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PHYSICAL REVIEW A VOLUME 25, NUMBER 2 FEBRUARY 1982
Hidden structure in liquids
Frank H. Stillinger and Thomas A. WeberBell Laboratories, Murray Hill, New Jersey 07974
(Received 24 August 1981)
The canonical partition function for classical many-body systems is transformed so that
the temperature-independent packing statistics and the thermal excitations are uniquely
separated. This requires classification of particle configurations according to multidimen-
sional potential-energy minima that can be reached by steepest-descent paths("quenches"). Such classifications have been constructed for several starting configura-
tions in the solid, fluid, and coexistence phases of the two-dimensional Gaussian core
model. These quenches reveal a remarkable degree of polycrystalline order hidden within
the fluid phase by "vibrational" distortion, and that order appears to have a large correla-
tion length. The results suggest that melting hinges upon defect softening in the
quenched packings, and a crude "theory" of melting for the Gaussian core model is
developed in the Appendix.
I. BACKGROUND
The melting of crystalline solids causes configu-rational regularity at the molecular level to be re-
placed by substantial disorder. This change is vi-
vidly revealed by x-ray- or neutron-diffraction ex-
periments wherein Bragg reflections characteristicof the crystal disappear. No less vivid is the hugeincrease in the self-diffusion constant upon melt-
ing, which depends upon the diversity of packingstructures available to molecules in the liquid
(compared to the crystal} and the ease of transitionbetween those structures.
The present paper follows the long tradition ofattempting to bring conceptual order to the disor-
dered liquid state. The key feature introduced in
this approach is a configurational mapping where-
by arbitrary sets of molecular positions are re-
ferred, essentially uniquely, to relative minima on
the potential-energy hypersurface for the many-
body system. ' This mapping is generated by a
quenching operation that follows steepest-descent
paths on the hypersurface; details appear in Sec. II.The configurational mapping separates the
statistical-mechanical description of the many-body
system fundamentally into two distinct parts. Inloose terminology these are the mechanically stable
packing part (all particles free of net force and
torque), and the vibrational part. Temperature ap-pears explicitly only in the latter. Section IIIshows how this separation causes the canonicalpartition function to transform formally from a
multidimensional integral over all configurationalcoordinates to a one-dimensional integral over po-tential energy. This new partition function repre-sentation is convenient in some respects for under-
standing the melting transition itself, as later re-
marks indicate.We have carried out portions of this general pro-
gram by specific numerical calculations on theGaussian core model ' in two dimensions. Re-sults are displayed and discussed in Sec. IV. Byundertaking the quenching operation from several
initial thermodynamic states we seem to have es-
tablished that a remarkable polycrystalline orderlies hidden in the fluid phase, heavily obscuredfrom view in the thermodynamic state by the dis-
tracting influence of vibrations. Not surprisingly,the quenching out of these vibrations causes thepair-correlation functions to sharpen and to re-
structure in a manner consistent with that polycry-stalline order. Barring the possibility of "hexatic"phases ' as the initial starting point, we believethat the hidden polycrystalline order is not peculiarto the Gaussian core model but would be revealedfor any two-dimensional system with central pairpotentials.
The idea of resolving observable order in liquidsinto a vibrational part and an inherent structuralpart is not new. It certainly underlies the study ofpair-correlation functions evaluated from time-averaged particle positions in computer simulationsusing molecular dynamics. ' It also underlies theexperimenta1 strategy that examines low-
25 978 1982 The American Physical Society
25 HIDDEN STRUCTURE IN LIQUIDS 979
temperature amorphous ice for clues to hydrogenbond order in liquid water. From an unremitting-
ly theoretical viewpoint the present quenching pro-cedure seems to be the most straightforward op-tion, and it is free of arbitrariness such as coordi-nate averaging times.
The specific results obtained for the Gaussiancore model suggest a simple view of two-dirnen-
sional melting that is explored in the Appendix.While the calculation offered there is admittedlycrude, it suggests (we believe correctly) that thefirst-order melting process depends fundamentallyon softening of the regular crystal by disorder.
II. CONFIGURATIONAL MAPPING
We consider a set of N rnolecules confined to afixed containment region V. For present purposesit is irrelevant whether containment results fromspecific wall forces or by imposition of periodicboundary conditions. For each molecule 1 &j &N,the vector rj will give the configurational coordi-nates; if j contains v nuclei and D is the space di-
mension rz will possess vD components.Let 4(r &, . . . ,rz) denote the system potential-
energy function. It will, in general, include in-
tramolecular bond potentials, intermolecular in-
teractions, and wall potentials (if any). We will
suppose that 4 is bounded and differentiable forall configurations r:—r &, . . . ,rz not involvingcoincidence of nuclei.
