Phase behavior of a system of particles with core collapseE. A. Jagla*Centro Ato´mico Bariloche, Comisio´n Nacional de Energı´a Ato´mica, (8400) S. C. de Bariloche, Rı´oNegro, Argentina~Received 17 February 1998!The pressure-temperature phase diagram of a one-component system, with particles interacting through aspherically symmetric pair potential in two dimensions, is studied. The interaction consists of a hard core plusan additional repulsion at low energies. It is shown that at zero temperature, instead of the expected isostruc-tural transition due to core collapse occurring when increasing pressure, the system passes through a series ofground states that are not triangular lattices. In particular, depending on parameters, structures with squares,chains, hexagons, and even quasicrystalline ground states are found. At finite temperatures the solid-fluidcoexistence line presents a zone with negative slope ~which implies melting with decreasing in volume! and thefluid phase has a temperature of maximum density, similar to that in water. @S1063-651X~98!05808-5#PACS number~s!: 64.60.2i, 64.70.Dv, 64.60.MyI. INTRODUCTIONDetermination of the phase structure of real materialsfrom first-principles calculations has been one of the aims ofstatistical mechanics for a long time. Although a qualitativeunderstanding of the processes leading to the different kindsof phase transitions ~between gas, liquid, and one or moresolid phases! in the pressure-temperature ~P-T! phase dia-gram of a classical system has been gained, it is clear that thequantitative fitting of the behavior of real materials requires adetailed knowledge of the interaction between particles and agreat deal of computational work, which only in recent yearshas become feasible.In addition to the usual materials in which atoms or mol-ecules are the basic constituents, in recent years colloidaldispersions have provided a different kind of system inwhich parameters such as particle size and interaction poten-tial can be varied greatly @1#. These systems consist of a setof latex spheres in colloidal suspension, with the aggregateof some amount of nonadsorbing polymer, which modifiesthe interaction potential between the particles. Their studyhas practical importance in relation to the properties of manycommon substances ~such as ink, paints, cosmetics, andblood!. It is clear that a knowledge of the phase behavior ofdifferent model systems is important in order to compare thetheoretical predictions with the experimental results.Much effort has been spent in the elucidation of the prop-erties of binary mixtures of particles of two different sizes,where segregation, flocculation, partial crystallization, andother phenomena may occur @2#. On the other hand, otherstudies have been directed towards the determination of thephase behavior of identical particles interacting through dif-ferent model potentials. In this case the possibilities for thebehavior of the system are not as wide as in the case ofbinary mixtures, but interesting phenomena occur. It wasshown, for instance, that the usual solid-liquid-gas phase dia-gram of particles interacting through a hard core repulsionplus a long-range attraction is modified when the range ofthe attraction is decreased @3#. More precisely, the liquid-gascoexistence curve disappears if the range of the attractivepotential is lower than about 30% of the hard core radius.More interestingly, when the range of the attractive potentialis reduced below about 8% of the repulsive range, a coexist-ence curve separating two isostructural solid phases appears.A more obvious isostructural transition occurs in the casein which the attractive well is replaced by a repulsive shoul-der. In this case, for low pressures, the repulsive shouldercan sustain a compact structure with a lattice parameter re-lated to its range. However, when applying enough pressurethe system must collapse to a new compact structure with alattice parameter given by the real hard core of the particles.This kind of model, whether with a square shoulder or alinear ramp soft core ~which is the one discussed in thispaper!, has been studied for a long time with the picture ofcore collapse in mind @4#. Extensions to a more general po-tential were also performed @5#. In recent papers the problemhas been revisited. In particular, the isostructural transitionhas been studied numerically @6# and analytical results haveshown that in three dimensions, the ground state of a systemwith a hard core plus a repulsive shoulder can be one ofvarious crystalline structures depending on parameters @7#.In this paper I show for the hard core plus linear rampmodel in two dimensions that even the stable zero tempera-ture structures may be very different from the expected tri-angular structures. The most stable configuration may be oneof a variety of crystalline structures, and even a quasicrystal.These structures melt with increasing temperature. The solid-fluid border in the P-T diagram has a zone with a negativeslope, which implies a melting with decreasing in volume,and in this region the fluid has an anomalous thermal expan-sion @8# up to a temperature at which a density maximum isattained.The paper is organized as follows. In Sec. II the model isintroduced and details of the simulation procedure to be usedin Sec. IV are provided. In Sec. III the ground state configu-rations are analyzed. In Sec. IV I present detailed results forthe P-T phase diagram for a particular value of the param-etera, which is defined below. In Sec. V the possible rel-evance to real systems is discussed and a summary of theresults is given.*Electronic address: [email protected] REVIEW E AUGUST 1998VOLUME 58, NUMBER 2PRE 581063-651X/98/58~2!/1478~9!/$15.00 1478 © 1998 The American Physical SocietyII. MODEL AND NUMERICAL TECHNIQUEThe model interaction U(r) between particles that will beused here consists of a hard core repulsion at a radius r0@U(r)ur,r05 `#, the interaction is zero for distances largerthan a value r1, and has a soft repulsive part for r0, r, r1ofthe form U(r)5 «0(r12 r)/(r12r0) ~Fig. 1!. This interac-tion gives a model that is a candidate to have an isostructuraltransition between compact configurations of lattice param-eter is r0and r1. Two particles interacting through this po-tential in the presence of an external force f trying to bringthem together will have a jump in the interparticle distancefrom r1to r0when f exceeds the critical value
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