the glass
frit in a given time by using Fick's law.

If f is the fraction of bound sodium ions then, (1' = (1(1 -f).
The results are shown in Figure l.

The term n(d/J/dn) in equation (2) is related to the osmotic

pressure Tl' by the Gibbs-Duhem equation:
n(du/dn) = - ns(duS/dns) (6)

where the subscript s designates the sovlent. The osmotic pressure

of the solution is:

O
S)

I : -(1/VS)(/JS -u (7)

where vS is the molar volume of the solvent. From (6) and (7):
n(d/J/dn) = nsvs(d1r/n) (8)
assuming a dilute solution, so that nSvs = 1 then,

n(du/dn) = (dTT/dn) (8a)

Katchalsky and Lifson27 have shown that the osmotic pressure

for a polyelectrolyte solution is given by the equation:

1T 2 (MU + l)nRT (9)

(1',%

40

30L

20-

10—

 

Figure 1.

20 40
C1, 070

Degree of Ionization a'

Neutralization a .

60 80

versus Degree of

100

 

where <1) is the osmotic coefficient and V is the number of ionized
groups.
Differentiating (9) with respect to n and substituting into

(2) one obtains the following:

 

_ kT 81n[¢(U+1)l—l
D_—p—¢(u+1) 1+ alnn _l (10)

In the case where a substantial number of groups on the poly-
ion are ionized (1/ > > 1) and since the self-ionization of a weak
acid is small within a wide range of concentration, so that dV/dn = 0,

then equation (10) becomes:

D=%Iu(¢+§—%) (11)

W. Kern3O determined the osmotic coefficient of polyacrylic
acid, in salt-free solution as a function of the degree of neutralization.
His dataare represented in Figure 2. This data in conjunction with (11)
allow a qualitative determination of the diffusion coefficient.

A more theoretical approach would be to express the chemical
potential of the solute in terms Of electrostatic free energy of the
solution. According to the interionic attraction theory it can be
assumed that the chemical potential of an electrolyte in dilute solution
may be separated into a term M e1 for the electrostatic interaction Of
the ions and a term M O +(1/ + l)RT 1n m for the ideal behavior of a
dilute solute which dissocated into (U + 1) particles per molecule and
where m is the molality.

The chemical potential of the solute and solvent are related to

Osmotic c oefficient, (I)

 

 

10

 

 

 

F-
L I I l l
0 20 40 60 80 100
(1', 070
Figure 2. Osmotic Coefficient of Counter Ions versus Degree

of Ionization. From Reference (30).

Osmotic coefficient, (1)

10

 

 

O 20 40 60 80 100
CL', 70

Figure 2. Osmotic Coefficient of Counter Ions versus Degree

of Ionization. From Reference (30).

11

the total electrostatic free energy by the equation:

Neglecting the contribution of the solvent to the electrostatic free
energy one obtains the following equation for the chemical potential

of the solute.

8F1 0 m
I1 = e +u +(I/+l)RT1n—— (12)
an P T 1000

 

Katchalsky and Lifson26 calculated the theoretical electro-
static free energy for a polyelectrolyte in dilute solution and further
modifications were made by Harris and Rice. 16 The polyion was
represented by a chain randomly coiled except for the electrostatic
forces between its elements which were assumed to change only its
mean square length. The reference state is assumed as the hypo-
thetical state in which all ions carry no charge instead of the infinitely
dilute state. This choice is made because the electrostatic inter-
action in the polyion does not vanish with dilution.

Debye's method of charging a system is used in the transition
from the reference to the final state. This process is divided into
three separate steps: (1) The macromolecules are stretched to their
final end-to-end distance h and the free energy increased by F1;

(2) the ionic atmospheres are introduced at constant h increasing
the free energy by F2; (3) the fixed charges on the polyelectrolyte
are allowed to interact which produces the repulsive free energy F3.

According to Harris and Rice16 the free energy of the first

step will be:

12

2
_ 3 h h
F1 _ 7 k1? (if -1 -3 len (ho) (13)

0

where h2 is the average square end-to-end length of the polyion and

hO is the unionized mean square length defined by Kuhn32 as
h: = Zst (14)

where Z is the degree of polymerization, b the hydrodynamic length
of a monomer, and s the numer of monomers per statistical element.
The free energy of the second step, which introduces the ionic

atmospheres, is given by the Debye -Hiickel expression:

 

F : - um

where 6 is the electronic charge, V is the number of fixed charged
groups per macromolecule, De is the dielectric constant of the

solvent, and K is the reciprocal Debye radius which is given by

 

41TEZZn.

2_ i 1
K ‘ D kTV - HQ

6 m

where, Vm is the volume per macromolecule and ni is the number
of ions of type 1 per macromolecule.

The free energy attributed to the third step is due to the inter-
action of the various charges on the polyion. If the interaction energy,
uij’ between a pair of charges on monomer units 1 and j a distance

rij apart, is:

 

l3

2
6 exp( -KYi.)
u.. = J (17)
1‘] De Yij

 

Katchalsky"26 has shown that for a randomly coiled polymer Of length
h:

11']
II
’M
C
I

 

ln(l + 2 ) , (l8)

1/ e 6 8 1/2 h 6
F = —— - (“') —(
3 De Kh: 31T 2 Kh

 

 

)+... . (19)

The over-all electrostatic free energy per macromolecule

is given by the sum of the three steps.

 

2
_ -2 h— J:
Fel—F1+F2+F3— 2 kT(2—l)-3len(h)
h o
O (20)
_ EZKV + V262 6 _(£)1/2h_( 6)2+
3D D 2 3w 2 °
e e Kh Kh

Since equation (20) represents the electrostatic free energy of
the solute, then NFel is the partial molar electrostatic free energy

which is also equal to the electrostatic chemical potential of the

solute. Therefore:

Mel : NFel

Using the chain rule for derivatives and substituting equation (12) into

14

equation (2) we obtain

8F
_ 1_ e1 a_11_ (v +1)RT
D — p n (N 8h n + T? > (21)
where
2 2
g _ 3kT 3kT v e (_§_)1/2( 6 ) (22)
dh ‘ 2 " h ' 2D 3w 2 °
ho hO

With the information available it is not now possible to determine the
derivative dh/dn.

One can obtain the equilibrium end-tO-end distance for the
macromolecule by minimizing the electrostatic free energy as given

in equation (20). Therefore setting

dF
el : 0
V

we obtain:

 

 

15
Electrical Anisotropy Effect

It is desirable to obtain the polyion dimensions by an independent
means. The method of measuring the electrical anisotropy effect was
chosen as described by Gotzl4 and Heckman.

Anisometric particles which are electrically charged exhibit
an anisotropic electrical conductivity if these particles are oriented
by means of mechanical shear. The constant velocity gradient q,
necessary to produce the shear field can be induced in the gap between
the two concentric cyclinders of a Couette apparatus (Figure 14). The
rotating cylinders produce an approximately uniform shear field between
two sliding planes.

For anisometric particles, the probability of finding the longest
principal axes of the ellipsoids in the direction of flow (Y) is a maximum,
whereas for the shortest principal axes the probability is maximal in the

direction of the velocity gradient (X). (Figure 3). Therefore as shown

Z Direction perpendicular
T to the XY-plane

 

 

Y Direction
of flow
/< d
Gap Direction Of the
V X Velocity Gradient

Figure 3. Vector Diagram in Gap of Couette Apparatus.

16

by Schwarzzlr2 the electrical conductivity of the solution will increase
in the direction of flow.

For rod shaped particles the conductivity will decrease in the
X and Z directions. In the case of complete orientation the
conductivity change in the direction parallel to the axis of symmetry
of the particle (i. e. , in the direction of flow) is always twice the
decrease in the perpendicular directions.

The smaller the particles, the greater the velocity gradient
q must be in order to produce a partial orientation against the Brownian,
rotational movement. In resting solutions the random movement is

described by the rotational diffusion constant Dr

Z
_ 9
Dr _ ‘2? (24)

2 . . . .
where a) 15 the mean square value of a suff1c1ently small rotation Cb
that occurs in the time t around any axis perpendicular to the axis
of symmetry.

The size of the particle can then be related to the rotational

40 derived an

5

diffusion constant. Kuhn31 and other investigators

approximate relationship for rods and discs. For rods Of length 1:
1 (cm) 2 (8kT/TrnODr)1/3 (25)

where k is the Boltzman constant, T the absolute temperature and
no the viscosity of the solvent.

The rotational diffusion coefficient is determined by measuring
the electrical conductivity Of the polyelectrolyte solution in the direction
of flow (Y) in the Couette apparatus as a function of the velocity

gradient. (See pages 45 - 47.)

17

An anisometric particle in solution can be described approximately
as a rotational ellipsoid with an axial ratio given by p. A rod corresponds
to a stretched rotational ellipsoid with one long major axis and two short
minor axes. Therefore p approaches infinity for a rod and b, the shape

factor, approaches + 1, where

b = (p2 -1)/(p‘2 +1). (26)

b is to be used later in determining the electrical anistropy.

The conductivity KO of a solution at rest is proportional to
the average particle mobility WO . 9 W0 is composed of W1 and
WH , which are the mobilities of the particle perpendicular and
parallel to the major axis. Therefore

W“ + 2W_l_

WO 2 3 (27)

 

If N is the number of polyions in a unit volume of solution
and Q its effective charge, then according to equation (27) the

greatest possible conductivity change is,

"IE—Kzi

Z
0 3 NO (WH -W_L ) . (27a)

The qualitative dependence of the relative conductivity anisotropy upon
0' , where 0' is the ratio of velocity gradient to the rotational diffusion

constant
0' = q/D (Z8)

is given in Figure 4.

l8

 

 

 

 

 

 

 

0 q
K-Ko . .
:-——- -0, 5..- X-d1rect10n
lK-K I
O
-1. 0-L
+1. 0T-
_ Y-direction
0 +0. 5 .-
K-K
o
0 <1
0 q
K-Ko . .
_ -0. 5 .. Z-directlon
lK- l
o
‘1 . 0 d)

 

Figure 4. Relative Anisotropy of Conductivity in X-, Y-, and

Z-directions as a Function of the Velocity Gradient

q, for Rod—Shaped Particles.

