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 beenstudied 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 apparentdivergence of 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 evidencefor pensionshave recently beenused 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 numerousother 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 someexperimental 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 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 (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 effectsof 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. simulation runs. The simulations are performed in the tra- The latter is equivalent to about 8 s of laboratory time. ditional, strictly 2D simulation setup. (Based on experi- NVT simulations are carried out, mostly using an isok- mental evidence concerning a dependency of the phase inetic constraint algorithm.31 In the very long runs the transition behavior. on the dimensionality of the system is alternatively coupled to--a heat bath by periodic system,3(d)‘29 one would expect-if anything-that this reinitialization of the velocities,3’ thus presumably improv- could imply a slight bias towards KTHNY-type melting ing the sampling efficiency even more. The runs are carried behavior.) out at room temperature (20 “C) and the The colloidal particles, suspendedin a medium of di- is explored via density variation, similarr- to the experimen- electric constant .sr, interact with the pairwise additive, tal setups.3-5 Yukawa-type potential function To characterize the translational order, we calculate the total pair correlation function 1 se-Kr U(r) =E, r (1) g(r) = @WEW--r)), The charge P is an effective, renormalized charge that is (2) much smaller than the actual charge Z. In Murray et al.‘s experiments two different kinds of colloidal spheres were where the ensemble brackets are understood to imply used whose renormalized charges have been estimated to proper normalization and averaging over all pairs of par- be SF=750 e. and F=loOO eo.3(d)We use P=750 e, ticles and all angles. At solid densities, we can also calcu-

J. Chem. Phys., Vol. 100, No. 4, 15 February 1994 3116 K. J. Naidoo and J. Schnitker: 2D melting of the Yukawa system late the individual translational correlation functions that 120 are obtainedfrom the Fourier componentsPo of the mi- croscopicdensity p(r) ,9 . 80 gG(r)=(p8(r')pG(r'-r)). (3) e&j Here pG is the translational order.parameter with respect Ccrml to a particular reciprocal lattice vector G of the underlying 40 crystal structure pG=@. b We only consider the first three vectors of the triangular lattice, IGel =Gc, IG,I =d3Go, and lGZl &2G,, where Gc =47r/( J3ao), with a0 being the lattice spacing.The corre- lation function (3) is alwaysaveraged over the three equiv- alent crystallographic directions; if a particular configuration was obtainedin a liquid-solid traverse (with possible “misorientation” relative to the computational unit cell) the exact lattice orientation is determinedlirst. The orientational order is characterizedby calculating the bond-orientationalcorrelation function g6(r) ,. 0.0 -1.05. 1.1 1.15 1.2 1.25 g6(r)=(yg(r’)y6(r’-r)), ‘35) whereY?e is the order.parameterfor the local orientation of PIlJ.m? the “grain” that is given by the “geometric bonds” (with angles& relative to an arbitrary referenceaxis) of a given RG. 1. (a) Correlation lengths 6 and & for assumed exponential decay particle with its six closet neighborsk, of translational and orientational correlation fknctions g(r) and g,(r), respectively. The lines drawn are only guides for the eye. (b) Power law exponents TJand o6 for assumed algebraic decay of the same two corre- (6) latioii fun&ions. The lines drawn are only guides for the eye. ” k=l . As a measureof the deviation from triangular order, Ys has an absolute value of one for a particle in a perfectly [Ref. 3(a)] and ao= 1.31 pm [Ref. 3(d)], for their two orderedenvironment and representsthe spatial orientation choicesof colloid. Our system thus lies about halfway be- of the local “grain” through the phasefactor. All correla- tween these two values. -. tion functions are calculated using fast-Fourier-transform (FIT) techniques,3’descretizing the relevant data on a A. Orientational and translational order 1024X 1024square lattice, l4 and usually averagingthe cor- Figure 1(a) gives the correlation lengths & and 6 for relation functions over a number of independentconfigu- orientational and translational order, respectively,as ob- ._ rations. tained by fitting the bond-orientational correlation func- Finally, the defect structure is determinedby Voronoi tion g6(r) and the pair correlation function g(r) to expo- tessellationwith a standard algorithm,3’ identifying discli- nential decay envelopes.In the two-step freezing scenario nations as particles with either more or lessthan six nearest of KTHNY theory thesecorrelation lengthsdiverge at dif- neighbors. ferent thermodynamic state points. If we analyzeour data the sameway as Murray and Van Winkle3@’and identify ‘the onset of orientational and translational order-with cor- III. RESULTS AND DISCUSSION relation lengths of &~30a, and c=: 15a,, respectively, then we have to conclude that there are two distinct tran- It is well-known32that there is no volume changeon sitions at densities of pz 1.161 and 1.172pm- ‘, respec- melting in a D-dimensional system that interacts via an tively. Thus there is an intermediate(and possiblyhexatic) inverse-power-lawpotential rmn provided that n

