382
Vol. 67
K O R I O ISS
SECOND VIRIAL COEFFICIENT OF POLYELECTROLYTES. 11' BY NORIOI S E ~ School of Chemistry, Rutgers, The State University, Kew Brunswick, New Jersey Received September 10, 196d
A previous treatment of the second virial coefficient of polyelectrolytes was applied to the experimental data of lithium polyphosphate and potassium polyvinylsulfonate. Qualitatively the same results were obtained, as in the case of sodium polyphosphate previously studied: (1) the macromolecular dimensions, calculated from the simple electrolyte-polyelectrolyte interaction parameter, were found t o be fairly close to those obtained from viscosity, and (2) the polymer-concentration dependence of the macromolecular dimension, estimated from the polyelectrolyte-polyelectrolyte interaction parameter, was positive. From comparison of polymer- and simple electrolyte-concentration dependences of the macromolecular dimension, it was suggested that the molecular does not depend on the dimension of polymeric electrolytes having a fairly small degree of polymerization (PO) polymer concentration. At PO,two interaction parameters mentioned above are not determined by intramacro-ion interaction, bat by interionic interactions only.
Introduction In a previous paper,3 the second virial coefficient of polyelectrolytes a t the limit of zero polymer concentration was discussed using a spherical model for macro ions. The treatment was applied to experimental data of sodium polyphosphate (YaPP). The molecular dimension of this polyelectrolyte was found t o be comparable to that estimated from viscosity. It was found furthermore that the polymer-concentration dependence of the molecular dimension was positive. In order to examine the validity of these results, it appears necessary to check whether the same results may be obtained for other polyelectrolyte^. The materials to be analyzed in this paper are lithium polyphosphate (LiPP)4s5 and potassium polyvinylsulfanate (KPVS).6 As in previous me shall calculate the macromolecular dimension from the simple electrolyte-polyelectrolyte interaction parameter, Pz3,which can be estimated from membrane equilibrium data, and then we shall determine the polymer-concentration dependence of the macromolecular dimension from the polyelectrolytepolyelectrolyte interaction parameter, &, which is estimated from both the membrane equilibrium data and light scattering data. The effects of the size of small ions coexisting, and of the degree of ionization, i, on the molecular dimension and its concentration dependence will be discussed also. Simple Electrolyte-Polyelectrolyte Interaction Parameter.-@23 was determined from the membrane equilibrium parameter r fby using the relationship
+
rf = io(i~
1B231323)/m3'w2
+ p3,m3)
(1)
where B33 (= b In y3/bms)is the simple electrolyte-simple electrolyte interaction parameter, which can be estimated from the literature value of the activity coefficient y3of the simple electrolytes, m8and m3' me the molar concentration of simple electrolyte in the phases with and without polymer, respectively, P is the degree of polymerization, and M is the molecular weight. On the other hand, it already was s h o ~ that n ~ ~ the~ parameter pZ3can be written (1) This work was done during the tenure of the Colgate Fund (19611962). (2) Department of Polymer Chemistry, Kyoto University, Icyoto, Japan. ( 3 ) N. Ise, J . Chem. Phys., 36, 3248 (1962). (4) U. P. Strauss and P. Ander, J . Am. Chem Soc., 80, 6494 (1958). (5) P. Snder, Doctoral Thesis, Rutgers, The State University, New Brunswick, N. J., 1961. (6) H. Eisenberz a n d D . Woodside, J . Chem. Phys., 36, 1844 (1962). The author is grateful to Dr. Eisenberg for sending his data before publication. (7) N. Ise, (bid., 36, 1145 (1961).
