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PHYSICAL REVIEW E MAY 1997VOLUME 55, NUMBER 5
Constraints, metastability, and inherent structures in liquids
D. S. Corti, P. G. Debenedetti, and S. SastryDepartment of Chemical Engineering, Princeton University, Princeton, New Jersey 08544
F. H. StillingerBell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974
and Princeton Materials Institute, Princeton University, Princeton, New Jersey 08544~Received 23 December 1996!
Void-size distributions have been calculated for the shifted-force Lennard-Jones fluid over substantial tem-perature and density ranges, both for the liquid-state configurations themselves, as well as for their inherentstructures~local potential energy minima!. The latter distribution is far more structured than the former,displaying fcc-like short-range order, and a large-void tail due to system-spanning cavities. Either void distri-bution can serve as the basis for constraints that retain the liquid in metastable states of superheating orstretching by eliminating configurations that contain voids beyond an adjustable cutoff size. While acceptablecutoff sizes differ substantially in the two versions, ranges of choices have been identified yielding metastableequations of state that agree between the two approaches. Our results suggest that the structure-magnifyingcharacter of configuration mapping to inherent structures may be a useful theoretical and computational tool toidentify the low-temperature mechanisms through which liquids and glasses lose their mechanical strength.@S1063-651X~97!03805-1#
PACS number~s!: 61.20.Gy, 64.60.My, 64.70.Fx, 05.70.2a
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I. INTRODUCTION
The formation, characterization, and utilization of mestable forms of matter present challenges and opportunthroughout science and technology. All phases are involvsolid, liquid, and vapor. The usefulness of a metastable sstance in any given application normally depends on the aity to frustrate kinetic processes of phase change that creversion to thermal equilibrium. The phenomenologiclassical nucleation theory has enjoyed substantial succepredicting such rates of phase change under many circstances of interest@1,2,3#, but its connection to the rigoroumolecular theory of matter remains incomplete. Indemany basic questions also remain about the nature of mstable states themselves@4#.
The present paper reports results of a theoretical stdesigned to enrich fundamental understanding of liquidsare superheated~i.e., metastable with respect to nucleatiand growth of the vapor phase!. Real-world connections involve explosive boiling@5#, cavitation in turbulent flow@5,6#,bubble-chamber detectors for elementary particles@7#, andthe tensile behavior of sap rising in trees@8#. The initial stageof sonoluminescence experiments also involves cavitatowing to ultrasonic excitation@9#.
Section II below discusses the general theoretical straof imposition of constraints designed to prevent vapor nucation in a superheated and/or stretched liquid and the inence of those constraints on measurable properties ofliquid. Primary interest in the present study focuses on sptaneously formed voids in the liquid medium, the largestwhich must be prevented to frustrate nucleation. Sectionshows how void-size constraints, both before and after cfiguration mapping to ‘‘inherent structures’’@10,11#, can beemployed to effect the desired metastable extensions; c
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parisons of the two constraint versions are provided. SecIV presents results of an analysis of the statistical geomof the shifted-force Lennard-Jones liquid, examined in bthe equilibrium region of its phase-plane, as well as inmetastable extension regime into the region of equilibriliquid-vapor coexistence. Void distributions have been otained for all of these states, both for the liquid itself, as was for the corresponding sets of inherent structures obtaby mapping particle configurations onto potential enerminima. The final section, Sec. V, summarizes our resustates our conclusions, and attempts to predict the mostductive directions for further study.
II. CONSTRAINTS AND THEIR EFFECTS
Under conditions of strict thermal equilibrium, the proerties of anN-body system can be represented by the canocal ensemble. It suffices for present purposes to supposethe system comprises point particles~no internal degrees ofreedom!, and that the classical limit applies. Consequenthe canonical partition functionQ in the following form pro-vides the thermodynamic properties at temperatureT andvolumeV @12#:
Q~N,V,T!5~N!L3N!21E •••E e2bFNdr1•••drN . ~1!
Here L is the mean thermal deBroglie wavelength;b is1/kT, k being Boltzmann’s constant; andFN(r1•••rN) is theinteraction potential for theN particles located atr1•••rN .The integrals in Eq.~1! are restricted to the interior of volumeV. The pressure equation of state emerges as a volderivative,
5522 © 1997 The American Physical Society
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55 5523CONSTRAINTS, METASTABILITY, AND INHERENT . . .
P51
b S ] ln Q
]V DN,T
; ~2!
in particular, this expression correctly captures the breathe pressure isotherms as the system crosses an equilibphase transition curve in the (V,T) plane.
Nucleating and growing a new phase at a first-order phtransition normally is kinetically difficult, provided the material under study is clean and mechanically undisturbConsequently, metastable extensions beyond the transpoint, in temperature or density, are commonplace. The pence of a kinetic barrier to first-order phase change is tamount to bimodality of the canonical probability distributioexp(2bFN), corresponding to virtually distinct and nonovelapping contributions that reside in different regions of tN-body configuration space. The ‘‘bottleneck’’ region btween them has a low probability of occupation, at leastmodest extensions into metastability.
In order to achieve a reformulation of the canonical esemble that accords with experimental observation of mstability, a mathematical device must be provided to clothe bottleneck, thereby trapping the system’s configuratiothe appropriate one-phase region even as the location oequilibrium phase change is passed. This constraint canways be attained in principle by imposing an additional ptential on theN-body system,WN(r1•••rN), that vanishes inthe desired one-phase region of the configuration spaceis positive and arbitrarily large in the region that displaysleast some portion of the alternative and thermodynamicfavored phase. The discontinuity locus forWN in the3N-dimensional configuration space is a (3N21)-dimensional hypersurface that dynamically acts as a refling boundary for the system point, returning it to the interof the one-phase region.
The choice of specific form for the constraint potentWN will be dictated by the metastable phase of interest.percooled liquids and the glasses they form must be freall but the smallest recognizable crystallites, soWN mustbecome large whenever any substantial local set of partidisplays crystalline order@11,13#. Supersaturated vapors requireWN to be large if any substantial cluster~droplet! ofparticles appears@2#. Superheated liquids, or liquids undtension, must be free of substantial voids or cavities,WN must become large if such patterns exist anywhere inliquid medium@14,15#. This last case is the one examinedthe following sections of this paper.
