One important reason for the failure of the PY approximation at higher densities is its ther-modynamic inconsistency : due to the approximate nature of the PY-g(r), the therther-modynamic routes in Eqs. (48-53) give pressure curves, p(ρ), which become increasingly different from each other with increasing density. The results for the thermodynamic properties obtained via the three routes are in general different for all OZ integral equations discussed so far. This lack
of thermodynamic consistency is a common feature of approximate theories.
To illustrate the thermodynamic inconsistency of the PY approximation for the case of hard spheres we integrate the compressibility in (133) with respect toΦ. This yields the compress-ibility equation of state
pc
pid = 1 + Φ + Φ2
(1−Φ)3 . (138)
Using instead the pressure (or virial) equation of state, Eq. (49), one obtains pv
pid = 1 + 4 Φg(r=σ+) = 1 + 2Φ + 3Φ2
(1−Φ)2 , (139)
which agrees with the pressure, pc, derived from the compressibility equation only up to third order in the volume fraction. The pressure Eq. (49) is referred to also as virial equation of state, since it can be derived from the virial theorem of classical mechanics. The first equality in Eq. (139) relating the pressure to the contact value ofg(r)is anexactrelation stating thatp= ρkBT [1 + 4 Φg(σ+; Φ)]. There is thus only a trivial temperature dependence of the pressure in the equation of state of hard spheres which arises solely from the ideal gas contribution.
The energy equation, Eq. (48), leads for hard spheres to the exact result E = Eid = N kBT, with zero excess internal energy coming from particle interactions. Thus, the excess pressure can not be derived using the energy equation, since the internal energy of hard spheres is of purely kinetic origin. As a consequence, the phase diagram of hard spheres is temperature in-dependent and depends only onΦ(athermal system). Note that the pressure and the elastic bulk and shear moduli are measures of the energy density and thus scale for hard spheres asρkBT times a function ofΦalone.
The phase transitions in a hard-sphere fluid are purely entropy driven, sinceAex =−T Sex, so that a stable phase at a givenΦhas maximal entropy. A system of monodisperse hard spheres is in a supercritical fluid state for Φ < Φf ≈ 0.494, where Φf is the freezing volume frac-tion, followed by a fluid-crystal coexistence region for Φf < Φ < Φm ≈ 0.54where Φm is the melting volume fraction. In thermodynamic equilibrium, a fully crystalline face-centered cubic (fcc) phase, characterized by 12 nearest neighbors, is found for Φm < Φ < Φcp. Here, Φcp = π/√
18 ≈ 0.7404 is the maximal packing density allowed for monodisperse spheres, as conjectured by Johannes Kepler already in the year 1611. However, the mathematical proof that fcc is the densest packing of monodisperse, non-deformable spheres was given no earlier than in 1998 by Thomas Hales, about three hundred years past its conjecture. For comparison, the closest-packed volume fractions for a simple cubic (sc) and a body-centered cubic (bcc) crystal are π/6 ≈ 0.52 and π√
3/8 ≈ 0.68, respectively. A hexagonal cubic crystal has the same closest packed volume fraction as a fcc crystal. However, a fcc hard-sphere crystal is thermodynamically more stable than a hc crystal [36]. While the pressure in the fluid branch of a hard-sphere fluid is well described by the Carnahan-Starling equation of state given in Eq.
(140), a decently good parametrizations of the fcc branch, valid forΦ>Φm, has been provided by Wood [37] and Hall [38]. The orientation-averaged pair distribution function,gf cc(r), of the fcc hard-sphere phase has been parameterized by Kincaid and Weis [39].
Real suspensions of colloidal hard spheres are always polydisperse to a certain degree, and nu-cleation often becomes exceedingly slow so that for0.58 < Φ < Φrcp ≈ 0.634 the system is
trapped in a glass-like, non-equilibrium phase without long-range ordering. Here,Φrcp is the volume fraction of random close packing [40] where a sphere has on the average six contacts with neighboring ones. Both at random close packing and crystal close packing the particles get immobilized so that the pressure at concentrations close to these two concentrations diverges likep∼3kBT /(Φ−Φrcp)andp∼3kBT /(Φ−Φcp), respectively.