when the simple electrolyte-concentration dependence of limiting viscosity number [v] can be given by Cox's relation.8 For convenience, the definitions of the notations are repeatedly given; A' is ,4vogadro's number, a is the number of electric charges of a macro-ion ( = iP),eo is the elementary charge, E , dielectric constant of water. IC, Boltzmann's constant, T , absolute temperature, n3is the number of simple electrolyte molecules per unit volume of solution, 3^n8, the excluded volume parameter of simple electrolyte ions (= 1 - nav3), ua, the excluded volume of simple electrolyte ions (= 4aS3/3), a3, the radius of simple electrolyte ion, II, extended Debye's reciprocal length (2 = 87rn33^n3eo2/&?' for simple electrolyte5 of 1-1 type), &, radius of gegenion and ionized group assumed to be a metallic sphere, R, radius of macro-ions assumed to be spherical, u (y) = (3/y3). y - 1/(1 y) - 2 In ( 1 y) 1 and 7 (Y) = (3/y3). Iln (1 Y) - u Y2/21. Thus, it is possible to estimate p23from r', and then R from the Pn3. The calculations were carried ?ut at 2.5' with further assumptions of S2 = S3 = 5 A. for HPVS-KC1 systems, 6 A. for LiPP-LiBr systems. and i = 0.25, unless otherwise specified, and the results are listed in Tables I-IV, where RE is the molecular dimension estimated from Einstein's viscosity law using the value of [v]determined from m30[q0]. From Tables I and 11, values9are seen to increase
+
+
+
+
+
(8) Using thelimiting viscosity number [ n ] (,../lo0 ml.) of LiPP (31 = 4.3 X 106) reported in ref 4, one finds t h a t ma"[q] (=mss"[no]) is constant for m = 1 and is equal to 0.910, where m is Cox's constant determined b y trial a n d error method, and [sa], the limiting viscoqity number in a reference solution of simple electrolyte of concentration msa. Since one can assume [ V I a M ' / ? a t m8 = 1.7 for this polymer (note that the Flory point is 1.8 ;M a t 25' for this system9, mso[qa]is found t o be 0.638 for our polymer (h!! 2.42 X 106). Converting the unit of the limiting nscosity number reported in ref. 6 into g./lOO ml., one obtains a n average value of 0.155 for mso[nol of KPVS. Here again Cox's constant W a s found t o be unity for this polymer. (9) I n the case of KPVS, pzs was calculated after interpolating the observed equilibrium parameter (at 23") given in Table I of Eisenherg and Casassa's work [ J . PolzJmer Scz., 47, 29 (l960)] to simple electrolyte concentrations nhich are given in the first column of Table I. The temperature difierence is not important according t o ref. 6.
SECOND VIRIALCOEFFICIENTOF
Peb., 1963
POLYELECTROLYTES
383
TABLEI polymer domain with a constant total number of p23, pzZrMOLECULAR DIMENSION AND ITS POLYMEI~-CONCENTRAcharges would cause a lowering of the electric potential T l O N DEPENDENCE around the macro-ion. Thus the activity coefficient of (KPVS-KCl-Hz0) simple electrolytes would not be much affected by addi(bPJbnz)/P tion of the macro-ion, or the absolute value of stays ma,
M
x $1.07 +1.17 +1.45 +1.62
@ZS
0.35 .50 .65 .72
R,
RE,
A. 78 68 58 58
A. 116 105 96 94
822
x
dR/an2,
x
cm.4
+4.26 f1.72 +1.23 -0.74
1011,
om.8
+1.4 X 10-'1 +8.8 X 10-2' +5.5 X lo-'' +6.0 X lo-''
-0.94 -0.81 -0.77 -0.78
TABLEII MOLECULAR DIMENSION AND p 2 3 (LiPP (M = 2.42 X 105)-LiBr-Hz0) ma, M 0.0201 .lo27 .4047
EFFECTSOF (KPVS-KC1-H20, 68,
A.
R, A.
Baa
R, A.
RE, A.
-1.00 x 104 -1.61 x 103 -7.05 X 1 0 0
262 162 92
494 287 246
TABLEIII MOLECULAR DIMENSIUS FROM T Y L ~ = 0.72, 623 = f1.62 X loa, RE := 94 6 7 4 5 49 39 61 57 ON
A~)
TABLE1V EFFECTSOF i O N AND MOLECULAR DIMENSION (RPVS-KCI-Hz0, W Z ~= 0.72, RE = 94 A.) i
R, A.