J. Chem. Phys., Vol. 100, No. 4, 15 February 1994 K. J. Naidoo and J. Schnitker: 2D melting of the Yukawa system lo"

Id: gG(r) loo

10 r Cpml IQ2

FIG. 2. Log-log plot of the orientational correlation function g6(r) at 1 IQ density p=1.164 pm--*. rbml

FIG. 3. Log-log plots of translational correlation functions go(r) for tirst a maximum occurs at g~90a, [Ref. 3(a)] and {z4Oa, three reciprocal lattice vectors Go, 6, , and G2, and at density p= 1.172 pm-*. The exponent of the algebraic decay law, rP0, is ~~0.7 in all [Ref. 3 (d)], and in the system of Tang et al. the transla- three cases. tional correlation length does not even rise to more than ,$~lOa, until far into the solid region.’ Hence our simu- individual correlation functions gG(r) . More specifically, lated system would in this regard appear to- be not much KTHNY theory predicts that the decay exponents776 de- worse than is the case under the best experimental condi- pend quadratically on the length of the Bragg vector G if tions. the system is at the solid to hexatic transition.’ If/z and p Figure 1(b) is obtained by alternatively fitting gs(r) denote the Lame elastic constants we have and g(r) to algebraic decay envelopesr-76 and rw9, re- spectively. The interpretation of the new figure is some- vo(solid-hexatic) =kT 4~~(2~+il)3p+A IG12. (7) what less clear; a particular impediment is the fact that the translational decay exponent 77barely falls below r] =: 1.0, Specifically for the first Bragg point, Go, the decay expo- againbecause of the only quasi-long-rangepositional order nent at the transition is predicted’ to fall iq the range and the microgranularity of the solid. Nevertheless, the r]o= l/4-1/3. picture is at least consistent with a delay betweenthe den- From the double-logarithmic representationof Fig. 3 it sities where orientational and translational order diverge. can be seenthat the decay at a density on the borderline A power-law decay is in fact expectedfor the bond- betweensolid and intermediate phaseis in fact algebraic orientational order in the presumedhexatic phase.Figure 2 over the entire distance range (r<40 ym) for each of the gives a log-log plot of the decayof the correlation function first three reciprocal lattice vectors. However, there is g6(r) at a density of p= 1.164pm- “, towards the liquid hardly any differencebetween the three cases,and the de- side of the intermediate region. Obviously, the correlation duced decay exponentsqo, vl, and q2 are all on the order function can be well describedby an algebraic decay law, of ~0.7, contrary to the prediction of the theory. Of at least up to a distance of ~20 pm. (The deviations seen course, our determination of. the melting density may be for larger distanceswould be consistent with the presence quite inaccurate. We therefore present in Fig. 4 data indi- of a finite fraction of a solidlike phase.) More specifically, cating the variation of the power law exponentsqo, vi, and KTHNY theory predicts that the decay exponent drops Q with density. Due to the lack of extensive statistical continuously from a value of vs= l/4 at the liquid to averaging,only qualitative trends can be deduced,but it is hexatic transition to Q=O at the hexatic to solid transi- neverthelessclear that there is in fact little difference be- tion.’ From Fig. l(b), we find q6z0.35 at the density of tween the first three lattice vectors, regardlessof the den- pz 1.161pm- ’ that correspondsto the former of the two sity range, and that the decay exponent r], for the first transitions. The predicted exponent of r/i= l/4 is not too Bragg point will definitely not be very close to the pre- far from this value and probably just outside the error bar dicted range of l/4-1/3. t of the simulation value. We thus seethat the results of a more detailed analysis We now turn to a more detailed analysis of the trans- do not really support the initial conjecture regarding pure lational order. Figure 3 shows its decay upon reaching the bond-orientational order in the intermediate region. We melting density of p= 1.172pm- ‘, but now resolved with have to conclude that a result such as shown in Fig. 1 is respect to the first few Fourier componentsof the micro- apparently much less telltale than expectedat the outset. scopic density. Just as for the overall pair correlation func- However, this does not necessarilycontradict the Murray tion, the quasi-long-rangeorder of the solid phase is ex- et al. claim of having successfullyidentified a hexatic phase pected to give rise to algebraic decays r-qG of the in the experimental colloidal spheresystem. In fact, if the

