Feb., 1959
SECOND VIRIALCOEFFICIENT FOR POLYELECTROLYTES
253
THE SECOND VIRIAL COEFFICIENT FOR POLYELECTROLYTES. THEORY AND EXPERIMENT' B Y T. A.
O R O F I N O Z s 3A N D
P. J. F L O R Y 3
Department of Chemistry, Cornell University, Ithaca, N . Y . Received Seplember 5 , 1958
The theory of the second virial coefficient An for non-ionic polymers has been extended to include polyelectrolytes in salt solutions. Each polymer molecule is assumed to be described satisfactorily by a gaussian distribution of chain elements about its center of gravity; each small region of volume within the molecular domains is considered to be in Donilan equilibrium with the esternal (salt) solution. Summation over all volume elements and subsequent integrat,ion over the coordinates of the cenkrs of gravity of two neighboring polpelectrolyt~moleculesyields an espression for A Zidentical in form with that found previously for non-ionic polymers, A Z = (const.)[ ( S ~ ) ~ / ' ~ / ~ ~ ~ ] but I Z (with X I , the X S )interaction , quantities X I and XZredefined as X I = 103(33/2/221r3!2)(1 / ~ ~ V 1 ) ( ~ ~ 2 ~ u / ~ ~ ~1/2 ) z (-s ~X I) - 3V1i2/4VUS*) /2( and X Z = 106(36/2/23~3). (1/N2Vv,)(~~V,/n2u)3(s")-3(1/3 - x.1 Vl(z- - ~ + ) i ~ / l 2 V , ~ S *Mu ~ ) .is the molecular weight of asegment, VI and Vu the molar volumes of solvent and segments, respectively, ST)^/^ the radius of gyration of the polymer coils, i their degree of neutraliz:ttion, and S* the ionic strength of the external solution. The parameters X I and x2 have been defined previously in the theory for non-ionic polymers. The function I 2 ( X 1 , X z )is a definite integral which takes into account the extent of interpenetration of polymer molecular domains. TIYO useful approximations, suitable for stipulated ranges of X I and X ? , have been previously given. Parameters calculated from the above relationships are compared with the results of light scattering measurements on the systems poly-(aci,ylic acid), NaC1, water and polystyrene-p-sulfonic acid (K-salt), KCI, water. The agreement is qualitatively good although the osmotic effects of the small ions are greatly depressed by the chnrges on the chain, particularly for high degrees of neutralization.
+
+
Introduction Theories of dilute polyelectrolyte solutions heretofore have been largely concerned with the expailsion of a single macromolecule. Before proceeding further, it mould be well therefore to examine briefly the progress made in this connection. Several treatments have been published in which the electrostatic free energy was directly considered in arriving a t expressions for the equilibrium size of the polyion in solution. Harris and for example, have applied this method t o a random chain model of the polymer. A survey of earlier theoretical treatments is given in their publications and we shall therefore not reiterate. An equivalent and niore direct theoretical interpretation' of polyelectrolyte solutions can be developed on the basis of the conditions for Donnan type osmotic equilibrium between mobile electrolyte within the domain of the coil and its external environment. Electrostatic interactions affect the equilibria, of course, but they need not be separately taken into account, provided that the net electrical charge within the polymer domain is negligible in comparison with the sum of the charged sites on the polyelectrolyte chain; this latter condition is almost universally met by high molecular weight polyelectrolyteq under experimental conditions. I n accordance with the method described above, the following relationship for the equilibriuni size of a high molecular weight polyelectrolyte molecule immersed in an infinite bath of salt solution was e~tablished'~~ 01~
- a 3 = 103(3ah/231r8/2)( ~/NVI)(MV~/M,,)~ (?2)-3/2[1/2
+
- xl + vli2/4~,zS*i . . .
(1)
(1) Presented before the 130th American Chemical Society Meeting, Atlantic City, N. J . , September, 1956. (2) U. S. Rubber Company Fellow, 1955-19513, (3) nIellon Institute, P ~ t t s b u r g h ,Pa. (4) F. E. Harris and S. A. Rice, THISJOURNAL, 68, 725 (1954). (5) S. A. Rice and F. E. Harris, a b d , 68, 733 (1954). (6) See also R. A. RIarcus, zbid., 68, 621 (1954). (7) P. J. Flory, J . Chem. Phys., 21, 162 (1953). (8) P. J. Flory and J. E. Osterheld, THIEJOURNAL, 68, 653 (1954).
