20 An Electrochemical Method for the Determination of the Effective Volume of Charged Polymers in Solution
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PETER SLOTA and JACOB A. MARINSKY Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14214
An electrochemical method projected for the determination of the effective volume of a colloidally dispersed polyelectrolyte phase in aqueous media was evaluated. Experiments with the highly flexible Sephadex (carboxymethyldextran) gel and the more rigidly cross-linked polymethacrylic acid resin were performed for this purpose. With the well-defined resin (gel) phase it was possible to measure the polymer volume as a function of every experimental condition used to test fully the fundamental concepts on which the method is based. The results substantiate the validity of concepts developed. Application of this method for estimating the effective volume of weakly acidic (basic) polymers in solution seems worthy of further consideration. However, some modification of the treatment of the electrochemical data is necessary for polymeric sols, and this aspect is discussed briefly.
^ p h e electrochemical method proposed here for the measurement of the •*· effective volume of charged polymers dispersed as colloidal suspensions is based on an observation b y Merle ( I ) . I n his recent study of the influence of the swelling of the weakly acidic Sephadex gel on its potentiometric properties during neutralization with standard base i n the presence of a fully dissociated polyelectrolyte, sodium polyvinyl sulfate, he discovered that over the complete neutralization range p H — p N a — log -r^ I
—
log a
= Constant
Kp
0-8412-0482-9/80/33-187-311$05.00/0 © 1980 American Chemical Society
Eisenberg; Ions in Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
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312
IONS
IN
POLYMERS
I n this equation a represents the degree of neutralization, ν the number of repeating functional units i n the polymer, and V the volume of the gel. W e sought the origin of this constant term i n the following theoreti cal analysis of Merle's observation. A t equilibrium, during each step of the potentiometric titration of a weakly acidic (or weakly basic) gel (HA)„, i n the presence of a simple electrolyte M X , the chemical potential μ of each diffusible component ( H X , M X , and H 0 ) is equal i n both phases; for example, p
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2
μηχ =
MHX
μΜχ
/ÂMX
=
and μπ ο = μπ ο 2
(2)
2
the bar placed above μ identifying the gel (resin) phase. In the gel (resin) phase / Z H X — μ°ηχ + RT In Ô H X + UVnx /XMX —
RT In Ô M X
/*°MX +
+
(3)
UV
UX
where μ° represents the chemical potential of each component i n the standard state, Π the osmotic pressure of the water i n the gel phase, and V the partial molar volume of the diffusible components. The osmotic pressure is related to the activity of water a i n the two phases by w
U=-{RT/V ) W
In ( â / a ) w
(4)
w
A t equilibrium, the distribution of H X and M X during each step of the potentiometric titration is defined by the reaction H X + MX^± M X + H X
(5)
Recalling that the activity of each component i n the solution phase is defined by μ
(6)
= μ° + RTlna
we have chosen the standard state to be the same i n both phases so that the sum of all μ° terms is zero; we obtain RT In [ (α χά χ)/(ΟΗχα χ) ] — U/RT(V x Η
Μ
Μ
H
- V ) MX
as the expression of this equilibrium.
Eisenberg; Ions in Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
(7)
20.
