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J. Phys. Chem. B 2001, 105, 5262-5270
Solvent Effects on Osmotic Second Virial Coefficient Studied Using Analytic Molecular Models. Application to Solutions of C60 Fullerene Jean-Pierre Simonin† Laboratoire Liquides Ioniques et Interfaces Charge´ es (LI2C-UMR 7612), UniVersite´ P.M. Curie, Bote no. 51, 4 Place Jussieu, 75252 Paris Cedex 05, France ReceiVed: NoVember 7, 2000; In Final Form: January 29, 2001
Osmotic second virial coefficients (OSVC’s) are calculated analytically for some molecular models in which the solvent is taken into account explicitly. The basic effect of solvent packing is examined first by applying the method to the reference case of a binary hard sphere mixture. Then, cohesion forces for the solvent and solute-solvent attractive forces are introduced successively into the description. The results are applied to interpret recent experimental data on solutions of C60 in nonaqueous solvents. With a plausible assumption in the model, a very good agreement is obtained with the experimental OSVC value for C60 in CS2. In the case of benzene as the solvent, the theoretical result is consistent with the experimental observation (formation of aggregates). No specific adjustable parameter was needed in the model to interpret the experimental results. The description suggests that the solute-solvent interaction has a determinant influence on the observed OSVC value.
Introduction In the past decade, there has been a renewed interest for the study of osmotic second virial coefficients (OSVC’s), commonly denoted by B2. Experimental values of the OSVC have been determined for colloids such as proteins1-7 (including bovin serum albumin, lysozyme, R-chymotrypsin, β-lactoglobulin, ...) in aqueous solutions and a fullerene,8,9 the buckminsterfullerene C60, in various nonaqueous solvents. The determination of B2 values is interesting because this quantity reflects the (attractive or repulsive) nature of the interaction between solute molecules in a solvent. Furthermore, the fact that it might be a good indicator of various properties seems to emerge. So, a correlation has been mentioned with C60 solubility in solvents9 and, in the case of proteins, it has been reported that quality crystals are formed when B2 falls within a narrow range.10 Models have been used1-7,11-15 to interpret OSVC values, and their variation, with the properties of the supporting medium (e.g., pH). Generally, the reference interaction is the colloid hard-sphere volume exclusion, with respect to which various solute-solute interactions are added. These include chargecharge, dipole-dipole, charge-dipole, dispersion forces, .... For solutions containing another (smaller) solute such as an electrolyte or a polymer, additive forces have been added to the description4,11,12 to account for attractive depletion forces between the colloid particles. This effect was introduced originally by Asakura et al.16 In these models, the solvent has been regarded as a continuum. Besides, fundamental theoretical studies have been developed in which the solvent was taken into account explicitly.17-20 The structural properties of the model systems considered were generally determined by using approximations derived from integral equation theory, such as Percus-Yevick or hypernetted chain (HNC) equations. Basically, this type of study has shown †
E-mail:
[email protected]. Fax: 33 (0)1 44 27 38 34.
that depletion of solvent between the colloid particles can be the source of specific forces. So, for a simple binary hard-sphere mixture, it was found that the small spheres tend to push the large spheres together at short separation of the large spheres.17 (Notice that this effect is absent in the classical DLVO theory.21) Some fundamental principles have emerged from these studies: it has been found that solvent-solvent cohesive forces make the solute-solute potential of mean force more attractive;20,22 on the other hand, solute-solvent attraction (as modeled by a Yukawa potential20) seems to reduce the depth of the depletion well. Besides, a very complex behavior has been observed in the case of charged macroparticles in the presence of an added electrolyte.22 In these studies, integral equations were solved numerically. No OSVC value was calculated. The purpose of this paper is to show that OSVC values may be calculated analytically for some simple systems in which the solvent is taken into account explicitly. The results are obtained directly via the thermodynamic route, using approximate molecular models of solutions. Interestingly, they allow one to examine solvent contributions in a simple way. Moreover, they are used for a discussion of recent experimental results on solutions of C60 fullerene in carbon disulfide and benzene.9 This type of system has the following noteworthy features: (i) the solvents used were nonaqueous solvents, (ii) the C60 molecule is a neutral spherical particle, and (iii) its size is not much larger than that of the solvent. The first two features provide relative simplicity as compared to aqueous solutions of charged molecules. The three features may allow one to use classic thermodynamic molecular models in a fairly realistic way. This paper is organized as follows. In the next section, some basic definitions and further remarks are given. In the third section, the general procedure for calculating B2 is presented. It is then applied to the case of a binary mixture of hard spheres. Next, solvent-solvent and solute-solvent interactions are added successively to this reference system. The last section is devoted
10.1021/jp004077l CCC: $20.00 © 2001 American Chemical Society Published on Web 05/12/2001
Solvent Effects on OSVC’s
J. Phys. Chem. B, Vol. 105, No. 22, 2001 5263
to an examination of the results, focusing on an interpretation of experiments on C60.