19

Heckman and Gotz18 developed a relationship for the electrical
conductivity anisotropy as a function of the rotational diffusion constant
and the velocity gradient.

The velocity gradient developed in a gap of width (:1 is given by

dV
..1 2 RPM
«sec > = a? = 4%; (29’

where r is the inner radius of the rotor. (Figure 14)

It is necessary to maintain laminar flow conditions but since
the centrifugal force strongly stabilizes the laminar flow for the
apparatus with the outer cylinder rotating, 43 the transition to
turbulent flow occurs at relatively high Reynolds number. The critical
gradient qmax up to which the flow remains laminar depends on the

dimensions of the apparatus. Jerrard‘Z3 showed that

qmax(sec‘1) = f(rIdM/p (30)

where n/p is the kinematic viscosity of the solvent in stokes, and the
function f(r, d) is approximately 106 cm”.2 for this system.

The particle is characterized by a shape factor b according
to equation (26). The position of its major axis is described by the
angles 4: and V (Figure 5) and its rotary movement by <1) and Y .
Therefore the axial flux, f, which flows in the volume angle
9 = siny dd) dy is proportional to the angular velocity sinY - (I) on
the one hand and y on the other hand. In the velocity gradient the
particle performs an unsymmetrical rotational movement since CI)
and V are dependent upon the angle. Jefferyzz derived this dependence

for rotational ellips oids:

Figure 5.

Figure 6.

20

 

 

Position of the Major Axis of the Particle,

Coordinates Y and 43 .

 

 

X

Position of the Particle Relative to the Directional
Measurement Of Conductivity, Coordinates 9

and (3.

 

 

 

21

q (l +bcos 24>)

qbsinZY sin2¢

Jeffery ccnsiders only the case where the Brownian movement is
negligible and the orientation of the molecule is determined only by
hydrodynamic forces.

According to Peterlin39 one can now introduce a directional
distribution function for the particle axes <I>(¢,y, t). If N is the
number of particles per unit volume, dN the portion of particles

found in the volume angle with orientation angles of <1), v , then

-— = <I>(¢IYIt) d9 (32)

From the axial flux f(siny - (139) and equation (32) there results the

rotational flux Of the particle, which results entirely from the flow:
IF = <I>(<1>.v,t)° f(sinvciv') (33)

The Brownian rotational movement Opposes the orientation due to

flow, the flux resulting from the Brownian movement is given by:

JD = — Dr N grad <I>(¢, y,t) (34)

The time dependence of the particle density in the direction (I), Y

is given by the continuity equation:
(35)

Since steady state is attained rapidly, d¢/dt = 0, which gives

the result that the two fluxes are equal but in opposite direction. There-

 

22

fore one obtains the solution to the directional distribution function as
only dependent on 0' = q/Dr (equation (28)) and the shape factor b
(equation (26)).

The relationship between the distribution function and
conductivity anisotropy was recently examined by Schwarz. 42 For
any directional measurement of conductivity the position of the

particle is given by 9 and (3 (Figure 6) and the relative change in

conductivity is given by the equation:

T: :— (1-3COSZG)F1+Sln2COSZB'F2

(36)
+ . 2 e . Z . “0
$111 5111 [3 F3 K

where FD (11 — 1, 2, 3) are exponential series of the type:

the coefficient lni in the general case depends on t and the ratio

0' = q/Dr, for the steady state case however, lni depends only on
0' .
1 42
For 0' _<_ 2- , Schwarz has shown that F3 2 b (7/10 and

F1, F2 : 0 . Therefore equation (36) reduces to

K "' Ko 2 b0' K ' Ko
T = sin 9 sin Zfi m ' ——K——— (37)
O O

Alsofor 0' —>oo F1: 1/2, F2: —3/Zb and F3:0.

23

Heckman and Gotz18 calculated the theoretical curves for the
electrical anisotropy from equations (36), (37), and (38). By
comparison of the experimental curve with the theoretical curve one

can determine the rotational diffusion constant (see pages 45-47) and

with equation (25) obtain the length of the particle.

EXPERIMENTAL METHOD

Diffusion Appa ratus

The translational diffusion coefficients were determined with

a Mach-Zehnder3 4’ 51

diffusiometer. The diffusiometer measures
the refractive index of the solutions in a glass-windowed diffusion

cell with an optical interferometer. The instrument was patterned
after a similar diffusiometer described by Anderson,1 Caldwell, Hall,
and Babb4’ 5 and was constructed by Bidlack2 for the study of the
diffusion of binary nonelectrolyte liquid solutions. No modifications
were necessary for the study of the diffusion of polyelectrolytes.

The Mach-Zehnder interferometer consists of a system of
four mirrors positioned at the corners of a parallelogram. A schematic
diagram in Figure 7 shows the location of the mirrors and other
component parts. Mirrors 1 and 4 are half silvered mirrors which
transmit one -half of the incident light beam and reflect the other one-
half. Mirrors 2 and 3 are full reflectors.

The light beam originates from a Cenco quartz mercury arc
lamp and is passed through a filter to isolate the 5461 A) green mercury
line. The monochromatic light beam is then focused on a point source
after which it passes through an achromatic lens to collimate the light
beam. The collimated, monochromatic light is then split into two
beams by mirror 1, half of the beam goes to the full-reflecting mirror

2 and the other half to the full-reflecting mirror 3. The two beams

pass through the diffusion cell and are recombined at mirror 4. If the

24

25

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light path 1 - 2 - 4 is slightly different from the light path 1 — 3 - 4,
interference between the two beams will take place.

The interferences pattern is adjusted by a small rotation of
mirrors 1 or 2 about their axis in the plane of the paper to produce
straight line interference bands. If the path length difference
between the two beams is small the bands will be parallel and
evenly spaced. The mirrors are designed to also rotate about their
other principal axes and a combination of rotation and translation
of the mirrors will yield a fringe pattern which consists of straight,
vertical, parallel lines which are evenly spaced.

The diffusion cell is positioned in such a manner that light
beam 1 - 3 - 4 passes through the boundary between the two solutions.
The light beam 1 - 2 - 4 passes through a region in the cell where the
solution is of constant concentration and is therefore a reference beam.
The vertical fringes are displaced horizontally a distance proportional
to the difference in refractive index of the two solutions. 5 The resulting
fringe displacement pattern (Figure 9) is a direct plot of refractive
index versus distance. Thus if there is a vertical concentration
gradient in the cell this gradient may be followed by the amount of
horizontal displacement of the fringes.

The diffusion cell (Figures 10 and 11) consists of a channel cut
into a stainless steel plate with two parallel optical windows clamped
over the slot to form a sealed chamber. In order to meet the conditions
for a direct solution of Fick's law for the case of free" diffusion it is

necessary that the channel be of uniform cross section.

28

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29

Solution Re servoir s

r... I 11‘ I

‘ Syringe

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1—
Valve 2 Valve 1
Cell Cell
Window _..z / Body
Valve 4
; Valve 3
Boundary /
Sharpening __/
Slits
K a J
Siphon Valve 5

Figure 10. Diagram of Diffusion Cell.

 

 

 

   

 

 

 

 

 

Figure 11' Photograph of Diffusion Cell for Measurement of Diffusion
Coefficients

31

The boundary between two solutions of slightly different

7, 33 by

concentration was formed by the flowing boundary technique
which the interface between the layers of liquid is formed by flowing
the two liquids simultaneously through two fine horizontal slits in the
cell wall. The denser solution is introduced at the bottom and the
lighter solution at the top to prevent convection due to a density
gradient.

The interferometer is so constructed that the fringe pattern
can be observed by eye through a telescope or photographed by a
camera. (See Figure 8) Thus it is possible to obtain a plot of
refractive index versus distance at a given time interval. This

information then allows us to calculate the diffusion coefficient.

(See Appendix 1)

“VS

1)

Z)

32

Procedure for Experimental Run of Diffusiometer

Two aqueous polyelectrolyte solutions of different concentration
are prepared. In the case of polymethacrylic acid the desired
degree of neutralization is obtained by addition of a known

amount of sodium hydroxide.

The diffusion cell is clamped in a rack outside of the water bath

with all valves closed.

Reservoir B (Figure 10) is filled with the more concentrated of
the two solutions. Valves 5 and l are opened to allow the liquid
to flow into the cell to approximately 3/4 inch above the exit

slits. Valves 1 and 3 are then closed.

Valves 4 and 2 are opened and the exit line is filled with liquid
by forcing the liquid out the exit with the syringe connected to
valve 2. Care must be taken so that the liquid level does not

drop below the slits.

Additional solution is allowed to flow into the cell as in step 3.
This time the liquid is forced through valve 3 into the exit line.
In order to assure complete removal of all air in the exit line
this step must be repeated several times until the liquid will
siphon out by itself. At this point the liquid level should be
1/8 inch above the slits. All valves are then closed and

reservoir B is refilled.

“V50

10)

11)

33

The syringe above valve 2 is filled with the less dense solution
and valve 1 is opened. Valve 2 is opened slightly and the

solution is allowed to trickle down the wall of the cell until the
cell is full and a small amount of solution appears in reservoir

A. Reservoir A is then filled and valves 2 and l are closed.

The cell is placed into position on the cell hanger in the constant
temperature water bath of the interferometer. Valves 1 and 5
are opened slightly until the rate of flow from the exit line is
about 5-6 drops per minute. Then valve 4 is opened until the

flow rate from the exit line is doubled (10-12 drops per minute).

The formation of the boundary can be Observed through the
telescope and when a satisfactory boundary is formed all the
valves are closed, first closing valves 3 and 4 and then valves

1 and 5.

The mirror reflecting the image into the telescope is then
removed so that the interference fringe pattern can be

photographed by the camera.

A series Of exposures are taken at a set time interval. A

reproduction of a photographic plate appears in Figure 9.

