Freezing and clustering transitions for penetrable spheresC. N. Likos,1M. Watzlawek,2and H. Lo¨wen1,21Insitut fu¨r Festko¨rperforschung, Forschungszentrum Ju¨lich GmbH, D-52425 Ju¨lich, Germany2Institut fu¨r Theoretische Physik II, Heinrich-Heine-Universita¨tDu¨sseldorf, Universita¨tsstraße 1, D-40225 Du¨sseldorf, Germany~Received 30 April 1998!We consider a system of spherical particles interacting by means of a pair potential equal to a finite constantfor interparticle distances smaller than the sphere diameter and zero outside. The model may be a prototype forthe interaction between micelles in a solvent @C. Marquest and T. A. Witten, J. Phys. ~France! 50, 1267~1989!#. The phase diagram of these penetrable spheres is investigated using a combination of cell and densityfunctional theory for the solid phase together with simulations for the fluid phase. The system displays unusualphase behavior due to the fact that, in the solid, the optimal configuration is achieved when certain fractions oflattice sites are occupied by more than one particle, a property that we call ‘‘clustering.’’ We find that freezingfrom the fluid is followed, by increasing density, by a cascade of second-order, clustering transitions in thecrystal. @S1063-651X~98!05309-4#PACS number~s!: 61.20.Gy, 64.70.Dv, 61.20.JaI. INTRODUCTIONMuch of our current understanding of the liquid-solidtransition from a microscopic point of view is based on thedensity-functional theory of inhomogeneous liquids @1–3#.This approach allows, in principle, the systematic calculationof the phase diagram of any system, once the pair potentialbetween its constituent particles is given. A number of pairinteractions of variable ‘‘hardness’’ ~hard spheres, inverse-power, Yukawa, etc.! have been studied, yielding the phasecoexistence between a fluid phase, which is stable up tomoderate densities, and a crystal, which is stable at higherdensities. For most of the systems that have been consideredin the literature, the assumed pair interaction between par-ticles has the property that it grows as the distance betweenthe particles decreases, and diverges at zero separation.These are the usual, unbounded interactions. For such inter-actions, a whole mechanism of liquid-state integral equationtheories has been developed, which allows one to calculatewith a high degree of accuracy the structure and thermody-namics of the fluid phase, which is in turn a necessary ingre-dient in any density-functional treatment of the freezing tran-sition.Much less is known about interactions that are bounded;i.e., they allow the particles to ‘‘sit on top of each other,’’imposing only a finite energy cost for a full overlap. This isnatural since a true, microscopic interaction always forbidsoverlaps. However, the situation may be different if, e.g., oneconsiders the ‘‘potential of mean force’’ between two poly-meric coil centroids in a good solvent, as suggested manyyears ago by Stillinger @4#. The two centroids may coincidewithout this resulting in a forbidden configuration. Stillingerthus introduced the ‘‘Gaussian core model,’’ consisting ofparticles that interact by means of a pair potentialf(r)5f0exp(2r2/s2), where r is the interparticle distance,f0isan energy scale, andsis a length scale. This model and itsphase diagram have been examined in Refs. @4,5#, followingan approach based on general mathematical properties par-ticular to the Gaussian potential and on computer simula-tions, for a review see Ref. @6#.In this paper we also consider a bounded potential, albeitan apparently simpler one. We take an interaction betweenspheres that is simply equal to some positive constant if thereis any overlap between them and zero otherwise. The studyof such a model is not of purely academic interest; a fewyears ago, Marquest and Witten @7# suggested that interac-tion potentials qualitatively similar to a step function areexpected for micelles in a solvent. We study the phase dia-gram of this model by using standard techniques ~integralequation theories for the fluid and a cell model for the solid!,also combined with computer simulations. We find, on theone hand, that the boundedness of the interaction makes thestandard integral equation theories inadequate to accuratelydescribe the dense liquid phase of the system. On the otherhand, the fact that the interaction is constant brings about anovel possibility for the crystal to lower its free energy,namely, the formation of groups of two or more particles~‘‘clusters’’! occupying the same lattice site, a property thatwe call clustering. As a result, there are second-order clus-tering transitions within the region of the phase diagram oc-cupied by the solid.The rest of the paper is organized as follows: In Sec. II wepresent our approach for the fluid phase and in Sec. III forthe solid phases. The results are combined in Sec. IV wherewe present the phase diagram of the model. Finally, in Sec.V we summarize and conclude.II. PENETRABLE SPHERE MODEL: THE FLUID PHASEWe consider a model of penetrable spheres, whose inter-actions are described by the pair potential:f~r!5H«,0<r,s0,s, r,~2.1!wheresis the diameter of the spheres and « is the height ofthe energy barrier («. 0). The packing fractionhand re-duced temperature t are defined ash5p6rs3, t5kBT«, ~2.2!PHYSICAL REVIEW E SEPTEMBER 1998VOLUME 58, NUMBER 3PRE 581063-651X/98/58~3!/3135~10!/$15.00 3135 © 1998 The American Physical Societywhereris the number density, T is the temperature, and kBis Boltzmann’s constant.Clearly, at zero temperature the model reduces to the hardsphere ~HS! potential. The first task is to investigate thestructure and thermodynamics of the fluid state. In a theoret-ical approach to the problem, typically one of the variousapproximate liquid-state integral equation theories is em-ployed, which yields the radial distribution function g(r)ofthe fluid together with the direct correlation function c(r)related to g(r) by means of the Ornstein-Zernicke ~OZ! re-lation @8#:g~r!2 15 c~r!1rEc~ur2 r8u!@g~r8!2 1#dr8. ~2.3!Another exact relation connecting g(r) with c(r) reads asg~r!5 exp$2bf~r!1g~r!212c~r!2B~r!%, ~2.4!where B(r) is the so-called bridge function @9#, the sum ofall elementary diagrams that are not nodal. Since B(r) is notknown, the various approximate liquid-state integral equa-tion theories can be regarded as approximations of this quan-tity. In this way, an additional equation or
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