R)1+3
Definitions and Further Comments The OSVC is the second virial coefficient of the osmotic pressure, Posm. In the virial series expansion of Posm, it is defined as23
β Posm ) Fs + B2(T)Fs2 + ...
2
B2 ≡
1 d βPosm (Fs ) 0) 2 dF 2
(2)
s
with Fs the solute number density (number of particles per unit volume), β ) (kBT)-1, kB the Boltzmann constant, and T the temperature, which is held constant. Hereafter, the indices “s” and “w” are chosen for convenience to designate quantities for the solute and for the solvent, respectively. It is known since the work of McMillan and Mayer24 that the OSVC is related to the potential of mean force, W, between two solute particles at infinite dilution. In the case of centrosymmetric interactions and spherical solute particles this relation reads
B2 ) 2π
∫0∞[1 - exp(-βW(r12))]r122 dr12
(3)
where r12 is the distance between the centers of two solute particles. The potential W is related to the solute-solute radial distribution function, gss, through
gss(r12) ) exp[-βW(r12)] Then one also has from eq 3
B2 ) 2π
∫0∞[1 - gss(r12)]r122 dr12
(4)
Equation 3 or 4 may be split into two terms, one of them being the hard core contribution (gss ) 0 for r smaller than the contact distance), which yields
B2 ) B(0) 2 + 2π
∫σ∞[1 - gss(r12)]r122 dr12
(5)
s
In this equation 3 B(0) 2 ) 2πσs /3
(6)
is the second virial coefficient for a pure fluid of particles of diameter σs. B(0) 2 is also the “covolume” for the solute (four times the volume of a particle). Equations 5 and 6 may be written because gss ) 0, or W ) +∞, for r12 < σs, if we assume that σs is also the closest distance of approach. Throughout this paper we will use the reduced OSVC defined as
R≡
B2 B(0) 2
with B(0) 2 given by eq 6. Then, from eqs 5-7, one gets
(7)
(8)
where jr12 ) r12/σs and gjss(rj12) ) gss(r12). In the early OSVC literature,25-27 one can find the statement that, in the case of hard spherical, otherwise noninteracting, solute molecules (denoted by the superscript “hs”) one should have
(1)
so that
∫1∞[1 - gjss(rj12)]rj122 drj12
(0) Bhs 2 ) B2
(9)
because the solute-solute potential of mean force should be zero outside the core (W ) 0, gss ) 1 for r12 > σs). Then, eq 9 was deduced by virtue of eq 5. In his well-known paper, Zimm25 assumed “an effectively continuous and structureless solvent” to establish this relation. Equation 9 has often been mentioned in the literature for “hs” solute molecules. However, eq 9 is certainly incorrect in general because it neglects effects arising from the solvent. Obviously, a great advantage of the McMillan-Mayer framework to deal with osmotic experiments is that the solvent chemical potential is implicitly kept constant.24 Nevertheless, the effective solutesolute potential W appearing in eq 3 or 5 is not a priori known since it results from an average over the states of the solvent. It therefore contains necessarily a contribution from the latter. This contribution is likely to be significant because, basically, the background medium (the solvent) is a dense phase. Hence, the fact that the solute molecules be much bigger than that of the solvent should not allow one to reduce the solvent to a continuous medium, as assumed by Zimm.25 These conclusions are confirmed by recent fundamental work.17-20,22,28,29 So, in the basic case of a hard sphere mixture in the colloidal limit (large size asymmetry and low colloid concentration), it has been shown28,29 that the contact value of gss is a rapidly increasing function of the solute/solvent size ratio. Consequently, because of eq 8, one may expect R to take low, possibly negative, values in this case for large size asymmetry. Negative values3,4,6 of B2 have also been found experimentally for spherical, or approximately spherical, proteins. In other noteworthy cases (e.g., for globular proteins2,25 and C609), measurements have led to realistic effective hard-sphere sizes for the solute by use of eqs 6 and 9. This observation suggests that a subtle compensation must occur between the effects resulting from the various interactions in the system. So, the effect of hydration forces opposing that of electrostatic forces has been proposed recently to interpret the non-DLVO variation of the OSVC with added salt concentration in the case of a protein, apoferritin.7 Some of the above-mentioned issues are addressed in the present paper. First, the influence of packing on the value of B2 is illustrated in the simple reference case of hard spheres. Then the additional effects of solvent-solvent and solutesolvent interactions are considered in the case of C60. Theory Method for Calculating B2. The OSVC was calculated using eq 2. One may also think of using the relation
B2 )
d(βPosm/Fs) (Fs ) 0) dFs
which stems from eq 1 but, although it involves only a first derivative, this procedure is more cumbersome because it leads to difficulties when taking the limit Fs f 0.
