Orientational environments in liquids and solids

Volume

107. number

CHEMICAL

1

ORIENTATIONAL

ENVIRQNMENTS

PHYSICS

18 M3.

LETTERS

1984

IN LIQUIDS AND SOLlDS

A.D.J. HXYMET

Received

3 February

1984

Local orientational order in liquids and solids is eumined using rhe average “distribution of cosines” of angles subtended at pxticlcs in the liquid by pairs of nearest-neighbour particles. a technique fit introdurrd by Scott and \ladrr. This average information is compared with recent, more complete theories and simulations of orientational order. The lowest-order approximation to the “distribution of cosines” is shown to fail for dense systems. but the true distribution m3y mawred wry efficiently in computer simulations.

I. Orientation

is shown tllat (i) the Scott-hlader

Liquids

and glasses in three dimensions

lack not

only the translation31 order of crystals but also the particular kind of long-r3nge orientation31 order which

accompanies translational order. Liquids do possess short-range orientational order. and recent studies [l] have esamined both the ex3ct n3ture of this order and whether or not a different kind of long-range orientations1 order (in the absense of translational order) can be induced in the supercooled liquid [2---I]. Here we discuss 3 procedure for routine esamination of local orientational order in computer simulations of simple liquids. supercooled liquids, glasses and hot crystals. The idea is to study the “distribution of cosines” of the angles subtended at a particular pxticle by p3irs of nearest-neighbour particles. This technique was developed by Scott and hlader [5J twenty years ago, and although it has been rediscovered or applied 3 few rimes. particularly in studies of supercooled liquids (6-81 * snd smorphous solids [9]. it has not been investigated systematically_ In particular, the adequacy of the natural first spproxi-

01 her useful

Note that hlountain nekhbours

and Basu [ 71 distinguish in the bee crystal.

locrtl orientational

emironmcnr

of particles

B.V

in dense

we discuss the kind of information

which could be obtained by routine nique in computer simulations.

first and

0 009-2613/84/S 03.000 Elsevier Science Publishers Worth-Holland Phvsics Publishino- Division)

113s

in cq~rilibrilm

C&lo computer simularions [ lo]. The distribution oicosines defined in section 2 bclow represents 3 certain average over the more complete loc3l orientsrional order obtained (3t vet mush prestcr effort) in the computer simulations oi Sreinhardr et al. [3] and of Young 3nd Alder [ 111. and rhe analysis of kyntet et 31. [I]. At low tempcrarure the orientational “signature” is very distinclive. and the distributions for a fxe-centered cubic (fee) ccs131. 3 bodycentered cubic (bee) ccstal and perfect local icosahsdral (ices) pxking are displayed hcrc. The results of t\vo short (2000 time steps) molecular dynamics simuktrions of the Lennard-Jones system. one of rite “hot” solid and rhe other of the liquid ne3r the rriplz point. arc rcporred in section 5 &spire the short Jura~ion. tllrsr siwplr simulations display clrarl~ the importanse of triplet correlations irl determining the phases. Finally,

second

applications

tribution ol‘cosincs can be calculatrtd by an estrenlel> ~forma~ion slresdy efficient alo,oritlm \vhicli c3lcul3ted in standard molcculxr dynsmicr and Slonte

mntion to the distribution of cosines. obtained from the superposition approximation to the pair correlation function [I], has never been investigated. Here it *

technique

systems (e.g. in the solid-liquid inlsrfxe). (ii) the distribution of cosines in dense systems is 11or well described by the lowest-order approximation, bur (iii) the true dis-

niany

use of this recli-

77

Volume 107. number 1

CHEMICAL

PHYSICS

I8 May 1984

LETTERS

2. The distribution of cosines We consider the set of nearest neighbours of each particle in turn, at each time step of a molecular dynamics simulation (or at each configuraGon of a Monte Carlo simulation). The fist task is to defitze the nearest neighbours of any given particle. One (very useful) rigorous defiiition involves the Voronoi polyhedron contruction [ I2-- 14J around each particle, but for the present purpose this definition is unnecesnrily cumbersome and time-consuming, and we adopt the following simple definition. A nearest neighbour of a given particle at position ri is any particlc with position ri such that Ifi - fil
environments