After a run is completed, the cell is removed from the water
bath, rinsed with acetone and dried by drawing air through the

cell with an aspirator.

“VB.

 

34

Calculation of Diffusion Coefficient

It is possible to apply Fick's second law, equation (1), to the

diffusion in the cell described in Figure 10.

second law

8c 1 8c

2‘D5’t‘

 

In order to solve Fick's

(1)

for the present case the following boundary conditions must be used:

Case I (x> 0) i) x
ii) t
iii) x
Case 11 (x < 0) iv) x
v) t =
vi) x =

where x is defined in Figure 12.

cell body

////'//

x<

 

Figure 12.

/1|'/////

’////

 

cell body

Diffusion Cell Coordinates .

[Vx ll

va 11

“VA:

35

In order to solve equation (1) with the given boundary conditions
it is necessary to assume that D is constant or is linear with c and
the diffusion gradient has the properties of normal distribution curves.

These assumptions are valid if c and c2 are nearly equal.

1

The solution to equation (1) may be obtained with Laplace

transforms. The resulting solution is

—:'— : — erf. (38)
C2 ‘1 2 m

for both Case I (x > 0) and Case 11 (x < 0).

 

is the c onc entrati on

c
at the zero position in the cell and is equal to %— (c + c

l 2) '
For small differences in concentration, the refractive index,

n, may be assumed proportional to the concentration, 50 and n can

be substituted for c in equation (38) to obtain:

n-n
O X

4Dt

 

531—1- 2 (39)

The fringe pattern Obtained from the interferometer is a plot
of the refractive index versus distance in the diffusion cell. The
distances are magnified by the camera by a factor M.

A representation of the fringe pattern is given in Figure 13.
In traversing from point A to point B, the total number of fringes
crossed will be the number of fringes displaced by the difference in
refractive index between the solutions at point A and B. Each fringe

will correspondto a change in refractive index by an amount An .

 

36

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T

 

 

 

1‘

 

_ lllll xx

O
X

r
‘l

 

Figure 13. Fringe Pattern.

37

Let J be the total number of fringes and let k be the local
fringe number in the top half of the cell and j the local fringe number
in the bottom half. xj and xk are the measured distances

corresponding to fringes j and k respectively.

Therefore, where x > O

1
n'no k‘Z‘J Zk-J

 

 

 

 

x
k : erf-l ZkJ- J (40)
4Dt
Also, where x< O
n - nO : J _ 2.
n2 -n1 ZJ
and
X1 —1 J — 21'
: erf . (41)
4Dt J

It is difficult to determine the midpoint of the diffusion; however,

the distance, + Xj’ is easily determined by difference measure-

Xk

ments . Therefore

 

 

x. x . E
—-L— + k = erf_1 (—-———J—J E Z) + erf_1 ZkJ- J . (42)
V4Dt V4Dt

The measurements taken from the photographic plate are different from
the cell distances because of the magnification by the camera lens. The

image is magnified by a factor, M. Therefore,

38

x.+x x!+x'

k 1 k
\f4Dt erf4Dt

where X} and x1; are distances on the photographic plate. Hence,

 

 

 

Z
x' + x'
l j k
Dt = —— . (43)
4M2 erf..1 (ii—El) +erf_1 (ZkJ- J)
The value
2
I I
xJ + xk
erf—l (J -JZ]) + rf-l (ZkJ-J)

is obtained for several j's and k‘s for each exposure and averaged.
The averages for several exposures are plotted versus exposure time.
The slope of the resulting line is determined and thus,

D __ slope (44)

4M2

The value for M was determined to be 1. 923. 2 The calculated

diffusion coefficient is assumed to be equal to the mutual diffusivity at
the average concentration of the two solutions.

An optical comparator made from a Gaertner microscope was
used to measure the distances on the photographic plate. The average
deviation for the measurement of diffusivities by the Mach-Zehnder

. . 2.
interferometer was determined to be 0. 5%.

3 9
C ouette Appa ratus

The electrical anisotropy effect was measured in a Couette
apparatus. The Couette apparatus produces steady flow conditions
with a constant velocity gradient perpendicular to the direction of
motion in solutions which are in the annular gap between two
concentric cylinders.

The apparatus was constructed according to the description

20, 21, 49 The

given by Heckman17 and several other investigators.
principal features of the apparatus are shown in Figure 14. It
consists essentially of a rotating outside cylinder (rotor) and a
stationary inner cylinder (stator) with a narrow gap between the two
cylinders. The gap width of the apparatus,which was constructed for
these measurements,was approximately 0. 05 cm.

Silver electrodes were imbedded in the surface of the stator
to allow the measurement of the conductivity of the solution in the
gap between the rotor and the stator. The electrodes were insulated
from each other and from the brass cylinder by a thin layer of epoxy
resin which covers the stator. A thin telfon lining was installed in
the rotor to prevent conduction along the rotor wall.

The stator and the rotor were maintained at a constant temper-
ature of 250C by circulating cooiling water from an Arthur H. Thomas
Co. constant temperature bath. The rotor was driven by a l/3 H. P.
Master direct current motor with a General Radio Co. variac speed

control, the combination of these two pieces of equipment allow an

infinitely variable speed drive. The rotational speed was determined

 

40

Stator
7

 

Measuring

7 electrodes

Gap ———-u- *—

\\

Rotor ——-

 

 

 

 

 

 

Figure 14. Couette Apparatus (outline).

41

by a Hewlett Packard electronic counter which counts the electrical
pulses generated by a magnetic pickup device activated by a series
of gear teeth cut into the drive pulley. The counter converts the
electrical pulses into a reading of R. P. M. Because of the required
dimensions of the teeth in relation to the size of the magnetic pickup,
it was only possible to fit 30 gear teeth into the drive pulley instead
of the normal 60 teeth; therefore the number displayed by the
electronic counter was one half the actual R. P. M. of the rotor.

The conductivities of the solutions were measured with an
Industrial Instrument Inc. RC-l 8 conductivity bridge operating at
l, 000 cycles/ sec. The conductivity bridge is fitted with a cathode-
ray oscillographic detector and the cell capacitance effects can be
balanced out by means of a variable condenser in parallel with the
variable resistance.

Figures 15 and 16 give a more detailed description of the
apparatus. The rotor is mounted on three precision ball bearings
to give stability and rotational accuracy. Two oil seals protect
the ball bearings from the cooling water.

The stator is positioned by several bolts in the top. It is
important that the stator is returned to the same position for each
run.

The stator consists of a brass cylinder with a thin layer of
epoxy resin. In order to apply the epoxy layer, a form was
constructed of plexiglass around the brass cylinder. The epoxy resin
was poured in between the plexiglass cylinder and the brass cylinder

and allowed to harden. After the epoxy was cured in an oven for

42

{If

     
     
 

Ball Bearing
Oil Seal

Water <~——-

Teflon Lining

Rotor

Cooling Fins -———-—Water

Oil Seal
Ball Bearings

Pulley —-———

Teflon

 

Filling -Cup

Figure 15. Couette. Apparatus without Stator.

4.3

Water <——- " " ""

l I
33
ML

Water ._____..

Epoxy—Coated Brass

 

 

 

 

 

 

 

 

 

 

 

Cylinder
fl Silver Electrodes
.1 , : imbedded in Epoxy
// Resin Layer
l“
1v.

Figure 16. Stator for Couette Apparatus.

 

44

several hours, the plexiglass was cut off and epoxy was machined to
the desired diameter.

The silver electrodes were placed into position before the
epoxy was poured,so that the surface of the electrodes could be
machined to the same diameter as the epoxy surface. This is
necessary to assure steady uniform flow in the annular gap.

The polyelectrolyte solutions were introduced into the gap
from below through a channel which was drilled through the rotor
axle. The solutions were allowed to flow into the gap by gravity
from a funnel connected by tygon tubing to a filling-cup. The
filling -cup had a hexagonal nut fitted into a slot in the bottom. The
rotor axle is threaded so that when the gap is filled with solution

the nut can be screwed into place by turning the filling-cup.

45

Determination of the Size of the Polyion

The anisotropic behavior of the conductivity of charged
particles in a conducting medium was calculated by Schwarz
(equation (36)). For the determination of the size of a rod shaped
particle these functions are given in Figure 17 in the form of the
normalized anisotr0pies Kw), for different abscissa scales,
where K is equal to the relative change in conductivity divided
by the value of the relative change for complete orientation of the
charged particles.

The rotational diffusion constant Dr’ was determined by
comparison of the change of conductivity Ay(q), measured in the
Y direction (see Figure 3) as a function of the velocity gradient q,
to the corresponding set of anisotropy curves calculated as functions
of 0‘ = q/Dr .

The experimental curve Ay(q) was plotted with the arbitrary
abscissa scale Ma(sec. -1/cm.) and ordinate scale Mo(fraction/cm. ).
The curve system in Figure 17 was projected upon the measured curves
in such a way that one curve out of the system fits optimally to the
measured values. The parameter P of this curve and the corresponding
enlargement V (the length of the V-scale shown on Figure 17 measured
in the plane of projection in decimeters) were determined. One obtains

from these values the rotational diffusion constant

Dr(sec‘1) = 10‘3 MaPV . (45)

46

The value of Dr inserted into equation (25) gives 1, the length of the
particles. Under ordinary conditions (T = 3000K; no 2 0. 01 poise)

one obtains approximately
1(cm) = 0.0022 (MaPV)-1/3 . (46)

From equation (46) and Figure 17 one observes that the
length of the charged particles is directly related to the height of

the relative change of conductivity curve.

-47

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48

Synthesis of Potassium Polyphosphate

Potassium polyphosphate was prepared by heating primary
potassium phosphate at 12000F for 24 hours. The polymerization
is a condensation reaction and is described by the following

equation:

nKH PO

2 4 -* (KPO3)n + nHZO

!

According to Van Wasser 5 the polyphosphates are straight

chain polymers and have the following structure:

OK “K+
”OI-LOT..-
c“)

The polyphosphate chain undergoes hydrolysis quite rapidly
in aqueous solution46 and therefore the solutions must be prepared
freshly for each run.