5264 J. Phys. Chem. B, Vol. 105, No. 22, 2001
Simonin
The osmotic pressure is the variation in pressure caused by addition of solute, with repect to pure solvent, at constant solvent chemical potential. This may be written as
Posm ) P - P0
are obtained by further differentiation of eq 15, which yields
(10) and
with the constraint
µw(Fs,Fw) ) µw(0,F(0) w )
f ≡ β[P(Fs,Fw) - P0]
(12)
For a given solvent, this is a function of Fs and Fw, which coincides with βPosm along the osmotic curve by virtue of eq 10. So, eq 2 shows that B2 may be obtained by differentiating f along the osmotic curve, in the vicinity of the point (Fs ) 0, Fw ) F(0) w ). Then, using the chain rule, we first obtain
df ∂f ∂f dFw ) + dFs ∂Fs ∂Fw dFs
φ ≡ βµw
dFw ) -φs/φw dFs
(15)
with the notation
φx ≡ ∂φ/∂Fx
[ ( ) ( ) ] ( )
∂f ∂ f dFw ∂f ∂ dFw df ) 2+2 + + 2 ∂Fs∂Fw dFs ∂Fw ∂Fs dFs dFs ∂Fs ∂ dFw dFw ∂2f dFw + ∂Fw dFs dFs ∂F 2 dFs 2
2
(18)
∂2φ ∂Fx∂Fy
Therefore, it stems from eqs 2 and 12 (and the comment following eq 12) that the lhs of eq 16 is equal to 2B2 provided it is calculated along the osmotic curve and its value is taken at the point (Fs ) 0, Fw ) F(0) w ), which is
B2 ≡
1 d2f (F ) 0, Fw ) F(0) w ) 2 dF2 s
(19)
s
Equation 19, together with eqs 12-18, permits the analytic calculation of B2 for a given model of solution. This derivation becomes rapidly heavy, however. So, most of the calculations were performed using Maple, a symbolic computation device. A simple way of checking the result for B2 found from eq 19 is as follows. Let us denote by η the volume fraction occupied by molecules in pure solvent. When the latter are modeled as hard spheres of diameter σw, one has
π σ 3 η ) F(0) 6 w w
(20)
In the model systems treated below, the solute particles are taken as otherwise noninteracting hard spheres. Therefore, in this case, one should have
B2 f B(0) 2 when η f 0, that is in the absence of solvent, with B(0) 2 defined through eq 6. Equivalently, this condition may be written as
lim R ) 1 ηf0
(21)
with R defined by eq 7. Equation 21 is a useful test for checking the validity of the expressions for R derived below. Binary Mixture of Hard Spheres. Here, both the solute and solvent molecules are taken as simple hard spheres of diameter σs and σw, respectively. The accurate Boublik-Mansoori-Carnahan-StarlingLeland (BMCSL) equation of state30,31 was used. Defining
Next, recursive differentiation of eq 13 leads to 2
w
φxy ≡
(14)
and writing that dφ ) 0, which is equivalent to eq 11 at constant temperature, we get
( )
(17)
with
(13)
in which the notation dX/dFs is used to denote that the quantity X is differentiated with respect to Fs along the osmotic curve (Fw, as well as Fs, varies along this curve). As usual, writing ∂/∂Fs implies that Fw is kept constant, and vice versa for ∂/∂Fw. Equation 11 allows calculation of the derivative dFw/dFs along the osmotic curve, in an implicit way. Setting for simplicity
w
φsw φww ∂ dFw )+ φs 2 ∂Fw dFs φw φ
(11)
In these relations P is the pressure over solution, P ) P(Fs,Fw); (0) P0 ) P(0,F(0) w ) is the pressure over pure solvent. Fw and Fw are the solvent number density for solution and for pure solvent, respectively. The condition for osmotic equilibrium, expressed by eq 11, provides an implicit relation between Fs and Fw. This defines a curve in the (Fs, Fw) plane, that we will call the “osmotic curve”. Then, Fw may be regarded as a function of Fs alone. Although Fw cannot be expressed explicitly as a function of Fs, B2 may be calculated from eq 2 using implicit differentiation, provided the pressure and the solvent chemical potential are known as a function of composition. The procedure is described now. Let us set
( )
φss φsw ∂ dFw )+ φs 2 ∂Fs dFs φw φ
Xn )
π 6
Fiσin ∑ i ) s,w
the excess Helmholtz energy per unit volume for a mixture of hard spheres (HS), ∆FHS, satisfies the relation32
2
(16)
w
The derivatives appearing in the square brackets of this equation
(
)
X23 3X1X2 X23 β∆FHS + ) - 1 ln x + Ft X0x X0X32 X0X3x2
(22)
Solvent Effects on OSVC’s
J. Phys. Chem. B, Vol. 105, No. 22, 2001 5265
ln(1 - η) AHS(r,η) 3 RHS(r,η) ) r(r - 1)2 (31) + 3 4 η 8r (1 - η)DHS(η)
with
Ft ≡ F s + F w
with
the total number density and
4
x ≡ 1 - X3
AHS(r,η) )
