May, 1961
SECOND VIRIALCOEFFICIENT FOR UNCHARGED SPHERICAL MACROMOLECULES 777
03 THE SECOKD VIRIAL COEFFICIENT FOR UNCHARGED SPHERICAL MACROMOLECULES BY ANDREWG. DE Rocco Department of Chemistry, University of Michigan, Ann AT^, Michigan Receiaed September $8,1960
The second virial coefficient haa been calculated for spherical particles obeying an intermolecular potential having the following properties: a hard sphere repulsion and an integrated R-8 attraction. The results are compared with the theory of Isihara and Koyama and with the data of Krigbaum and Flory on polyisobutylene in benzene. The linearity of A2 veTs'sus temperature in the region (2'/8) = 1, a result discussed by Isihara and Koyama, was not found in this approach. The general appearance of' A2 veTsus temperature, however, wm in good agreement with experiment as well as the previous theoretical results. The Flory temperature was found to be independent of molecular weight.
Introduction In a recent paper Stigterl has discussed the osmotic pressure and the osmotic coefficient of sucrose and glucose solutions. Treating the cases of spheres and ellipsoids, Stigter computed the second and third virial coefficients in the expansion of the osmotic pressure in terms of powers of the concentration. The intermolecular potential employed was the well-known (m-m), which for the special case of m = 6 is more often called the Sutherland potential. Such a potential has been used extensively in the discussion of the equilibrium and transport properties of simple gases. Experience has shown that the values of well depth needed to fit experimental properties, for example, second virial coefficients, are generally much larger than the corresponding values for a Lennard-Jones or (6-12) potential.2 Nonetheless the (6- a) potential has continued to be of theoretical interest largely because it is fairly flexible and. does reflect in its attractive component the leading term in the London dispersion energy. For particles of intermediate and colloidal sizes, numerous attempts have been made to compute the integrated (6-12) p ~ t e n t i a l . ~I n all cases two facts stand out: (1) the essential additivity of the dispersion potential leads to an enhanced interaction for larger particles; (2) the integrated repulsion becomes more nearly approximated by a hard sphere cutoff. Qualitatively, these considerations lead lis to expect interactions having a range of the order of the particle sizes with a concomitant effect on the virial coefficients. On the basis of his analysis of the second and third virial coefficients Stigter concluded that the London-van der Waals attraction alone was insufficient to account for the observed osmotic pressures of glucose and sucrose, and suggested, therefore, specific interactions probably due to hydrogen-bonding. It is not unreasonable, con(1) D. Stigter, J . Phys. Chem., 64, 118 (1960). (2) It is interesting that surprising agreement can be obtained for a triangular-well potential if tho cutoff is chosen to make the reduced Boyle temperature. TR* = k T / c . conform t o the value asaociated with the (6-12)potential. These results will soon be published. (3) H. C. Hamaker, Ph,/sico, 4, 1058 (1937); M.Atoji and W. N. Lipscomb. J . Chem. Phys., 2 1 , 1480 (1953); A. Isihara and R. Koyama, J . Phya. SOC.Japan, 12, 3:? (1957); A. G. De Rocco, J. Phys. Chem., 62, 890 (1958); C. J. Bouwkamp, Kon. Nederlond Akod. Wetenschop., 50, 1071 (1947); G.J . Thomaes. J. chim. phys., 49,323 (1952): K.S. Piteer, J. A m . Chem. Sac.. '77, 3427 (1955); J. A. Lambert, A d r d i a n J. Chem., 12, 109 (1958); A. G. De Rooco and W. G. Hoover. Proc. No& Acod. Sci., 46, 1057 (1960).
sidering the significant role played by hydrogenbonding elsewhere, to suggest such interactions for glucose and sucrose. I n this communication we shall investigate the related problem of the second virial coefficient for the case of an integrated (6-12) potential in the limit of infinite dilution. The results should have relevance to solutions of uncharged spherical macromolecules. Review of Theory Using an appropriately generalized grand partition function McMillan and M a ~ e r ,and ~ subsequently Kirkwood and Buff,6 were able to derive rigorously an expansion for the osmotic pressure in terms of the concentration which bears a formal resemblance to the virial expansion employed in the theory of imperfect gases. The osmotic pressure can be written as - 77
RT
- M-'C
+ AzC* + A3CZ + . . .