each angle adopts

one of a

small number of values. In fii. 1 is plotted the number of angles as a function of the cosine of the angle, for three perfect (“zero temperature”) local orient3tional environments. Using this signature, a local icosahedral environment is easily distinguished from any cubic environment by the value of s2(cos 0 = 0), namely the number of 90” angles subtended at particles by nearest neighbours. For computer simulations of uniform systems at non-zero temperature, the final step is to average the distribution of cosines over particles and time (or configurations). (A notable exception to this step is in the non-uniform system discussed below.) The distribution of cosines is then a valuable measure of the average local environment seen by particles in the system. As the temperature of a crystal is increased, the distribution of fig. lb broadens due to excursions of particles away from their lattice sites, in a manner 78

- 0.5 COSlIIe

00 Of

the

SuDlended

05 Angle

Fig. 1. The number of andes subtended at a particle by pairs of its nearest nekhbours as a function of the msine of the angle. for three perfect (zero temperature) local orientational environments: (a) icosahedral (iws), (b) face-centred cubic (fee) and (c) body-centred cubic (Ixc).

which is not yet known in full detail. nIlen the crystal melts the distribution of cosines adoptsa characteristic form shown below. As the liquid is supercooled, the liquid-like distribution “sharpens” and, according to the simulations of Steinhardt et al. [3], may show features consistent with local icosahedral order. If the

metastable liquid happens to crystallize, rhe distribution of cosines suddenly reverts to a crystal-like distribution. As shown in the simulations of Hsu and Rahman [63, and of Mountain [7], monitoring the onset of nucleation is a highly successful application of this technique. We discuss further applications here. The distrl%ution of cosines represents an average

over the more complete local orientational information obtained in two recent studies [3.1 J, and in the original work by Scott and Mader [S ] (who despite a limited statistical sample, were able to measure azimuthal angular distriiutions). Steinhardt et al. 131 use a similar definition of nearest neighbours in their molecular dynamics simulations but study the orientational environment by assigning a (finite) set of spherical harmonics YI, [O(rfj), ~~i~)l to each neighbour and averaging over particles, neighbours and time. The angles B(ri,.) and Qfrij) are the polar angles which the line joining the nearest neighbour to the

Volume 107. number 1

particle makes with respect to an arbitrary reference coordinate system. The unique feature of this approach is that it can go beyond local order to look for long-range orientational order [24]. Local orientational order has been studied in great detail by Haymet et al. [I] using both theory and computer simulation. The triplet correlation function gt3)(r, s. t) is studied as a functionof all threedistances. without any averaging over “nearest neighbours.” A vast amount of information is contained in a complete an3lysis of the triplet correlation function [I .I 51. Since the triplet correlation function gf3)(r, S. t) is just the (unnormalized) probability of finding three particles at the vertices of a triangle with sides of length r. s. and 4 the distribution of cosines studies here is esactly

Q(cosfI)=C;

drr/

dssr’dr 0

0 x q:"

fgt3’(r,

s. I)

r--J

-r?' - s3 +zrscose).

0)

where C is a normalization constant and we have adopted the “delta-function” form to display explicitly the symmetry of the inlegrand. The goal of ref. [ 11 is to calculate from lirst principles the corrections r(r. S. I) to the superposition approximation, where g(3)@. s, t) = g(r)g(s)g(t)T(r

9,s f) .