the free-space volume fraction. Then, the excess solvent chemical potential, ∆µHS w , is
∆µHS w )
∑An(r)ηn n)0
A0 ) 2r3(3r3 - 6r2 + 3r + 4)
∂∆FHS ∂Fw
A1 ) 21r6 - 42r5 + 21r4 + 12r3 - 15r2 - 6r - 1 A2 ) 11r6 - 24r5 + 9r4 - 12r3 + 3r2 + 12r + 3
yielding33
A3 ) -(r - 1)(40r5 - 38r4 - 11r3 + 9r2 - 9r - 3)
2 β∆µHS w ) - ln x(σwX2/X3 - 1) (2σwX2/X3 + 1) +
3σwX2/x + 3σw2[X1/x + X22/(X3x2)] + σw3[(X0 X23/X32)/x + (3X1X2 - X23/X32)/x2 + 2X23/(X3x3)] (23) The solvent chemical potential, µHS w , is the sum of the ideal and HS excess contributions34 id HS βµHS w ) βµw + β∆µw
(24)
3 βµid w ) ln(FwΛw )
(25)
with
in which Λw is the de Bro¨glie wavelength of a solvent molecule: Λw(T) ) h/x2πmwkBT, with h Planck’s constant and mw the mass of a solvent molecule. Since Λw is a function of T alone, one may take
φ ≡ ln Fw + β∆µHS w
(26)
instead of φ ≡ βµw, as defined in eq 14. Besides, the excess pressure can be obtained from the general relation
P)
∑ Fiµi - F
(27)
i ) s,w
or by direct differentiation of eq 22. One finds33 3 β∆PHS X3 3X1X2 X2 (3 - X3) ) + + Ft x X x2 X x3 0
(28)
0
The total pressure, PHS, is given by
βPHS ) Ft + β∆PHS
(29)
This expression can be used to express the function f (eq 12). Then the OSVC can be calculated by plugging the expressions for µHS w (eq 24) and f into eq 19, using eqs 13-18. Let
r≡
σs σw
(30)
Then, the result for R (eq 7) in the case of a binary hard sphere mixture is
A4 ) (r - 1)2(14r4 + 4r3 - 3r2 + 2r + 1) DHS(η) ) η4 - 4η3 + 4η2 + 4η + 1 in which η is the volume fraction occupied by molecules of pure solvent (eq 20). Equation 31 is a new result. This expression was tested in two ways as follows. First, it was checked that it verifies eq 21. Second, a value for the integral appearing in eq 4 has been given previously35 in the particular case r ) 1, following a different route. With the present notations, the value given for this integral leads to
RPY )
(4 - η)(2 + η2) 8(1 + 2η)2
which result was derived from Wertheim’s solution36 of the Percus-Yevick (PY) equation. Besides, for r ) 1, eq 31 gives
RHS(r ) 1) )
4-η 4(η - 4η + 4η2 + 4η + 1) 4
3
A plot of these two expressions vs η shows an excellent agreement, with a maximum relative deviation of 0.4% for η = 0.24. It must be pointed out that accuracy of eq 31 becomes uncertain when the size ratio r is large. This restriction is motivated by recent theoretical investigations28 that have shown that the contact value of the BMCSL37 solute-solute distribution function, gss(σs), becomes inaccurate in the limit where r is large and Fs is small. Now one notes that the condition Fs f 0 is included in eq 2. Hence, since B2 is also defined through eq 5 and because the two routes for computing B2 must be equivalent, use of the BMCSL framework in the present derivation may become questionable for high values of r. Although this issue is still unclear,38-40 application of eq 31 certainly needs to be restricted to below some critical value of r, denoted by rc (possibly a function of η). Considering the information available, one might expect rc not to exceed a value on the order of 4 in any case. This point is further discussed below in the Results section. Introduction of Cohesive Forces for the Solvent. In the Application section we wish to consider solutions of C60 in carbon disulfide and in benzene. In a first approximation, we will assume that these solvents may be modeled as a collection of quadrupolar hard spheres. (The importance of these electrostatic interactions has been pointed out recently for benzene.41)
5266 J. Phys. Chem. B, Vol. 105, No. 22, 2001
Simonin
Considering these molecules as hard spheres may seem a crude approximation at first sight. However, it has been shown42 that, even in the case of markedly aspherical molecules, this representation allowed good correlation with compressibility data, over a wide range of pressure and temperature. We adopt here a model based on the Pade´ approximation of Stell et al.,43 that was extended to mixtures by Gubbins et al.44 This model is a special version of thermodynamic perturbation theory. It was tested by Pitzer et al.45 for mixtures of dipolar and quadrupolar hard spheres. By setting the dipolar moment to zero, one can particularize the equations to the case of a mixture of hard spheres and quadrupolar hard spheres. Then, the excess Helmholtz energy (per unit volume, over the hard sphere reference energy) may be written as45,46