Having selectedWN for the application of interest, themodified canonical partition functionQ* has the obviousform
Q* ~N,V,T!5~N!L3N!21E •••E e2b~FN1WN!dr1•••drN ,
~3!
with the same integration limits as before. BecauseWN hasbeen chosen to be large and positive somewhere in thefiguration space, but is non-negative everywhere, we ha
0,Q*,Q. ~4!
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The metastable-state pressure corresponding toQ* involvesthe direct analog of Eq.~2!:
P*51
b S ] ln Q*
]V DN,T
. ~5!
This constrained-ensemble pressure contains an idealcontribution, as well as contributions from the interparticinteractions (FN) and from the constraint (WN).
Most statistical-mechanical modeling of liquids~includ-ing the present study! assume thatFN consists only of addi-tive pair terms, which for structureless particles requires
FN5(i. j
(j51
N21
f~r i j !, ~6!
where r i j is the distance separating the centers of partici and j . For some applications one might wish to assignsimilar form toWN . But as indicated above, attaining metstability generally requiresWN to realize a specific patternrecognition capacity in much larger groups of particles thjust pairs.
The volume derivative appearing in Eq.~5! can behandled by the same volume scaling tactic that has becstandard practice for the unconstrained canonical ensem@16#. Presuming that the volumeV is cubical, the coordinatetransformation
r5V1/3s ~7!
causes the integrations inQ* , Eq. ~3!, to span a unit cube ins space, while permitting the volume derivative to be carrout by the chain rule. The result of these operations mayexpressed in the following way:
P*5r
b22pr2
3 E0
`
r 3f8~r !g* ~r !dr2C. ~8!
Herer is the number densityN/V andg* is the pair corre-lation function for the constrained system,
r2g* ~r 12!5N~N21!E •••E e2b~FN1WN!dr3•••drN
E •••E e2b~FN1WN!dr1•••drN
,
~9!
and generallyg* will differ from the unconstrained-systempair correlation functiong. The constraint also affects thpressure by generating the new termC,
5524 55CORTI, DEBENEDETTI, SASTRY, AND STILLINGER
C5
E0
1
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dsNze2b~FN1WN!~]/]V!WN~V1/3s1x•••V
1/3sNz!
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1
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1
dsNze2b~FN1WN!
. ~10!
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If WN(r1•••rN) depends on the relative positions of thN particles, but has no explicit dependence on volumeV,then the integrand in the numerator of expression~10! forC will vanish everywhere except at the discontinuity hypsurface ofWN . Consequently, the numerator integral reducto an integral over that (3N21)-dimensional hypersurfacof theN-particle probability function weighted by an amoudetermined by the specific form ofWN . When this type ofconstraint is applied to the metastable-state problem, thecontinuity hypersurface should occur where theN-particleprobability is small, at least for the modest extent of mestability, as argued above. Then the contributionC to thepressure should likewise be small.
An alternative class of constraint functionsWN has vol-ume appearing explicitly as a scaling factor. Specifically,
WN[F~V21/3r1•••V21/3rN!, ~11!
and in this event the key factor in the numerator integrandEq. ~10! is
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55 5525CONSTRAINTS, METASTABILITY, AND INHERENT . . .
For densities less thanro , however, two types of inherenstructures were possible. The periodic arrangement ofticles can be maintained belowro , causing the potential energy local minimum to rise in value as the systemstretched. However, another configuration of particles alocal energy minimum can be formed, corresponding toarrangement of broken chains. The broken chains yieldsolute minima, while the unbroken periodic arrays yiehigher-lying relative potential-energy minima. The stretchperiodic array loses its mechanical stability at a density,r1,ro . Hence, the one-dimensional Lennard-Jones fluidr1<r<ro that has an unbroken periodic inherent structis equivalent to a stretched metastable state. Therefore,reasonable to assume that removing from the partition fution configurations that have inherent structures containlarge voids, for systems capable of exhibiting a first-ordliquid-vapor phase transition, should allow one to rigorouobtain the properties of the metastable liquid@15#. Disallow-ing the liquid to sample those configurations that map oinherent structures that contain voids larger than some spfied size serves to suppress the vaporization transitionkeep the liquid homogeneous within the metastable regTherefore, following the suggestion outlined in@15#, we uti-lize the inherent structure formalism to obtain the equilrium properties of a model superheated liquid and analthe effect of the severity of the constraint, consisting of spressing the formation of voids in the inherent structure,its equation of state.
Mathematically, the various local potential enerminima, or inherent structures, are solutions to@10#:
¹FN~r1•••rN!50, ~13!
where¹FN is the gradient of the potential energyFN . Eachinherent structure is included in the set of stable packinwithin which the force on every particle vanishes.@Note thatsaddle points and potential-energy maxima are also solutto Eq.~13!.# The number of distinctFN minima is enormous,of the order ofN!eN distinguishable particle packings@18#.N! accounts for particle permutations that yield minimaidentical potential energy;eN estimates the number of distinct ways of arranging particles in mechanically stable paings. Most of these packings are amorphous; others will crespond to defective crystals at slightly lower valuesFN . Those inherent structures that are perfect crystalsbe the absolute minima in the potential energy.
Each configuration ofN particles can be assigneuniquely ~mapped! to its own inherent structure. The procdure for finding the proper local minimum is to move thparticles along the steepest direction on theFN hypersurfaceuntil the forces on each particle vanish. This procedureequivalent to quenching the system ofN particles from aninitial temperatureT to a final value ofT50. As a result, allkinetic energy is removed, leaving the system at rest aFN minimum. In the case of identical structureless spherparticles, the appropriate mapping is generated by thelowing steepest-descent equations for each particlei @19#:
dr i~s!
ds52¹FN@r1~s!•••rN~s!#, ~14!
r-
anb-
d
reisc-gry
oci-ndn.
