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292 J.M. Brader, R. Evans / Physica A 306 (2002) 287 – 300 and r p ≡ a r p is the packing fraction of polymer in the reservoir. Eqs. (9) and(10) provide an implicit osmotic equation of state and, while it is not possible to invert this expression analytically andobtain ∗ ≡ ∗ ( c ; r p;q) in closedform, the inversion is easily performednumerically. It is important to note that is the osmotic pressure, i.e., the contribution to the pressure arising from the coordinate dependent terms in H e , in this case the pair potential term. In order to obtain the total pressure P tot of the binary mixture we must take into account the coordinate independent terms 0 and 1 in Eq. (7). It is straightforwardto show generally [8] that P tot (N c ;L;z p )=(N c ;L;z p )+P r p(z p ) ; (11) i.e., we must add the pressure of the polymer reservoir. Similarly the total chemical potential of the colloidis given by c tot (N c ;L;z p )= 1 (z p )=N c + c (N c ;L;z p ) ; (12) where c is the chemical potential of the one-component uidof particles interacting via the eective pair potential [8]. In the next section we compare results from this exact treatment of the equation of state with the predictions of the approximate (one-dimensional) free-volume theory. 3. Results of calculations and comparison with the free-volume approximation The free-volume theory [5] is a widely used theory of the bulk phase equilibria in models of colloid–polymer mixtures. It is dicult to assess the accuracy of the free-volume theory in 3D for anything other than highly asymmetric mixtures (q¡0:1547), where the mapping from the binary AO mixture to an eective onecomponent Hamiltonian is exact—making simulation possible [7]. For AO mixtures with q ¿ 0:32 the free-volume theory predicts stable uid–uid phase separation but simulation results for the full binary mixture are not yet available. We are now in a position to test the 1D version of the free-volume theory against our exact result in order to establish where the approximation is accurate, and where it breaks down. It is expectedthat the observedtrends will be preservedin 3D. We rst give a brief derivation of the 1D version of the free-volume theory. 3.1. The one-dimensional free-volume approximation The key quantity which enters the free-volume theory is the statistically averaged free volume fraction . This gives information about the volume which is accessible to a polymer particle in a system of colloids at some given packing fraction. The approximation in the theory [7] is to assume that this free-volume is unalteredwhen more than one polymer particle enters the system, i.e., that the colloidstructure is unaltered by addition of polymer. If only a single polymer particle is placed as a test particle into the system, it makes no dierence whether we are dealing with the colloid–polymer model or with a binary hard sphere (hard rod) mixture; the unlike interactions are the same in both models. The congurational integral for an additive
J.M. Brader, R. Evans / Physica A 306 (2002) 287 – 300 293 binary hardrodmixture can be evaluateddirectly andyields the following expression for the Helmholtz free energy [10] ˜F(N 1 ;N 2 ;L)=−(N 1 + N 2 )(ln(1 − 1 − 2 )+1) + N 1 ln( 1 1 )+N 2 ln( 2 2 ) ; (13) where N ; and ≡ N =L are the number, thermal de Broglie wavelength and density of species , with =1; 2. 1 =b 1 and 2 =a 2 are the packing fractions. The chemical potential of species 2 is thus given by ( ) 2 2 2 =ln + q 1 + 2 ; (14) 1 − 1 − 2 1 − 1 − 2 where q = a=b is the size ratio. We now wish to calculate the free-volume which is available to a single rodof species 2 if it is insertedinto a system which only contains species 1. This single rod, when inserted into the system, will behave like an ideal gas but is only free to move in a reduced ‘volume’ L, where is the free-volume fraction we seek. Thus in this limit ( ) 2 N 2 2 =ln =ln( 2 2 ) − ln() ; (15) L which must be identical to Eq. (14) in the limit 2 → 0. This yields the following exact result for the free-volume fraction [5,10,12] ( =(1− 1 ) exp − q ) 1 : (16) 1 − 1 is the average fraction of the ‘volume’ L which is available to a single test rodof length a in a system of hardrods at a packing fraction 1 . The free-volume theory can be derived [7] from the following identity for the thermodynamic potential of a binary (colloid–polymer) mixture: F(N c ;L;z p )=F(N c ;L;z p =0)− L ∫ zp 0 dz ′ p (z ′ p; c ) ; (17) where (z p ; c )=〈N p 〉 zp =L r p is the ratio of the density of polymer in the mixture to that in the reservoir. The brackets 〈〉 zp denote an average in the (N c ;L;z p ) ensemble. In an exact treatment this ratio of densities depends upon z p as the chemical potential of the polymer reservoir aects the distribution of the colloids. The free-volume approximation is to neglect this dependence on z p andset (z p ; c )=(0; c ) ≡ . It follows that within this approximation F(N c ;L;z p ) L = F(N c;L;0) L − z p : (18) Since the rst term is simply the Helmholtz free energy density of the pure hard roduidwe can substitute the known result andwrite the total thermodynamic potential as F(N c ;L;z p ) = c (ln( c c ) − 1) − c ln(1 − c ) − z p ; (19) L
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