&a
0.10 .17 .25 .30 .50
+5.00 +3.42 +l.62 +4.95 -4.00
X lo2 X 10% X lo2 x 101 X 10'
54 57 58 58
58
TABLEV COMPARISON OF VALUESAND MOLECULAR DINENSIONS OF KPVS, LiPP, AND NaPP AT m3 = 0.4 KPVS
m30[7ol
M
P pZ3
R, A. RE,A.
0.155 2.38 x 105 1.63 x 103 +l.lO X lo2 75 113
small.11 On the other hand, the absolute value of P 2 3 of KPVS can be found to have to be one fourth of that of NaPP, because is pmportiohal to P 2 , through cy2, according to eq. 2 : in the factir of the second braces (in the right hand side of the equation) that is sensitive t o M , [TI, and k,the decreases in J l and mao[qo] are approximately cancelled out by the increase in l / R 4 , Actually the observed value of ,823 is not far from this expectation. The positive p23 for KPVS indicates that the molecular dimension is fairly small for a polymer having the observed value of limiting viscosity number. This appears to be in line with Eisenberg and Woodside's observation6 that KPVS has a value of 1.39 X for a, a dimensionless constant in the Flory theory of viscosity, which is significantly higher than an average value of 0.9 X loz3found for a number of polyelectrolyte systems. Polyelectrolyte-Polyelectrolyte Interaction Parameter.-When the second virial coefficient, B, is measured and 8 2 8 is knowii, pZ2can be obtained by the relationship
LiPP
0.638 2.42 x 105 2.82 x 103 -7.05 X 10" 97 246
NaPP
0.177 3 . 3 2 X 106 3.26 x 103 -5.92 X 10'
85 122
with increasing simple electrolyte concentration for both KPVS and LiPP, as was discussed b e f ~ r e ,and ~ the value of R, smaller than RE but still compatible, deereases with increasing simple electrolyte concentration as it should. I n Table 111, the R values obtained with various a3 and with i = 0.25 are shown. It is clear that the R value obtained from ,&decreases with increasing a3. Table IV shows the effect of i. I n this case, not only the observed value of pzabut also the calculated value depends on the assumed value of i and the R value itself hardly varies with i. For the comparative purpose, the R values of KPVS, LiPP, and NaPP are summarized in Table V together with the relevant parameter values. The corrcehtration of simple electrolytes is about 0.4 M. It is seen that the R value is on the order of RE,as it should be. ,!Iis z in 3 the order of KPVS > LiPP > NaPP. The examihation of the magnitude of each term of eq. 2 reveals that the small absolute value of ,823 of the lithium salt is due to a larger value of m30[r]o]than that of the sodium salt.'* This means that the larger molecular dimension of the lithium salt is respoinsible for the small absolute value of ,823. This is quite possible, since the enlargement of
Using the ,823 values given in the second column of Table I and the observed values of B,6 one can obtain the values of p22 for KPVS, which are listed in the fifth column of Table I. Putting the Pz2 thus obtained and the R values given in the third column of Table I into the formula as
P2z = - {a4e02x/12ekTm3m3)X { ~ u ( x R )~ u ( H ~ z ) / ( Y 2U(H&)/f% 3/23?&(1 ~ 8 3) ~ 4n3u(x&) uZp/ cy2) - (Na2e02/10 ek TR)(bR/dn2) X
+
+
+
{ D / O + NR)- d
4 I x
+ '/&I
(4)
where vZp = 4nR3/3and n2is the number of macro-ions per unit volume of solu+ion, the polymep-concentration dependence of the molecular dimension, dR/dn2, can be estimated. The results are listed in the sixth column of Table I. As for LiPP, B values of the polymer designated as LiPP-2 (M = 4.3 X lo6)in ref. 5 were used. pZ3was obtained frdm the membrane equilibrium data of a LiPP (A4 =t 2.42 X lo5) by assuming the molecular meight independence of the data, and is shown in the second column of Table VI. From these B and pZ3 values. one can estimate pZ2,which is listed in the fourth column of Table VI. The R values for bhis LiPP were determined by assuming R cc &P'Zfrom the R value given in Table 11. The values of bR/bn2 calculated with the use of these Pz2 and R values are shown in the fifkh column of Table VI. (10) The right hand side of eq. 2 is composed of the first braces with a negative factor and th- second braces u i t h a positive factor, a n d these two braces are positive under the piesent experrmental condition. Therefore, t h e second faotdr containinp mso[si] may determine the sign and magnitude of 8 2 3 . It should be noted furthermore that the choice of different 8a values for lithium salt a n d alkali salt does not affect the statement in the text. (11) This stetainent is Juetified only so far as the volume excluded by the polymer is not influential. afi IS the case for the present systems.
