54 THERMODYNAMICS OF MACROMOLECULAR SYSTEMS and n 1 = 0 leading to (S co ) 2 = kn 2 [ln X −X +1 +ln z +(X −2) ln(z −1)] (4.10) Hence the entropy of mixing (S co ) for mixtures with low H mix is simply expressed as S mix = S co −[(S co ) 1 +(S co ) 2 ] S mix =−k  n 1 ln n 1 n t +n 2 ln n 2 X n t  (4.11) The same expression can be rewritten as a function of the volume fractions of the two components, with  1 being the volume fraction of the solvent and  2 being that of the polymer:  1 = n 1 n t and  2 = n 2 X n t which results in a very simple expression for S mix : S mix =−k(n 1 ln  1 +n 2 ln  2 ) Substituting in this expression of S mix the number of moles for the number of molecules gives (with R =kN a , R being the gas constant) S mix =−R(N 1 ln  1 +N 2 ln  2 ) (4.12) where N 1 and N 2 are the number of moles of solvent and macromolecules, respectively. It should be emphasized that this expression of the entropy of mixing is appli- cable only to athermic systems or to mixtures exhibiting only weak interactions between molecules—that is, solutions with low enthalpy of mixing. Deviations from ideality could arise in particular in the following situations, which will be discussed later on: • When molecules interact strongly, the assumption of a random placement of the components in the solution is not realistic. Strong interactions induce a short-range order, which leads to a lower entropy of mixing; • In dilute solutions, polymers are subjected to excluded volume. Excluded vol- ume prohibits the access of any other homolog into the vicinity of a chain segment and thus causes a lower entropy of mixing; • At high temperature, the density of a mixture can considerably decrease and the contribution of the “free volume” to the entropy of mixing cannot be neglected. FLORY–HUGGINS THEORY 55 4.2.2. Enthalpy and Free Energy of Mixing Macromolecular solutions deviate generally from ideality and are characterized by a nonzero enthalpy of mixing. In the Flory–Huggins theory the calculation of H mix is inspired by that of the enthalpy of mixing of regular solutions. Three types of interactions between nearest neighbor pairs of molecules are considered: solvent–solvent, solvent–monomeric segment, and segment–segment interactions characterized by ε 11 , ε 12 , ε 22 ; ε ij corresponds to the potential energy of an ij pair contact or the energy to dissociate it. The proportion of the various interactions depends on the relative proportions of solvent and solute. The enthalpy of mixing of such a system can be calculated starting from the relation: H mix = H − (H 1 +H 2 ) where H , H 1 ,andH 2 are the energies of the interactions which develop within the mixture and in the pure components (solvent and polymer), respectively. The energy required to break the n 1 /2 solvent–solvent interactions that occur in a lattice exclusively constituted of n 1 molecules of solvent is equal to H 1 = z(n 1 /2)ε 11 where z is the number of immediate neighbors. In the same way, the enthalpy required to break segment–segment interactions in a medium containing n 2 macro- molecules having degree of polymerization X can be written as H 2 = z(n 2 X/2)ε 22 In the case of a binary mixture, each solvent molecule is surrounded by zn 1 /(n 1 +n 2 X) molecules of solvent and zn 2 X/(n 1 +n 2 X) repetitive units. The energy corresponding to the interactions of the n 1 solvent molecules involved in solvent–solvent and solvent–segment contacts is given by 1 2 z  n 2 1 n 1 +n 2 X  ε 11 + 1 2 z  n 1 n 2 X n 1 +n 2 X  ε 12 the factor of 1/2 is necessary in the above expression since each solvent–solvent contact is counted twice. In the same manner, the energy of segment–segment and segment–solvent contacts involving the polymer repeating units can be expressed as 1 2 z  n 2 2 X 2 n 1 +n 2 X  ε 22 + 1 2 z  n 1 n 2 X n 1 +n 2 X  ε 12 The sum of all these energies H is equal to H = 1 2 z  n 2 1 n 1 +n 2 X  ε 11 + 1 2 z  n 2 2 X 2 n 1 +n 2 X  ε 22 +z  n 1 n 2 X n 1 +n 2 X  ε 12 56 