(1)
where M is the molecular weight of the solute, C the concentration in grams per unit volume of solution, and the coefficients, Ai, are the virial coefficients. Introducing a generalized distribution function in the coordinates of N particles such that V - N F ~(1,2,. . .,N) d(1,2,. . .,Ar) is the probability that N specific solute molecules lie in generalized configuration space between (1,2,. . . , N ) and (1,2, . .,N) d(1,2,. . , N ) , the second virial coefficient was shown to be
.
+
AZ=
.
F$&~
[Fdl)Fd2) - F2(1,2)1 d(1,2) ( 2 )
(One of the earliest applications of this result to problems of physical interest was made by Zimm,6 and the subsequent literature has been both abundant and varied). In equation 2, F1(l) and F1(2) are the distribution functions of single molecules, and F2(1,2) is the pair distribution function: all distribution functions are those of the solute molecules a t infinite dilution. It proves convenient to introduce the function g2 { 2 \ = F2(1,2) - F1(1)F1(2) to measure the interaction of the particles on one another by the difference between the pair distribution function and the corresponding product of singlet distribution functions for the case of independent probabilities. Equation 2 can then be written as (4) W. G. McMillan and J. E. Mayer, J. Chem. Phus.. 13, 276 (1945). 15) J. G.Kirkwood and F. P. Buff, ibid., 19, 774 (1951). (6) B. H.Zirnm, ibid., 14, I64 (1946).
- 4 2 = ~ T N o M - ~ ZJ I: l ,,~ (I
-e
B p j ' ( ~ ~ 1 () I )
+ z)*ds
(8)
(Sotice that ( N ) = (1,2,. . .,,Y)). F(1,2) can be where Do = 2Ro is the diameter of the sphere. decomposed into the product of F1(l),F1(2) and Integrating first from minus one to zero, we obtain the appropriate pair correlation function; the latter, in turn, can be approximated by the radial distribution function, neglecting a term O(l ',v),~ (1 - eBHu*(Z,l))(l + z)zdz] (9) y(r), permitting eq. 3 to be written in the convenient form The leading term is exactly the hard-sphere second virial coefficient, ao, and may be combined with -4, = 2 ~ ~ V J l - l [l - I / ( T ) ] T ' ~ T (4) A2,in a fashion analogous to the case in imperfect where No is the Avogadro number and g(r) is the gas theory, yielding a reduced second virial coradial distribution function at infinite dilution for efficient Az* = As/ao: thus two solute molecules. Kext the radial distribution - t 2 * = I + 3 J m (1 - e p H w * ( z . l ) ) ( I + r)2dr (10) function is approximated by exp[-pW], where p = 'kT and TV represents the potential of the mean The integral of eq. 10 can be accomplished force! We wish to consider here spherical particles numerically, but below we shall offer a first-order and we will use for such particles an integrated theory for weak interactions in order to gain some(6-1 2 ) potential, thus preserving central symmetry ; what more physical insight into the problem. The this integrated potential will be taken as W . trouble with a first-order theory, of course, is that Consider two spheres of diameters Dl and D2 upon expansion of the exponential we get an inte(for convenience set D, < 0 2 ) having a distance of grand that explodes a t the limits of integration. closest separation d = r - I / z X (Dl Dz), where To obviate this annoyance we consider that around r is the internuclear separation. Define 5 = d/D1 the hard core there occurs a well of energy - H and y = D2/D1so that x measures d in terms of the extending to x = 6 , where 6 is a small quantity. smaller sphere and y describes the relative size of Beyond 6 we permit eq. 5 to hold as written. For the two spheres. It is known, then, that W(x,y) simplicity we introduce p = (z 1) = R/2Ro can be expressed as9 and notice that It'(z,l~)= - H ~- Y A,* = 1 + 3 Jla [I - eBHf(d]pz d p
J:
+
+
[9
+ y: +
5 +52
2 In
It'(5,y)
+ ry + z + y +
+ + + -1
52
+ zy +
9 X?/ -1