-e-n
s,
ns
f
-r-fill
is
all-
where s>0 is a progress variable indicating the extentwhich the descent trajectory has been followed. Startfrom an initial configuration (s50), a positive value ofsdisplaces the configuration along the direction of the netive of the potential energy gradient until it comes to re(s→`) at the appropriate minimum. Given an initial configuration of N particles, the simultaneous solution of thN equations in Eq.~14! quenches the system into a potentenergy minimum, yielding the appropriate inherent structu
Steepest-descent trajectories@Eq. ~14!# of the shifted-force Lennard-Jones fluid~discussed below! were generatedusing a conjugate gradient method@20#. The conjugate gra-dient method is highly efficient during the initial part of thquench, rapidly decreasing the potential energy to withinof the true minimum. At this point, however, a large comptational effort is required to further decrease the potenenergy towards its value at the local minimum. Fortunatean important simplification results over the range of densiexplored in this work, once the potential energy is quenchto within approximately 2% of the true local minimum. Under these conditions, we found that the configuration of pticles is essentially identical to the inherent structure. Tremainder of the descent trajectory involves only minor ajustments of particle positions so that the force on each pticle vanishes exactly. Over the range of densities exploin this work, we found that stopping the quench via the cojugate gradient method when the potential energy was wi1–2 % of the true potential energy minimum yieldequenched configurations that were virtually indistinguishafrom the inherent structures. Each iteration of the conjuggradient method causes a decrease in the potential enThe quench was terminated when a new iteration yieldedecrease in the potential energy of less than 0.001%. Atpoint, the energy was within 1–2 % of the true local minmum. Using this approach, the time required to convergea local minimum for a system composed of 256 particles was short as 20 min on a HP715 workstation.
The computer simulation studies discussed in this pawere performed using the shifted-force Lennard-Jones 1potential. The Lennard-Jones 12-6 potential is defined by
uLJ~r !54eF S s
r D12
2S s
r D6G , ~15!
wherer is the distance between two particles,s the distanceat which the potential is zero, ande the well depth. Nor-mally, the Lennard-Jones 12-6 potential is truncated beya given cutoff distancer c . Numerical instabilities, howeverare introduced during a descent trajectory using the truncLennard-Jones potential, since the force between twoticles changes discontinuously atr c . To avoid this problem,we instead simulated a fluid whose particles interact viashifted-force Lennard-Jones potentialusf(r ), in which @28#
usf~r !5H uLJ~r !2uLJ~r c!2~r2r c!uLJ8 ~r c!, r<r c
0, r.r c~16!
and uLJ8 (r c) is the value of the derivative of the LennardJones 12-6 potentialuLJ at r c @Eq. ~15!#. The shifted-forcepotential is now such that the potential and force both
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5526 55CORTI, DEBENEDETTI, SASTRY, AND STILLINGER
smoothly to zero at the cutoffr c , eliminating any problemsin the convergence towards the appropriate inherent stture. Of course, the properties of the shifted-force LennaJones fluid are different from the Lennard-Jones fluid. Fexample, since the well depth of the shifted-force potentiashifted slightly upwards to20.93e ~as opposed to2e forthe Lennard-Jones potential! for r c52.5s, one expects thevalue of the critical temperature to be lower than that ofLennard-Jones fluid. Table I compares the critical and trippoint parameters of the shifted-force fluid withr c52.5s tothose of the Lennard-Jones fluid. The equation of state ofshifted-force Lennard-Jones fluid was estimated using acedure suggested elsewhere@21#: if one assumes that thradial distribution functions of the shifted-force anLennard-Jones fluid are identical, then the pressure ofshifted-force fluidPsf is equal to@22#
Psf5rkT22pr2
3 E0
r cr 3
dusfdr
g~r !dr, ~17!
whereg(r ) is the radial distribution function of the LennardJones, calculated by using perturbation theory@23,24#.Vapor-liquid equilibrium properties were obtained by equing pressures and chemical potentials, the latter calculateintegration of Eq.~17!. Triple-point parameters were obtained from Eq.~17! and a pressure-temperature expressof the liquid-solid coexistence curve determined from simlation @25#.
B. Simulation method
We now compare the results of applying the void costraint on the unquenched and quenched configurationthe superheated shifted-force Lennard-Jones liquid~void-constrained and inherent-structure void-constrainedsembles, respectively!. In the former, voids exceedinggiven size are prevented from forming within any instanneous configuration of the liquid. In the latter, the liquidprevented from sampling those configurations thatmapped to inherent structures containing voids that excsome specified size. The success of both sets of simulahinges upon the development of an efficient void-countalgorithm. As described in an earlier study@14#, we quantifythe size of voids within a given configuration by performina Voronoi-Delaunay tessellation@26,27#. The Voronoi-
TABLE I. Comparison of the critical and triple-point parameteof the r c52.5s shifted-force Lennard-Jones fluid~SF! and theLennard-Jones fluid~LJ!. The triple-point parameters were obtaineusing the solid-liquid phase equilibrium data of Kofke@25#. r tr, liq* isthe density of the liquid at the triple point.r*5rs3, T*5kT/e,andP*5Ps3/e. The numbers in parentheses indicate the errothe last one or two digits of the data.
SF LJ
rc* 0.247 0.304~6!
Tc* 1.16 1.316~4!
Pc* 0.109 0.130~10!
Ttr* 0.687 0.685~3!
r tr, liq* 0.670 0.85~2!
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Delaunay tessellation is defined as follows: a pointx ~wherex is a vector denoting location in space with respect to soorigin! belongs to the Voronoi cell of atomi located at po-sition xi if it is closer toxi than to any other pointxj of thesystem. Mathematically, this can be represented by@27#
xPVi⇔ux2xi u<ux2xj u, ; j , ~18!
whereVi denotes the Voronoi polyhedron which surrounatom i . The dual Delaunay construction is a tiling of spaby simplices~d-dimensional tetrahedra whered is the sys-tem’s dimension!, whose vertices are the atom positionsxi ,while the centers of the spheres circumscribing these splices are the Voronoi vertices. The diameter of the circusphere minus the Lennard-Jones diameters is taken as theeffective size of a void about a Lennard-Jones particle.convenience, these voids were visualized as sphericalticles with diameters equal to their effective lengths. Figurillustrates the dual tessellation and the definition of a vsize.