Within the purview of classical mechanics thetime evolution of the system (in mass-reduced timeunits) normally follows the Newtonian equations
r= —V4. (2.l)
We shall be concerned instead with the correspond-ing first order evolution e-quations
r= —V4, (2.2)
which describe steepest-descent paths on the mul-
tidirnensional 4 hypersurface. Excepting certainunimportant cases with zero measure, Eq. (2.2)connects any starting point r in the configurationspace uniquely to a local minimum of 4. By thismeans we generate a mapping M(r ) from the con-tinuum r onto the discrete set of minima whichcan be indexed by a:
constant. It is appropriate to view the correspond-
ing dissipative motion as a quench since bothkinetic and potential energy are being removed un-
til the system comes to rest. In the large dampinglimit [i.e., Eq. (2.2)] the quench is so effective thatthe system is never able to surmount a potential-
energy barrier. Instead, the system is forever
trapped in the neighborhood of the same relativeminimum a where it found itself at the beginning
of the quench. Since no annealing (barrier hop-
ping) is possible during the course of steepest des-
cents, the mapping M can informally be regarded
as the operation of an infinitely rapid quench toabsolute zero temperature.
Let R (a) denote the set of system configurationsr which quench to local minimum a; that is, the
region in the vDN-dimensional configuration spacewhich satisfies mapping (2.3) for given a. Obvi-
ously, each R (a) is connected, for any two points
in its interior are each connected to the minimum
a and, therefore, to one another by a path within
R (a). We cannot say under general circumstancesthat the R (a) are convex. Indeed, it may happenthat they are multiply connected, though that canoccur only if a saddle point exists within the interi-
or of R (a) at which the quench paths bifurcate.Boundaries separating distinct quench regions
are (vDN-I)-dimensional hypersurfaces. Their un-
ion constitutes the zero measure set of system con-
figurations for which mapping M is undefined.
Many pairs of mechanically stable particle pack-ings will be identical except for particle permuta-tion. For conceptual simplicity it makes sense togroup packings and their associated quench regionsinto equivalence classes. We can write the number
of members in any such equivalence class as N!/cr.The symmetry number o. will be unity for all pack-ings that are rigidly confined by wall forces. Bycontrast, periodic boundary conditions permit freetranslation of packings that for a periodic crystal-line array allow any one particle to be moved tothe position of any other without changing 4, and
for this case o.=N. Yet other circumstances canbe devised for which 0. would lie between these ex-treme values 1 and N.
III. TRANSFORMED PARTITIONFUNCTION
M(r)=a . (2.3)
The steepest-descent paths are limiting solutionsfor the Newtonian equations (2.1) augmented by afirst-order damping term with very large damping (3.1)
The canonical partition function for N structure-less particles at inverse temperature P= I lkii T is
Ziv=(A. N!) ' f exp[ —P@(r)]dr .
980 FRANK H. STILLINGER AND THOMAS A. WEBER 25
Here, A, is the mean thermal de Broglie wavelength.We shall now transform Zz from this conventionalrepresentation to a less familiar but conceptuallyuseful alternative, based on the considerations ofSec. II.
First, we break the configurational integral inEq. (3.1) into separate contributions from each ofthe quench regions R (a):
Z~ ——(A, N!) 'g f exp[ —P@(r)]dr .
@(r}=4+6 4(r}, (3.4)
where 4~ is the value of the potential-energy func-tion at local minimum a, and the non-negativequantity 6 4 measures potential energy from thatminimum. Consequently,
Z~ A—g—',[cr( a)] 'exp( —P4 )
X f exp[ —Pb, 4(r)]dr .
(3.5)
Notice that the evaluation of Z~ here has beenseparated into two parts, namely, the identificationof all distinct (temperature-independent) packings,and the thermal excitation of each of those pack-ings within their own regions R (a).
The most effective way to describe the large sys-tem limit utilizes the potential energy per particle
P as the basic variable. Insofar as the mechanical-ly stable packings are concerned the system willhave its potential per particle confined between fin-ite limits:
(3.6)
The lower limit corresponds to the ordered crystalthat has absolute stability at zero absolute tempera-ture. At the other extreme there exists presumablysome "worst" packing that puts particles in amechanically stable configuration with P„as thepotential per particle. The vast majority of pack-
(3.2)As remarked earlier, it is convenient to treat to-gether members of R (a}equivalence classes. If weselect one quench region from each equivalenceclass to sum over (indicated by a primed summa-tion) then Eq. (3.2) is trivially modified to the fol-lowing:
Z~=A, g'[o(a}] ' f exp[ P4(r)]d—r .a
(3.3)
Within any region R (a) we can write
ings lie between these extremes with some distribu-tion of ((l values which we will denote by G(P), adensity of packing states along the P axis. Itshould be kept in mind that this density of statesenumerates only distinct packings, i.e., packingequivalence classes. Obviously, we have the fol-lowing formal expression:
G((('i}=g 5(!{i—@ /N} ja(a) . (3.7)
lim g(P) = —~,4'p
lim g(t())= —~ .(3.9)
The collection of thermal excitation integrals ap-pearing in the Zz expression (3.5) can be treated inthe same asymptotic order. We can group togeth-er, and average, results for packings whose minimahave energies lying within a narrow range aboutany preselected value NP. The corresponding
In the event that the total potential energy 4(r )
is composed of short-range particle interactions itis reasonable to suppose that conversions from onepacking to another could be effected by sequencesof localized particle rearrangements. If a verylarge system were imagined subdivided into cells,each large compared to the particle neighbor spac-ing, then rearrangements within each cell couldlargely be carried out independently of those in theother cells. The overall number of packings couldthen be reckoned as multiplicative over the cells, atleast in leading order. One thus concludes that forlarge N the total number of distinguishable pack-ings (equivalence classes} should be exponential inN. A more accurate version of this argumentwould account for interactions between neighbor-ing cells, but since the effect of such interactionsloses relative importance as cell size increases, itseems difficult to avoid the same conclusion thatthe total number of distinguishable packings risesexponentially with increasing N at fixed density.