KORIOISE
384
Vol. 67
the purely electrostatic intera~tion.'~Therefore we suspect that the solvent-polymer segment interactions iiz the systems of KPVS would not be negligible. This sort of interaction appears to manifest itself also in the temperature dependence of the limiting viscosBza F, Pza X 10-8 ma X 10-8 A. aR/anz, cm.4 (bP/bnz)/P ity number. l4 1.15 +1.26 50 f 1 . 1 6 +8.0 X lodz1 -3 7 X lo-'' Discussion 1.50 +1.36 35 +1.56 +9.2 X -1.3 X 10-'6 From the foregoing results, it is made clear that the 1.70 +1.38 29 f 1 . 7 2 f 2 . 4 X lo-'' -7.3 X lo-'' second virial coefficients of KPVS and LiPP can be Tables I and VI show that bR/dn2is positive as was accounted for by two parameters, R and bR/bnp, as reported b e f ~ r e . The ~ derivative of KPVS appears to was the case for T\'aPP.3 The value of R calculated have a tendency of decreasing with increasing ms as from pZ3 was found t o be practically independent of the was the case for KaPP,3 whereas that of LiPP increases. assumed value of i, and to change slowly with that of &. Following the reasoning given in ref. 3, one can calculate This means that it is difficult to obtain conclusive inthe polymer-concentration dependence of the osmotic formation of high accuracy pertaining to the state of effects of sihlple electrolyte ions, (bP/bnz)/P,which is dissociation of macro-ions from flZ8or membrane equilibshown in the seventh column of Table I and in the sixth rium data. The polymer concentration dependence of column of Table VI for KPVS and LiPP, respectively. molecular dimension was found to depend on the asWe can understand that as the absolute value of (bP/ sumed values of i and &, but the sign of the derivative bn2)/Pdecreases or increases, bR/bnz decreases or inwas always positive for LiPP and KPVS, as in the case creases. This is wholly in line with the inference in of NaPP. terms of electrostatic interaction developed in r&f.3. It has to be mentioned that Cox's relation assumed in the derivation of eq. 2 gives a negative simple electrolyteTABLE VI1 concentration dependence of the macromolecular diEFFECTS OF h3 o s aR/dnz mension. Although this negative dependence appears (KPVS-KC1-H20, m3 = 0.72, pZ2 = -7.40 X l o 4 ) a t first sight to be incompatible with the positive polyaa (b.) 4 5 6 7 mer-concentration dependence found from f122, the difbR/bna" (cm.4) 5.1 X 10-22 6 . 0 X lo-%*8.0 X 10-32 1.4 X 10-21 ference is due to the presence of numerous electric a Calculated with an R value of 58 A. irrespective of 63. charges on the polymer chain. As was pointed Table VI1 gives the results of calculations carried out the positive polymer-concentration dependence has its with different 8 8 values and R = 58 A. As 88 increases, origin in lowering of the osmotic effectiveness of simple dR/bn2increases, and remains positive. l 2 Table T'III electrolyte ions by a newly added macro-ion. Addition presents the effect of i. The larger the assumed value of a simple electrolyte ion, instead of a macro-ion, on the of i, the larger &R/bn2becomes and remains positive. other hand, easily can be shown to increase the osmotic This was the case at all other concentrations of KCI and pressure outside the polymer domain so that the macrofor LiPP as well, Thus one could conclude that the ion shrinks,15i.e., the simple electrolyte-concentration result of bR/bnp> 0 is not due t o the improper choices dependence is negative. Thus it mould be expected of the i and SI values. that there is a degree of polymerization, Po, a t which the concentration dependence of molecular