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1
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0003625785
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For recent reviews of the density-functional method and its applications to classical fluids, see, edited by, J. P. Hansen, D. Levesque, J. Zinn-Justin, Elsevier, New York
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(1990)
Liquids, Freezing, and the Glass Transition, 1989 Les Houches Lectures, Session 51
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Oxtoby, D.W.1
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23
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84926823087
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The results reported in this paper were obtained using erroneous direct correlation functions as input, See the Erratum, J. Chem. Phys. 88, 4104 (1988).
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26
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84926866728
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Mol. Phys. (to be published);
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27
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84926844859
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W. G. T. Kranendonk, Ph.D. thesis, University of Utrecht, 1990.
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33
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84926823086
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Note that in Eq. (8) the same free-energy function f0 has been used for both components. Although in principle one could define different functions for the two components, in practice the definitions would not be unique and would lead to greater computational complexity.
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34
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84926866727
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Note that the symmetry relation c12(2)= c21(2) implies that w12= w21. As a consequence, there are only three independent weight functions.
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35
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84926801321
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Contrary to statements in Refs. 7 and 9, the self-consistency in Eq. (9) is not required to satisfy the hierarchy relation [Eq. (15)]. It can be shown, however, that it is necessary to ensure that in Fourier space the third- and higher-order DCF's derived from the approximate excess free energy are smooth functions of their wave-vector arguments.
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42
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84926823085
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Besides perfectly ordered and perfectly disordered crystal structures, a third type of structure would be characterized by partial compositional order. Although we have not studied freezing into this type of structure, in another application of the theory we have studied the intrinsically solid-phase phenomenon of order-disorder transitions in binary hard-sphere alloys: A. R. Denton and N. W. Ashcroft (unpublished).
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51
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84926801320
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An important conclusion of this work is that the equilibrium vacancy concentration in this system is very small (roughly 10-9).
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56
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84926823084
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In computing the weighted densities from Eq. (26) it is extremely important to include a sufficient number of RLV's to ensure convergence of the sum. We have included a conservatively high number, in the range of 100–150 RLV shells (a shell comprising all vectors of the same magnitude). Further, we find that usually no more than 30 iterations are necessary to ensure convergence to a self-consistent solution, and that the technique of mixing the old iteration with the new often speeds convergence.
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58
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84926823083
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The Fourier transforms of cij(2) have been given explicitly by Ashcroft and Langreth (see Ref. 24).
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62
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84926801319
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Note that the MWDA weight functions [Eqs. (20)] include constants proportional to 1/V, which vanish in the thermodynamic limit (V -> inf). These constants, though not reflected in Fig. 2(a), are nevertheless essential in the computation of the weighted densities, as they result in finite contributions when integrated over the volume.
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63
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84926823082
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For the hard-sphere potential φHS(r) the partition function Z is independent of temperature T, since the Boltzmann factor e-ΦHS(r)/kBT can be only zero inside the hard core (r < sigma), or unity outside (r > sigma). Consequently, the free energy F = - kBT ln Z, and the pressure P = - ( partial F / partial V ), are both simply proportional to T.
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65
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84926823081
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Note that in applying the MWDA we choose the PY approximation for the uniform-state excess free energy over the more accurate expression of Mansoori et al. (Ref. 43). This is done to be consistent with our use of the PY approximations for the two-particle DCF's. For the latter there exist, to our knowledge, no alternatives consistent with the excess free energy of Mansoori et al.
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66
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84926844858
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In solving Eqs. (32), the term [3(1-x) ln λ1+ 3 x ln λ2] in the expression for Fid [Eq. (27)] may be ignored, since it appears in both gf and gs, and is linear in x, and therefore does not affect the placement of the common tangent.
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67
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84926844857
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Alternatively, the temperature may be kept fixed and the pressure varied, resulting in a P versus x phase diagram (see Ref. 16).
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68
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84926823080
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In Ref. 9, in order to permit direct comparison between the MWDA and the WDA on freezing of the one-component hard-sphere fluid, only 24 RLV shells were included in the computations. As a result, the freezing parameters reported there are slightly inaccurate. When more RLV shells are included, the correct results for the coexisting fluid and solid densities, the latent heat per particle, and the Lindemann ratio are, respectively, ρfσ3= 0.9117, ρsσ3= 1.0436, Δ ( S/N) =1.4131, and L =0.097.
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69
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84926844856
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Note that for hard spheres the free energy is completely entropic.
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70
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84926823079
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Note that for freezing into an ordered crystal, temperature is irrelevant in hard-sphere systems, since it simply scales out of the Helmholtz free energy (see Ref. 42). On the other hand, for freezing into a disordered crystal, temperature is relevant, since the Gibbs free energy—regarded as a function of P/T and x mdashhas a nontrivial T dependence.
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75
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0006226446
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For a critical examination of the truncated-expansion approximation in the context of one-component systems, see
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(1988)
J. Chem. Phys.
, vol.88
, pp. 7050
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Curtin, W.A.1
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