In both constrained ensembles, the relevant parametthe diameter of the maximum allowed void,dmax* 5dmax/s. Itis convenient to referencedmax* to the average interparticleseparation (r* )21/35(rs3)21/3 to obtain the dimensionlesvariableb5dmaxr
1/3. The parameterb simply quantifies the
FIG. 1. ~top! Two-dimensional illustration of the VoronoiDelaunay dual construction. The central atomi is surrounded byatoms j . The solid lines form the Voronoi polygon about atomi .The dashed lines form the Delaunay triangles whose circumsphare centered at the corresponding vertices of the Voronoi polyg~bottom! The determination of the size~diameter! of ‘‘void par-ticles’’ about a Lennard-Jones particle in the superheated liqThe Lennard-Jones particles lie on the vertices of the Delautriangles~smaller shaded circles!. The void-particles are centereon the vertices of the Voronoi polygon. For clarity, only one voparticle is shown~larger shaded circle!. Its diameter equals that othe sphere circumscribing the Delaunay triangle and centered onvertex of the Voronoi polygon, minus the Lennard-Jones diames. ~Figure taken from@14#.!
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55 5527CONSTRAINTS, METASTABILITY, AND INHERENT . . .
severity of the constraint; the smaller the value ofb, themore severely constrained is the superheated liquid. Anditional advantage is gained by expressing the constrainterms ofb rather thandmax* . Sinceb scales with (r* )21/3, orequivalentlyV1/3, the nonstructural contribution to the presure arising from the application of the constraint@that is tosay, the quantityC in Eq. ~10!# is identically zero. The voidconstraint applied to inherent structures, however, isstrictly of the form shown in Eq.~10!, since its positionvariables are not those of the canonical distribution thaused to derive the equation of state, but rather a functionthe mapped position variables. Consequently, one cannotorously appeal to Eq.~10! to argue that the termC vanishesidentically. Yet, it is not unreasonable to assume, as aapproximation, that the mean linear dimension of tsteepest-descent basins scales asV1/3; therefore the inherentstructure void constraint~which eliminates entire basins!should involve a negligibly small termC in the equation ofstate @Eq. ~8!#. In what follows, we calculate inherenstructure void constraint pressures using Eq.~8! with C50; the effect of the constraint should only be felt throuthe virial, since this term is a function of the radial distribtion function of the constrained liquid.
Simulations of the superheated shifted-force LennaJones liquid in both the void-constrained and inherestructure void-constrained ensembles were performed uthe canonical Monte Carlo algorithm@28# ~constantN, V,and T!. The equation of state was determined in bothsembles for a system size ofN5256 and for two subcriticatemperatures: T*50.70 and T*50.90. In the void-constrained ensemble, normal Monte Carlo procedures wfollowed except that a particle move was rejected if thmove created a void larger than some specified size.semble averages were calculated over 4000 Monte Csteps per particle~MCS! after an initial equilibration periodof 1000 MCS.
In the inherent-structure void-constrained ensemble, pventing the sampling of liquid configurations having inherestructures containing large voids requires that the liquidfrequently quenched to the local potential energy minimaprinciple, that would require that, after each particle mothe inherent structure of the new liquid configuration andmaximum void size be determined. Such a large numbequenches, even for a system size ofN5256, is computation-ally prohibitive. Fortunately, there is no need to perfoquenches so frequently. The dynamics of transitions betwsubstantially different inherent structures is relatively sloDuring a molecular-dynamics simulation of a 32-particsystem, Stillinger and Weber@18# found that the number otransitions between distinct potential energy minima wasproximately 250 out of a total run of 104 time steps at atemperature 50% greater than the melting temperatThough they simulated systems at high densities~at or nearthe liquid-solid transition!, it is not unreasonable to assumthat the dynamics of transitions between inherent structuof superheated liquids, at densities lower than the triple-pdensity of the liquid and at subcritical temperatures, is asluggish. Therefore, during a Monte Carlo simulatioquenches were performed every 100 MCS. To check thelidity of the proposed simulation results, we calculatedpressure for simulations in which quenches were perform
d-in
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-
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every 50 MCS. The results were statistically indistinguisable. Simulations at a higher frequency of quenches thanMCS were not performed, since, as we will discuss infollowing subsection, pressures obtained while quenchevery 100 MCS within the inherent-structure voiconstrained ensemble were consistent with pressurestained in the void-constrained ensemble for approprichoices of the severity of the constraint.
Simulations within the inherent-structure void-constrainensemble are performed with a Monte Carlo simulationgorithm. The simulation begins with an initial configuratiothat is mapped to an inherent structure that has no voexceeding some specified size. For 100 MCS, configuratof the liquid are sampled according to normal Metropoguidelines. At the end of the 100-MCS block, the liquidquenched to its inherent structure. A Voronoi-Delaunay tsellation indicates the largest void size within the inherstructure. If the maximum void is less than the specifivalue, then the properties of the liquid~e.g., pressure andconfigurational energy! accumulated over the 100-MCblock are counted towards the simulation’s ensemble aages. If the void exceeds the given size, then the accumulaverages are discarded. The simulation proceeds withother 100 MCS but starts with the initial configurationparticles used at the beginning of the previous 100-Mblock. Each simulation began with 1000 MCS of equilibrtion followed by 3000 MCS of ensemble averaging. Thus,quenches were performed in determining the equilibriproperties of the constrained superheated liquid. Durequilibration, the total number of MCS was increasedmore than half of the quenches yielded rejections. Therefwe were assured that at least 500 MCS were used to reequilibration. After equilibration, we found that, for looseconstrained systems~the diameter of voids allowed to formin inherent structures was equal to five or six interpartidistances!, on the order of 0–2 quenches resulted in a viotion of the constraint~only a small fraction of 100-MCSblocks were rejected!. When the liquid was severely constrained~the diameter of voids was less than or equal to fointerparticle distances!, at most ten quenches resulted inviolation of the constraint~i.e., twenty 100-MCS blocks werecounted towards the system’s ensemble averages!.