extension is TABLE VI11 equal to zero, when the simple electrolyte is considered EFFECTSOF i O N pgzAND POLYMER-CONCENTRATIOX DEPENDENCE a special case of polyelectrolytes ( P = 1). Assuming OF MOLECULAR DIMENSION the molecular weight independence of membrane equilib(KPVS-KCl-H20, m3 = 0.72) rium data and of the second virial coefficient of KPVS, i Pza Pzz bR/anaa (cm.4) one can calculate bR/bnzof K P V s of various degrees of 0.10 +5.oo x i o 2 +4 34 x 104 $9.1 x 1 0 - 2 3 polymerization. The results are listed in Table IX, .17 f 3 . 4 2 X lo2 -2.56 X lo4 f2.5 X showing that bR/bnz decreases with decreasing P and .25 f1.62 X lo2 -7.40 X lo4 +6 0 X becomes zero about a t P = 24 (=Po) and thereafter .30 +4.95 X 10' -2.70 X lo6 +8.0 X lo-" assumes negative values." Although the PO value .50 -4.00 X lo2 -8.33 X l o 5 +2.2 X has to be re-examined,18 one may assert that, a t P = Calculated with an R value of 58 A. irrespective of i. Po, the flexible macromolecule behaves as a rigid one. Comparison of the bR/bn2 value for NaPP reported (14) I n view of the electrostatic interaction, the viscosity number of in ref. 3 with that for KPVS reveals that the former is polyelectrolytes is expected to decrease uwth increasing temperature [N. Ise, J . Polymer Sci., 39, 413 (1959)j. The limiting viscosity number also is exlarger than the latter. It would be clear that, as the pected to decrease with temperature because the contractive force of the degree of polymerization increases, the effect of a newly polymer chain becomes larger. The limiting v1scoslty number of KPVS, added macro-ion becomes larger so that larger change however, increases with temperature.6 (15) The change of the osmotic pressure can be calculated using eq. 10 in R is observed.la The degree of polymerization of the in a previous work [N. Ise and M. Hosono, J . P o l y m e r Scz., 39, 389 (19S9)l NaPP in question is about 7 X lo3 whereas that of and eq. 12 in ref. 3. (16) The insensitivity of membrane equilibrium data toward P was KPVS is 1,63 X 103. established experimentally. See, for example, IT. Eisenberg and E. F. It has to be noted that pZzdecreases with increasing Casassa, J . P o l y m e r Scz., 47, 29 (1960). Table IX of ref. 6 shows t h a t m3 for KPVS and increases for LiPP. As was menthe second virial coefficient can be regarded as molecular weight-independent when P is large. tioned already in the case of WaPP,a the interaction (17) This tendency appears to be in agreement with a previous observaparameter is expected to increase with ma in view of tion [H. Fujita a n d T. Homma, J . Colloid Scz., 9, 591 (1954)l that the posiTABLE VI
P23,
&,
MOLECULAR DIMENSION AND ITS POLYMER-COXCENTR.4TION DEPENDENCE (LiPP (M = 4.3 X 106)-LiBr-H,0)
(12) Qualitatively the same results were obtained with LiPP. (13) It might be possible t h a t DR/bnz depends also on the intrinsic flexibility of the chain. This derivative would be zero, if the polymer chain is rigid, a n d become larger a8 the chain becomes more flexible.
tive slope of viscosity number polymer concentration curve a t high dilutions decreases with decreasing degree of polymerization. (18) It is really difficult t o assume that the spherical model holds for such a low molecular weight polymer.
COKDUCTASCE OF TETRAPHENYLBORIDE IONI N XIETHANOL
Feb., 1963
It is clear furthermore from eq. 4 that the electrostatic
TABLE IX CONCENTRATION DEPENDENCE OF MOLECULAR DIMIE?JSION us. DEGREEOF POLYMERIZATION
(KPVS-KCI-HSO, P
3.25 X lo3 1.d X I O 3 8.13 X lo2 2.4 X 10
1
+1.71 $4.26 +1.06 +9.3
X X X X
...