∆FP ) F2/(1 - F3/F2)
(32)
Association was described using a thermodynamic perturbation theory due to Wertheim49 and developed by Gubbins et al.50,51 In the present work, it is supposed that the C60 possesses a certain number of sites onto which the solvent molecules can bind. This solvation number depends on the nature of the solvent. In this framework, the excess energy (over the nonassociating reference fluid) is given by51
β∆Fassoc )
2
Yi-1 ) 1 + njFj∆swYj
(35)
(36)
for i * j. For the quantity ∆sw, we adopt an approximate expression50,51 that may be written in short form as
7 2 βF2 ) - xw2Q/w4σw3IHS 10 (F*)Ft 10 36 2 / 6 3 HS 1 βF3 ) x Q σ I (F*)Ft2 + x 3Q/ 6σ 6IHS (F*)Ft3 245 w w w 15 6400 w w w qqq xixjσij3 ∑ i,j
σsw ) (σs + σw)/2
∆sw ) σsw3 gref sw(σsw)K
(33)
HS in which 0 is the permittivity of a vacuum. Besides, IHS 10 , I15 , HS and Iqqq are given by
(37)
where σsw ≡ (σs + σw)/2 is the solute-solvent closest approach distance, gref sw(σsw) is the contact value of the solute-solvent pair distribution function for the reference fluid, and K is a constant without dimension, which characterizes the strength and range of the interaction potential.50 We will further assume that
Ft ≡ Fs + Fw, xi is the mole fraction of component i, Q/w is the solvent reduced quadrupolar moment defined as
Q/w2 ≡ βQw2/(4π0σw5)
]
Yi - 1
where Yi stands for the fraction of free (not bonded) component i and ni is the number of association sites on molecule i. The Yi’s verify the following equations51
where
F* ) Ftσ3 ) Ft
[
∑ niFi ln Yi i ) w,s
HS gref sw = gsw
since we deal with closely packed fluids.52 The BMCSL version37 of gHS sw was used. Equation 36 (written for i ) s, j ) w, and i ) w, j ) s) can be solved for Ys and Yw. It is convenient for subsequent calculations to write the Yi’s in the form of a series expansion in powers of Fs. Then we obtain
5
IHS k (x) )
∑Jn,kxn n)0
Ys =
with the coefficients Jn,k listed elsewhere.45,47 The full expression for R, denoted by RHS+Q (the superscript Q being used to denote quadrupolar interaction), is quite large and so it cannot be given here. However, numerical results are given in the Results. Moreover, it must be mentioned that, as in eq 31, the result is again of the form
ln(1 - η) 3 + ... RHS+Q ) r(r - 1)2 4 η
(34)
This peculiarity comes from the term containing ln x in eq 22, that also cancels for r ) 1. Introduction of Solute-Solvent Association. Experimental evidence9,48 supports the conclusion that strong attractive interactions occur between C60 and the solvents CS2 and benzene. Solute-solvent association was introduced in the present description to account for this phenomenon. As shown in the Results, solvent cohesion causes OSVC values to drop significantly. On the contrary, solute-solvent interaction may be expected to separate solute molecules and thus to counterbalance the effect of attractive solvation forces on B2.
Rw Rw(Rw - 1) 2 1 + R + Rs 1 + Rw (1 + R )3 s (1 + Rw)5 w Yw = 1 -
1 1 R + R2 1 + Rw s (1 + R )3 s w
(38)
(39)
where
Ri ≡ niFi∆sw and in which it is sufficient to limit the expansions to second order in Rs because of eqs 10-18, 27, and 35 and ∆µassoc ) w ∂∆Fassoc/∂Fw. Here, this association model was added to the system studied in the preceding section (collection of hard spheres for the solute and quadrupolar hard spheres for the solvent). Of course, the full expression for RHS+Q+assoc is bigger than that for RHS+Q and therefore it is not given explicitly either. An explicit result is given in the Results in the case of the C60 fullerene. However, it is remarkable that, after simplification and similarly to eqs 31 and 34, the expression is again of the form
ln(1 - η) 3 + ... RHS+Q+assoc ) r(r - 1)2 4 η
(40)
Solvent Effects on OSVC’s
J. Phys. Chem. B, Vol. 105, No. 22, 2001 5267
Figure 1. RHS (eq 31) as a function of η, the volume fraction occupied by molecules in pure solvent, for four values of r (the solute-to-solvent size ratio): r ) 1 (solid line), r ) 2 (dashed line), r ) 3 (dotted line), and r ) 4 (dash-dotted line).