We take this last result to indicate that thedensity-of-states function G (P) likewise is essen-tially exponential in N. First, understanding thatsome suitable coarse graining is applied to smoothout the Dirac 5 functions appearing in Eq. (3.7),we write for large N,
InG(((l) -Ng(P) . (3.8)
It seems reasonable to suppose that g (P) is at leastcontinuous within the interior of the interval (3.6),and that rarity of possible packings near the end-
points of that interval requires
25 HIDDEN STRUCTURE IN LIQUIDS 981
free-energy per particle f is defined by
f(P,P) = —lim (NP) 'lnN~oo
X(f exp[ —pb 4(r)]dr) .
It suffices to employ a maximum-integrandevaluation in Eq. (3.11) in order to obtain thesystem's free-energy correct to order N. In otherwords, if P is the solution to
g(P) PP Pf—(P,P—) =max
for given density and temperature, then
InZiv -N[g(P ) I3$ Pf(—P,P )]—.
(3.12)
(3.13)
It is worth stressing at this stage that P is theaverage potential per particle that would be ob-tained by quenching a collection of system configu-rations randomly selected from the thermodynamicequilibrium state at P and the fixed density. Es-tablishing how quenching causes the thermo-dynamic average potential per particle P(P) to maponto the corresponding P for the starting tem-
perature is one of the obvious goals of this generalapproach. The nature of this latter mappingthrough the melting transition clearly has consider-able importance.
(3.10)
Just as will be the case for the coarse graining usedto define g above, the width of the P integral overwhich the average in this last equation is to be car-ried out can go to zero as N diverges.
In view of these asymptotic considerations thepartition function expression (3.5) may be rewritten
ZN &-' f, ' exp[ &[K((() W' Pf(P 4)1 ]d04'p
(3.11)
sional ' '" and two-dimensional versions. Weshall rely on the latter as a starting point for test-ing the quenching procedure advocated in thepreceding sections. Specifically, N =780 particlesare involved in our two-dimensional calculations,subject to periodic boundary conditions. The fun-damental cell containing the 780 particles is rec-tangular, with a side ratio of 15i/3/26=0.99926. . . so that an unstrained triangular ar-
ray with 30 rows of 26 particles will just fit. Thereduced density throughout this work has the fixedvalue
2—1/2 (4.2)
The single most important attribute of the D =2Gaussian core model is that it exhibits a first-ordermelting transition. Under the fixed density condi-tion (4.2) we have previously identified the coex-istence region as (T'= I/P)
6.6X10 '&T'&7.2X10 '. (4.3)
0 500—
0 499 & CRYSTAL
o HEATING
o COOLING
0.498—
0.497—
0.496—
In the course of our simulation studies we havefound that hysteresis effects (superheated crystal,supercooled fluid) could be generated by relativelyrapid temperature changes through region (4.3).However, it is within the capacity of available
computing power to avoid such effects with suffi-
ciently slow temperature variation. Figure 1 showssome pressure results through the transition regioncalculated during one cooling and reheating se-
IV. TWO-DIMENSIONAL GAUSSIAN
CORE SYSTEM
The Gaussian core model is defined by thepotential-energy function
4(r )=+exp( r,j ) . — (4.1)
Using the method of molecular dynamics the clas-sical statistical mechanics of this system has beennumerically investigated in both three-dimen-
0.495— 0FLUID
0494 i I i I i I & I i I i I & I
0 2 4 6 8 10 12 14 16
10' T"
FIG. 1. Pressure vs temperature for the two-dimensional Gaussian core model. Data only from asingle cooling-heating sequence are shown explicitly.Dashed curves show previous results inferred for thetwo pure phases. The reduced density is 3 ' . Pres-sure drop upon melting is characteristic of the Gaussiancore model in both two and three dimensions if the den-sity is sufficiently high.
982 FRANK H. STILLINGER AND THOMAS A. WEBER 25
ht* =1.25 X 10 (4.4)
While it is possible to continue this integrationprocedure essentially to the quench end point, ex-perience shows this to be inefficient. Thus, afteran initial phase which exhibits most of thepotential-energy reduction we convert to an effi-cient function minimization routine involving theconjugate gradient technique. '
By monitoring the course of the quenches fromstart to finish it becomes clear that complex sys-tem reorganization can be involved. Figure 2shows for starting configuration J how the "tem-perature" [i.e., mean kinetic energy per particle as
quence. While numerical fluctuations obviouslyare present it appears that substantially the samecurve has been traced in both directions.