THERMODYNAMICS OF MACROMOLECULAR SYSTEMS and the enthalpy of mixing is established as follows: H mix = z  n 2 1 ε 11 2(n 1 +n 2 X) + n 2 2 X 2 ε 22 2(n 1 +n 2 X) + n 1 n 2 Xε 12 (n 1 +n 2 X)  −  n 1 ε 11 2 + n 2 Xε 22 2  or H mix = zX n 1 n 2 (n 1 +n 2 X) ε 12 with ε 12 = ε 12 −  ε 11 2 + ε 22 2  Strong interactions result in negative values of ε. If interactions of 1–1 and 2–2 types are stronger than 1–2 type, ε 12 and H mix are positive and the mixture is then endothermic. The mixing will be exothermic in the opposite case. H mix can also be written as a function of the volume fraction of polymer ( 2 ): zn 1  2 ε 12 This corresponds to H mix per unit of volume; to obtain the molar enthalpy of mixing, it must be multiplied by V 1 , the molar volume. Defining χ 12 = z ε 12 V 1 RT , the expression for the enthalpy of mixing becomes H mix = RTχ 12 N 1  2 (4.13) where χ 12 is called polymer–solvent interaction parameter. The free energy of mixing G mix is then easily established as G mix = RT(N 1 ln  1 +N 2 ln  2 +χ 12 N 1  2 ) (4.14) χ 12 is also frequently associated with the Hildebrand solubility parameters through the relation χ 12 ≡ V m (δ 1 −δ 2 ) 2 /RT where δ 1 and δ 2 are the solubility parameters of the two components and V m is their molar volume taken identical for solvent molecules and monomer units. 4.2.3. Miscibility Conditions and Phase Separation In the case of an athermic solution, the replacement of a contact between similar species by a “hetero-contact” (ε 12 ) between solvent and repetitive unit (segment) FLORY–HUGGINS THEORY 57 does not cause any modification of G mix , since, in this case, χ 12 is 0. Except for some rare cases of athermic solutions, solvent–polymer mixtures are gener- ally endothermic, characterized by a positive enthalpy of mixing. The interactions developed within such solutions are intermolecular repulsive forces of the van der Waals type. In the case of nonpolar polymer–solvent mixtures, which are the only ones considered here, these interactions are indeed controlled by the polarizability of the components and are described by the relation ε ij =−3/2[I i I j α i α j /(I i +I j )]r −6 where I i and I j are the ionization potentials and α i , α j are the polarizabilities of components i and j. Hence, the interaction parameter that reflects the whole of these interactions can be written as χ 12 = A(α 1 α 2 ) 2 where A is a constant and indices 1 and 2 correspond to solvent and polymer, respectively. This expression, which is established considering only London-type van der Waals interactions, shows that interactions between dissimilar units are necessarily repulsive (or zero); hence, χ 12 should be positive. Even in the case of toluene–polystyrene solutions, χ 12 is in the range 0.3–0.4. Thus a positive enthalpy of mixing tends to oppose the polymer dissolution in a solvent (G > 0). In the Flory–Huggins model, two components can mix with each other only if the positive enthalpy term is compensated by the entropy term (−TS ), which is always negative. In the particular case of specific interactions of higher energy—such as hydrogen bonding—the interaction parameter can take negative values. Solvent and polymer are then miscible in all proportions, but the Flory–Huggins theory does not account for this case, which implies a completely different calculation of the entropy of mixing. Hence, the interaction parameter χ 12 is a measure of the quality of a solvent, and its knowledge is essential to the prediction of the domains of concentration corresponding either to the miscibility or to the phase separation of the components. As a matter of fact, it is possible, using the Flory–Huggins theory, to delimit these domains as a function of the concentration of the species and of the interaction parameter. Figure 4.2 shows the variation of the free energy of mixing (G mix ) as a func- tion of the volume fraction of component 2 for various values of the interaction parameter χ 12 . A symmetrical variation