J. Chem. Phys., Vol. 100, No. 4; 15 February 1994 3118 K. J. Naidoo and J. Schnitker: 2D melting of the Yukawa system

0.9 r 0 t \ 0.8 ! \” 1 q0 \ 0.2 \ 0.7 f & --\ -AZ FD 0 0.6 1.14 1.22 1.3 0.1 0.9

0.0-.- 0 5 10 15 20 million time steps

FIG. 5. Fluctuation of defect density F, ( =fraction of particles that are not sixfold coordinated in a Voronoi tessellation) during 22 million step run at density p= 1.170 pm-‘.

simulations of 2D melting by at least one, if not several 0.8 : orders of magnitude (regardlessof the kind of system). q2 i i The pattern seenis quite dramatic. The defect density \” does not only fluctuate betweenextreme values of ~0% 0.7 - ‘>- 0 ---_ and ~30%, but one can also infer fluctuation cycles of considerablelength, some of them apparently extending over millions of time steps. In addition, up to about 20 million time steps there is an apparent drift towards an P lp-2l overall higher defect density. However, the trend during the last few million time steps make this conclusion less PIG. 4. Power law exponents qo, as a function of density, obtained by definite, and the pattern seenwould also be consistentwith fitting translational correlation functions go(r) for first three reciprocal the presenceof one long oscillation of >30 million time lattice vectors to algebraic decay laws. The lines drawn are only guides for steps. the eye. We do not have enough data for a detailed analysis, but we can still conclude that correlation times of r>lO s are contributing to the dynamics of our system. The cor- latter detailed analysis of translational order is applied to some of the colloidal sphere data the predictions of relation times for translational and orientational motion in KTHNY theory appearto be reasonablywell obeyed,3(d)‘11 the experimentalcolloidal spheresystem (as obtainedfrom in contrast to what is found here. Obviously there must a analysis of van Hove correlation functions) were found to significant differencebetween experiment and simulation, a range between0.1 and 10 s.3(b) This seemsto be qualita- point to which we will return in Sec. IV. tively consistent with our findings. Murray puts strong emphasison the needfor sufficient equilibration.” The colloidal sphere experiments were in- B. Topological defects deedcarried out after waiting for up to a few days, ratio- The dissociation of defect pairs is at the heart of nalized with the suggestionthat climb acrossa KTHNY theory. Figure 5 shows the evolution of the total translational correlation length of the system is important defectdensity (i.e., the fraction of all particles with ~5 and for equilibration.3J1 ’ Murray has estimated that the time 27 neighbors) in a 22 million step run at a system density for dislocation climb is about z 10 h in a system of 0.305 of p= 1.170prne2. This is a density in the intermediate ym sized spheres.” Following this argument, proper equil- region--slightly on the solid side-and the run corre- ibration of the Yukawa system in the present simulation sponds to ~8 s of laboratory time. (We note that the setup would require enormous runs, vastly exceedingthe dynamics in our system are not strictly Newtonian because length of our 22 million step run. Even if the latter argu- of the use of an algorithm for canonical ensemblesam- ment should not be valid, we are still left with a drastic pling.) This is a time range that is not very far from the illustration of the sampling problem in simulations of 2D equilibration time that is implicit in the experimentalpro- phase transitions. It is likely that a comparable problem cedure of Tang ef aLS It is also clear that the 22 million also affects the experiments of Tang et aI.’ This would time stepsexceed the sampling-whether by molecular dy- naturally explain the generally great resemblanceof their namics or Monte Carlo-in virtually all previous computer observationswith ours.