The molecular weight of the polyphosphate was determined
from a viscosity correlation derived by Van Wasser45 which relates
the intrinsic viscosity of the polymer solutions to the average

molecular weight, MV , T = 25 0C.
[n] = K.M + K (47)

where: [n] : intrinsic vicosity

1.2.5x10‘5

0.085

 

49

A Cannon—Fenske10 capillary tube viscosimeter was used to
determine the viscosity of polyphosphate solutions of various
concentrations of polymer dissolved in a 10% solution of tetramethyl-
ammonium bromide. The intrinsic viscosity is obtained by plotting
specific viscosity divided by concentration versus concentration and
extrapolating to zero concentration. Thus,

Tl
[n] 2 limit —%9-
C-*0

According to Billmeyer, 3 nsp : (t - tO)/to, where t is the efflux
time for a given volume of polymer solution and t0 is the corresponding
efflux time for the solvent.

Figure 18 is the plot of nSp/C versus C for the polyphosphate
sample that was used in the diffusion experiments. When extrapolated

to zero concentration the value for the intrinsic viscosity is equal to

8.0. Substituting [n] = 8.0 into equation (47), gives "MV = 640,000.

50

 

 

 

6 _.

4)-

2r—

0 l l l I
0 0.05 0.10 0.15 0.2.0

concentration, g/100cc

Figure 18. Plot of 718 C versus Concentration for Potassium
Polyphosphate in 10% Solution of Te tramethylammonium

Bromide.

51
Synthesis of Polymethacrylic Acid

Polymethacrylic acid was synthesized from the monomer
methacrylic acid by a free radical type polymerization. The
monomer, purchased from Eastman Chemicals, was first distilled
under a vacuum to remove the inhibitor and then polymerized in a
water-methanol solution using 0.5% benzoyl peroxide as the free
radical initiator. The mixture was agitated by bubbling nitrogen
through the solution and the temperature was maintained at 70°C
by immersing the rector flask in a constant temperature bath. The
appearance of a cloudy and viscous solution indicated completion of
the reaction. 48

The swollen polymer was dissolved in methanol and precipitated
with ether to remove the monomer and catalyst. This procedure was
repeated several times to ensure the complete removal of monomer
and catalyst.

The polymethacrylic acid was then redissolved in methanol
and fractionally precipitated by the gradual addition of ether to obtain
different molecular weights. The fractionation procedure used was
essentially that proposed by Flory. 12 The fractions were dried in a
vacuum at 1100C for 48 hours and then ground to a fine powder.

The resulting polymer has the formula

52

where n represents the degree of polymerization.

The molecular weights were determined by viscosity measure-
ments of polymer solutions in 2N sodium hydroxide according to the
procedure outlined by Katchalsky and Eisenberg. 25

Figure 19 is a plot of nsp/C versus C for the different
polymer samples used in the experiments. By extrapolation to zero
concentration, the intrinsic viscosity is obtained. The intrinsic
viscosity for the polymer solutions is related to the degree of

polymerization (DP) and molecular weight as given in Figure 20. 25

 

53

10"

 

 

 

 

 

2 _ 206,000

1 1x71 =127,000
"" V

0 I I l I I
0 0.10 0.20 0.30 0.40 0.50

c oncentration, g/l 00cc

Figure 19. Plot of Reduced Viscosity versus Concentration for
R>1ymethacry1ic Acid in 2N Sodium Hydroxide.
T = 30°C.

10.

54

l

 

I l I l l l l

.0 1.5 2.0 3.0 4.0 6.0 8.010 15 20
3

HU'IO‘
_

degree of polymerization x 10-

Figure 20. Plot of Intrinsic Viscosity versus Degree of Polyv
merization of Polymethacrylic Acid in 2N Sodium

Hydroxide. From Reference 25.

 

RESULTS AND DISCUSSION

The experimental results for the diffusion of polymethacrylic
acid are tabulated in Appendix 11 along with viscosity measurements
of the polymer solutions. Diffusion coefficients, D, were obtained
as a function of the degree of neutralization, a, of the acid groups
for different molecular weights and different average concentrations.

Figures 21 to 24 are plots of the diffusion coefficients of
polymethacrylic acid. These plots show the dependence of D upon
a for several different molecular weights and different average
concentrations.

It can be seen that the molecular weight of the polymer has
very little effect upon the plot of D versus a, but the curve is
markedly effected by a change in the average concentration of the
polymer solutions.

The initial increase of D is readily explained by the increase

in the number of counterions, which according to equation (11)

 

D=——v<¢+3¢) (11)

and the theory of the Nernst potential should cause an increase in D.
Beyond 10%, D decreases in all cases because of the increase in
the hydrodynamic resistance of the polyion. From 0 - 10% neutral-
ization the molecules are hyperc oiled by intramolecular hydrogen
bonds and their hydrodynamic resistance, p, is relatively small.
Beyond 10% neutralization the molecules open up by virtue of the

intramolecular electrostatic repulsion (equation 19), so that p

55

 

 

 

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60

increases rather sharply. The increase in p, which is more rapid
than that in v( <1) + 8¢/8 1n n) causes a decrease in D, as would
be expected from equation (11).

Kedem and Katchalskyzg observed this same phenomenon, but
this investigation obtained a second decrease in D at lower concen-
trations. One can clearly see in Figures 23 and 24 that a two step
process is occurring in the uncoiling and opening of the polyion at
the lower concentration.

In order to verify that the decrease in D was due to a rapid
increase in the size of the polyion for both the depressions in D
versus 0 curves that are observed in Figures 23 and 24, it was
necessary to study the size of the polyion as a function of a by an
independent means. The electrical anisotropy effect gives an
approximate measure of the size of charged molecules in a conducting
medium.

The experimental results from the Couette apparatus for the
relative change of conductivity, A = (K - Ko)/Ko’ as a function of
R. P. M. for different degrees of neutralization and concentration are
tabulated in Appendix IV and plotted in Figure 25 to 28.

As noted on page 46 , the size of the polyion is proportional
to the cube root of the height of the relative change of conductivity
curve. Therefore a plot of the relative change of conductivity, A at
a particular R. P. M. versus a should give a qualitative picture of
the size dependence of the polyion upon the degree of neutralization.

Figures 29 and 30 are plots of A versus a for different

400

concentrations. Again it can be seen that there are two abrupt changes

61

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NM .0

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O
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om I so .0
I00 .0
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65

 

 

O 16 "' g 1
0.14 P
" o
0.12 "’
I O
0.10 ""
O I 2
O
0 08 - I
400 0
° n
0.06 - 0
I
O
O 04 "' I I
O C = 0. 0058 monomoles/liter
0. 02 .. C — 0. 011 monomoles/liter
O
I
0 " L I 1 1 I
0 10 20 3O 4O 50

C1, 70

Figure 29. Plot of the Relative Change in Conductivity at 400 R. P. M.
versus Degree of (Neutralization. (I) C = 0. 0058

(monomoles/liter. C = 0. 0116 monomoles/liter.

KI : 206,000.
V

66

 

 

0. 05 P n 3
0. 04 I- ~ g o 4
O
I
0.03 - n
0
A400
a O
0.02 e o
[3 C = 0.0213 monomoles/liter
O 01 O C = 0. 0291 monomoles/liter
I
0
a
o
O " L l 1 1 i

0:70

Figure 30. Plot of the R‘glative Change in Conductivity at 400 R. P. M.
versus Degree of Neutralization. C = O. 0213
monomoles/liter. @ C = 0. 0291 monomoles/liter.

M = 206, 000.
V

67

in the size of the polyion at the lower concentrations and only one
abrupt change in the size of the polyion at the two higher concentrations.
Also the abrupt changes in size occur at approximately the same values
of a as do the depressions in the D versus 0 curves of Figures

21, 22, 23 and 24.

The data from the electrical anisotropy effect do indeed verify
that the depressions in the D versus CL curves are due to the sudden
increase of the size of the polyion and indicates that a two step process
is involved in the opening and uncoiling of the polyion.

The viscosity measurements are represented in Figures 31 to
34. The kinematic viscosity is plotted versus <1 for different
concentrations and molecular weights corresponding to the solutions
used in the diffusion measurement. Each curve indicates a rapid
increase in viscosity between 10% and 30% neutralization. This increase
in viscosity is assumed to be due to the Opening of the coiled polymers.

Originally the increase in viscosity was attributed to electro-
viscous effects, but a quantitative estimate showed that the increase
is much greater than might be expected from the electroviscous
behavior of the solutions. 24

The curve in Figure 32 for the concentration of 0. 0213 monomoles
per liter has a rather steep slope between 10 and 15%, the slope then
decreases until at 20% and then increases again to beyond 25%. Thus
if the increase in viscosity is due to the opening of the polymer chain
as assumed, then this curve also indicates a two step process in the
opening and uncoiling of the polymer chain. The concentration of 0. 0213
monomoles/liter is the same as the average concentration in Figures 23

and 24.

Vi sc osity, centi stoke s

24

20

t--'
0\

g...
N

 

68

C = 0. 0388

   

C = 0.0213

 

 

 

8
C : O. 0039
4
0 1 I 1 I __l
0 10 20 30 4O 50

(1970

Figure 31. Plot of Apparent Viscosity versus Degree of

Neutralization for PMA Solutions, Molecular

Weight 2 357, 000.

Visc osity, centistokes

180 ..

160 -

 

69

C = 0. 0388

140 -

120 '-

100 "

C = 0.0213

80-

40)-

20 -
C = 0.0039

 

(l, (70

Figure 32. Plot of Apparent Viscosity versus Degree of
Neutralization for PMA Solutions. Molecular

Weight = 860, 000.

Visc osity, centistokes

100(-

 

 

70

 

 

C = 0. 0582
80 )-
60 r c = 0.0320
40 t-
20 *'
C = 0. 0058
O L 1 L l J
0 10 20 30 40 50
(1, (70
Figure 33. Plot of Apparent Viscosity versus Degree of

Neutralization for PMA Solutions. Molecular

Weight = 470, 000.