of G mix is observed when the components 1 and 2 have the same size. The situation for a solution containing a polymer with a degree of polymerization X is different. The variation of G mix becomes strongly asymmetrical in this case. The effect of χ 12 can be seen in the form of the curves depicting the variation of G mix with  2 (Figure 4.2). When concave, these curves indicate a total miscibility of the two components. This occurs for χ 12 values lower than 0.5. When χ 12 is 58 THERMODYNAMICS OF MACROMOLECULAR SYSTEMS 1.8 1.8 X >>1 0 Φ 2 ∆G mix /RT ~1 >>1 ~1 χ 12 P 0.5 0.2 0 −0.2 −0.4 0.5 1 ∆m 1 ∆m 2 Figure 4.2. Variation of the average free energy of mixing (in RT units) as the function of volume fraction of the aqueous solution for various values of χ 12 and the degree of polymerization X. higher than 0.5, these curves show a maximum and two minima and therefore two inflection points, characteristic of the presence of two phases in equilibrium. These inflection points, also called spinodal points, correspond to ∂ 2 (G mix )/∂ 2 2 = 0 They define the thermodynamic limits of metastability. For concentrations corre- sponding to the spinodal points, the system is unstable and demixes spontaneously into two distinct continuous phases which form an “interpenetrating system.” This type of phase separation characteristic of spinodal regions, is also called spinodal decomposition. As for the minima, they are called binodal points and a common tangent line passes through them. The chemical potential of a component at these binodal points is the same in each of the two phases in equilibrium (called prime and double prime): µ  1 = µ  1 et µ  2 = µ  2 which gives µ  1 = µ  1 −µ ◦ 1 = µ  1 −µ ◦ 1 = µ  1 and µ  2 = µ  2 −µ ◦ 2 = µ  2 −µ ◦ 2 = µ  2 FLORY–HUGGINS THEORY 59 The chemical potential (µ i ) of a component i is by definition the variation of the free energy of mixing G mix resulting from the introduction of N moles of i: (∂G mix /∂N) T,P = µ i (4.15) According to the Gibbs–Duhem relation  i N i dµ i = 0 (4.16) and for a mixture of components 1 and 2, one obtains G mix = N 1 (µ 1 −µ 1 ◦ ) +N 2 (µ 2 −µ 2 ◦ ) a relation that can also be written as G mix = [µ 1 +(µ 2 −µ 1 ) 2 ]N t where N t = N 1 +N 2 G mix thus varies linearly with  2 with a slope equal to (µ 2 −µ 1 ), and the chemical potentials are given by the intercepts of the function G mix =f ( 2 )for  2 →0and 2 →1; this function also corresponds to the tangent (P)tothecurve shown in Figure 4.2 for a given composition  2 . When a polymer of degree of polymerization X is one of the two components, one obtains G mix =  µ 1 −  µ 1 − µ 2 X   2  N t (4.17) The slope of the common tangent that passes through the two binodal points is equal to (µ 1 −µ 2 /X), and its intercept for  2 =0 is equal to µ 1 . Insofar as the two binodal points possess a common tangent, the chemical potentials of the two components are identical in both phases for p  and p  compositions. As for the compositions located between the spinodal and binodal points, the free energy of mixing of the corresponding systems, albeit negative, is higher than those of bimodal compositions. These systems will thus demix into two phases with compositions equal to those of the binodal points in order to minimize their free energy. Indeed, even a negative energy of mixing is not necessarily synonymous with miscibility: should a lower free energy be accessible, a system will tend to it even if it requires that it demixes into two phases. To summarize, three areas can be distinguished at a given temperature: • Between  2 =0 or 1 and the binodal points, a system forms homogeneous solutions and only one stable phase. 60 THERMODYNAMICS OF MACROMOLECULAR SYSTEMS • Between the binodal points, two phases coexist whose composition is given by the contact points of the tangent to the curve; the regions between binodal and spinodal points form metastable solutions; in this case, the phase separation is kinetically controlled by the nucleation and the growth of nuclei leading to the dispersion of one phase into the other. • The regions between the spinodal points lead to unstable solutions that demix spontaneously into two phases. ∆Gmix homogeneous mixture T 2 T 1 Φ 2 Φ 2 0 binodal spinodal p' p" I SMS 1 2 ∆m 1 −∆m 2 /X T M Figure 4.3. 