J. Chem. Phys., Vol. 100, No. 4, 15 February 1994 K. J. Naidoo and J. Schnitker: 2D melting of the Yukawa system 3119

. .’ ...... os ...... have not analyzed these in any detail. We tinally did not ...... ‘*-a; ...... ~...... -0,...... : :...... D .:~.~.-e~-~~-(.~.~.~.-.o.~:.~.~.~~.~~ ...... ~~~~o~:.~.~.~.-~.~ ..... investigate whether pairs of dislocations dissociateand re- .:.: ...... ~..;:;.~.~~ ...... ::...... :::...... ~:.-.:~.-.~.~.~.~.‘.~.‘.~.“, . . ‘...‘.“*~ ...... combine during the course of the simulation. Such events ...... -.~:..*::; ...... r ...” o ...... -.p ...... have not beenseen in the experimental system of Tang et ...... ~..:o -,,:.-.:.I.* ...... ;.’ ...... al., and this absencehas then been used as an argument ...... o ...... against the applicability of KTHNY theory.5 However, as ...... a.-...... a.-.:: ...... :*..“,...: ...... pointed out by Murray et aZ.,3(d) the theory is basedon a ...... group treatment and the defect dissocia- ...... ~. .: ... & ...... LLm :...... *.‘-‘.-I tion may take place on an arbitrarily large length scale. Hence, this does not provide a very sensitive test. -% .:yz-TT;-.-.. 0. . ..,. . ...a*~.‘..o~*o.,‘. •~o-.:-.~*..‘~ .9:~..‘.‘.‘.‘.“: ...m;‘ ~~..:-..~:;s..,...:: - 0 ..-*-. . :::;*.:::;*. *.:.‘.‘.‘.:‘.;:-. ..a~=.‘.-~“.. . ..,.e.e.... .*.:.:: ..:.:-.-0.-o. “‘p’.‘.-.-.‘. .: *.*.. . . . :*: ...... *9*-. . . ..2 ...... ‘to. . . . . -: _.... -.:‘.‘...... *.. “...a..:.*. : . .:o.:.-o~~. ‘.:g...:‘_... ..‘*.*. . ..__ p.o:. *--..*a.. .-~:-o-o...-:‘- a_ _ .:~-~..o-“~.-..“... .“...:$..:..,’ IV. CONCLUSIONS .O. . . I... O..-.-. o-..o:. -... ***e.o. ..*.. o.,..:~~..o.o...... ‘.“.+..:.*,:: .-..o’.~:~‘.‘.*..~~. :;.. : . ...*:p.y.=. ‘.~.-.-:.: b:;: ..:‘..p; . .. .-. *...... - -..o. Does the melting transition in our simulated Yukawa . ..* * . ..*. .o;‘:: ...... *-.O.. *Z...’ Q,. 1.:’ *.e....*:.m..m Qm’.O...... *.. .-‘..‘o..:-.::::* . P.0...... **.. . ‘...... -.: . . . ** .p..-.‘.. . . .‘.;;z? ...... system really follow the scenario of KTHNY theory? ..~“.....‘.-. ._.. .~~:d..0:.9::I....0...‘.‘~~::‘.y.o:~:: .*.:.. :.‘.‘.-‘.‘.‘,.‘....‘:r...‘..*. ~.:....‘.;~~.O:.:=o-:. -“.. Based on the evidenceshown we do not believe that the ..:.:-..::;... --..a..0 ...... o.::.:;:.::,:.,: . 10. ‘,‘....*. . . . , ...... *&*.‘.*o- ‘.:&..o*. ._...... -.-..b. ‘,.I. intermediate phaseus obtained in this set of simulations can ..*D ._I..... ,.:~.,:....-..4~..~.‘.‘.4;..:~~.. _ .; i ..*. .o.o....~.rl - . . . . _ .*...... -.*-.*.. . . be called a true hexatic. This is so despitethe fact that two central expectations of KTHNY theory are easily con- -*;x.. .: .I.*.* .-i ::::...-:.:m-o- * . -. . . :..::..:.:::.:.‘.d,‘.P :.-.*. . . ,:.*_.... . :::.6.;“:.‘::.::,..:.:..:::::~~~.z’. firmed, namely a delay betweenthe divergencesof transla- I:.:-.I-... ‘.‘..:“.d.o.” ...... O....-. .“..*.*. .::::: . . . . :‘:‘.....::.-..:::..::::.::. ..‘::.:p*.. tional and orientational order upon melting and the pres- .-_. .%.:.:Q.. ..:: ..:::.::::: . . . . :~:::::::q..:.:..::::...... --0. ence of “free” dislocations in the intermediate-phase.It is ::...* *:::*.:: . ..*. .:..“~.::.:..::::..;;“=“..:dG.‘. ----I ..:.*:-.:...... i.::::..-..:.~. .-... -: .:;:.:j:. 0. . . only upon closer inspection that disagreementsbetween . . ,‘*:~:::~*~.:‘...... *.~.. :.:.::::.::::.:...... : :.:::-::.‘..:::o,.:.:;o=,:: -_- . . ..*. ...*... . . :::.:.::._-: ‘:.*.I.. . theory and simulation become obvious. In this sense,we .,-.. ‘.:.’ ::::: . ...**... .*.:“‘.- ::‘::::.-.‘.‘...‘.::....~p 1 .~.‘..-*-.‘..~. -:..:.::‘.....::.I.:::. *: .:-:- *._.... fully agreewith the skepticism of Tang et aL5 concerning ..* ... ‘..... :.*.:...;..: ...... *..... ::;.*.:. ..a..“.. ‘~.~.~.‘;Q;.-..‘.~...... _. ‘.J.‘.‘.’ I..‘.... .:.:. .*.:.*. .:..*, .*.‘o’~.“:~:..:...:.:.