Thermodynamic Stability. For each model, one has to check for the stability of the system constituted by pure solvent. This condition is expressed by53
∂µw >0 ∂Fw
(41)
We remark that this quantity appears in the denominator of eqs 15, 17, and 18 in which φ is used to denote βµw. Thus, a singularity found for B2 (infinite value) corresponds to a boundary for stability. Then the following stability parameter
S ≡ F(0) w
∂µw (0,F(0) w ) ∂Fw
(42)
was calculated in each case for pure solvent. This definition is taken for S because it has the property that S f 1 when F(0) w f 0. This is due to the fact that, in this case, µw reduces to the ideal contribution (eq 25). Results and Discussion Binary Mixture of Hard Spheres. Equation 31 giving RHS shows that it is a function of r and η, the volume fraction occupied by the solvent molecules in pure solvent. This quantity is plotted in Figure 1 as a function of η, ranging from 0 to 0.5 (η = 0.54 being the value for hard sphere fluid-solid transition) for four values of r: r ) 1-4. For r ) 1 (the solute being supposed to be distinguishable from the solvent by some other property) and for r ) 2, RHS is a monotonically decreasing function of η. For the other values of r, it exhibits a minimum. It is somewhat surprising that RHS is not much dependent on r for values of r greater than 2, up to η ) 0.3. This fact is likely due to a compensation in eq 5, in which the oscillations of gss around 1 (due to packing18) are amplified with increasing r. The fact that RHS < 1 is in agreement with the notion of potential depletion16 leading to an effective attraction between neighboring solute particles (in eq 8, R < 1 implies gss > 1 or W < 0 near contact). It illustrates the above statement that the packing effect causes B2 to differ from B(0) 2 , contrary to the often-made assumption of eq 9. However, it will be seen below that the effect of packing alone is less pronounced than the combined effects of packing and solvent cohesion, at least in the range of r values studied here.
In Figure 1, one notices a clear rising portion of RHS for r ) 4. This might be an artifact. This conclusion may be drawn from recent results54 showing that the BMCSL contact value of gss (and therefore gss itself) is significantly inaccurate for r ) 5. Then, because of eq 8, RHS may be evaluated incorrectly for r ) 4. Other equations of state, e.g., see refs 55 and 56, may be more appropriate in this case. Nevertheless, this point is not quite clear at the present time and it will be reserved for subsequent examination. Study of Experiments on C60 Solutions. The subject of this section is the interpretation of recent OSVC measurements9 in the CS2 and benzene solvents. In the latter work, an experimental value of 4 was determined for R in the case of carbon disulfide. Besides, it is known that C60 forms reversible aggregates in benzene.57 It is the purpose of this section to attempt an interpretation of these results. The interaction models presented above all suppose that, besides hard sphere volume exclusion, there is no direct interaction between the solute particles. This assumption was adopted because there is experimental evidence48 that interactions between two C60 molecules were irrelevant in a CS2 solution. As a consequence, eq 21 should be obeyed by the expressions for R derived above, that is
lim R ) 1 ηf0
for RHS+Q and RHS+Q+assoc. It was found that both expressions indeed verify this relationship. We now examine the influence of solvent-solvent and solute-solvent interactions on the value of B2. Let us recall that attractive interactions between C60 and CS2 have been reported in two independent studies.9,48 The present description is developed as follows. The C60 solute molecules are modeled as hard spheres; the CS2 and benzene solvent molecules are modeled as quadrupolar hard spheres. Using the theoretical results found above, numerical values of R are calculated according to whether solute-solvent attractive forces are taken into account or not. Prior to these results, the determination of parameter values and the stability of the model taken for pure solvent are considered. Determination of Parameter Values. The parameters of the model are the sizes of the solute and solvent molecules, the solvent quadrupole moments and the parameter K, characteristic of the solute-solvent interaction, appearing in eq 37. The choice for the value of K is discussed below. For the size of the C60 molecule, various sources seem to agree58,59 on a value for the van der Waals diameter, namely, σC60 = 10.0 Å. This value was adopted in the present work. The effective size and quadrupolar moment of the solvent molecules may be chosen at least in two ways. The first one consists of taking a particular size and then determining the value of Q/w such that the pressure of the system (pure solvent) is 1 atm. The second one would be to impose the additional requirement that the excess chemical potential of the solvent molecule, ∆µHS+Q , be equal to the “coupling work”60 correw sponding to the passage of one solvent molecule from the gas to the liquid phase (the latter quantity being estimated from vapor-liquid equilibrium data60). However, this procedure supposes that the internal degrees of freedom of the molecule are not too different in the two phases. This condition is likely not fulfilled in the case of CS2 and benzene.