1: Phase diagram of a macromolecular solution whose phase separation occurs through a decrease of temperature (UCST). 2: Variation of the average free energy of mixing as a function of the volume fraction of the solute: (a) formation of a homogeneous solution at T 2 ; (b) demixing in two phases for compositions between p  and p  at T 1 .S,M,Iindicatethe regions of stability, metastability, and instability, respectively. Curves of binodal and spinodal points can be drawn as a function of the temper- ature up to the critical temperature (T c )(T 2 in Figure 4.3) where these two curves meet. Beyond T c , the system forms only one phase. At this critical temperature, the partial first- and second-order derivatives of the chemical potential (µ 1 )are equal to zero; the chemical potential is the derivative of the free energy of mixing (G mix ) relative to the number of moles (N 1 ): µ 1 = ∂G mix ∂N 1 = RT  ln(1 − 2 ) +  1 − 1 X   2 +χ 1,2  2 2  (4.18) FLORY–HUGGINS THEORY 61 ∂µ 1 /∂ 2 = RT  2χ 1,2  2 −(1 − 2 ) −1 +  1 − 1 X  = 0 (4.19) ∂ 2 µ 1 /∂ 2 = RT2χ 1,2 −(1 − 2 ) −2 = 0 (4.20) which gives the following relation for the critical volume fraction of phase 2:  2,crit = 1 1 + √ X (4.21) The critical parameter of interaction of demixing is given by χ 1,2,crit = 1 2  1 + 1 √ X  2 ∼ 1 2 + 1 √ X (4.22) For values of the degree of polymerization tending to infinity,  2,crit thus tends to 0andχ 1,2,crit tends to 0.5. 4.2.4. Determination of the Interaction Parameter (χ 12 ) χ 12 can be determined through osmometry measurements (see Section 6.1.2.1). The osmotic pressure, which is the pressure to apply to stop the flow of the sol- vent molecules through a semipermeable membrane (permeable to the solvent and impermeable to the macromolecules), is related to the solution activity and to the chemical potential by the relation −V 1 = RT ln a 1 = µ 1 (4.23) V 1 being the molar volume of the pure solvent; taking into account (4.18), one obtains the following relation for the solvent activity: ln  a 1  1  −  1 − 1 X   2 = χ 12  2 2 (4.24) χ 12 is the slope of the variation of the term of the left member of the aforementioned equation versus  2 2 . The osmotic pressure can be easily deduced from equations (4.20) and (4.23) after expressing them as functions of the concentration C 2 . Considering that  1 =(1 − 2 ) and that ln (1 − 2 ) can be developed into a series,  − 2 − 1 2  2 2 − 1 3  3 2 −···  equation (4.18) becomes µ 1 =−RT   2 X +  1 2 −χ 12   2 2 +···  (4.25) 62 THERMODYNAMICS OF MACROMOLECULAR SYSTEMS After observing that  2 can also be written as C 2 V 2 or C 2 V 2 0 /M 2 ,whereC 2 is the concentration of polymer, V 2 the partial specific volume of the polymer, and V 2 0 its molar volume, and considering that X is also the ratio of the molar volume of polymer to that of the solvent (V 2 0 /V 1 0 ), one obtains µ 1 =−RTV 0 1  C 2 M 2 +  1 2 −χ 12  V 2 2 V 0 1 C 2 2 + V 2 3 3V 0 1 C 3 2 +···  (4.25a) The expression for the osmotic pressure can be deduced as follows:  = RT  C 2 M 2 +  (1/2 −χ 12 )V 2 2 V 0 1  C 2 2 + V 3 2 3V 0 1 C 3 2 +···  = RT(A 1 C 2 +A 2 C 2 2 +A 3 C 3 2 +···) (4.26) where A 1 , A 2 ,andA 3 , the virial coefficients, correspond to A 1 = 1 M 2 ,A 2 = [1/2 −χ 12 ]V 2 2 V 0 1 ,A 3 = V 3 2 3V 0 1 (4.27) Knowing the osmotic pressure () and hence the chemical potential, one can determine the interaction parameter (χ 12 ). 