Twelve system configurations were selected andquenched. They are listed in Table I as A —L,where the thermodynamic state that supplied eachone is identified. Those thermodynamic states in-clude both crystalline and fluid cases, as well asthree examples (C, D, and E) chosen from thecoexistence region. We have found in the courseof the molecular-dynamics study that phase coex-istence is relatively easy to prepare and to maintainin this two-dimensional system.
Table II shows the potential energies achieved byquenching each of the 12 configurations. Briefdescriptions are supplied for the final system struc-tures, some details of which are discussed below(including definition of disorder parameter n5).
Our numerical procedure for carrying out thequenching begins by solving Eq. (2.1) directly, us-
ing a fourth-order algorithm due to Gear' and atime increment +nj 6N, —— (4.5)
where n& is the number of nearest-neighbor po-lygons with j sides. Thus, any disruption of theperfect triangular crystal which creates, say, a setof pentagons must compensate by simultaneouslycreating an appropriately weighted set of polygonshaving more than six sides.
Figure 3 shows configuration H (see Table I),which was selected at random from a moleculardynamics run for the homogeneous fluid phase justabove its melting point. Each of the 780 particlesin the fundamental cell is shown in a way whichindicates how many sides its nearest-neighbor po-lygon possesses. For simplicity, all "sixes" havebeen rendered simply as asterisks. The picture
determined by the solution to Eq. (2.2)] varies asquenching causes 4 to decline. Instead of a simplerelaxation behavior which would produce a linearcurve in the given 4 vs T* representation, a com-plicated meander emerges. This kind of behaviorunfortunately makes the numerical contruction ofquenches for the given system a rather demandingtask regardless of the algorithm used, and it hasseverely limited the number of quenches we havebeen able to construct.
The assignment of nearest-neighbor polygons toparticles is a convenient way to reveal and toanalyze structural disorder in two-dimensional sys-tems. ' All particles inhabit hexagonal polygonsin the defect-free triangular lattice that obtains atthe low-temperature limit. The presence ofnonhexagonal nearest-neighbor polygons (disclina-tions) indicates disorder. It is a topological necessi-
ty ' that for any particle configuration in the sys-tem
Name
TABLE I. Configurations selected for quenching.
p'X 10 Phase
ABCDEF6HIJKL
4.9950X 101.5579 X 106.9402X10 3
6.9584X 106.9726 X 106.9939X 10-'7.0010X10 '7.4083X10 3
8.7026 X 101.5429 X 10-'3.4700 X 10
1.7378
4.9894704.992 5814.957 3884.968 4004.964 7384.954 6834.980 1904.954 7784.952 7114.952 5224.995 280
14.876 84
323.48044323.395 12330.540 26329.866 29330.11934330.750 20329.17923331.11366332.342 44337.959 16351.577 43
590.599 30
Rotated crystal, two interstitialsAligned crystal
CoexistenceCoexistenceCoexistence
Slightly supercooled fluidSuperheated aligned crystal
FluidFluidFluid
Hot fluid
Nearly ideal gas
25 HIDDEN STRUCTURE IN LIQUIDS 983
TABLE II. Quenched configurations.
Name n5 Structure
ABCDEFGHIJKL
323.442 65323.273 43323.273 43323.392 98323.41800323.442 02323.273 43324.093 27323.509 46323.514 84323.709 75323.947 11
4002240
21449
16
Rotated crystal, two interstitialsPerfect aligned crystalPerfect aligned crystal
Two nearby dislocationsTwo separated dislocations
Rotated crystal, two interstitialsPerfect aligned crystal
Two antiparallel grain boundariesFour dislocations
Four dislocations, slightly rotated grainTwo antiparallel grain boundaries
Enclosed grain
presented is typical of the fluid in this temperaturerange. While the majority of particles have sixneighbors, substantial numbers have five or seven
and, infrequently, cases of four or eight will be en-
countered. While the spatial distribution of thesevarious coordination species is clearly inhomogene-ous at any given instant, no simple description ofthe topological texture immediately suggests itself.
Figure 4 presents the structure that developsfrom configuration H in Fig. 3 as a result ofquenching. The simplification produced is dramat-ic. The number of nonhexagonal particles dropssharply, and those that are left show a rather clearpattern of alternating five's and seven's along pathsacross the periodic unit cell. A close 5-7 pair ofdisclinations constitutes a dislocation, ' and acurvilinear sequence of oriented (heat-to-tail) dislo-
324 40—
324 30—
324 20—
324 10—e
~ 32400—Z
c 32390—I-Z+ 32380—0
323.70
32360~323.50
0
FIG. 2. Potential energy vs temperature duringquench of the fluid configuration J.
cations constitutes a grain boundary between cry-stalline domains. ' The angle of rotation betweenneighboring domains is related to the mean separa-tion between dislocations on the boundary. Evi-dently, the quenching operation has managed to re-veal a polycrystalline structure that lay hidden inthe starting configuration due to thermal disrup-tion.