-- 0.35)
ma
R,
Pa2
IO6 lo6 lo5 10'
38 5
A.
110 78 55 9.5
...
bR/bnea
+i.3 x 10-70 +1.4 X +1.4 X 10-2a 0 -4.3 x 10-27
ne: polyelectrolyte or simple electrolyte concentration in numbers per unit volume of solution.
intramacro-ion interaction, which is proportional to 3/gR,plays no role a t P = Po and only contributions from the interionic interactions are important as far as fizz is concerned. Moreover, one can see from eq. 2 that, at P = Po,P 2 3 also is determined by interionic interactions only, since the result from b R / b ~ z = Oisobtainedby [l/(l xR) - ~ ( l i R ) 3/5&] = 0. Acknowledgment.-The author wishes to thank Professor H. Hertder of the Computation Center, Rutgers, The State University of New Jersey, for kind help, and Professors I. Sakurada, A. Nakajima, and P. Ander for interesting criticisms.
+
+
CONDUCTAKCE OF THE TETRA4PHENYLBORIDE ION IS n/lETHANOL1,2 BY ROBERT W. KUNZEAKD RAYMOSD M. Fuoss Contribution No. 1717 from the Sterling Chemistry Laboratory of Yale University, New Haven, Connecticut Received September 19, 1962 The conductances of lithium, sodium, and potassium tetraphenylborides in methanol a t 25" have been measured. Using these data and literature values for the conductances of lithium, sodium, and potassium chlorides in methanol and the transference numbers of the chloride ion in sodium and potassium chlorides, the limiting conductance of the tetraphenylboride ion Is found t o be 36.50 f 0.05.
Tn order to study ion-solvent interaction, it is necessary to have single ion conductances. These are in general difficult (or impossible) to obtain, because reversible electrodes, essential for transport measurements, are known in only a few solvents. I n order to circumvent this difficulty, Fowler and Kraus3 made the assumption that the two ions of tetrabutylammonium triphenylborofluoride and of tetrabutylammonium triphenylborohydroxide had equal mobilities, so that XO+ = XO- = A0,12. Later, Fuoss and Hirsch4 made the same assumption concerning the ions of tetrabutylammonium tetraphenylboride. Since the transport numbers of the chloride ion in sodium and potassium chlorides in methanol are known6 and likewise the conductances of the chlorides of lithium, sodium, and potassium6in the same solvent, the conductance of the tetraphenylborido ion can be obtained from conductance measurements on the alkali tetraphenylborides in methanol. I n this paper, we report data which give Ao(B(C~H&-) = 36.50 rt 0.05. Preliminary measurements on tetrabutylammonium salts give ho(BuaiT+)= 39 in methanol, whence n + = 0.513 instead of 0.500 as assumed by Fuoss and Hirsch. Experimental Sodium tetraphenylboride (Aldrich) as received is brownish and smells of phenol. It was dissolved in dry acetone (5% solution); on the addition of toluene, the salt begins to separate. The initial precipitate is gelatinous and adsorbs most of the brown impurities. Thiq material was filtered off and discarded. Addi(1) This paper is based on part of a thesis which will be presented by Robert W. Kunze t o the Giaduate School of Yale University in partial fulfillment of the requiiements for the degree of Dootor of Philosophy. (2) Grateful aoknowledgment is made t o the donors of The Petroleum Research Fund, administered by the American Chemical Sooiety, for support of this research. (3) D. L. Fowler and C. A. Kraus, J . A m . Chem. SOC.,62, 2237 (1940). (4) R. M. Fuoss and E. Hirsch, ibzd., 82, 1013 (1960). ( 5 ) J. A. Davies, R. L. Kay, and A. R. Gordon, J . Chem. PhUB., 19, 749 (195 1). ( 6 ) J. P. Butler, H. I. Schiff, and A. R. Gordon, ib