Viscosity, centistokes

 

 

Figure 34.

 

 

 

 

 

71

10 20 '40 40 50

Plot of Apparent. Vis-t 0311',“ vet‘s'us Degr...: ol
Neutralization for PMA. Solunorfisi Molt.» Iii Ir

Weight: 127,000.

_‘.

72

The size of the polyion was determined by the procedure out-
lined on page 45. The results are tabulated in Table I for different
degrees of ionization and different concentrations. These values
for the size of the polyion can be compared to the fully extended length
of 6. 500 X. The fully extended length of the polyion is the maximum
extension of the chain, which is calculated from the bond angles and
bond distances between the carbon atoms in the polyion chain. It is
observed that at low concentration and high degree of neutralization
the measured length of the polyion approaches the fully extended length
of the polyion. This suggests that the chain is nearly completely

extended at low concentration and high degree of neutralization.

73

Table I

Polyion Size of Polymethacrylic Acid and Rotational
Diffusion Constants of Polymethacrylic AcidSolutions

 

a, % Concentration D (sec-1) Length (X)
(monomoles/liter) r
50 O. 0058 38. 3 6, 000
50 O. 0116 76. 5 5, l 80
50 O. 0213 191 3, 83.0
50 0. 0291 383 3, .040
10 0.0058 3.800 1, .410

0
Fully extended length = 6, 500 A

CONCLUSIONS

The diffusion coefficient of polyelectrolytes has been shown to
depend greatly upon the size and shape of the polyion and the degree
of ionization. The dependence upon the size and shape is attributed
to the change in the hydrodynamic resistance of the polyion and the
dependence upon the degree of ionization is the result of the Nernst
potential of the counterions. The Nernst potential, as explained
earlier, is due to the difference in mobility between the counterions
and the polyion. The counterions have a greater mobility and there-
fore actually drag. the polyions with them in the diffusion process.
The effect of the Nernst potential was illustrated by the suppression
of the ionization of the cations by the addition of sodium thiocyanate.
This addition of an electrolyte decreased the diffusion coefficient of
potassium polyphosphate by a factor of 60.

In the diffusion of polymethacrylic acid it was observed that
the polyion rapidly increased in size as the degree of neutralization
was increased. The increase in size of the polyion caused a decrease
in the diffusion coefficient because of the increase in the hydrodynamic
resistance of the polyion. In the case of low concentration,..two
depressions in the diffusion coefficient were observed, which indicates
a two step opening of the coiled polyion chain. The two step process
might be due to a different ionization constant of the acid groups on
the inside of the polyion from that of the acid groups on the outside of
the polyion. Originally it was assumed that the ionization constant was

the same for all the acid groups.

74

75

Another possible explanation for the data presented, is that
the first abrupt change in the size of the polyion is due to the breaking
of hydrogen bonds, resulting in an expansion of the coiled polyion; and
the second abrupt increase in size is due to an uncoiling of the polyion,
in which the polyion approaches a rod-like structure.

The size and shape of the polyion was determined by measuring
the electrical anisotropy effect of aqueous polymethacrylic acid
solutions. The results of these measurements verified that the
depression in the diffusion coefficient was due to the increase in
size of the polyion. It was also noted that the measured length of
the po’lyion at high degrees of neutralization and low concentration
approaches the fully extended length of the polyion for polymethacrylic
acid. Therefore the polyion is nearly completely extended at low

concentration and high degree of neutralization.

10.

11.
12.
13.
14.
15.
16.

17.

 

 

 

 

 

 

 

 

BIBLIOGRAPHY

Anderson, D. K., Ph. D. Thesis, University of Washington (1960).
Bidlack, D. L., Ph. D. Thesis, Michigan State University (1964).

Billmeyer, F. W. , Textbook of Pollmer Chemistry, Interscience

 

Publishers, Inc., N. Y., 1957, pp. 128-138.

Caldwell, C. S., Ell—D— Thesis, University of Washington(1955).
Caldwell, C. S., Hall, J. R., and Babb, A. L., Rev. Sci. Instr.,
28, 816 (1957).

Callis, C. F., Van Wasser, J. R., Arvan, P. (3., Chem. Revs.,
54, 777 (1954).

Coulson, C., Cox, J., Ogston, A., and Philpot, J., Proc. Roy.

Soc. (London), A122, 382 (1948).

Cohn, J. E. and Edsall, J. T., Proteins, Anino Acids and Pegtids

 

as Ions and Dipolar Ions, Reinhold, N. Y. , 1943, p. 624.

 

Eigen, M. and Schwarz, C., J. Colloid Sci., 12, 181 (1957).
Fenske, M. and Cannon, M., Ind. Eng. Chem., Anal. Ed., 1_(_)_,
297 (1938).

Fick, A., Ann. Phys. Chem., 94, 59 (1855).

Flory, P. J., J. Am. Chem. Soc., §_5_, 375 (1943).

Gans, R., Ann. Physik (4F) 86, 628 (1928).

Gotz, K. C., J. Colloid Sci., 2_(_)_, 289 (1965).

Gotz, K. and Heckman, K., J. Colloid Sci., 13, 266 (1958).
Harris, F. E., and Rice, S. A., J. Polymer Sci., L5, 151 (1955),

Heckman, K., Z. Phys. Chem., 9, 318 (1956).

76

l8.

19.

20.

21.
22.
23.
24.

25.

26.
27.

28.

29.
30.
31.
32.
33.
34.
35.
36.
37.

38.

 

77

Heckman, K., and Gotz, K., Z. Electrochem., 88, 281, (1958).
Huizenga, J. R., Grieger, P. F., and Wall, F. T., J. Am.
Chem. Soc., Z8, 4228 (1950).

Jaenicke, R., Z. Physik. Chem. (Frankfurt) (N.F.), E2, 315
(1959)-

Jacobson, B., Rev. Sci. Instr., 84, 949 (1953).

Jeffery, G. B., Proc. Roy. Soc. (London), 8182, 171 (1923).
Jerrard, H. G., J. Appl. Phys., §_1_, 1007 (1950).

Katchalsky, A., J. Polymer Sci., _7_, 393 (1950).

Katchalsky, A., and Eisenberg, H., J. Polymer Sci., 8, 145
(1951).

Katchalsky, A., and Lifson, S., J. Polymer Sci., 18, 409 (1953).
Katchalsky, A., and Lifson, S., J. Polymer Sci., 18, 52 (1954).
Katchman, B. J., and Smith, H. E., Archives of Biochemistry
and Biophysics, 18, 396 (1958).

Kedem, O., and Katchalsky, A., J. Polymer Sci., 1_5, 321 (1955).
Kern, W., Z. Physik. Chem., 1281, 249 (1938).

Kuhn, W., Z. Physik. Chem. (Leipzig), A868, 1 (1932).

Kuhn, W., and Kuhn, H., Helv. Chim. Acta., _2__6_, 1394(1936).
Longsworth, L. G., Rev. Sci. Instr.,2_l, 524(1950).

Mach, L., Sitz. Wiener Akad., _1__1_, HA, 5 (1892).

Malmgren, H., Actachemica Scandinanica, 8, 147 (1948).
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Oth, A. and Doty, P., J. Phys. Chem., 88, 43 (1952).

39.
40.

41.

42.
43.

44.

45.

46.

47.
48.
49.
50.

51.

 

78

Peterlin, A., Z. Physik, 111, 232 (1938).
Perrin, F., J. Phys. Radium(7), 8, 497 (1934).

Rice, S. A. and Nagasawa, M., Polyelectrobrte Solutions,

 

Academic Press, N. Y., 1961.

Schwarz, G., Z. Physik, L412, 563 (1956).

Taylor, G. J., Proc. Roy. Soc. (London), _A_1_§_7_, 546 (1936).
Van Wasser, J. R., and Holst, K. A., J. Am. Chem. Soc.,
7_2_, 639 (1950).

Van Wasser, J. R., J. Am. Chem. Soc., Z8, 906 (1950).
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Vink, H., Makro. Chem., 6_Z, 105 (1963).

Vithani, K. K., M. S. Thesis, Michigan State University(l960).

 

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Wiener, O., Wied. Ann., :12, 105 (1893).

Zehnder, L., Z. Instrumentenk., l_1_, 275 (1891).

APPENDICES

79

 

Appendix I
Sample Calculation of Diffusion Coefficient

Experimental run number: A-6
Date: July 29, 1963
Diffusion of potassium polyphosphate in aqueous solution.

Solution A (For upper level of diffusion cell)

C1 = 6 gm (KPO3)n/11ter

Solution B (For lower level of diffusion cell)

C

2 = 3 gm (KPO3)n/11ter

See Figure 9 for actual phOtographic plate.