In all of the quenches we have carried out the fi-nal result has contained only polygons with five,six, or seven sides. The conservation condition(4.5) thence requires equal numbers of pentagons(n5) and heptagons (n7). We have found it con-venient to classify quench packings by their n5values as a measure of overall disorder. These
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FIG. 3. Particle arrangement for configuration H.This was selected at random from the fluid just aboveits freezing point. Particles are classified according tonearest-neighbor polygons; asterisks imply hexagonswhile integers specify polygon side counts for nonhexag-onal cases.
984 FRANK H. STILLINGER AND THOMAS A. WEBER 25
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FIG 4 Quenched stmcture pr~uc~ from ~nfiguration H.
values have been entered in Table II.Figure 9 exhibits the quench packing that em-
erges from configuration L. The starting point inthis case was an extremely hot gas at about 240times the freezing temperature. At his high tem-perature, particles interpenetrate deeply upon collitt
sion, and we find for this thermodynamic statethat the pair-correlation function at zero separationhas the value
g"'(r =0)=0.65 . (4.6)
From Fig. 5 one concludes that hidden under thenearly ideal-gas configuration is once again a po-lycrystalline structure. In contrast to the anti-parallel pair of grain boundaries that ran across thesystem in the packing shown in Fig. 4, the grainboundary closes on itself to yield an enclosed crys-tallite in a surrounding matrix, rather than asystem-spanning strip.
The particle packings produced from fluid statesdo not always obviously have polycrystalline char-acter. Figure 6 shows the result obtained byquenching configuration I, wherein four disloca-tions appear. When the slightly supercooled fluidconfiguration F is quenched, the result, shown in
Fig. 7, consists of a rotated crystal that in defect-free form would contain 778 particles; the two ex-
cess particles are present as locally relaxed intersti-tials whose positions are revealed by compact 7-5-7-5 tetrads.
While the aligned perfect crystal can emergefrom the positive-temperature equilibrium crystal(B), the superheated crystal (G), and even the coex-istence state (C), this is not the only option. First-
ly, a distinguishable (but nearly isoenergetic) cry-stal rotated by 90' can spontaneously form byfreezing the fluid phase to the solid; subsequent
quenching of the solid will only damp out phononmodes in such a structurally perfect but still rotat-ed crystal. Secondly, there is a good chance (we
estimate about —,) that spontaneous freezing will
create a rnisaligned solid, the quenched version ofwhich has already been shown in Fig. 7. Whetheran aligned or misaligned crystal forms in amolecular-dynamics sequence which cools the sys-tem slowly through the transition region is arnatter of chance. But once the choice has beenmade it is very difficult for the system to convertto the other option by solid-phase annealing. In-stead, it is, practically speaking, always necessaryto remelt and refreeze.
Figure 8 shows the quenched structure stemmingfrom confguration A. The latter was spontaneous-
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FIG. 5. Quenched structure produced from the veryhot fluid configuration L.
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~ ~
~ ~~ ~
~~
t~ ~
~ t
I
~I
tI ~ ~
~~
~I
~t
~ ~ ~ ~
I~7 ~
I I
I
~ ~
~t
~~
I~ ~
t
I ~ ~
t~ ~
~ ~
~ I
I~
~
It
~ I~ ~
I~ ~
~ ~I
I~ ~
~ ~~
t~
tt
~~
I~ ~
~ II
~~
I
~ ~~ ~t~
~~ I
~ ~
~ ~t~ ~
~~
~ ~
~ ~~ ~I
~ ~
~~
~ ~~ ~
~ I
~ t
~ ~ I
I1 ~
I
~~
~~~ ~
~ ~~
~
~ ~~ ~t~ ~
I~ ~~
~~ I
FIG. 6. Structure produced by quenching configura-tion I. The separated 5 and 7 disclinations at lower-and upper-right corners are actually neighbors due tothe periodic boundary conditions.
25 HIDDEN STRUCTURE IN LIQUIDS 985
I I t tI t It I I tI t t tI tt tt I I t tI I t I It t tt tI It I
t I ~ I I I II t II tt II II 7 ~ II I 7t I I It I tt II II
t I t It t II t It It tI I
tI t tt tt II t II
I II Itt I tt I tI tI I t tt II I It I
t tI tI
I II II tI I I
tt tII tI
ItII t
t~ ~
tI
~7 5
tI I
It I tt t I I ~t III t ~ t II I tI I ~ ~ 5 7
~ I I I~ I I ~
7 5 ItI t ~ ~II I ~ ~ ~
I tI I It t III
t t
t
Itt tt I I
It I
tIt I tt I I
tI tt
FIG. 7. Rotated crystal containing two interstitialsthat is produced by quenching supercooled fluid config-uration F.
ly frozen by slow cooling from the homogeneousfluid, and was then cooled in stages to its thermo-dynamic temperature listed in Table I. The struc-ture shown in Fig. 8 has the same misalignedN =778 crystal character as thai shown earlier in
Fig. 7. However, now the two interstitials havecome to rest in a slightly different position withslightly higher energy. Evidently, there is a bandof energies corresponding to various stable arrange-ments of the interstitials, for each one of whichthere is a distinct configuration-space region R (a}.
