Fluid-solid phase-separation in hard-sphere mixtures is unrelated to bond-percolation

1 Pages·0.06 MB·English

by  A. A. Louis

“Fluid-solidphase-separationinhard-spheremix- bond-percolation induced fluid-solid line at n =2.4 de-
b
tures is unrelated to bond-percolation.” rived from the approximation to g (r) used by Buhot.
ll
0.2
In a recent letter, A. Buhot [1] proposes that entropy 10
8
driven phase-separation in hard-core binary mixtures is
g(r) 6
directly related to a bond-percolationtransition. In par- 0.15
4
ticular, Buhot suggests that a phase-instability occurs
2
when the coordination number n , defined as:
0 b n=4.8 0
η r 0.1 b 0.9 1 1.1 1.2 1.3
0 n=2.4 r/σ
0 n =ρ g (r)dr, (1) s b l
b lZ ll n=1.2
2 σl≤r≤σl(1+R) b
0.05

n is equal to zp , where z is the coordination number of a n=0.6
c b
a particular crystal lattice, and p is its bond-percolation
J c
8 threshold. Here ρl is the number density of the larger 00 0.1 0.2 0.3 η0.4 0.5 0.6 0.7
particles, g (r) is the radial distribution function of the
1 ll l
larger particles, and R = σ /σ < 1 is the ratio of
s l FIG. 1. Solid lines: the fluid-solid and fluid-fluid
t] the diameters σi. However, for binary hard-sphere mix- phase lines from the simulations of Dijkstra et. al.
f tures, calculations based on an accurate approximation [2] for R = 1/30. Dotted lines and dash-dotted
o
to g (r) demonstrate that n varies widely along the
s ll b lines: the coordination number calculated as in the
. phase-boundaries calculated directly by simulations, im- text using the AO potential and the potential from
at plying that bond-percolation is unrelated to the phase- the simulations respectively. Long-dashed line: the
m separation in these systems. nb = 2.4 line proposed by Buhot [1]. Inset: gll(r) for
- Forhighlyasymmetricbinaryhard-spheresystems,Di- R = 0.2,ηl = 0.25,ηsr = 0.25 with AO potential. Solid
d jkstra et. al. [2] conclusively demonstrated that an ef- line: from gll(r) = exp(−βVdep(r)); dashed line: from
n
fective one-component description based on a depletion PY integral equation.
o
c potential picture quantitatively describes the fluid-solid Clearly: (a) As expected, the approximation used by
[ transition. This in turn implies that the one-component Buhot for g (r) breaks down as η increases. (b) The
ll l
description should give a fair representation of the ra- lines of constant coordination number are not related to
1
v dial distribution function gll(r). Recent simulations [3] either the fluid-fluid or the fluid-solid phase lines, im-
1 oftheAsakura-Oosawa(AO)depletionpotential[4]show plying that there is no direct relation between bond-
4 that the Percus-Yevick (PY) approximation quantita- percolationandphase-separation. Thesameresultswere
2
tivelydescribesthepair-correlationsalongthe fluid-solid foundforothersizeratios,anditishardtoseehowmore
1
transition line. In the inset of Fig. 1, the simple form: accurateapproximationsfor g (r) could changethis pic-
0 ll
0 gll(r) = exp(−βVdep(r)), where Vdep(r) is the effective ture.
0 depletion potential, is compared to more the accurate Thebreakdownofthebond-percolationpictureforthis
t/ PY integral equation results. Typically for packing frac- archetypical hard-core mixture model implies a similar
ma tionηl =πρlσl3/6≤0.25alongthephaseboundariesthis breakdownforother,morecomplex,mixtures. Thegood
formgivesnearquantitativeagreementforr ≤σl(1+R), agreementfound[1]atafew statepointsforhard-square
d- which is not surprising since for small ηl the potential is systemsprobablyresultsfromeithertheir2-dnature,the
n typically at least 2.5kBT along the fluid-solid transition imposed parallel symmetry, or the rather unusual pur-
o linewhilethehard-coreinducedcorrelationsaresmallso ported2ndorderfluid-solidtransition. Itdoesnotimply
c that the exponential form dominates. In fact, Buhot’s that bond-percolationis generally relevantfor fluid-solid
:
v treatment of binary hard-spheres reduces exactly to this phase-separationin binary hard-core mixtures.
i simple formbut withanAOdepletionpotentialwhichis A.A.Louis
X
valid only when η →0. If one replaces η with the ηr of Department of Chemistry, Lensfield Rd,
r l s s
a smallspheresinareservoirkeptatconstantchemicalpo- Cambridge CB2 1EW, UK
tential, the correctformofthe AO potentialis recovered PACS numbers: 64.75.+g, 61.20.Gy, 64.60.Ak
[2]. In Fig. 1, the metastable fluid-fluid and the stable
fluid-solidphaselinestakenfromsimulations[2]arecom-
paredtolinesofconstantcoordinationnumber,whichare
[1] A. Buhot,Phys. Rev.Lett. 82, 960 (1999).
calculated with eqn. (1) and g (r) = exp(−βV (r)),
ll dep [2] M.Dijkstra,R.vanRoijandR.Evans,Phys.Rev.E.59,
together with the depletion potential used for the simu-
5744 (1999), and references therein.
lations as well the simpler AO potential. The difference
[3] M. Dijkstra, J.M. Brader and R. Evans, to be published
between the two is very small, implying that the coor- J. Phys.: Condens. Matter (2000)
dination number is not very sensitive to the exact form [4] S. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255
of the depletion potential. Also included is the proposed (1954), A. Vrij, PureAppl. Chem. 48, 471 (1976).
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