SLOTA
Charged Polymers in Solution
A N D MARINSKY
313
In Merle's examination of carboxymethyldextran, polyvinyl sulfate (a fully dissociated poly electrolyte) was used i n place of a simple salt to control the ionic strength. Invasion of the gel phase b y coion X was thus avoided. T h e electrochemical potential of the mobile counterions are equal throughout such a system at equilibrium and Equation 7 trans forms to (U/RT) (V
In [ ( α ά ) / ( ά Η α ) ] Η
Μ
Μ
K
-
V) u
or
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pM -
pH — pM -
pîï +
0 . 4 3 4 3 (V
(U/RT)
-
u
(8)
V) u
A t equilibrium the p H of the gel (resin) phase is given by Equation 9
P
H = pZ^ „ A )
+
l
o
g
? L - + logy-A-
r
(9)
where pH
log C
H
- log y
(10)
H
K J ^ i s the intrinsic dissociation constant of the repeating acidic group in the macromolecule, a is the degree of neutralization, C denotes the concentration of H i n the gel, and y A- and t/ are activity coefficients of the designated species i n the gel phase. Correction for deviation from ideality of the associated species H A is considered negligible and activity coefficients are assigned only to the charged species H and A " . W i t h this representation of p H Equation 8 becomes H
+
H
+
p M - p H — p M - vK £ l
A)v
~ log
- log y - + Δ
1^(0.4343) ( V H - V I I )
(ID
- log 2/M
(12)
B y definition p
M
log C
M
and Equation 11 becomes pM - pH
log C
M
- logT/n - pK\ ^ n
log y A" +
A)v
- log
χ
" ^ -
(0.434) ( V
H
-
V) u
Eisenberg; Ions in Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
(13)
314
IONS IN
POLYMERS
F o r each molecular unit of H A dissociated, one molecule of M the gel (resin) phase and
+
enters
C M = «v/V
(14)
P
where ν denotes the number of ionizable groups i n (HA)„ and V is the effective volume of the gel domain. Substituting this definition of C in Equation 1 3 P
M
pH
-
pM
-
log γ ^ —
+
log ^
+
- jSL
( 0 . 4 3 4 3 ) (V
Downloaded by FUDAN UNIV on March 23, 2017 | http://pubs.acs.org Publication Date: June 1, 1980 | doi: 10.1021/ba-1980-0187.ch020
P^(HA) +
u
1 θ
δ » A - +•
-
l 0
V )
=
H
ë
(15)
In the experiments of Merle, the p H , p N a , and V of carboxymethyldextran were measured as a function of a during titration with standard N a O H i n the presence of sodium polyvinyl sulfate. W e can therefore substitute N a for M i n Equation 1 5 . In addition, the U/RT (0.4343) term is small enough in this system to neglect so that the following equation should adequately describe the potentiometric data. p
2 pH -
pNa
-
log γ ^ 1
_ fog
—a
— P-KIHA)^ + K
log y A - +
log
y
Na
p
(16)
The left-hand side of this equation is identical to the left-hand side of Equation 1 found by Merle to yield a constant value of 3 . 7 6 ± 0 . 0 7 over the complete neutralization range of his study; the preceeding anal ysis does not lead to prediction of the observed constancy. Instead, the right-hand side of the equation is equal to p K as shown later, and should vary with the degree of neutralization by increasing slowly in magnitude with increasing a. There is experimental evidence to support the estimate that j / = i/Na i n this system. In the research of Travers and one of the present authors ( 2 ) , a study of the complexation of divalent metal ions by polymethacrylic acid ( P M A ) showed that deviation from ideality with a of the divalent ions exposed to the same potential as the H ion was described exactly by the deviation term deduced from the potentiometric properties of the P M A . Additional evidence for this estimate is available i n the ion-exchange literature ( 3 , 4 , 5 ) . A t a relatively low cross-linking percentage ( 2 wt % divinyl benzene) {/ is about equal to f/ a> as evidenced by the ion-exchange distribution of N a and H between a polystyrene sulfonate-based resin and a simple dilute electrolyte mixture of N a and H ( N a , H , X " ) . The selectivity coefficient measured over the complete composition range of the resin deviates very little from unity to demonstrate this as an experimental fact ( Κ Η = 1.02 ± 0.02 H
+
N
H
+
+
+
+
+
+
Ν Λ
Eisenberg; Ions in Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
20.
SLOTA
at X «
— 0, Κ Η » = ν
N
315
Charged Polymers in Solution
AND MARINSKY
1.07
±
0.02
at X
N a
=
0.5,
and K
H
N
A
=
1.12
±
0 . 0 3 a t X = l ) (6). _ Since 1/ t/ the right-hand side of Equation 16 can be written as P ^ ( H A ) P + 1°S F A - + log t/ . F r o m Equations 9 and 10 N a
Na
H
H
P *