5268 J. Phys. Chem. B, Vol. 105, No. 22, 2001
Figure 2. Stability parameter S (eq 42; S > 0 means stability) vs η for three values of the solvent reduced quadrupolar moment. From bottom to top: Q/w ) 1.422 (value taken for CS2), Q/w ) 1.267 (critical value), and Q/w ) 1.2.
Therefore, it was preferred to take values for the size of CS2 and benzene molecules from a compilation,42 in which effective hard sphere diameters were estimated for molecular fluids by a fit of compressibility data. This leads to values of ca. 4.5 and 5.3 Å for CS2 and benzene, respectively, at 25 °C. The corresponding values for r are 2.22 and 1.89. These small values of r should allow one to use the BMCSL equations with confidence. Using the value for the density of liquid CS2, we (0) get FCS = 9.98 × 1027 m-3, which yields η = 0.48. For 2 benzene we obtain FC(0)6H6 = 6.77 × 1027 m-3 and η = 0.53. Next, by adjusting the pressure to 1 atm, it results that the value for the quadrupole moment is Q/w = 1.422 for CS2 and 1.517 for benzene. From the value for CS2 one obtains, using eq 33, QCS2 = 4.13 × 10-39 C m2. Notice that the experimental values are quite scattered since they lie in the range61 (-0.6 to +2.3) × 10-39 C m2 for the isolated CS2 molecule. For benzene one gets QC6H6 ) 6.63 × 10-39 C m2 and an experimental value62 is 3.33 × 10-39 C m2. Stability of the Quadrupolar Hard Sphere System. The thermodynamic stability of this model for the solvent is examined using eq 42. It turns out that the stability parameter S is a function of η and Q/w only. The result is shown in Figure 2. The curve for S is tangent to the axis S ) 0, at η = 0.22, in the case Q/,cr w ) 1.267, which defines the critical value of Q/w for stability. Below this value the model system is always stable (S > 0 for any η). For Q/w > Q/,cr w there is an interval of η on which S < 0 and the system is unstable. It is seen in Figure 2 that, for Q/w = 1.422 (value taken for CS2) the curve for S (lower curve) recrosses the η-axis at η ) 0.39, which is indeed smaller than the packing fraction η = 0.48 corresponding to the size adopted for CS2. OSVC without Solute-SolVent AttractiVe Interaction. Using these values for σw, Q/w and σs in eq 34 (in which solvent cohesion is accounted for) yields
RHS+Q = -36.2 for CS2, and
RHS+Q = -24.7 for benzene. So, contrary to the pure hard-sphere case, a negative value is obtained for R in both cases. This result is consistent with the conclusions of previous studies.20,22
Simonin
Figure 3. RHS+Q (eq 43) vs r (the solute-to-solvent size ratio) for the model taken to represent CS2.
Figure 4. RHS+Q+assoc (eq 40) as a function of the parameter K (appearing in eq 37) for C60 in CS2, with ns ) 24 (see text).
It may be understood by an enhanced depletion of solvent between neighboring solute particles, caused by the solvent cohesive forces. Thus, as compared to the case of the pure hard sphere system, the depletion force is of longer range. This phenomenon increases both the width and depth of the depletion well, W; besides, oscillations in W are damped and can disappear.20 Then, since gss ≡ exp(-βW), a higher and broader peak for gss is produced near contact of the solute spheres. Consequently, R is significantly reduced, according to eq 8, and can become negative. It is informative to look at the variation of RHS+Q as a function of the solute size. In the case of model CS2, one obtains the following result
RHS+Q ) -4.480/r3 + 4.699/r2 - 0.3282/r - 6.205 + 3.617r - 1.038r2 - 3.039r3 (43) which is represented in Figure 3. It is seen that RHS+Q has a maximum near r ) 1 (just below this value), and then decreases for r > 1. A similar variation is found for benzene. This behavior contrasts with the one observed in the absence of solventsolvent attractions. OSVC with Solute-SolVent AttractiVe Interaction. First, the influence of K (appearing in eq 37) is examined in the case of CS2, for a proposed value9 of the number of association sites on C60, ns ) 24. The corresponding result for RHS+Q+assoc (eq 40) vs K, using the other parameter values for C60 in CS2, is depicted in Figure 4. The curve reaches a plateau value for K of the order of 10. Besides, a fit of the properties of water in the same framework leads to a value of the order of 15 to account for the effect of hydrogen bonding.63 Therefore, assuming that solute-solvent
Solvent Effects on OSVC’s
J. Phys. Chem. B, Vol. 105, No. 22, 2001 5269
Figure 5. Variation of RHS+Q+assoc (eq 44) vs ns (number of association sites on C60) in the case of CS2, for σCS2 ) 4.5 Å (solid line, eq 44), σCS2 ) 4.45 Å (dashed line), and σCS2 ) 4.55 Å (dotted line). The black dot (b) and the vertical bar indicate the experimental result and the experimental uncertainty,9 respectively.