Since both constrained ensembles are concerned withprevention of voids of appreciable size, it is useful to analythe distribution of void sizes in both the unquenched aquenched liquids. Figure 2 displays the void-size distributof the unquenched and quenched superheated shifted-Lennard-Jones liquid atr*50.70 andT*50.70. Althoughour approximate equation of state@Eq. ~17!# predicts that thedensity of the liquid at the triple point isr*50.670 (Tcr*50.687), we obtained a slightly negative pressure for simlations atT*50.7 andr*50.725. From simulation data, westimate that the density of the liquid at the triple po(T*50.687) is closer tor*50.730. Voids forming withinthe liquid~unquenched! rarely exceed two particle diameterfurthermore, the distribution is unimodal and almost symetrical about the most probable void diameter~approxi-mately 0.7s!. In contrast, the void-size distribution of thcorresponding inherent structure exhibits three distinct ftures: two sharp peaks at small void sizes (,1.0s), a broadshoulder spanning intermediate void sizes~ca. 1.0s–2.5s!,
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5528 55CORTI, DEBENEDETTI, SASTRY, AND STILLINGER
and a slowly decaying large-void-size tail (.2.5s). The in-herent structures of a superheated liquid contain voids whsizes exceed that of the largest voids seen within the uquenched liquid. Therefore, the sizes of voids that mustprevented from forming within inherent structures of the sperheated liquid will greatly exceed the sizes of voids prhibited from developing within the unquenched liquid.
FIG. 2. Probability density of the diameter of voids found in thinherent structure and unquenched liquid of the shifted-forLennard-Jones fluid atT*50.7 andr*50.7. The dashed line is thedensity distribution function of the unquenched liquid.
TABLE II. Pressure and configurational energy per particlethe shifted-force Lennard-Jones liquid within the inherent-structuvoid-constrained ensemble for various values ofb at T*50.7.From Eq. ~17!, the density of the liquid at coexistence isr*50.663; the density of the liquid at the spinodal is approximater*50.506. The numbers in parentheses indicate the error in thetwo or three decimal places.
r* b P* U* /N
0.55 4.0 20.5429(478) 23.528(167)0.55 5.0 20.2376(424) 23.117(52)0.55 6.0 20.2772(504) 23.194(49)
0.60 4.0a 21.897(888) 21.956(241)0.60 5.0 20.2370(444) 23.407(68)0.60 6.0 20.2451(429) 23.387(48)
0.65 4.0 20.3966(638) 23.602(58)0.65 5.0 20.2723(489) 23.575(37)0.65 6.0 20.2711(467) 23.577(39)
0.70 4.0 20.3792(463) 23.859(39)0.70 5.0 20.1385(223) 23.848(40)0.70 6.0 20.1880(497) 23.843(38)
0.725 4.0 20.06983(472) 23.977(39)0.725 5.0 20.01233(465) 23.982(37)0.725 6.0 20.01619(471) 23.969(31)
aLess than ten 100-MCS blocks used to calculate ensemble aages, due to frequent violation of the constraint.
sen-e--
C. Results
Simulations at two subcritical temperatures~T*50.7 and0.9! were performed at various densities forb51.0, 1.5, 2.0within the void-constrained ensemble and forb54.0, 5.0,6.0 within the inherent-structure void-constrained ensemValues of the pressureP* and configurational energy peparticleU* /N were determined at each density and seveof the constraintb. Simulations atT*50.9 were performedfor densities well within the metastable region and for desities just below that of the liquid at the triple point. Simlations atT*50.7 were performed within the metastable rgion and for some densities inside the unstable region.
At each temperature studied, the pressure withininherent-structure void-constrained ensemble is extremsensitive to the severity of the constraintb ~see Tables II andIII !. For most densities simulated, pressures obtained fob55.0 andb56.0 are statistically indistinguishable. Howevewhenb54.0 the pressure is drastically reduced, the chabeing dependent upon the proximity to the superheateduid spinodal. For example, atT*50.9 andr*50.55, a statepoint within the predicted metastable region, the artificiahigh tension produced forb54.0 is almost two orders omagnitude larger than the pressures obtained forb55.0 andb56.0. Similar trends, though not as pronounced, are sfor the configurational energy per particle. This extreme ssitivity of the thermophysical properties of the stable ametastable liquid is quite remarkable, given the fact thatare preventing infrequently visited configurations with larvoids in the inherent structures. Thus, we see that large vin inherent structures are extremely important in determinthe correct equilibrium properties of the liquid state.
e
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yst
er-
TABLE III. Pressure and configurational energy per particlethe shifted-force Lennard-Jones liquid within the inherent-structvoid-constrained ensemble for various values ofb at T*50.9.From Eq. ~17!, the density of the liquid at coexistence isr*50.545; the density of the liquid at the spinodal isr*50.427. Thenumbers in parentheses indicate the error in the last two or tdecimal places.
r* b P* U* /N
0.55 4.0 215.19(39) 27.968(47)0.55 5.0 0.06622~5091! 22.934(44)0.55 6.0 0.06555~4939! 22.931(46)
0.60 4.0 24.825(458) 24.465(127)0.60 5.0 0.1483~516! 23.181(42)0.60 6.0 0.1505~531! 23.179(44)
0.65 4.0 0.1114~792! 23.426(59)0.65 5.0 0.2951~555! 23.431(45)0.65 6.0 0.3038~514! 23.433(43)
0.70 4.0 0.4349~568! 23.750(55)0.70 5.0 0.5530~514! 23.688(44)0.70 6.0 0.5547~548! 23.694(58)
0.725 4.0 0.7121~542! 23.831(47)0.725 5.0 0.7498~544! 23.820(46)0.725 6.0 0.7403~567! 23.821(54)
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55 5529CONSTRAINTS, METASTABILITY, AND INHERENT . . .