It is obvious that hidden structure uncovered byquenching should produce sharpening of themolecular distribution function. Figure 9 showsthe pair-correlation functions g' '(r} before andafter the quench of configuration H. The lowercurve is simply the smooth function appropriate tothe equilibrium fluid at a reduced temperature T'of approximately 7.41)& 10 . In principle, theupper curve should consist of a discontinuous sum
1 I I I I I I I ' I ' I
I i I i I & I i I 1 I
1 2 3 4 5 6r
FIG. 9. Pair-correlation functions for fluid atT =7.41)& 10 and the corresponding quench (fromconfiguration H).
where
4p ——323.273 43
of closely spaced Dirac 5 functions since all 780particles are strictly at rest. However, data were
collected in bins of radial width hr* equal to 0.08,and we show only the corresponding smoothedcurve. Even so, the sharpening of peaks uponquenching is very clear. Successive peaks in thequench function correlate strongly with those thathave been calculated before for the low-tempera-ture crystal, but increasing peak width with in-creasing r' arises now from built-in strain due todislocations, not to phonon motion. The sharpnessof the quench g' 's correlates inversely with theparameter n5 listed for each case in Table II.
The quench energies are plotted against thecorresponding n5 values in Fig. 10. The smoothcurve shown represents the function
H(n5}=Cto+Ans+B (B +C—ns)'7
(4.7)
~ It
~~
7 ~ ~
55 I
7~ tt
t
75
5 ~
7 ~
tt
II tt
~ ttI It
t II tt II
It t
t tI
t tI II t
I
tI IIIt
t I Ittt tI
t
t tt t It t tt t tt I I tI
I I tt ItII It t I
324.5
324.0
323 5
3230 1 I ( I I I I I I I I I i I I I
0 4 8 12 16 20 24 28 32 36ns
FIG. 8. Quenched configuration produced from con-figuration A in Table I.
FIG. 10. Potential energy of the quenched configura-tion vs ns I The smooth curve is specified in Eq. (4.7).
986 FRANK H. STILLINGER AND THOMAS A. WEBER 25
is the potential energy of the perfect crystal, andwhere constants A, B, and C were obtained by aleast-squares fit:
A =0.0623,
B =0.2145,
C =0.0314 .
(4 9)
The rms deviation is 2.723)(10 for the pointswith n» 0. One sees from Fig. 10 that on theaverage the potential energy of the quenched pack-ings rises with n5. But the downward curvature il-lustrated with the smooth-fit function H impliesthat dislocations soften the crystal and tend tomake it easier to insert yet further dislocations. Itmay be noted that H'(0) is over twice as large asthe limiting slope of this function for large n5.
Although the small number of quenches actuallycarried out makes such inferences risky, we believe
that P increases only by about 2)& 10 " across themelting transition. This is a rather small fractionof the latent heat change per particle 2.1X10The remainder must be attributed to increasinganharmonicity.
V. DISCUSSION
Even though we have been able to investigate
only a modest number of quenched configurations,several conclusions can be drawn. Most important,perhaps, is that the correlation length in the ran-
dom packings generated (i.e., the mean diameter ofcrystalline grain regions and/or the mean separa-tion of isolated dislocations} is comparable to the
size of the system used. This causes the quenchenergies to scatter rather widely, making it diffi-cult to extract ((i accurately from our results. Inorder to avoid such scatter it may be necessary to
employ two-dimensional systems of at least 10particles and/or to perform many more quenches.
The large correlation length present in thequench structures probably reflects a generalsparseness of stable packings in two dimensions,compared to three dimensions. In the one-dimensional case there is only one packing, the reg-ular periodic array; two dimensions permits a lim-
ited diversity of packings with, at most, a ratherlow concentration of disorder elements (disclina-tions). It is reasonable to suppose that for fixed Nthe number of distinct packings continues to in-crease with dimension beyond two.
The structure presented earlier in Fig. 5 shows
that large correlation lengths arise even when onestarts the quench with the very high-temperaturefluid. It seems obvious that the correlation lengthsin all thermodynamic fluid states are bounded
from above by the corresponding quench correla-tion lengths.
The presence of the boundaries for the multidi-mensional regions R (a) makes it unlikely that asystem confined to any one such region could un-
dergo a melting transition. ' Rather, it should bethe shift from low-disorder R (a}'s to higher-
disorder R (a)'s that characterizes melting. Acorollary statement is that for fixed P the free-
energy function f(P,P) [Eq. (3.10)] will be analyticin P throughout the transition region of this vari-able. The nonanalytic character of the phase tran-sition then resides in the quantity P as a functionof P. The Appendix supports this viewpoint with
a simple (and crude) estimation of the partitionfunction that predicts the existence of the first-order melting transition.