I n this equation, cy is the factor by which the unperturbed linear dimensions of the molecules are expanded due t o segment-solvent interactions, VI and Vu are the molar volumes of solvent and segments, respectively, N is Avogadro's number, M and Muare the molecular weights of the polymer and its repeating unit, and ( 2 ) ' / 2 is the radius of gyration of the unperturbed molecule; i and X" are, respectively, the degree of ionization of the fixed charges on the chain and the ionic strength of the medium surrounding the polymer molecule. The parameter x1 has been defined p r e v i o ~ s l y . ~ Higher terms in the series of eq. 1 are negligible in most cases of interest. The second virial coefficient A , in the semiempirical equation for the reduced osmotic pressure T/C
=
+ AZC+ A & + .
f?T[l/Mo
.]
(2)
where c is the polymer concentration and M the molecular weight, or in the equation for the excess scattering intensity Kc/Ro
+ 2.42~+ 3.43~*+ . . .
= 1/av
(3)
where R is the optical constant and Ro is Rnyleigh's ratio a t zero angle, bears an intimate relationship to the sizes of the dissolved polymer molecules. It is the object of the present communication to extend the foregoing treatment of the isolated polyelectrolyte molecule to the interaction of two identical polyions in salt solution. We shall employ the methods used previouslylO,ll in the treatment of the second virial coefficient for non-ionic polymers. The theory will be compared with light scattering results on the systems poly-(acrylic acid), NaC1, water and polystyrene-p-sulfonic acid (K-salt) KCI, water. Theory Consider a single polyelectrolyte molecule im(9) P. J. Flory, "Principles of Polymer Chemistry," Cornell University Press, Ithaca, N. Y., 1953: J . Chem. Phys., 17, 303 (1949). (10) P. J. Flory, ibid.. 17, 1347 (1949): P. J. Flory and W. R. Krigbaum, ? b i d . , I S , 1086 (1950). (11) T. A. Orofino and P. J. Flory, ibid., 26, 1067 (1957).
T. A. OROFINOAND P. J. FLORY
284
mersed in an infinite bath of solution of strong electrolyte (salt), &!I A;; ;: at a concentration cs* nioles/l. The polymer chain bears ionizable substituents which are taken to be ident*icalwith M2+.The domain of the polymer is considered to be divided into a succession of spherical shells of volume 6V (volume elements), each of which coiltains segment,s, ionized gegenions from the chain, and salt which has migrated into the volume element'. The volume elements are assumed to be in Donnan equilibrium12 with the external salt solution. If mean ionic activity coefficients of all mobile ion species are taken as unity, the difference in solvent chemical potential between the jth shell and the external solution in this approxiniation13may then be written' ~ i j PI* = --RT[I/Z - xi ViP/4Vu2S*]z)2j2 [1/3
- XI
+
+
+ V I ( Z -- ~ + ) i ' / 1 2 V u ' S * ~ l t ~+~ j '. . .
