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Melting of Two-Dimensional Colloidal Crystals: a Simulation Study of The Melting of two-dimensional coliloidal crystals: A simulation study of the Yukawa system Kevin J. Naidoo and Jurgen Schnitker Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055 (Received 13 July 1993;accepted 3 November 1993) The two-dimensional melting transition of charged polystyrene spheres in aqueous colloidal suspensionhas been studied by molecular dynamics simulation of a screenedCoulomb system. Some central predictions of the Kosterhtz-Thouless-Halperin-Nelson-Young theory of defect-mediatedmelting are confirmed, such as an apparent divergenceof the correlation lengths for translational and bond-orientational order at different thermodynamic state points, but there are also predictions of the theory that are violated. The defect topology is very complex, with oscillation periods of the defect density of many million time steps duration. The need for extensive sampling and, to a lesser degree,the choice of potential function continue to be the crucial issuesfor any attempt to generatea hexatic structure by meansof computer simulation. 1. INTRODUCTlON KTHNY theory, no consensushas emergedconcerning its applicability to real systems.Nevertheless, it seemsthat at Monodispersecolloidal particles in aqueoussuspension least some experimental systems, such as certain smectic provide a model system of unique versatility for the study liquid crystals,lo have provided good indications in support of classical condensedphase phenomena. “’ Thus such sus- of the theory. Probably the most compelling evidence for pensionshave recently been used in severalkinds of exper- KTHNY-type melting has been obtained by Murray et al. imental setupsto study the nature of the melting transition in experiments with colloidal suspensionsof simple poly- in two-dimensionally (2D) restricted geometries.3-5Sur- styrene spheres that are confined between glass plates.3 prisingly, the conclusions drawn from the various experi- There are also other experimental systemswith more indi- ments are conflicting, adding to a multitude of contradic- rect indications for two-step melting as prescribed by tory data that have been obtained over the last decadeand KTHNY theory.6 a half in numerous other investigationsof 2D melting, both Even the experimental evidencefor 2D colloidal sus- experimental and theoretical.6*7Motivated by these unre- pensions is not uncontested4,5(see Ref. 11 for a recent solved controversies, we here present a computer simula- comprehensivereview), reminiscent of the highly contro- tion study of the 2D Yukawa system as a simple model for versial interpretations that have beengiven of many of the charged colloidal suspensions.The rationale is that simu- numerous computer simulation studies of 2D melting. A lation results will not only complement the experimental good example of the latter predicamentis provided by two information, but may also help to clarify the origin and the recent papers on the hard-disk system that contain quite essenceof the ongoing disputes regarding the interpreta- different inferencesabout the nature of the melting transi- tion of the experiments.. tion.t213 Intricate issuesalso arise in the analysis of simu- The strong interest that has beenshown in 2D melting lations of the inverse-twelfth-power-lawsystem that we re- for more than a decadeemanates from the theory of defect- examined recently.t4 mediated melting by Kosterlitz, Thouless, Halperin, Nel- Nevertheless,it appearsthat a majority of simulation son, and Young (KTHNY) . This famous theory predicts a results has been interpreted as indicative of a first-order peculiar two-step melting process.First, a continuous tran- transition, or at most a weak first-order transition,6’15 sition leads from the solid to a bond-orientationally or- whereasat least some experimental data appear to provide dered phase, a so-called “hexatic”; then a second transi- definite hints in favor of KTHNY-type melting. It is not tion, also continuous, leads from the hexatic phase to the clear if this apparent disparity between experiment and liquid.8,gThis is to be opposedto the conventional view of simulation has to be attributed to ( 1) either fundamental melting, for systems in both 3D and 2D, where a single differences between the types of systems that have been first-order transition takes the system from the long-range studied, or (2) inherent problemswith the setup of current (or quasi-long-range) ordered state of the solid phase to simulations, in particular regarding system size and degree the disordered state of the liquid phase.The physical driv- of sampling, or (3) different methods for data analysis and ing force behind the KTHNY transitions are long- interpretation. wavelength fluctuations that lead to the unbinding of de- Even though it can be arguedthat all of theseissues are fect pairs, specifically pairs of dislocations (solid -+hexatic ) potentially relevant, the true significanceof each point still and pairs of disclinations (hexatic *liquid). No account is needsto be determined. What appearsto be called for is a made of short-wavelength fluctuations, related to effects of systematic investigation of the same type of system with local packing and such phenomenaas the possible occur- both experimental and simulation methods. This is the goal rence of grain boundaries. of the present paper. We conduct a series of very long Because of the just mentioned incompleteness of molecular dynamics simulations of a crude model for 3114 J. Chem. Phys. 100 (4), 15 February 1994 0021-9606/94/l 00(4)/3114/8/$6.00 0 1994 American institute of Physics K. J. Naidoo and J. Schnitker: 20 melting of the Yukawa system 3115 charged colloidal suspensions,employing a simple, pair- and the dielectric constant of water (~,=78) in all simu- wise additive model potential. We then look for evidence lations. For the Debye screening length I/K we generally similar to that of the experimental information.3-5 In par- use 1/~=0.20 pm. It will later be seen that this yields a ticular we try to determine-just as in the experimental melting density about halfway between the melting densi- case-if structures from the intermediate region of the ties of the Murray et al. two colloids. Some data were also phasediagram exhibit all the features of a true hexatic or if collected for a screeninglength of 1/~=0.15 pm; no qual- they have to be attributed to conventional two-phase coex- itatively different observations were made and results of istence. these simulations will not be presented. While several modest simulation studies of the (un- Of course the potential function ( 1) can only be ex- screened) 2D Coulomb system have been reported,16-‘9 pected to give a very crude and qualitative representation there is only one previous simulation study of the 2D of the actual interactions in the experimental system under. Yukawa system” and this study was mostly limited to a study.2121(b)P23*30 For example, the potential is purely repul- characterization of its thermodynamics. This is in stark sive without a Hamaker-type attractive term, and we do contrast to the 3D Yukawa system which has receiveda lot not have an explicit geometric factor that could account of attention21-26and whose melting transition has repeat- for finite sphere size effects. These omissionsshould not be edly been addressedfrom the viewpoint of phenomenolog- of concern for simulations in a fairly narrow phase transi- ical theories of melting.2626In this paper, we will ignore tion region where we can simply take the charge Z?as a the 2D analogs of these theories and only focus on argu- renormalized and practically density-independent value ments for and against KTHNY-type melting, as obtained that representsall of the more subtle effectsin an empirical from a direct analysis of the microscopic structure. We also and summary fashion. In general, one would expect the do not study the entire phase diagram, but rather work sphere-sphere interactions in a realistic model for the ex- with one specific set of potential parameters-basically perimental 2D system to be even more complex than in the chosen ad hoc-such that the simulated system resembles 3D bulk system, due to the image charges induces in the in some ways the experimental colloidal sphere systems3” confining walls. We do not attempt to account for such and in particular the systems of Murray et aL3 effects. ’ _ The organization of this paper is as follows. In Sec. II, The pair interactions are evaluated up to a cutoff dis- we describe the computational methods. The results are tance of r-,=2.3 pm. Since our simulations are in a sense presented and discussed in Sec. III, first addressing the loosely modeled on the experimental studies of Murray et orientational and translational correlation functions, and al3 we use a particle mass of 8.95X 10’ amu. For an ap- then examining the defect density and defect topology. The proximate density of 1.0 g/cm3, this would correspond to final Sec. IV contains the conclusions. spheres of 0.305 pm diam, i.e., exactly the sphere size of both monodispersecolloids that have been studied experi- mentally. 3 II. METHODS The equations of motion are integrated with the leap- A system of 1225 particles in a periodically replicated frog Verlet algorithm,31 A roughly linear increase of the rectangular unit cell of approximate area A = (25 x42) energy fluctuation with increasing time step shows a clear pm2 was studied, corresponding to a 25 X49 supercell of an change in slope for time steps >0.45 ,XS;therefore a time unstrained triangular lattice. This is a fairly small system, step of At=0.35 ,us (about l/40 of the Einstein vibrational but given the fact that size dependencieshave to some period) is used. Runs of at least 130 000 and sometimes 2 extent been studied before ‘?12’we decided to use the avail- million steps are carried out. Certain runs are carried on able computational reso&ces for performing very long even longer, for as long as 8 million or 22 million steps.
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