‘:..:::.-.~:..~~ hasty confirmations of the theoretical predictions, whether 0. .‘:.a.-. *: . . ..I.. ::.*.:.. i:...... :..*.a *._* *..._... ‘.‘.‘.‘:.:... *. . . . . ] in an experimental or in a simulated system. ~~.‘:‘::::...o~o’...... ::...-..::.:::::.p::. . -_- i IL:ylA-, ‘.>.‘.:. ‘.‘.m’. **. .- . It may neverthelessbe going too far, for more than one reason,to call the agreementsbetween theory and simula- FIG. 6. Defect structure in three representative configuratidiis for.density tion entirely fortuitous. First, indications are that the first- ~=I.170 pm-*. Open and filled circles denote five- and sevenfold coor- order and KTHNY transitions are very close, such that dinated particles, respectively, as determined by Voronoi analysis. one of them barely pre-empts the other. Most likely, the actual melting mechanism-whichever one that may be- will still be strongly influenced by the other.35Second, we We now turn to the defect structure. Within KTHNY cannot exclude the possibility that much more extensive theory, we expect 5-7-5-7 quartets in the crystal, 5-7 pairs sampling will still give us an intermediate phase that is in the hexatic, and isolated 5’s and isolated 7’s in the liquid. genuinely hexatic in character. This seemsto be a partic- Figure 6 displays the defect pattern in three arbitrary con- ularly intriguing possibility considering the similarity of figurations of the long run at an intermediate system den- our simulation results with the experimental data of Tang sity. An identifiable fraction of free, or incipiently free, et aL5 As mentionedearlier; the latter may well suffer from dislocations was observedin the intermediate region of the an equilibration problem if compared with the experimen- colloidal suspensionexperimentsB5 and also in simulation tal procedure of Murray et aL3 studies of the r-l2 and Coulomb systems.14*i9 According to Now proceeding to the experimental data, does the the figure, the Yukawa system is no exception to this rule, melting transition in any of the colloidal sphere systems i.e., occasionally we do tind “free” dislocations that are really follow the scenarioof KTHNY theory? Although we separatedby at least a few nearest-neighbordistances. It have shown that the crucial initial observation3(‘)of a de- can also be seenthat the defect structure is in generalvery lay in the divergence for translational and orientational complex and that clustering of dislocations is the norm, order is not a strong indicator for the presenceof a hexatic similar to previous observationsin the other systems.335P’4219phase, we still believethat there are good reasonsto answer Overall, the “clumping” of defectsin the pictures looks the question in the affirmative, at least with respect to the more similar to what is seenin the colloidal sphereexper- data of Murray et al3 The latter have beenshown to be in iments of Tang et al.5 than to what is seenby Murray et agreementwith subtle predictions of the theory,3(d)and at aL3(c)13(d)V11333It is tempting to associate this clumping the sametime there are no observationsthat would literally with lack of equilibration, .aspreviously suggestedby Mur- and unequivocally contradict any predictions. The experi- ray.” Still, there is not really a pronounced tendency for ments of Tang et al,’ on the other hand, appear to pose the clumping to occur in the form of grain boundaries more of a problem. Since their data resembleso much our which are so central to Chui’s theory of first-order melt- simulation data it would not be too surprising if all our ing.34There are other differenceswith respect to the pre- reservations concerning the applicability of the theory cise defectstructure betweenthe various systems, but we would equally apply to this set of experiments.