Clearly, it is desirable to apply the quench pro-cedure to three-dimensional systems to see what
type of hidden order emerges. If our presumptionis correct that three dimensions permit many morerandom packings and that they have shorter corre-lation lengths, then useful and reproducible resultsshould be attainable with reasonable system sizes(N & 10'). For the case of monatomic substancesin three dimensions, one potentially valuable wayto classify random packings (in addition to usingnearest-neighbor polyhedra} would be to followRivier's procedure for identifying disclinationlines. '
Great interest also surrounds the study of polya-tornic substances in three dimensions. The case ofwater is particularly important in regards to thehydrogen bond topology of its quenches from thestable liquid state. ' Specifically, one would seekto identify local regions similar in structure to theknown ices' or clathrate hydrate crystals. In ad-
dition it might be useful to examine quenches ofstrongly supercooled liquid water, since emergenceof hidder order could help to explain the super-cooling anomalies that appear to exist at approxi-mately —45 C. '
We express thanks to Dr. Linda Kaufman ofBell Telephone Laboratories for suggesting the"conjugate gradient method" which leads to drasticimprovement in convergence rate for the quenchprocedure, and for providing the necessary com-
puter software.
25 HIDDEN STRUCTURE IN LIQUIDS 987
APPENDIX
NP(xg—) .
Here we have set
(A1)
We now show how the preceding observations onthe Gaussian core model can be assembled into a"theory" of the first-order melting transition. Webegin by accepting the fitting function shown inFig. 10 and Eq. (4.7) as a proper description (in thelarge system limit) for packing energies with vari-ous degrees of disorder. For present purposes it ismost convenient to rewrite 8 in a form whichmakes its extensive character explicit:
H(n5)=N[gc+Axq+b (b +—C x~)' ]
x 2
O 0
0 1 2 3 4 5 6 7 8 9 10
10 XS3
FIG. 11. Graphical determination of the integrandmaximum in Eq. (A7). Below the melting point(p& 147.65), x, sticks at the origin; above the meltingpoint (p & 147.65), x5 has discontinuously jumped to theposition of a new maximum.
x5 ——n5/N,
Pc——4c/780=0. 414453,
b =B/780=2. 750K 10
(A2)
(E —Fx, )""', (A3)The parameters A and C were specified earlier inEq. (4.9).
In estimating the number of distinct packingsthat can be formed when precisely n5 ——Nx5 pen-tagonal disclinations exist in the N-particle system,we must keep in mind that the disclinations typi-cally must be arranged in linear grain boundarieswith some average separation. In laying down agrain boundary across an initially defect-free sys-tem one can reckon that on average some numberE of choices will exist at each stage for lengthen-ing the boundary by one 5-7 disclination pair. Onthis basis E ' would be the appropriate estimatefor the total number of n5 packings. But the factis that grain boundaries can interfere with oneanother, requiring a reduction in this estimate forlarge n5 (i.e., large xq). This is borne out by thefact that our numerical procedure above fails toproduce packings with high defect concentrations(x5 approaching —,).
Consequently, we postulate a linear reductionwith x& in the factor E. The modified estimate forthe number of packings at fixed n5 is thus taken tobe
where E and F are suitable positive constants.Next, we need an expression for the vibrational
free-energy function f [P,P(x5)]. When x5 van-
ishes so the perfect crystal obtains, the free-energyis indeed attributable to harmonic phonon modes
coj' at low temperature; hence,
2N
exp[ —NPf(P, No)]= Q(P~,'") '.j=l
(A4)
[H'(0)/H'(Nx5)]'/ (A5)
If we let a stand for the fraction of modes subjectto this rescaling, then
Normal modes coj. in defective packings will differfrom the crystal coj. ', obviously. We suggest thatthe predominant effect stems from the defectsoftening of the medium conveyed by downwardcurvature of H(n5). The derivative H' provides ameasure of the rigidity of the medium, and itseems reasonable that at least some of the normalmodes (principally, those of low frequency and ofshear character) ought to be rescaled by a softeningfactor
Cx5exp{ NPf [P,P(xq)] I =—1—
A(b +C x )'
—aN2N
II (Pe '")-' .j=l j (A6)
These expressions can be used to provide an approximate expression for the partition function ZN alongthe lines indicated earlier by Eq. (3.11). In that representation P was treated as the fundamental intensivevariable, but in view of Eq. (Al) we can transform to x& as the more convenient choice. By using expres-sions (A1), (A3), and (A6) we arrive at the following expression for ZN ..