(4)
where v2j is the volume fraction of segments in the volume element and 8" is the ionic strength of the external solution. The parameters x1 and xz also appear in the treatment" of non-ionic polymers; they retain their formal significance in the present case . The free energy difference 6AFj between the jth volume element and the salt solution follows from integration of eq. 4 8AFj = kT( 103/N)( V,*SVj/Vl) [( 1/2
+
- x1 +
- +
Vii2/4VuzS*)pj2 (1O3/2N)VU(1/3 XI V ~ ( Z~ + ) i ~ / 1 2 V ~ ~ S * ~ ). p. .j]' ( 5 )
+
In this equation the volume fraction of polymer in the volume element has been replaced by its equivalent in terms of the segment density pj. In analogy with the previous treatment for uncharged polymers, me compute the total free energy AFa associated with the process of bringing two identical polymer molecules, initially well separated, within a separation of centers a. The distribution of segments of each molecule is assumed to be gaussian. The result is 417~ = k~ ill
xi exp( -uiuz)
(6 )
x
1
eq. 7 consists of two additive terms: the first ones, represented by 1/2 - XI an X1 and 1/3 - xz in X 2 , are identical with the corresponding XI and X2 for non-ionic systems"; the second represent the ionic contributions to X1 and X2 and will generally assume dominant importance in determining the values of these quantities, although the contribution of the non-ionic terms (e.9. 1 / 2 - XI) inny become appreciable for n sufficiently small ratio i3/S* (under which conditions the ionic characteristics of the polyelectrolyte molecules are considerably suppressed). The excluded volume is given by u = 4~
- XI
+ Vli2/4V2S*) -'
(7)
Xa = IO8(3'/2/2'?r3)( 1/N2Vi)(AfVu/Mu) a(?) X~ vl(z- z+)ia/12vU3~*2) (1/3
-
+
-
u1
= 3/4F
u2
a Z [ l- exp(ilF,/kt)]da
(9)
and in accordance with the relationship A2 = =
J
(IG~N/~% [ ( ?) ) ~ / z / M ~ I 0
m
1 2 1 1 - exp(-xle-t2
-
X?e-4t22'3) ] d t = (16~N/3'/2)[(s")'/~/nlz]r,(x,,x,) (10) The above equation is identical in form with the expression developed previously for non-ionic systems.l1 If i mere set equal to zero, or S' equal to a , the total mobile ion concentration inside and outside the polymer coils would be equal; under these conditions the system could be regarded as consisting of only two components, L e . , polymer and salt solution, and accordingly the eq. 7 would reduce to their previous definitions for uncharged polymers. For the case of a symmetrical added electrolyte, x+ = z-, and according to the defining relation in eq. 7, the ionic contribution to X2 vanishes. I n the general case, the total contribution of XZto the second virial coefficientis usually small. As in the intramolecular treatment,s the higher terms in the series of eq. 6 appear to be negligible in all cases of interest. The expression for Az in eq. 10 may be simplified through use of the approximations to I ~ ( X I , X ~ ) given previously." There it was shown
= 105( 3' / ~ / 2?r ~ '1%) ( 1/ N V , ) (Af Vu/Mu)2(>) -*Is
(l/2
0Jm
N u / 2 M 2yields
A2
where
Vol. 63
(161rN/3*/z)[(~)~/~/M2] In 11 + - ( T ~ / z / ~-!)-~ I (?r'/r3'/n/32)X*] (11) XI < 100
or, in accordance with the limiting approximation for 12(X1,X , ) for large XI AZ
(IBTN/~'/Z) [(?)a/z/MZ][ln (XI
+ X&'a) + yla/a
(12) (8)
= I/>
where 2 is the square of the radius of gyration for each polymer molecule. Each of the interaction quantities X1 and X2 in ( 1 2 ) An expression for the second virial coefficient for polyelectrolytes in salt solutions has been given previously by D. T. G. Pals and J. J. Hermans, Rec. trau. chim., 71, 458 (1952). They considered the Donnan equilibrium between two phnses, one of which contained a uniform dispersion of polyelectrolyte segments (with ionizable groups) and the other a solution of simple, strong electrolyte. The model treated by them is formally identical with that employed here on a microscopio scale. Their treatment, however, neglects the discontinuous nature of the polyelectrolyte solution in the dilute region. (13) The results of the present treatment would be unaffected however if we were to impose instead the less stringent requirement of equality of mean ionic activity coefficients in both phases.