4.2.5. Real Macromolecular Solutions As already shown above, the Flory–Huggins theory appears particularly well-suited to the case of regular macromolecular solutions whose components are nonpolar; indeed, their enthalpy of mixing is slightly positive and their entropy of mixing has mainly a conformational origin. Solutions of polyisobutene in benzene or natural rubber in benzene belong to such category. The phase separation in such sys- tems occurs upon decreasing the temperature (upper critical solution temperature (UCST)) (Figure 4.4), which means that their enthalpy of mixing is independent of the temperature and χ 12 is inversely proportional to the temperature. These are two basic assumptions of the Flory–Huggins theory. On the other hand, the same model does not predict the lower critical solution temperature case (LCST), which is observed in macromolecular solutions with polar components. Indeed, demixing upon an increase of the temperature (Figure 4.4) is a well-known phenomenon for solutions of polar polymers that are characterized by a high enthalpy of mixing. Conscious of this shortcoming, Flory modified the initial version of his model and reconsidered the assumptions of an enthalpy of mixing and of an energy of contact independent of the temperature. Observing that in solutions containing polar components, segment–segment interactions can be favored and perturb a random distribution of macromolecu- lar chains, Flory proposed to take into account the existence of such interactions FLORY–HUGGINS THEORY 63 1 1 1 2 2 U U 2 2 Φ 2 Φ 2 Φ 2 T L L Figure 4.4. Phase diagrams showing the temperatures of demixing (T) as a function of the volume fraction of polymer, with one-phase region (1) and two-phase regions (2). U is used for ‘‘UCST’’; it means that phase separation occurs upon decreasing the temperature. L is used for ‘‘LCST’’; in the latter case, phase separation occurs upon increasing the temperature. in the calculation of the entropy of mixing. In addition to the traditional conforma- tional component, he introduced a term reflecting such interactions or associations in the expression of the entropy of mixing. In such an event, the expression of the exchange (or contact) energy ( ε 12 ) also necessitates a reformulation with an entropy term introduced in complement to the enthalpy term, the latter reflecting the variation of enthalpy due to solvent–solute contact: ε 12 = ε 12,H −Tε 12,S (4.28) The parameter of interaction in such a case becomes χ 12 = (z −2) ε 12,H RT −(z −2) ε 12,S R = χ 12,H +χ 12,S (4.29) where χ 12,H and χ 12,S represent the enthalpic and entropic contributions, respec- tively. The entropy and the enthalpy of mixing can be expressed as S mix =−R[N 1 ln  1 +N 2 ln  2 + ∂(χ 12 T) ∂T N 1  2 ] (4.30) H mix =−RT 2 N 1  2  ∂χ 12 ∂T  (4.31) Only for contact energies really independent of the temperature and χ 12 varying in a proportional manner to temperature—which are two assumptions of the theory of Flory–Huggins—is the traditional expression (4.13) for H mix valid. One can also observe that the term 1/2 in the expression (4.25) for the chemi- cal potential ( µ 1 ) originates from the entropic term ln(1 − 2 ) and results from the development into a power series of the latter. Since χ 12 can also be written as (χ 12,H +χ 12,S ), this last term can be regrouped with 1/2 to define an entropic param- eter:  = 1 2 −χ 12,S . Because χ 12,H represents the enthalpic contribution—and thus has the dimension of an energy—the  term should also have the dimension of an [...]... M2 F (Y ) V10 1 − χ 12 u = (2/ Na ) 2 (4.49a) and Y = [2/ Na ] 1 − χ 12 2 2 2 V 2 M2 0 V1 3 4πs 2 3 /2 (4.51a) where V1 0 is the molar volume of the solvent in the three last expressions Hence, u the second virial coefficient, A2 = 2M 2 Na , can be written as follows: 2 A2 = 1 − χ 12 2 V 22 F (Y ) V10 (4. 