Exposure number Time, minutes
1 0
2 5
3 10
4 15
5 20
6 25
7 30
8 40
9 45
10 51
ll 55
12 60
13 65
14 70
J = 15

80

 

81

Exposure 2, t = 5 minutes

(For definition of measurements see Figurel3)

j (x:D - x3), cm. k (x:) + XL), cm. [(x'O + XL) - (x:D - x5” , cm.
2 2.703 8 2. 921 0.218
3 2.756 9 2.951 0.195
4 2.797 10 2.982 0.185
5 2.833 11 3.014 0.181
6 2. 863 12 3,054 0.191
7 2.893 13 3.103 0.210

Exposure 4, t = 15 minutes

j (x2) - x3), cm. k (x2) + XL), cm. [(x2) + XL) - (x:3 - 1(3)] , cm.
2 2.616 8 2.923 0.307

3 2.692 9 2.965 0.273

4 2.747 10 3.008 0.261

5 2.795 11 3.055 0.260

6 2.841 12 3.108 0.267

7 2.883 13 3.180 0.297

Exposure 7, t = 30 minutes

3' (x23 - x5), cm. k (x2) + xic), cm. [(x2) + xii) -(xé) - x5” , cm.
2 2. 539 8 2. 930 0.391

3 2.631 9 2.987 0.356

4 2.702 10 3.039 0.337

5 2.763 11 3.101 0.338

6 2.822 12 3.177 0.355

7 2.879 13 3.266 0.387

 

82

.Exposure 10, t = 51 minutes

j (X2) - XE), cm. k (XL) + XL), cm. [(x2) + XL) - (x2) - 1(3)] , cm,
2 2.408 8 2. 892 0.484
3 2. 517 9 2.957 0.440
4 2.606 10 3.024 0.418
5 2.687 11 3.101 0.414
6 2.757 12 3.192 0.435
7 2. 827 13 3.304 0.477

Exposure 12, t = 60 minutes

j (x;3 — le), cm. k (x; + xic), cm. ~ [(x2) + XL) — (x2) - xJ!)] , cm.
2 2.355 8 2.880 0.525

3 2.477 9 2.950 0.473

4 2.575 10 3.025 0.450

5 2.656 11 3.107 0.451

6 2.735 12 3.197 0.462

7 2.812 13 3.315 0.503

Exposure 14, : 70 minutes

j (x2) - x3), cm. k (X:3 + xl'(), cm. [(Xé) + XL) - (x2) - X3” , cm.
2 2.329 8 2.872 0.543

3 2. 442 9 2. 949 0. 507

4 2. 548 10 3.028 0.480

5 2.638 11 3.111 0.473

6 2. 718 12 3.208 0. 490

7 2.801 13 3.335 0.534

L—I.

~J o~<n ¢i quy

10
11
12
13

Expgsure}

3d +-x!, an
(1..)

0.218
0.195
0.185
0.181
0.191
0.210

average

OOOOOO

JI-Zj

HWU‘ldon

2k -J

~02.me

A.+ B

.8442
.7742
.7451
.7451
.7742
.8442

0.0667
0.2000
0.3333
0.4667
0.6000
0.7333

2.1..

A.+.B

1.1846
1.2916
1.3421
1.3421
1.2916
1.1846

xL-+Jd
‘ A+B )'an

.2582
.2519
.2483
.2429
.2467
.2488
.2495

0000000

(

I
“k

i¥+B

erf'1 (5.3.2.1)

erf

+Jd

2
avg.

.1

0.7850
0.5950
0.4405
0.3046
0.1793
0.0592
0.0592
0.1792
0.3046
0.4405
0L5950
0.7850

.A

0. 06225 cm

2

Exposure 4

 

' +Jd ,an
(xk J)

.307
.273
.261
.260
.267
.297

000000

average

Exposure'7

 

X'4-x!,an
(k JI

.391
.356
.337
.338
.355
.387

average

OOOOOO

.Exposure 10

 

'+-x!,an
(xk J)

.484
.440
.418
.414
.435
.477

average

OOOOOO

X'+-x!
___k____.l_
( .A+B

.4632
.4598
.4523
.4536
.4585
.4584
.4526

0000000

+16

x' .
(_1_<__1) ,Cm,
.363.
.352.

.A+B
0
0
0.
0
0
0
0

350.

.348.
.344.
.351.
.352.

x'i—x!
__k__.L
(A...)

0000000

.5733
.5683
.5610
.5556
.5618
.5651
.5642

84

O®\O\OUJO\\I

) ,an

,an

 

' +'X!. Z
.A+B
avg.
(XLi‘X1)2
.A+B
avg.
x'+-X! Z
_1£__.L
( AAI3 )
avg.

0.1239 cmz

0. 2094 cm2

0. 3183 cm2

 

85

Exposure 12

 

 

XL + x'.)
I I
(Xk + xj), cm. (——LA+B , cm.
0. 525 0. 6195
0. 473 0. 6109
' I I 2
0.450 0.6039 xk + X. 2
————-—-L = . 5
0.451 0.6053 ( A+B 0 366 cm
avg.
O. 462 0. 5967
0. 503 0. 5959
average 0.6054
Exposure 14
x1; + X!
I I
(xj + xk), cm. (——J—A+B ) , cm.
0. 543 0. 6432
0. 507 0. 6548
X' +X! 2
0. 480 0. 6442 k 1 = 0. 4101 cm?"
0. 473 0. 6348 A+B avg.
0. 490 0.6329
0. 534 0.6326
average 0.6404
X12 + x'. Z
Slope of the plot of fil- versus time, t is
avg
0. 922 x10-4 cm. 2/sec. See Figure 35.
510 e 0 922x10“4 -6 2
D———P— — ' = 6.23x10 cm. /sec.

 

4M2 14. 792

50

45

40

35
30
OD
>25
N <6
A
-34
>4 ,0 20
+ +
i <1
10
5
0

Figure 35. Plot of (

 

I
10

 

86

J I I I I I I I l
20 30 40 50 6O 70 80 90 100
time (min.)

x!+xi<

2
A+B versus Time.

Appendix II

Table II

Diffusion Data for Potassium Polyphosphate Solutions

 

Concentration Concentration D X 106
Run No. (In lower half of cell) (In upper half of cell) 2
cm /sec
8/1 g/l
A-2 5 0 6. 28
A-3 5 0 6. 60
A-4 5 0.4M NaSCN 0.10
(In 0. 4 M NaSCN)
A-5 5 l 4. 90
A-6 6 3 6. 23
Table III

Diffusion Data for Polymethacrylic Acid Solutions at an Average
Concentration = 0. 0320 Monomoles/liter, Molecular Weight

 

= 127, 000.
Run No. a, % D X 106
Lcmzlsec)
C-1 0 1.772
C-2 5 2. 430
C-3 10 4.185
C-4 15 3. 663
C-5 20 4.617
C-6 30 4. 816
B-2 50 5. 039

87

88

Table IV

Diffusion Data for Polymethacrylic Acid Solutions at an Average

Concentration = 0. 0213 Monomoles/liter, Molecular Weight 2 357, 000.

 

Run No. a, % D x2106
(cm /sec)
D-l 0 2.384
D-2 5 4.190
D-3 10 4. 931
D-12 11.25 4.725
D-8 12. 5 4. 553
D-4 15 4. 904
D-13 17. 5 5. 000
D-5 20 5. 032
D-ll 22. 5 4.683
D-6 25 4. 896
D-10 30 5. 055
D-7 40 5. 284
D-14 50 5. 498

89

Table V

Diffusion Data for Polymethacrylic Acid Solutions at an Average

Concentration = 0. 0213 Monomoles/liter, Molecular Weight = 860, 000.

 

Run No. a', % D x2106
(cm /sec)
153-11 0 2.668
E-Z 5 4. 081
13-3 10 4.926
13-14 12.5 4.569
153-5 15 4.912
E-6 17.5 5.087
E-7 20 4.676
13—8 22.5 4.875
13-9 25 5.193
E-lO 30 5.408
E-12 40 5.530
13-13 50 5.762

90

Table VI

Diffusion Data for Polymethacrylic Acid Solutions at an Average

Concentration : 0. 0320 Monomoles/liter, Molecular Weight = 470, 000.

 

Run No. c1, % D X2106
(cm /sec)
F-l 0 l. 810
F-Z 5 3. 906
F-3 10 4.754
F-4 12.5 4.562
F-5 15 4.432
F-6 17. 5 4.733
F-7 20 4. 993
F-8 25 4.167
F-9 30 5.162
F-10 40 5.278

F-11 50 5.516

91

Table VII

Apparent Viscosities of Polymethacrylic Acid Solutions
Molecular Weight = 127, 000

 

 

n (centistokes) n (centistokes) n (centistokes)
0.,% C = 0.0582 C = 0. 0291 C = 0.0058
(monomoles/liter) (monomoles/liter) (monomoles/liter)
0 1.037 0.984 0.939
5 1.236 1.123 1.035
10 1.730 1.530 1.177
15 2.260 1.976 1.499
20 3.207 2.708 1.880
30 5.102 4.007 2.383
Table VIII
Apparent Viscosities of Polymethacrylic Acid Solutions
Molecular Weight 2 357, 000
n (centistokes n (centistokes) r) (centistokes)
c1,% C = 0.0388 C = 0.0213 C = 0.0039
(monomoles/liter) (monomoles/liter) (monomoles/liter)
0 1.406 1.286 1.176
5 3. 204 -- 1. 899
10 4.946 3.880 2.465
15, 7.994 6.430 3.693
17'. 5 9. 247 7. 349 4. 099
20 11.744 9.159 4.635
22. 5 12. 854 10.080 5.058
25 14.965 . 10.738 5.562
30 19.169 13.338 5.790
40 22.377 15.535 6.199
50 23.296 16.147 6.214

92

Table IX

Apparent Viscosities of Polymethacrylic Acid Solutions
Molecular Weight = 860, 000

n (centistokes)
C = 0.0039
(monomoles/liter)

n (centistokes)
C = 0. 0213
(monomoles/liter)

r) (centistokes)
C = 0. 0388
(monomOles/liter)

 

10

12.

15

17.

20

22.

25
30
40
50

3.800 3.408 1.561
8.263 6.845 4.171
13.216 11.631 5.264
16.452 14.890 7.205
28.874 23.052 7.844
39.690 29.865 9.591
51.278 34.962 10.092
62.924 41.352 10.776
77.707 48.313 11.184
115.717 65.598 12.576
155.160 80.237 13.012
176.297 86.405 14.197

93

Table X

Apparent Viscosities of Polymethacrylic Acid Solutions
Molecular Weight = 470, 000

n (centistokes)
C = 0.0058
(monomole s/liter)

n (centistokes)
C = 0. 0320
(monomoles/liter)

r) (centistokes)
C = 0. 0582
(monomoles/liter)

 

10

12.

15

17.