attraction is strong enough, the limit K f ∞ was taken in the expression of RHS+Q+assoc. It must be noticed that this procedure is different from taking a larger size for the solute (solvated solute) because then a negative value results for R: for instance, taking σs = 17 Å, as proposed elsewhere,9 leads to RHS+Q = -170. In this case solvent molecules would be implicitly grafted onto C60 and would not exchange with molecules from the bulk. In contrast, the model used here allows solvent exchange, even if K is ascribed a large value. The final numerical result for RHS+Q+assoc, as a function of ns and nw, is
RHS+Q+assoc ) -36.25 + 1.596ns +
(
)
1 0.02394 - 0.02136 ns2 (44) nw
in the case of CS2. The variation of RHS+Q+assoc vs ns (eq 44) is plotted as a solid line in Figure 5, taking the most plausible value nw ) 1 (one solvent molecule being able to bind to one solute molecule at a maximum), together with the results for σs ) 4.50 ( 0.05 Å, and the experimental value9 Rexp = 4.0 ( 0.7. Measurements of the gyration radius of C60 in CS2 have suggested9 that ns is close to 24. An excellent agreement between the calculated and the experimental values is observed in Figure 5 for this very value of ns. In the present framework, this result means that the potential of mean force W can become repulsive (outside the hard core) when the solvation number is sufficiently high. This is so because B2 ) 4B(0) 2 for C60 in CS2; then eq 5 implies that, on average, gss < 1 and W > 0 for r12 > σs. In the case of benzene as the solvent, one has
RHS+Q+assoc ) -24.69 + 1.252ns +
(
)
1 0.03526 - 0.01988 ns2 (45) nw
A plot of this quantity vs ns is shown in Figure 6 for nw ) 1. One notices, in eqs 44 and 45, that the coefficients are of comparable order of magnitude, and that RHS+Q+assoc is an increasing function of ns for nw ) 1. The case of benzene mainly differs from that of CS2 through the fact that ns is seemingly much smaller: a value in the range 5-8 may be deduced from a measurement of the hydrodynamic radius of C60 in benzene57
Figure 6. Variation of RHS+Q+assoc (eq 45) vs ns (number of sites on C60) in the case of benzene.
(assuming that the difference with the radius of bare C60 results from the binding of ns benzene molecules on C60), from which one finds that RHS+Q+assoc would be on the order of -15. This significantly negative value indicates overall attractive solutesolute forces. This fact is consistent with the aggregation process observed for C60 in benzene.57 These results purport that the solute-solvent attraction may have a determinant influence on the observed B2 value. This conclusion is further supported by the discussion below. Besides the bare solute-solute interaction, three contributions have been taken into account: solvent packing, solvent cohesion and solute-solvent association. These contributions may be separated into two categories according to their effect: (i) Solvent packing and cohesion lower the value of B2 below B(0) 2 ; (ii) Solute-solvent attractive forces cause B2 to be increased. These effects may be understood according to the following molecular picture. As compared to the reference case of hard spheres, the introduction of solvent-solvent attractive forces enhances the depletion of solvent, and therefore the attraction, between neighboring solute molecules. Then, a significant well for W (a peak for gss) emerges near contact. It results from eq 5 that B2 is smaller than B(0) 2 (R < 1) and can be negative. In contrast with the effect of solvent cohesion, the further inclusion of solute-solvent attractive forces favors the reintroduction of solvent molecules between the solute particles that undergo depletion forces. As a consequence, the effective solute-solute interaction becomes less attractive and B2 is increased. The potential of mean force may switch from attractive to repulsive under the influence of high solvation. Although the present description may be thought to capture the main relevant physical features, the agreement observed in the case of C60 in carbon disulfide is clearly surprising in view of the approximations50,51 involved in the association model. Moreover, simple models have been adopted to represent the molecules. Hence, the present results, and in particular the role of solutesolvent forces, would have to be confirmed using other techniques, allowing more realistic representation of molecules and of their interactions. Acknowledgment. I thank E. Dubois and J.-F. Dufreˆche for useful comments. References and Notes (1) Vilker, V. L.; Colton, C. K.; Smith, K. A. J. Colloid Interface Sci. 1981, 79, 548. (2) Wang, L.; Bloomfield, V. A. Macromolecules 1990, 23, 194.
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