As found for the Lennard-Jones liquid in a previous stud@14#, the equation of state of the superheated shifted-forLennard-Jones fluid within the void-constrained ensembwas insensitive to values ofb.1.5, but yielded artificialtensions whenb,1.5. Figure 3 compares the equation ostate of the shifted-force liquid forb52.0 ~prevention ofvoids in the instantaneous configurations! with the equationof state forb56.0 ~prevention of voids in the correspondinginherent structures!. We see that the two sets of simulationswithin error bars, yield similar results. Though pressures fb55.0 are not shown in Figure 3, we conclude, sinceb55.0 andb56.0 produce statistically indistinguishable results ~see Tables II and III!, that in order to obtain the trueequation of state of the superheated liquid, voids of at leafive interparticle separations (b.5.0) must be allowed toform in the inherent structure. In addition, we obtain consitency between two different kinds of constraints: preventioof voids within the liquid and within the corresponding inherent structures serve to suppress cavitation and allowsuperheated liquid to reach equilibrium.
In @14#, the distribution of voids within the instantaneousunquenched superheated liquid provided a way in which ocould objectively determine whether a system was overcostrained~i.e., yielded artificially high tensions!. As long asthe void distribution has a well-developed large-void ta~i.e., the liquid is able to sample those configurations thhave voids with diameters greater than the average interpticle distance!, the pressure is independent of the constrainConstraints that result in abruptly interrupted void distributions, in which the second inflection point associated withe tail of the distribution is absent, lead to artificially hightensions. The appearance or disappearance of the seconflection point of the void-size distribution provides a cleageometric criterion for determining whether a constraintunphysical. Unfortunately, there is no correspondingsimple criterion for simulations in the inherent-structur
FIG. 3. Comparison of the equation of state of shifted-forcLennard-Jones fluid in both the void-constrained and inherestructure void-constrained ensembles. The solid lines~filled circles!are the equation of state within the void-constrained ensembleb52.0. The dashed lines~filled squares! are the equation of statewithin the inherent-structure void-constrained ensemble forb56.0.
yele
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he
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void-constrained ensemble. Simulations forb54.0 produceartificially low pressures, whileb55.0 andb56.0 yield sta-tistically indistinguishable results. Yet, simulations withb54.0 do not prevent the formation of a well-developlarge-void tail~see Fig. 2!. Except for the largest void sizesampled, the void-size distributions for all choices ofb areequivalent. Thus, there does not appear to be a simplemetric criterion for determininga priori whether the super-heated liquid within the inherent-structure void-constrainensemble is overconstrained.
IV. STATISTICAL GEOMETRY OF INHERENTSTRUCTURES
The distribution of void sizes within inherent structuresquite different from that of the instantaneous, unquenchliquid ~see Fig. 2!. Inherent structures contain large voidthat never form within the unquenched liquid. The narrodouble-peaked profile at small void sizes, along with tbroad shoulder spanning moderate void diameters, is clenot seen in the unquenched system. Yet, the samplinglocal potential energy minima containing large voids~in ex-cess of five average interparticle separations! is important indetermining the equilibrium properties of both stable asuperheated liquids. We are therefore interested in studthe statistical geometry of voids within inherent structuresboth stable and metastable liquids. Consequently, we addseveral questions: Is the void-size distribution a functionthe temperature~particularly as one moves, at constant desity, from a point within the metastable region to one lyingthe stable region of the phase diagram!, in addition to den-sity? Do the large voids disappear as the range of the inmolecular potential is increased? Are the large voids witinherent structures isolated, or are they connected? In intigating these questions, we analyze the importance of lopotential energy minima in determining the equilibriuproperties of both stable and metastable liquids for densup to the density of the liquid at the triple point.
Inherent structures~which are equivalent to infinitelyrapid quenches of the system toT50! in Lennard-Jones-likesystems have been found to be substantially invariant wrespect to the starting temperature@10#. To confirm this, wecalculated the distribution of void sizes for several stpoints along an isochore, increasing the temperature frominitial point within the metastable region towards a finpoint well above the critical temperature of the liquid. Figu4 shows the void-size distribution of inherent structuresr*50.725~a density just below that of the liquid at the trippoint! for several values of the temperature, ranging frojust above the triple-point temperature to approximattwice the critical temperature. We clearly see that the disbution of void sizes is independent of temperature. Nevertless, what is remarkable about Fig. 4 is that forT*50.7 thesystem is metastable and atT*50.9 and 1.84 the system iin the one-phase region. Therefore, we find that a stableuid, slightly below its triple-point density, has inherent strutures that contain extremely large voids. The appearanclarge cavities in the inherent structure of a highly metastaliquid ~high tension! might be expected, since a stretchliquid relieves tension by forming cavities~i.e., the meta-stable liquid phase-separates via cavitation!. Yet, it is sur-
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5530 55CORTI, DEBENEDETTI, SASTRY, AND STILLINGER
prising to find that a thermodynamically stable liquid, whequenched, contains the same large cavities as a metastsystem. In other words, a stable liquid samples potentiaenergy basins that contain voids in excess of several molelar diameters.
The void-size distributions of inherent structures in Fig.provide no information on the spatial distribution of cavitiesHence, we are interested in understanding how the larvoids are arranged: are they isolated or connected, possispanning the length of the simulation cell? Figure 5 shows
FIG. 4. Void-size distribution of the inherent structures of thshifted-force Lennard-Jones liquid for three temperatures at a desity of r*50.725. The solid line is the distribution atT*50.7; theopen circles are forT*50.9; the open squares are forT*51.84.
FIG. 5. Snapshot of a configuration ofN5256 particles inter-acting via the shifted-force Lennard-Jones intermolecular potentquenched to the potential energy minimum forT*50.9 andr*50.725. For clarity, particles adjacent to a void exceeding 1.0s areshaded darker; the remaining particles are lighter. The solid linindicate the edges of the simulation cell.
blel-u-
.elyaconfiguration of particles quenched to a potential-eneminimum forN5256 andr*50.725. In the figure, particlesthat are adjacent to a void exceeding 1.0s in diameter arecolored dark grey. We clearly see that forN5256, takinginto account the periodic boundary conditions, there issingle nonconvex cavity that spans the entire simulation cThe dark particles form an interface, dividing the system ina compact region~dark and light particles! and a void region~empty space!. Due to the appearance of the void, the copact region has a higher density than the original systemis solely responsible for the narrow double peak found invoid-size distributions~Fig. 4!.