J. Chem. Phys., Vol. 100, No. 4, 15 February 1994 3120 K. J. Naidoo and J. Schnitker: 2D melting of the Yukawa system

Thus there is apparently a discrepancy between the longer runs, possibly using different ensemblesand new data of Murray et ai. on the one hand and the data of sampling schemes,are called for. Naturally, there are ad- Tang et al5 and the present simulation data on the other ditional problems with the setup of 2D computer simula- hand. What is the origin of this discrepancy?We find it tions, such as related to system size,27boundary and initial helpful to againsuccessively address the points put forth in conditions,l4 substrate effects, and the possibly non- the Introduction of this paper. negligible 3D character that is left by the experimental ( 1) It has occasionally beenconjectured that the type confinement. of transition may depend on the nature of the potential (3) A tinal sourceof discrepancybetween expe$ment function, with “soft” interaction potentials favoring a and simulation is the data analysis and interpretation. The KTHNY-type phasetransition since only in the latter type results presentedin Sec. III illustrate vividly how much of system short wavelength effects (neglected in the the- care has to be exercisedto identify evidencefor KTHNY- ory) are not expectedto be quite that dominant. Examples type melting that is only superficial (or even deceptive). for a systematic variation in the interaction potential can We find that there is often little differencebetween the data be given for both experiment and simulation. Thus the two from experiment and simulation if these are only analyzed setsof experimentsby Murray et aLs and Tang et a1.3 were in a uniform mariner.... carried out with spheresof 0.305and 1.0pm diam, respec- In summary, we suspect that a true hexatic has not tively, while it has beenestimated that the screeninglength been obtained in any simulation study to date, including l/~ should be comparable in the two cases.3(d)Hence, the present study of the Yukawa system. This is primarily there is an implicit differencein the rangeof the interaction a problem of insufficient sampling. Choice of the “right” potential, with the Tang et al. system being more hard- potential function might considerably facilitate the search sphere-like.Murray et al. have speculatedthat this could for evidenceof KTHNY melting. Even if such evidence explain some of the differencesin the melting behavior of has beenobtained, careful analysis has to be carried out in the two cases.3(d)4 similar variation is encounteredif we order to recognizeagreements with theoretical predictions comparethe present simulation of a Yukawa ( = truncated that are only coincidental. Truly penetrating tests have to Coulomb) system with the simulation of the full Coulomb be devised, such as related to the predicted temperature systemby Bedanovet al. l9 Unfortunately, no run times are dependenceof the various correlation lengths. Becauseof given for the latter, rather old study and it is difficult to simultaneous and unavoidable changes in the screening assessits apparently “easy” confirmation of the predictions length of the colloidal interactions such a test cannot easily of KTHNY theory.19Because of the true long-rangechar- be carried out experimentally, but can still be accom- acter of the interactions, sampling in the