0.577.. . XI > 35
y =
The closed form used for Iz(X1, XJ in eq. 11 becomes increasingly accurate as the argument of 12 approaches zero, whereas the approximation to this integral employed in eq. 12 becomes a better representation for large values of the argument. In the neighborhood of XI = 35, the two approximations are equally satisfactory. Although the limiting expression for APin eq. 12 is to be preferred for ca. X1 > 35, the approximation in eq. 11 may be used for values of XI as large as 100 without introducing appreciable error. The latter has the advantage that it permits representation of Az by a, single simple expression applicable to both
SECOND VIRIALCOEFFICIENTFOR POLYELECTROLYTES
Feb., 1939
lion-ionic polymers aiid polyelectrolytes, provided only that XI is not excessively large. Comparison of the defining relations for XI and Xzin eq. 7 with the expression for the intraiiiolecular expaiision factor CY in eq. 1 yields a2
-
1 = x1/2
+ x2/3
(13)
With an error which is negligible, me may incorporate this result in eq. 11 to yield another expression for the second virial ~oefficieiit'~ 8 2 N (1GnN/3a/z)[(s2)3/1/U2]In [I + (lr1/z/2)(a2 - I)] (14
The above equation succinctly conibiiies the interand intramolecular theories; it is applicable to both polyelectrolytes and non-ionic polymers and, moreover, in the former case, circumvents the iiecessity of assigning a particular value to the degree of ionization (cf. seq.). The equations 11, 12 aiid 14 express the second virial coefficient, in accordance with the model chosen, in approximate closed form over the entire range of segmeiit-solvent internction. It should be borne in m i d , however, that although the iiiathematical represent,ations of 12(X1,X2)for all values of the argument d o n o t in any sense invalidate these equations, other limitations of theory (e.g., assumption of spherical symmetry) proba.bly place an upper limit on the range of usefulness of the theoretical expressions for A z . Experimental
285
1.4 p ) , using 5 rinses. The procedure described mas found to be satisfactory when care was taken to exclude dust during the operations. Upon completion of the light scattering, the same solutions were used in the determination of the refractive index increments. At the t,ermination of these runs the polyacid concentrat>ionand degree of neutralization were checked by dry weight analysis and titration. No significant changes from the original values were ever observed. The poly-( acrylic acid) conctmtrations in the sections which follow are all expressed in terms of the pure polyacid. Light Scattering.-The light scattwing apparatus and differential refractometer employed in these measurements have been described prc~iously.17:1~Proper cell alignment was confirmed by measuring the light iutensity from a fluorescein solution as a function of angle. Measurements of theo scattered intensities, using unpolarized light ( A = 4358 A . ) , were made a t angles from 45 to 135' on each of the solutions. The temperature of the solut'ion in the cell was 27.5 0.5'. Before and after each series of readings, the galvanometer deflection was observed for scattering at 90" by one of two secondary standards, depending on the intensity of the scattered light exhibited by the solutions: a polished block of poly-(methyl methacrylat8e),or a sealed solution of polystyrene in toluene. The ratios of the secondary standard wattering intensities to that of the Cornell Standard Styron (0.5 g. of polystyrene in 100 cc. of toluene solution) a t 00' were determined. The known absolute scattering power of the latter permits expression of Rayleigh's ratio as
+
+
Re = (&/& - i o / ~ c ) ( i c / ~ ~ p ~ ) ~e/(c p1 ~ [ scos2 i n e)] n ~ ~ / / t ~ ~ ~ ~
(15) where is, io and i, arc, respectively, the galvanometer readings for the solution, the solvent and the secondary standard. R,,, for the Cornell Styron a t 90" was taken18as 2.08 X The quantity ic/icp8 is the ratio of the galvanometer reading at 00' for the secondary standard to the (excess) reading Cornell Styron when the latter is in the same cell used Polymer Samples.-The poly-(acrylic acid) fraction used for the the scattering measurements, no is tlie refractive index in this investigation was prepared by Newman, et a1.16 for the solution (solvent) and ncllethat for the Styron (toluLight scattering measurements on the polysalt in NaC1 of ene). solutions gave A[," = i . 7 X 105. Depolarization measurenients were made at ewh concenIn order to ascertain the acid content, the sample V,VM ti- tration for each system studied. The ratios of the horizontrated with carbonate-free 0.05 A I NaOH, using phenol- tal and vertical exceRs polarizwtion readings were eometimes phthalein as indicator, i n the presence of approxiniat.ely 0.1 found to vary elmtically with concentration, due to tlie d l NaC1. The sample assayed about 92% purity. Similar u~iavoitlal~ly large experimental error involved in this ineasassays were found for other fract,ions of the same polymer urement (the galvanometer readings for horizontally poby IX'ewmnn, et al. They ran Karl Fischer tit,ratioiis on larized light a t 90' were often less than 0.1 the smallest the samples and found a water content of ca. 6yo. The ye- division on the galvanometer scale). I t was found, howniaining impurity is probably dioxane (fractionation sol- ever, that either the lowest concent,ration used for each vent). system gave no depolarimtion, or, when this was not,.the Light scattering nieasurements on n ~ ~ o l ~ s t ~ ~ r e n e - p - scsse, r i l - that the measured values for a series of conccritrittions fonic acid (I