The compact region is a dense amorphous arrangemeparticles exhibiting a narrow bimodal distribution of vosizes. The origin of this double-peaked structure canfound in the distribution of voids of the face-centered-cub~fcc! crystal, since the shifted-force Lennard-Jones liqucrystallizes into an fcc arrangement@25#. In other words, theinherent structure located at the global minimum of tpotential-energy hypersurface is a fcc crystal. Figure 6 copares the void-size distribution of the fcc crystal at a densof r*51.0577 with the void-size distribution of an inherestructure atr*50.725. The shifted-force Lennard-Jones fcrystal exhibits its lowest potential energy per particler*51.0577. The void-size distribution of the fcc crystal ehibits only two spikes at void sizes ofd/s50.35 and 0.55.These spikes correspond to the presence of tetrahedraoctahedral voids, respectively@29#, in the fcc crystal. A tet-rahedral void is formed when four particles are arrangedthe corners of a tetrahedron. An octahedral void is s
n-
al
s
FIG. 6. Void-size distribution of the inherent structure atr*50.725 and of the face-centered-cubic crystal atr*51.0577 forthe shifted-force Lennard-Jones potential. The two spikes atd/s50.35 andd/s50.55 constitute the void-size distribution of thface-centered-cubic crystal. They are scaled by a factor of 4.
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55 5531CONSTRAINTS, METASTABILITY, AND INHERENT . . .
rounded by six spheres, located at the corners of an octdron. Though the first and second peaks of the distributiothe liquid’s inherent structure are not aligned exactly wthe fcc crystal’s void-size distribution~indicating some pack-ing disorder, and that the density of the compact regiondifferent from the chosen density of the fcc crystal!, it isclear that the inherent structure’s bimodal distribution islated to the voids in the fcc crystal. In fact, the doubpeaked distribution is caused by the formation of distortetrahedra and octahedra. Using the algorithm of Sastryet al.@30# to calculate the volume of the void regions in the inhent structures for a bulk density ofr*50.725, we find thatthe density of the compact region of inherent structuresr*50.83860.034. Although it is still less dense than thcrystal at the global minimum, the compact region exhibparticle arrangements similar to those found in the fcc crtal. Of course, the tetrahedra and octahedra comprisingcompact region show continuously varying degrees of distion. This is an expected result of dense amorphous packwhere short-range order is present but long-range ordeabsent.
It is interesting to note that the characteristic double pfound in the void-size distributions of inherent structures walso seen by Finney and Wallace@31# in a study of denserandom packings of soft spheres. Finney and Wallace alyzed the size of voids within a dense arrangement of hspheres near random close packing. The distribution of vsizes for this system was unimodal. However, as thisrangement was allowed to relax under a soft potentiadistinct bimodality developed at small void sizes. Thdouble-peaked structure is very similar to ours, in whichfirst peak is larger than the second, indicating that the copact region is forming an amorphous structure.
We have seen that a single large void, spanning the silation cell, is found forN5256. It is not known whethersystem-spanning voids persist for largerN values. To ad-dress this question, we performed two separate quenchessystem composed ofN51372 particles, equilibrated atT*50.9 andr*50.725. The resulting void-size distributionshown in Fig. 7. Also included, for comparison, is the distbution forN5256 at the same density. The void-size distbutions both display the same characteristic form: a nardouble peak at small void sizes, a broad shoulder at interdiate void diameters, and a slowly decaying large-void tIn fact, the void-size distribution is approximately systesize independent; minor deviations between the two distrtions are found only at small void sizes. The inherent strture of a single quench forN51372 is shown in Fig. 8. Asbefore, particles adjacent to a void greater than 1.0s areshaded dark grey and the remaining particles are light gWe again see, accounting for periodic boundary conditiothat the cavities are indeed connected, forming a single chnel that percolates throughout the entire simulation cell.definition, the dark particles again separate the inhestructure into two distinct regions: a compact region~lightand dark! and a void, separated by an interface~dark!. Thecompact and interfacial regions are solely responsible fordouble peak of the void-size distribution. The dark particcompletely enclose the void region, which gives rise tobroad shoulder and slowly decaying large-void tail.
Figures 5 and 8 have interesting implications. The pr
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ence of large voids in the inherent structure means that, froa strictly energetic viewpoint, a thermodynamically stabliquid, even at its triple point, is unstable with respect to thappearance of large voids~‘‘boiling’’ !. Thermal motion pre-vents the stable fluid from phase-separating. Neverthelesslow enough temperatures, inherent structures should begindominate the properties of the liquid, becoming ultimateresponsible for the mechanical weakness of liquids. The p
FIG. 7. Void-size distribution of the inherent structure of thshifted-force Lennard-Jones fluid for bothN5256 andN51372 atr*50.725. Solid line is the distribution forN5256; open circlesdescribe the distribution, obtained from two separate quenches,N51372.
FIG. 8. Snapshot of a configuration ofN51372 particles inter-acting via the shifted-force Lennard-Jones intermolecular potenquenched to the potential energy minimum forT*50.9 andr*50.725. For clarity, particles adjacent to a void exceeding 1.0s areshaded darker; the remaining particles are lighter. The solid linindicate the edges of the simulation cell.
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5532 55CORTI, DEBENEDETTI, SASTRY, AND STILLINGER
sibility that ‘‘weak spots’’~small voids! in the instantaneousunquenched liquid are related to the large cavities seeninherent structures may, if confirmed, shed light on the roof the potential energy hypersurface in determining the tesile strength of liquids at low enough temperatures.
Figure 4 shows that inherent structures are independentemperature. We now investigate the effect of density on tvoid-size distribution of inherent structures. Figure 9 diplays void-size distributions for various densities, ranginfrom approximately twice the critical density to just belowthe triple-point density. The distribution of voids is sensitivto the system density. The large-void tail decays more slowas the density is decreased~i.e., larger voids appear at lowerdensities!. The distribution of voids within the compact region is also affected by the density. The location and heigof the first peak remains invariant to changes in density, bthe second peak clearly decreases with a decrease in denSince the first peak’s location does not change, the decrein the height of the second peak upon decreasing the bdensity indicates a corresponding decrease in the numbeoctahedral voids in the compact region. In contrast, the nuber of tetrahedral voids is independent of density. The fewappearances of octahedral voids, along with a decrease indecay rate of the large-void tail, suggests that the densitythe compact region is increasing as the system densitydecreased.