true Coulomb plished in straightforward computer simulations with a system is naturally expectedto be even more difficult than suitable potential function. in the Yukawa system. A more complex issue is the difference between the ACKNOWLEDGMENTS interaction potential that is actually effective in the exper- Partial support of this researchby a Rackham Faculty iment vs the one in the simulation. We note that the width Grant (No. 386294)is gratefully acknowledged.The com- in density of the intermediate region is only z 1% in the putations were performed with resources provided by a current study of the Yukawa system, as opposedto the NSF Instrumentation Grant -(No. CHE-8917309) to the about 4%-8% (dependingon the identity of the colloid) Department of Chemistry of the University of Michigan. in the experimental case.3(a)y3(d)15 Many-body potentials may be necessaryto realistically model colloidal spheres ’ R. Pieranski, Contemp. Phys. 24, 25 ( 1983). betweenglass plates with induced image charges. In this *D. Thirumalai, J. Phys. Chem. 93, 5637 (1989). context, we mention that the introduction of explicit po- 3(a) C. A. Murray and D. H. Van Wiie, Phys. Rev. Lett. 58, 1200 (1987); (b) C. A. Murray and R. A. Wenk, ibid. 62, 1643 (1989); (c) larizability has more recently been one of the major inno- C. A. Murray, D..H. Van Winkle, and R. A. Wenk, Phase Trans. 21,93 vations in the routine computer simulation of bulk aqueous (1990); (d) C. A. Murray, W. 0. Sprenger, and R. A. Wenk, Phys. systems. Implementation of similar techniquesin simula- Rev. B 42, 688 (1990). 4A. 5. Armstrong, R. C. Mockler, and W. J. O’Sullivan, J. Phys. Con- tions of 2D melting would be a most interesting develop- densed Matter 1, 1707 (1989). ment. ‘Y. Tang, A. J. Armstrong, R. C. Mockler, and W. J. O’Sullivan, Phys. (2) A major issue identified in the present study con- Rev. Lett. 62, 2401 (1989); Phase Trans. 21, 75 (1990). cerns the amount of sampling. It is the extremely long 6K. J. Strandburg, Rev. Mod. Phys. 60, 161 (1988). ‘Bond Orientational Order in Condensed Matter Systems, edited by K. J. equilibration time that sets the experiments of Murray et Strandburg (Springer, New York, 1992). al3 most prominently apart from the simulations carried ‘J. M. Kosterlitz and J. D. Thouless, J. Phvs. C 6, 1181 (1973). out here and from the experiments of Tang et al5 In a 9D. R. Nelson and B. I. Halperin, Phys. Rev. B 19, 2457 (1979); A. P. simulation study of the r-t2 system, we already presented Young, ibid. 19, 1855 (1979); A Zippelius, B. I. Halperin, and D. R. Nelson, ibid. 22, 2514 (1980). some evidence that kinetic bottlenecks are a big impedi- ‘OR. Geer, T. Stoebe, C. C. Huang, R. Pindak, G. Srajer, J. W. Goodby, ment for the exploration of configuration spaceduring a M. Cheug, J. T. Ho, and S. W. Hui, Phys. Rev. Lett. 66, 1322 (1991); 2D phasetransition.i4 It is easily conceivablethat only this R. Geer, T. Stoebe, C. C. Huang, R. Pindak, J. W. Goodby, M. Cheng, prevents us from generatinga hexatic phase.Of course,our 3. T. Ho, and S. W. Hui, Nature 355, 152 (1992). ‘IC!. A. Murray, in Ref. 7, p. 137. findings also put in doubt virtually all of the numerous “J. A. Zollweg and G. V. Chester, Phys. Rev. B 46, 11 186 (1992). other published simulation studies of 2D melting. Much 13J. Lee and K. J. Strandburg, Phys. Rev. B 46, 11 190 (1992).