Contrary to the three-dimensional case whereΦrcp < Φcp, and consistent with the occurrence of a glassy phase, the local rule of starting from a single sphere and placing the next one such that the (areal) density is maximised leads precisely to the areal volume fraction, Φ2D, of a two-dimensional hexagonal lattice. The latter is the one of the maximal packing configuration withΦ2Dcp =π/√
12≈0.9069.
p 1 p
id
−
Φ
FS’02_22
Fig. 16: Hard-sphere compressibility and pressure (virial) equations of states in PY and HNC approximations. Dashed line: exact pressure curve. From Ref. [1].
The PY and HNC compressibility and pressure (virial) equation of states for hard spheres are plotted in Fig. 16, in comparison with the ”exact” pressure curve obtained from computer simulations. The exact hard-sphere equation of state is very well described in the fluid regime (Φ≤0.494) by the Carnahan-Starling (CS) formula [1, 2]
pCS pid = 1
pid [1
3pv+ 2 3pc
]
= 1 + Φ + Φ2−Φ3
(1−Φ)3 . (140)
As can be noticed from Fig. 16, the exact pressure is bracketed bypcandpv, with the difference betweenpcandpv increasing for increasing Φ. The PY solution is obviously a better approxi-mation for hard spheres than the HNC. The fluid-solid coexistence pressure obtained from the CS formula ispCS/pid(Φf) = 12.48corresponding tog(σ+; Φf) = 5.81.
Rogers-Young (RY) approximation:
Several hybrid integral equation schemes have been proposed in the past which partially restore thermodynamic consistency. Out of these schemes, we discuss here only the one proposed by Rogers and Young [41], which interpolates between the PY and HNC approximations and removes part of their thermodynamic inconsistencies. It is based on the observation (cf. Fig.
17) that computer simulation data for theS(q)of particles with purely repulsive pair potentials are bracketed, nearqm, by the PY and HNC structure factors. The RY closure relation is given by
g(r)≈e−βu(r) {
1 + 1 f(r)
[ef(r)[h(r)−c(r)]−1]}
(141) with a mixing function,
f(r) = 1−e−αr , (142)
including a mixing parameterα ∈ {0,∞}. The closure relation is constructed in such a way that for
r or α→0 : f(r)→0 RY → PY (143)
r or α→ ∞: f(r)→1 RY → HNC .
Hence the RYy(r)reduces to its PY value forr →0and to its HNC value forr → ∞. This
1 2 3
r/ σ
0 1 2 3
g(r)
PY RMSA HNC RY MC
Fig. 17: Radial distribution function of a charge-stabilized, Yukawa-like suspension withΦ = 0.2andZ = 100. Comparison of RY, HNC, RMSA and PY predictions with MC simulations.
From Ref. [28].
is consistent with the observation that, while the HNC closure is correct at large separations, the PY approximation is expected to be more reliable at smallr, at least for strongly repulsive potentials. The parameter α determines the proportion in which HNC and PY are mixed at intermediater. Its numerical value follows from requiring partial thermodynamic consistency by demanding the equality,χpT = χcT, of the compressibilities obtained from the pressure and compressibility equations od state. SinceχT is directly related to the long-wavelength limit of the static structure factor, one may expect that the RY approximation will provide quite reliable results forS(q)at finiteq.
The RY mixing scheme has been found to perform very well for three-dimensional liquids with purely repulsive pair potentials, unless the system is very close to the freezing transition line.
Its predictions of the pair structure are less precise for two-dimensional systems. Moreover, the RY scheme is less accurate for ultra-soft potentials such as the one describing star polymers, since its closure is PY-like at small particle separations [26].
Fig. 17 includes a comparison of RY, HNC and PY results forg(r)with Monte Carlo (MC) sim-ulations of a three-dimensional colloidal dispersion of Yukawa spheres. The true Monte-Carlo g(r) is strongly overestimated by the PY approximation, whereas the fluid g(r) is somewhat underestimated by the HNC approximation, and to an even larger extent by the RMSA. The RY approximation, on the other hand, reproduces the MC data quite well.