20
25
30
40
50

1.871 1.608 1.322

4.986 4.233 2.660

8.005 7.175 4.061
10.751 9.438 5.371
15.757 13.093 6.374
23.244 18.384 8.193
37.278 26.029 10.119
48.489 34.310 12.030
62.555 41.779 12.855
84.146 51.147 14.748
93.783 57.361 15.787

 

93

Table X

v.‘y.. .0- -v- w_ _ _‘_ 'n'm—"w"‘

Apparent Viscosities of Polymethacrylic Acid Solutions

17 (centistokes)

Molecular Weight = 470, 000

n (centistokes)

n (centistokes)

 

a, % C = 0. 0582 C = 0. 0320 C = 0.0058
(monomoles/liter) (monomoles/ liter) (monomole s/liter)

0 1.871 1.608 1.322

5 4. 986 4. 233 2.660
10 8.005 7.175 4. 061
12. 10.751 9. 438 5.371
15 15.757 13. 093 6.374
17. 23.244 18.384 8.193
20 37.278 26. 029 10.119
25 48.489 34.310 12.030
30 62.555 41.779 12.855
40 84.146 51.147 14.748

50 93.783 57. 361 15.787

Appendix III

Table XI

Conductivity Data from Couette Apparatus
Concentration = 0. 0291 monomoles P. M. A. /liter

0% Neutralization

 

 

RPM K x 105 mhos (K — KO)/KO x102
0 1.134 0
100 1.135 0.088
200 1.135 0.088
300 1.135 0.088
400 1.132 -0.176
500 1.135 0.088
600 1.135 0.088
700 1.136 0.176
800 1.135 0.088
5% Neutralization
RPM K x105 mhos (K — KO)/KO x102
0 1.028 0

100 1.029 0.097
200 1.030 0.194
300 1.030 0.194
400 1.030 0.194
500 1.031 0.292
600 1.032 0.389
700 1.033 0.486
800 1.035 0.681

94

95

10% Neutralization

 

 

RPM K x105 mhos (K - KO)/KO x102
0 1.724 0
100 1.727 0.174
200 1.730 0.348
300 1.733 0.522
400 1.733 0.522
500 1.730 0.348
600 1. 736 0. 696
700 1.742 1.044
800 1.742 1.044
15% Neutralization
RPM K x105 mhos (K - KO)/KO x102
0 2.439 0

100 2.451 0.741
200 2.463 0.984
300 2.469 1.230
400 2.475 1.476
500 2.481 1.722
600 2.488 2.009
700 2.488 2.009
800 2.494 2.255

20% Neutralization

K x105 mhos

96

 

 

RPM: (K.K;O)/Kox102
0 3.164 0
100 3.185 0.664
200 3.195 0.980
300 3.215 1.612
400 3.226 1.960
500 3.236 2.276
600 3.247 2.623
700 3.257 2.939
800 3.268 3.287
25% Neutralization
RPM K x105 mhos (K - KO)/KO X102
0 3.846 0

100 3.876 0.780
200 3.906 1.560
300 3.922 1 976
400 3.937 2.366
500 3.937 2 366
600 3.937 2.366
700 3.937 2 366
800 3.937 2.366
900 3.952 2.756

 

3 0% Neutralization

97

 

 

RPM K x105 mhos (K - Ko)/KO x102
0 4.484 0
100 4.525 0.914
200 4.566 1.829
300 4.587 2.297
400 4.608 2.765
500 4.608 2.765
600 4. 608 2. 765
700 4.630 3.256
800 4.651 3.724
40% Neutralization
RPM K x105 mhos (K - KO)/Ko X102
0 4.854 0

100 4.901 0.968
200 4.950 1.978
300 5.000 3.008
400 5.025 3.523
500 5.050 4.038
600 5.076 4.574
700 5.102 5.109
800 5.128 5.645
900 5.155 6.201
1000 5 155 6.201

98

50% Neutralization

 

RPM K x105 mhos (K — KO)/KO x102
0 6.452 0

100 6.562 1.565
200 6.614 2.511
300 6.667 3.332
400 6.711 4.014
500 6.748 4.588
600 6.780 5.084
700 6.807 5.502
800 6.831 5.874
900 6.863 6.370
1000 6.887 6.742

99

Table XII

Conductivity Data from Couette Apparatus

Concentration = 0. 0116 monomoles P. M. A. /liter

0% Neutralization

 

RPM K x 106 mhos (K - KO)/KO
0 5.84 0
100 5.84 0
200 5.84 0
300 5.84 0
400 5.84 0
500 5.84 0
600 5.84 0
700 5.84 0
800 5.84 0
900 5.84 0
100 5.84 0

5% Neutralization

 

RPM K x 106 mhos (K — KO)/KO
0 6.11 0
100 6.13 0.0032
200 6.14 0.0049
300 6.15 0.0065
400 6.16 0.0081
500 6.17 0.0098
600 6.18 0.0114
700 6.19 0.0131
800 6.20 0.0147
900 6.21 0.0164
1000 6.22 0.0180

100

10% Neutralization

 

 

RPM K x 106 mhos (K - KO)/Ko
0 12.30 0
100 12.40 0.0081
200 12.50 0.0163
300 12.55 0.0203
400 12.60 0.0244
500 12.65 0.0285
600 12.70 0.0325
700 12.75 0.0366
800 12.80 0.0406
900 12.83 0.0431
1000 12.85 0.0447
15% Neutralization
RPM K x105 mhos (K — Ko)/K-O
0 1 727 0
100 1.770 0.0249
200 1.776 0.0284
300 1.786 0.0342
400 1.795 0.0393
500 1.805 0.0452
600 1.812 0.0492
700 1.818 0.0527
800 1.825 0.0567
900 1.832 0.0608
1000 1.838 0.0643

101

20% Neutralization

 

 

RPM K X105 mhos (K - KO)/KO
0 1.282 0
100 1.302 0.0156
200 1.312 0.0234
300 1.326 0.0328
400 1.335 0.0413
500 1.342 0.0468
600 1.351 0.0538
700 1.355 0.0569
800 1.360 0.0608
900 1.366 0.0655 ‘
1000 1.370 0.0686
25% Neutralization
RPM K x105 mhos (K - KO)/KO
0 1.565 0
100 1.600 0.0224
200 1.618 0.0339
300 1.634 0.0441
400 1.645 0.0511
500 1.650 0.0543
600 1.656 0.0581
700 1.661 0.0613
800 1.667 0.0652
900 1.672 0.0684
1000 1.675 0.0703

102

3 0% Neutralization

 

 

RPM K x 105 mhos (K — KO)-/Ko
0 1.798 0
100 1.862 0.0356
200 1.880 0.0456
300 1.898 0.0556
400 1.912 0.0634
500 1.927 0.0717
600 1.934 0.0756
700 1.941 0.0795
800 1.949 0.0840
900 1.957 0.0884
1000 1.965 0.0929
40% Neutralization
RPM K x 105 mhos (K - KO)/KO
0 3.279 0

100 3.401 0.0372
200 3.448 0.0515
300 3.497 0.0665
400 3.534 0.0777
500 3.571 0.0890
600 3.597 0.0970
700 3.623 0.1049
800 3.650 0.1131
900 3.663 0.1171

1000 3.690 0.1253

103

5 0% Neutralization

 

RPM K x105 mhos (K — Ko)/Ko
0 2.577 0

100 2.667 0.0349
200 2.725 0.0574
300 2.770 0.0749
400 2.809 0.0900
500 2.841 0.1204
600 2.865 0.1118
700 2.890 0.1214
800 2.907 0.1280
900 2.924 0.1346
1000 2.941 0.1412

104

Table XIII

Conductivity Data from Couette Apparatus
Concentration = 0. 0213 monomoles P. M. A./1iter

0% Neutralization

 

 

RPM: IKXlOénmos (Kn—KO)/Kox102
0 8.60 0
100 8.60 0
200 8.60 0
300 8.60 0
400 8.60 0
500 8.60 0
600 8.60 0
700 8.60 0
800 8.60 0
900 8. 60 0
1000 8.60 0
5% Neutralization
RPM K X105 mhos (K - KO)/KO 31:102
0 1.063 0
100 1.064 0.094
200 1.065 0.188
300 1.066 0.282
400 1.066 0.282
500 1.067 0.376
600 1.068 0.470
700 1.068 0.470
800 1.070 0.658
900 1.071 0.753
1000 1.071 0.753

105

10% Neutralization

 

 

 

RPM K x105 mhos (K — Ko)/Ko 3:102
0 1.887 0
100 1.887 0
200 1.894 0.371
300 1 898 0.583
400 1.901 0.742
500 1.905 0.954
600 1.908 1.113
700 1.912 1.325
800 1.916 1.537
900 1.916 1 537
1000 1.919 1.696
15% Neutralization
RPM: :Kx105nmes (Km—KO)/Kox102
0 1.760 0
100 1.770 0.568
200 1.779 1.080
300 1.786 1.477
400 1.792 1.818
500 1.798 2.159
600 1.805 2.557
700 1.812 2.955
800 1.815 3.125
900 1.821 3.466
1000 1.825 3.693

106

2 0% Neutralization

 

 

RPM K 31105 mhos (K - KO)/Ko x102
0 2.481 0
100 2.506 1.008
200 2.525 1.773
300 2.538 2 297
400 2.544 2.539
500 2.551 2.821
600 2.564 3.345
700 2.571 3.628
800 2.577 3.869
900 2.584 4.152
1000 2.591 4.434
25% Neutralization
RPM: :Kx105nmes (Km—KO)/Kox102
0 2.817 0

100 2.849 1.136
200 2.865 1.704
300 2.882 2.307
400 2.898 2.875
500 2.907 3.195
600 2.924 3.798
700 2.932 4.082
800 2.941 4.402
900 2.950 4.721

1000 2.958 5.005

3 0% Neutralization

107

 

 

 

RPM K x105 mhos (K - KO)/Ko x102
0 3.401 0
100 3.448 1.382
200 3.472 2.088
300 3.496 2.793
400 3.509 3.176
500 3.521 3.528
600 3.534 3.911
700 3.546 4.263
800 3.559 4.646
900 3.565 4.822
1000 3.571 4.998
40% Neutralization
RPM K x105 mhos (K - KO)/KO x102
0 4.115 0

100 4.184 1.677
200 4.219 2.527
300 4.255 3.402
400 4.292 4.301
500 4.310 4.739
600 4.329 5.200
700 4.348 5.662
800 4.367 6.124
900 4.386 6.586
1000 4.405 7.047