988 FRANK H. STILLINGER AND THOMAS A. WEBER 25
Zz-NA, exp{ N—P[gp+f(P, Pp)] )
CxX J dx5exp{ NP—[Ax5+b (b—+C x5)'~ ] J(E Fxs—) ' 1—
0 A(b +C x )'
' —aN
(A7}
In the large-N limit the free-energy per particle will be determined by that value of x5 which maximizes the
integrand. Equivalently, we require that xq at each P which satisfies
2
L(x5)=xs ln(E —Fx5)—aln 1—g ( b 2+ ( 2x 2
)1 /2
—P[Ax5+b (b —+C x5)' ]=max. (AS)
Three unevaluated parameters appear in the lastequation: E, F, and a. We now see that choicesfor each that are probably reasonable lead to aprediction of a first-order melting transition. Inparticular, we select
E=10, F=200, a=0. 10 . (A9)
P= 147.65,
T*=1/P=6.772 XS10(A10)
at which the integrand-maximizing value of x5jumps from 0 (low-temperature crystal) to5.918&10 (fluid at the transition point). Forcontinued temperature increase the local maximumcontinues to predominate, finally approaching thex5 value 3.065 X 10 at infinite temperature.
The first of these asserts that on the average, ten
choices exist for extending a grain boundary by a5-7 pair in a pristine medium. The second then
states that random packings become negligibly rarewhen the fractional disclination density x5 reaches0.05. The a value shown subjects only a small
fraction of the normal modes to the full softeningeffect.
Figure 11 shows L (x5) curves vs x5 for several
choices of the temperature parameter P. When
P & 147.65 the curves monotonically decline with
increasing x& from their maximum value 0 at theorigin. However, when P & 147.65, L (x q ) achieves
its greatest value at a local maximum displaced topositive x5. There is thus a discontinuity at themelting point
That the predicted melting temperature (A10)agrees well with the one determined by moleculardynamics for the Gaussian core model [Eq. (4.3)]primarily results from adroit choice of a. Howev-
er, the mean value predicted for n5 in quenchesfrom the N =780 fluid at its freezing point,
(ns) =4.612, (A 1 1}
seems roughly correct in comparison with entries
displayed in Table II.The latent heat predicted for the transition is
0.0461 with the parameter set (A9). This is toolarge by a factor of approximately 2, but doubtless
agreement could be improved by varying a, E, andF.
It is important to stress that the predicted ex-
istence of a first-order transition rests fundamen-
tally on the nonlinearity with x5 of the function H.That is, the softening phenomenon appears to be aprerequisite, and paves the way for an avalanche ofdefects to enter the system at the transition point.
Of the major deficiencies of the present simpleview of constant-density melting is that it fails toproduce a coexistence interval [see Eq. (4.3) above],as is required when the densities of the equal-
pressure phases differ. Correcting this would re-
quire accounting for correlation of density fluctua-tions with disclination concentration fluctuationsin the stable packings. It is worth pointing out inthis connection, however, that at one special densi-
ty for the Gaussian core model the molar densitiesare equal and the coexistence interval shrinks to apoint.
25 HIDDEN STRUCTURE IN LIQUIDS 989
'An outline of some of the ideas presented here has beenpreviously published: F. H. Stillinger and T. A.Weber, Kinam 3A, 159 (1981).
2F. H. Stillinger, J. Chem. Phys. 65, 3968 (1976).3F. H. Stillinger and T. A. Weber, J. Chem. Phys. 68,
3837 (1978); 70, 1074(E) (1979).4F. H. Stillinger and T. A. Weber, J. Chem. Phys. 74,
4105 (1981);74, 4020 (1981).5B. I. Halperin and D. R. Nelson, Phys. Rev. Lett. 41,
121 (1978); 41, 519(E) (1978).D. R. Nelson and B. I. Halperin, Phys. Rev. B 19,
2457 (1979).7G. Jacucci (private communication).(a) A. C. Belch, S. A. Rice, and M. G. Sceats, Chem.
Phys. Lett. 77, 455 (1981); (b) F. Hirata and P. J.Rossky, J. Chem. Phys. 74, 6867 (1981).
S. A. Rice, Top. Curr. Chem. 60, 109 (1975).F. H. Stillinger and T. A. Weber, J. Chem. Phys. 70,4879 (1979).
"F.H. Stillinger and T. A. Weber, Phys. Rev. B 22,3790 (1980).C. W. Gear, Argonne National Laboratory Report No.ANL-7126 (unpublished).
~3R. Fletcher, Practical Methods of Optimization (Wiley,
New York, 1980).' J. P. McTague, D. Frenkel, and M. P. Allen, in Order-
ing in Two Dimensions, edited by S. K. Sinha(Elsevier-North-Holland, New York, 1980), pp. 147-151.
'5W. T. Read and W. Schockley, Phys. Rev. 78, 275(1950).
' For particles with attractive forces, however, confine-ment to an R (a) many not be sufficient to inhibit avaporization transition if the overall system density issmall.N. Rivier, Philos. Mag. A40, 859 (1979).A. Rahman and F. H. Stillinger, J. Am. Chem. Soc.95, 7943 (1973).
~9N. H. Fletcher, The Chemical Physics of Ice (Cam-
bridge University Press, Cambridge, 1970), Chaps. 2and 3.D. W. Davidson, in Water, A Comprehensive Treatise,edited by F. Franks (Plenum, New York, 1973), Vol.II, pp. 115-234.C. A. Angell, in Water, A Comprehensive Treatise,
edited by F. Franks (Plenum, New York), Vol. VII (tobe published).