The above discussion pertains to a system with a potenrange ofr c52.5s. In principle, the range of the intermolecular potentialr c should affect the void-size distribution. Toinvestigate this effect, we performed several simulationswhich the value ofr c was increased from 2.5s to one-halfthe simulation box length. During each simulation severquenches were performed. Figure 10 displays the resultdistribution of void sizes in the inherent structures. As witchanges in density, the height of the first peak is independof the range of the potential but the second peak increawith r c . In addition, the large-void tail decays more slowlas r c is increased; a longer-ranged intermolecular potentallows for the formation of larger cavities. The effect of thrange of the intermolecular potential on the structure of loc
FIG. 9. Inherent-structure void-size distribution of the shiftedforce Lennard-Jones liquid for various values of the density.
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potential energy minima may offer insight into the effect oattractive forces on the properties of liquids and amorphomaterials. The attractive potential is responsible for the tesile strength of liquids, providing the cohesion necessaryresist stretching~negative pressures cannot be attained influid whose constituent molecules interact via a purely repusive potential!. At the same time, however, the attractive parof the potential is solely responsible for the appearancelarge cavities in inherent structures, possibly an importafactor in the low-temperature fracture of materials. To gaifurther insight into this interesting concept, it is important tostudy the effect of the well depth and the curvature of thintermolecular potential on the void-size distribution in boththe instantaneous and quenched configurations of model luids.
V. CONCLUSIONS
The inherent structure formalism is a useful way of describing molecular dynamics in liquids. The properties of thliquid phase are determined by the sampling of various locpotential energy minima~mechanically stable particle pack-ings! and anharmonic thermal vibrations about these minimInherent structures also provide a means by which unwantconfigurations can be removed from the partition functio~imposition of a constraint!. Here, we have studied one suchconstraint: the inherent-structure void-constrained ensembIn this constrained ensemble, limits are placed on the mamum size of voids allowed to form in the inherent structureof the superheated liquid. The equation of state of the staband superheated shifted-force Lennard-Jones liquid is etremely sensitive to the severity of the constraint. Howeveas long as the corresponding inherent structures contavoids in excess of five average interparticle separations, tresulting equation of state is consistent with simulation results in the void-constrained ensemble, in which voidgreater than 1.5 times the average interparticle separationnot allowed to form within the instantaneous unquencheliquid. Therefore, the sampling of potential energy basin
- FIG. 10. Inherent-structure void-size distribution of the shiftedforce Lennard-Jones potential atr*50.55 for various values of thecutoff r c . L is the length of the simulation box.
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55 5533CONSTRAINTS, METASTABILITY, AND INHERENT . . .
that have inherent structures containing voids greater thaaverage interparticle separations is important in determinthe equilibrium properties of liquids near the triple point.
The distribution of voids within inherent structures wanalyzed to determine whether there is an objective criteto determine when the system is overconstrained. Forvoid-constrained ensemble, where voids are prevented fforming within the unquenched liquid, the pressure is indpendent of the constraint if the void distribution has a wedeveloped large-void tail. The void-size distribution mustclude a second inflection point, sampling the infrequenvisited yet important large cavities, in order for the liquidbe ‘‘naturally’’ constrained. Within the inherent-structuvoid-constrained ensemble, there is no corresponding gmetric criterion to determine if the liquid is overconstraineFor all severities of constraint studied, the slowly decaylarge-void tail is always present; the second inflection poa prerequisite for the development of the large-void tail,curs at void sizes well below that of the maximum-allowvoid diameter~even for the most severely constrained stem!.
Even at the triple-point density, the presence of lavoids ~in excess of five average interparticle separatio!within the liquid’s inherent structures is necessary forproperties of the liquid to attain their true equilibrium valueWe therefore analyzed the statistical geometry of voidsinherent structures for stable and superheated liquids atsities between the critical and triple points. The distributiof voids within inherent structures of both the stable ametastable liquids are identical; a thermodynamically staliquid, even at its triple-point density, samples locpotential-energy minima that contain very large cavities. Wfound that these large cavities are connected, spanninglength of the simulation cell. Hence, strictly from an enegetic viewpoint, even a thermodynamically stable liquidunstable with respect to boiling. This result has potensignificance in the study of the fracture of liquids and amphous solids at low temperatures. In order to investigateconnection, a useful starting point is to study the temperaand tension-dependent correspondence between inhstructure cavities and voids in the unquenched fluid.
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The inherent structures of liquids above the triple-potemperature and below their triple-point density are serated into two distinct regions: a compact region, containall the particles at a density higher than the mean sysdensity, and the void region~a single, particle-free cavitypercolating throughout the system!. The compact region isresponsible for the bimodal distribution of voids at smvoid sizes, caused by the formation of distorted tetrahe~first peak! and octahedra~second peak!, found undistortedin the fcc crystal. The void region accounts for the broshoulder in the distribution, at intermediate void sizes, athe slowly decaying large-void tail. The void-size distribtion is a function of the system density and the range ofintermolecular potential. The first peak of the distributioninvariant with changes in the density and the range ofpotential; the second peak and large-void tail are sensitivchanges in these variables.
The present work offers insight into the dual role of atractive forces in determining the properties of liquids aamorphous materials. Attractive forces are responsiblethe cohesive strength of liquids. Yet, paradoxically, attrations are also~and solely! responsible for the existence olarge voids in configurations corresponding to potentienergy minima. Clearly, mapping the range of temperatand tensions where attraction is stabilizing or destabilizingimportant.
More generally, this study points to the importanceunderstanding and investigating the intimate relationtween metastability and constraints. Simulations arenatural method for studying the effects of constraints micscopically and hence should become a basic tool in thedamental investigation of metastability. We believe thmuch can be learned about the liquid state of matter unboth stable and metastable conditions from this type ofvestigation.
ACKNOWLEDGMENT
P.G.D. gratefully acknowledges the support of the UDepartment of Energy, Division of Chemical Sciences, Dvision of Basic Energy Sciences~Grant No. DE-FG02-87ER13714!.
l
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