108

5 0% Neutralization

 

RPM K x105 mhos (K — KO)/Ko 31102
0 4.237 0

100 4.329 2.171
200 4.367 3.068
300 4.405 3.965
400 4.444 4.886
500 4.484 5.830
600 4.525 6.797
700 4.556 7.529
800 4.587 8.260
900 4.620 9.039

1000 4.651 9.771

 

Table XIV

109

Conductivity Data from Couette Apparatus

Concentration = 0. 0058 monomoles P. M. S./1iter

0% Neutralization

 

 

RPM K x106 mhos (K — KO)/KO x103

0 3.16 0

100 3.17 3.153

200 3.16 0

300 3.17 3.153

400 3.16 0

500 3.17 3.153

600 3.17 3.153

700 3.17 3.153

800 3.17 3.153

5% Neutralization
RPM K x106 mhos (K _ KO)/KO x102

0 3.48 0

100 3.50 0.575

200 3.51 0.862

300 3.52 1.149

400 3.53 1.437

500 3.54 1.724

600 3.55 2.011

700 3.57 2.586

800 3.58 2.874

10% Neutralization

110

 

 

RPM K x106 mhos (K - KO)/KO x102
0 5.71 0
100 5.83 2.102
200 5.89 3.152
300 5.93 3.853
400 5.97 4.553
500 6.01 5.254
600 6.05 5.954
700 6.09 6.655
800 6.11 7.005
900 6.13 7.356
1000 6.13 7.356
15% Neutralization
RPM K x106 mhos (K - KO)/KO X102
0 86.3 0

100 88.7 2.781
200 90.0 4.287
300 91.0 5.446
400 91.8 6.373
500 92.3 6.952
600 92.8 7.532
700 93.3 8.111
800 93.8 8.691
900 94.2 9.154

111

12. 5% Neutralization

 

 

RPM K x 106 mhos (K — KO)/KO
0 6.35 0
100 6.53 0.0283
200 6.62 0.0472
300 6.69 0.0535
400 6.75 0.0630
500 6.80 0.0709
600 6.85 0.0874
700 6.88 0.0835
800 6.91 0.0882
900 6.95 0.0945
1000 6.98 0.0992
17. 5% Neutralization
RPM K 31106 mhos (K — KO)/KO
0 8.44 0

100 8.78 0.0403
200 8.90 0.0545
300 8.97 0.0628
400 9.03 0.0699
500 9.05 0.0723
600 9.10 0.0782
700 9.16 0.0853
800 9.21 0.0912
900 9.26 0.0972
1000 9.30 0.1019

112

2 0% Neutralization

 

 

RPM K x105 mhos (K - Ko)/Ko
0 1.072 0
100 1.111 0.0364
200 1.136 0.0597
300 1.149 0.0718
400 1.163 0.0849
500 1.168 0.0896
600 1.176 0.0970
700 1.183 0.1035
800 1.190 0.1101
900 1.198 0.1175
1000 1.205 0.1241
25% Neutralization
RPM K X105 mhos (K — KO)/KO
0 1.360 0
100 1.425 0.0478
200 1.447 0.0640
300 1.466 0.0779
400 1.483 0.0904
500 1.497 0.1007
600 1.510 0.1103
700 1.520 0.1176
800 1.529 0.1243
900 1.538 0.1309
1000 1.546 0.1368

113

30% Neutralization

 

 

 

RPM K x105 mhos (K — K0)/KO
0 1.460 0
100 1.534 0.0507
200 1.562 0.0699
300 1.592 0.0904
400 1.610 0.1027
500 1.629 0.1158
600 1.642 0.1247
700 1.653 0.1322
800 1.667 0.1418
900 1.675 0.1473
1000 1.684 0.1534
40% Neutralization
RPM K x105 mhos (K - KO)/KO
0 1.550
100 1.650 0.0645
200 1.681 0.0845
300 1.715 0.1065
400 1.742 0.1239
500 1.760 0.1355
600 1.776 0.1458
700 1.792 0.1561
800 1.805 0.1645
900 1.815 0.1710
1000 1.825 0.1774

114

5 0% Neutralization

 

RPM K x105 mhos (K — KO)/KO
0 1.852 0
100 2.016 0.0885
200 2.075 0.1204
300 2.119 0.1442
400 2.150 0.1609
500 2.183 0.1787
600 2.203 0.1895
700 2.222 0.1998
800 2.237 0.2079
900 2.257 0.2187
1000 2.268 0.2246

 

115

Table XV

Relative Change in Conductivity of Polymethacrylic Acid

Solutions in Couette Apparatus at 400 R. P. M. , A400

C = 0. 0116 Monomoles/Liter

 

“ A400
0 0

5 0.008
10 0.024
15 0.039
20 0.041
25 0.051
30 0.063
40 0.078
50 0.090

c ==0.0213InononnfleslPhL4/l

 

o. A400x102
0 0

5 0.35

10 0.75

15 1.80

20 2.35

25 2.90

30 3.20

40 4.10

50 4.90

116

C = 0. 0291 monomoles P. M.A./liter

 

“ A400
0 0.0008
5 0.0025

10 0.0055

15 0.0150

20 0.0195

25 0.0235

30 0.0275

40 0.0350

50 0.0400

.c = 0.0058 monomoles P.-M.A./1iter

 

a A400
0 0.002
5 0.014

10 0.046

15 0.064

20 0.085

25 0.090

30 0.102

40 0.124

50 0.161

11>

400

oo‘o‘o‘w

O

DUO

P

U

111910-

el

H's

I'h

Appendix IV
NOMENCLATURE

relative change in conductivity

relative change in conductivity at 400 R. P. M.

one -half the length of rod-shaped particles in Stoke's law, cm.
thickness of rod-shaped particles in Stoke's law, cm.

shape factor of rotational ellipsoid, equation (26)
hydrodynamic length of a monomer, cm. , equation (14)
concentration, moles/cm.

concentration at zero position in diffusion cell, moles/cm.
mutual diffusion coefficient, cm. 2/sec.

dielectric constant of solvent

rotational diffusion constant, sec.
degree of polymerization
gap width in Couette apparatus, cm.

free energy, ca1./mole

electrostatic free energy, cal./mole
fraction of ion binding, equation (5)

axial flux, equation (33)

average end-to-end distance

mean square end-to-end distance of polyion
unionized mean square length

diffusional flux, moles/ cm. 2/sec.

total number of interference fringes

rotational flux of particles due to Brownian movement

117

 

 

L

W W ‘-|. LI.

WI

KW)

K

”ZZZIZ

"U

)0th

118

rotational flux of particles due to flow

fringe number in lower level of diffusion cell
conductivity of solution measured in Couette apparatus
conductivity of solution at rest

conductivity of solution at complete orientation of polyions
in Couette apparatus

normalized anisotropy

fringe number in upper level of diffusion cell
Boltzman's constant, 1. 380 X 10"16 erg/deg.
length of polyion, cm. or angstroms

magnification factor of diffusiometer, 1. 923

arbitrary abscissa scale in electrical anisotropy measurements,
sec. ' /cm. '

arbitrar ordinate scale in electrical anisotropy measurements,
fraction cm.

average molecular weight
Avogadro's number, 6. 023 X 102'3
number of particles per unit volume, equation (32)
refractive index, equation (39)

moles of solute per milliliter, equation (2)
number of ions of type i per macromolecule
parameter used in Figure 17

pressure, atm.

effective charge of polyion

velocity gradient, s ec .

gas c onstant

R. P. M. revolutions per minute

N

N N K1

119

radius of sphere in Stoke's law

inner radius of rotor in Couette apparatus
distance between charges on monomer units i and j
number of monomers per Kuhn Statistical element
absolute temperature, 0K

time, sec.

interaction energy between charges

enlargement scale in Figure 17, decimeters
volume of macromolecule

velocity in the Y-direction cm./sec.

average particle molbility of a solution at rest

average particle mobility perpendicular to the major axis of
the particle

average particle mobility parallel to the major axis of the
particle

direction of the velocity gradient in Couette apparatus
distance, cm.

direction of flow in Couette apparatus

direction perpendicular to the XY-plane in Couette apparatus

degree of polymerization

degree of neutralization
degree of ionization

angular position of a particle relative to the directional
measurement of conductivity in Couette apparatus

angular position of the major axis of a particle in Couette
apparatus

electronic charge, e. s.u.

120

r) apparent viscosity, centistokes
no solvent viscosity, poise
sp specific viscosity
[ n] intrinsic viscosity
0 angular position of a particle relative to the directional
measurement of conductivity in Couette apparatus
K reciprocal Debye radius
0 chemical potential
11 e1 electrostatic chemical potential
1/ number of ionized groups per polyion
TT osmotic pressure
p hydrodynamic resistance
p density, gm./cm.
0' q/Dr
(I) directional distribution function for the particle axes
<1) osmotic pressure coefficient, equation (9)
<19 angular position of the major axis of a particle in Couette
apparatus, Figure 5
Subscripts
3 refers to solvent
1 refers to upper level of diffusion cell
2 refers to lower level of diffusion cell
0 refers to center of diffusion interface in diffusion cell
Superscripts

I

refers to measurements on photographic plate

 

120

n apparent viscosity, centistokes
nO solvent viscosity, poise
sp specific viscosity
[ n] intrinsic viscosity
0 angular position of a particle relative to the directional
measurement of conductivity in Couette apparatus
K reciprocal Debye radius
M chemical potential
[.1 e1 electrostatic chemical potential
V number of ionized groups per polyion
Tr osmotic pressure
p hydrodynamic resistance
p density, gm. /cm.
0' q/Dr
(I) directional distribution function for the particle axes
ct) osmotic pressure coefficient, equation (9)
(I) angular position of the major axis of a particle in Couette
apparatus, Figure 5
Subscripts
s refers to solvent
1 refers to upper level of diffusion cell
2 refers to lower level of diffusion cell
0 refers to center of diffusion interface in diffusion cell
Superscripts

refers to measurements on photographic plate

 

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