52) The Flory–Krigbaum model leads to an expression for the second virial coefficient (A2 ) that differs by... series of the form F (Y ) = 1 − Y 23 /2 2! + Y2 33 /2 3! with Y =2 1 − χ 12 2 v1 2 3 4πs 2 (4.50) 3 /2 Introduction of the specific partial volume of the polymer (V 2 ), 0 V 2 = Xv1 Na /M2 (4.51) DILUTE MACROMOLECULAR SOLUTIONS 73 1 F(Y) 0, 5 0 2 4 6 8 10 Y Figure 4.9 Variation of F(Y) versus Y obtained by graphic integration (according to Flory) into the preceding expressions delivers 2 2 V 2 M2 F (Y... Since n2 j =1 ln[1 u ] − ln n2 !] V (4.36) u − (j − 1) V ] can also be written in the form n2 ln 1 − j j =1 u V ∼− u V n2 (4.37) j j =1 and since the sum of the first integers is equal to 1 n2 (n2 − 1), Sco is equal to 2 Sco = kn2 ln A + kn2 ln V − k ln n2 ! − ku 2 n 2V 2 (4.36a) 68 THERMODYNAMICS OF MACROMOLECULAR SYSTEMS Expressed as a function of the numbers of moles of solvent (N1 ) and polymer (N2 )... variation of the chemical potential ( µ1 ) can be written as µ1 = G1 = ∂ Gm = −RT ∂N1 2 X + u 2 2 Na + · · · 2V10 X (4.39) Substituting the concentration of polymer for its volume fraction gives 2 Thus, = C2 V10 X M2 µ1 becomes µ1 = −C2 RTV0 1 1 uC2 + N + ··· 2 a M2 2M2 (4.39a) The model of excluded volume thus predicts that the second virial coefficient u A2 = 2M 2 Na decreases as the molar mass of the... be written as µ1 − µ0 = RT ln(1 − 1 and that ln(1 − 2) + 2 ∼− = 2 2) 1− 1 X + χ 12 2 2 (4.58) reduces to ln(1 − 2) − 1 2 2 2 with 1/X → 0 for X →∞, one obtains the following for (µ1 − µ1 0 ): (µ1 − µ0 ) ÷ 1 1 2 − χ 12 2 (4.58a) which gives for Fosm Fosm ÷ 1 2 − χ 12 X1 /2 v1 α4 (4.57a) since s0 2 is proportional to X This force is counterbalanced by an elastic force of entropic origin which can be written... + XN2 ), Sco becomes = Sco = kNa N2 ln A + kNa N2 ln[V1 0 (N1 + XN2 )] − k ln(Na N2 !) − N2 2 k u N2 0 (N + XN ) a 2 V1 1 2 (4.36b) where V1 0 is the molar volume of pure solvent and Na is Avogadro’s number For the calculation of the entropy of mixing, it is necessary to consider the conformational entropies of pure solvent, of pure polymer, and of the solute previously calculated [equation (4. 12) ]:... as RTC2 = 1 2 + A2 C2 + A3 C2 + · · · M2 (4 .26 a) The second virial coefficient (A2 ), which reflects the quality of solvent and is measured by taking the slope of the right-hand side of the equation /RTC2 = f (C2 ), is assumed to be independent of the molar mass of the sample Experimentally, one observes in contrast a decrease of A2 with the sample molar mass, which is 66 THERMODYNAMICS OF MACROMOLECULAR... total number of possible ways (P2 ) to arrange n2 macromolecules can be deduced easily: n2 P2 = n2 νi /n2 ! = (AV )n2 j =1 [1 − (j − 1)u/V ]/n2 ! (4.35) j =1 The n2 ! factor takes into consideration the number of ways of permuting n2 indistinguishable chains From the relation between P2 and the conformational entropy (Sco ), Sco = k ln P2 the following relation can be established: n2 Sco = k[n2 ln AV +... 2 1 /2 and hence to the radius of gyration s 2 1 /2 and R is the distance separating the point considered from the center of gravity: 3 β= r 2 1 /2 = 31 /2 s 2 1 /2 21 /2 (4.41) 1 Ω(d) Ω(d = ∞) 0 d Figure 4.7 Curve describing the variation of the probability of placement the distance separating the macromolecular barycenters of two coils as the function of 70 THERMODYNAMICS OF MACROMOLECULAR SYSTEMS ρ(R)... )2 (4.38) (Sco )2 can be deduced from the expression (4.36b) with N1 = 0; as for (Sco )1 , it is equal to 0, and Sm can be written as Sm = −kNa N2 ln N1 XN2 k u N2 + Na2 0 X N + XN N1 + XN2 2 V1 1 2 (4.38a) Expressed as a function of the volume fractions and substituting the gas constant (R) for kN a , Sm reduces to Smix = −RN2 (ln 2 − u Na 2V10 X 1) (4.38b) Under these conditions, the free energy of . z  n 2 1 ε 11 2( n 1 +n 2 X) + n 2 2 X 2 ε 22 2( n 1 +n 2 X) + n 1 n 2 Xε 12 (n 1 +n 2 X)  −  n 1 ε 11 2 + n 2 Xε 22 2  or H mix = zX n 1 n 2 (n 1 +n 2 X) ε 12 with ε 12 = ε 12 −  ε 11 2 + ε 22 2  Strong. −χ 12 )V 2 2 V 0 1  C 2 2 + V 3 2 3V 0 1 C 3 2 +···  = RT(A 1 C 2 +A 2 C 2 2 +A 3 C 3 2 +···) (4 .26 ) where A 1 , A 2 ,andA 3 , the virial coefficients, correspond to A 1 = 1 M 2 ,A 2 = [1 /2 −χ 12 ]V 2 2 V 0 1 ,A 3 = V 3 2 3V 0 1 (4 .27 ) Knowing. expressions delivers u = (2/ N a )  1 2 −χ 12  V 2 2 M 2 2 V 0 1 F(Y) (4.49a) and Y = [2/ N a ]  1 2 −χ 12  V 2 2 M 2 2 V 0 1  3 4πs 2  3 /2 (4.51a) where V 1 0 is the molar volume of the solvent in