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14K. J. Naidoo, J. Schnitker, and J. D. Weeks, Mol. Phys. 80, 1 (1993). 24E. J. Meijer and D, Frenkel, J. Chem. Phys. 94, 2269 (1991). *‘K. J. Strandburg, in Ref. 7, p. 32. zJ C. Zahorchak, R. Kesavamoorthy, R. D. Coalson, and S. A. Asher, 16R. C. Qann, S. Chakravarty, and G. V. Chester, Phys. Rev. B 20, 326 J: Chem. Phys. 96, 6873 ( 1992): (1979). 26M. J. Stevens and M. 0. Robbins, J. Chem. Phys. 98, 2319 (1993). “R. Morf, Phys. Rev. Lett. 43, 931 (1979). “S. Toxvaerd, Phys. Rev. Lett. 51, 1971 (1983). ‘sR K. Kalia, P. Vashista, and S. W. de Leeuw, Phys. Rev. B 23, 4794 28J. A. Zollweg, G. V. Chester, and P. W. Leung, Phys. Rev. B 39, 9518 (1981). (1989). “V. M. Bedanov, G. V. Gadiyak, and Y. E. Lozovik, Sov. Phys. JETP ‘9C. A. Murray, W. 0. Sprenger, and R. A. Wenk, J. Phys. Condensed 61, 967 (1985). Matter 2, SA385 (1990). “H. Cheng, P. Dutta, D. E. Ellis, and R. Kalia, J. Chem. Phys. 85, 2232 30H. Gwen, J.-P. Hansen; and P. A. Madden, J. Chem. Phys. 98, 3275 (1986). (1993). “(a) D. Hone, S. Alexander, P. M. Chaikin, and P. Pinkus, J. Chem. “M. P. Allen and D. Tildesley, Computer Simulation of Liquids (Clar- Phys. 79, 1474 (1983); (b) S. Alexander, P. M. Chaikin, P. Grant, G. endon, Oxford, 1987). J. Morales, P. Pinkus, and D. Hone, ibid. 80, 5776 (1984).. 32J. D. Weeks, Phys. Rev. B 24, 1530 (1981). “(a) K. Kremer, M. 0. Robbins, and G. S. Great, Phys. Rev. Lett. 57, 33Note that the defect patterns in Ref. 3(a) tire incorrect because of a 2694 (1986); (b) K. Kremer, G. S. Grest, and M. 0. Robbins, J. Phys. faulty Voronoi construction. The defect patterns in Refs. 3(c), 3(d), A 20, L181 (1987); (c) M. 0. Robbins, K. Kremer, and G. S. Grest, J. and 11 are correct. Chem. Phys. 88, 3286 (1988). 34S. T. Chui, Phys. ‘Rev. B 28, 178 (1983). ?3R. 0. Rosenberg and D. Thirumulai, Phys. Rev. A 36, 5690 ( 1987). 35C. Udink and D. Frenkel, Phys. Rev. B 35, 6933 (1987).

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