dynamics

The distribution molecular dynamics

conditions

zero. For the liquid were chosen to appros-

imate the empirically determined triple point of the Lennard-Jones system. namely a density p* = poj = 0.85 and an average temperature 7” = X-BT/E = 0.70. The simulation of the solid ~3s performed 31 the same density but 3~ 3 lower temperature T* = 0.50. Staninp from previously equilibrated samples. eac!l system was propagated in time for 1000 steps using 3 flt’th-

order predictor-corrector algorithm [6] with a time step Ar * = 0.005. where I* = I(E/IIIu~)~/~ and 111is the mass of each particle. For values of the parameters 1~. E and u appropriate

for argon,

this rime step

corr+

sponds to approsimately 1.082 X IO-” s. The pair correlation function ~3s measured b) standard techniques [l] and the result for the liquid is displayed in fig. _. 1 The distance R which constitutes the definition of “nearest neighbour” was taken to be 1.5Oo. close to the first minimum of,&). Using.this definition a histogram of the number of nearest neighbours ~3s constructed and is shown as the inset on fig. 2. The m*eruge number of nearest neighbours for this simulation is 12.0, which agrees. 3s ir musr. wirh

35

30

i

25r 9(‘)

/

201

simulations

of cosines has been measured in simulations of 500 classical par-

ticles confined in a cube and subject to periodic boundary conditions. The particles interacted according to the Lennard-Jones potential (3) truncated

tential well and u the finite-range the thermodynamic

(2)

The distribution of cosines provides an efficient way to summarize the magnitude of these effects as shown below. Boil the recent studies of orientational order are probably too time-consuming to perform on 3 routine basis. However. the distribution of cosines may be obtained with almost no extra calculation from a standard computer simulation.

3. Molecular

18 May 198-l

CHEMICAL PHYSICS LETTERS

at r= 2.50, where E is the depth of the po-

00’

05

J IO

4

20

15 D~stonce

25

r/o.

Fig. 1. The pair correlation function g(r) for the LewdJones liquid at density p* = 0.85 and tempemturr T* = 0.70. The dismnce R which defmes %CX~R nrighbours” is 1.5. The inset shows ZIhisto_gmm of the numb+r of naest neighhours, usins this definition 79

Volume 107. number 1 the value obtained

from

CHEMICAL PHYSICS LETTERS the expression

p_$

dr 4~2

GO6

x &). Analysis of local orientational order was performed after every tenth time step. This is important for two reasons. First, more frequent analysis does not produce better statistics because each confiiuration is of course nearly identical to the previous configuration [ 11. Secondly, the information needed to construct the distribution of cosines is just a list of the nearen neighbours of each particle, and in efficient simuiations [IO] just such a list is constructed every 10 or so steps. Equivalent information is available in simulations which use the “cell method” instead of neighbour tables. To be explicit. the angle 0 subtended at a particle with position ‘i by nearest neigbours at fj and Q is given by cosine rule

Cos 8 = (~~ + ~~ - ~~)/?-Tij~i~ .

(4)

where rv = [ri - ‘il. In ~ptima~imulation codes only the squared distances r~ and q, are available. Hence, the calculation of the distribution of cosines requires for each angle the minimal extra work of taking two square roots, and calculating the squared distance between each pair of nearest neighbours (taking the usual care with the “minimum image” convention). A histogram

of the

distribution

of cosines

was

mea-

sured as a function of cos 0 by dividing the interval from -1 to +I into 50 equal subintervals and collecting the number of angles in each cosine subinterval. For the liquid the result is displayed in fig. 3. There are three broad peaks in the distribution Q(cos e), around the angles SS”, 110” and 180”. The distribution is naturally trucated at small angles due to the repulsive core in the pair potential_ Fig. 3 may be re-

garded as providing rhe “lowest level” orientational information about the liquid in the same sense that the pair correlation function (fig. 2) provides translational information. Also shown in fii. 3 is a histogram constructed from eq. (I) and the pair corre!ation function of fig. 2, using the superposition approximation for the triplet correlation function (that is, eq. (2) with r = 1). This is a striking demonstration that the superposition approximation is inaccurate in liquids near the triple point, and moreover it is a very easy demonstration to calculate. The shaded (unshaded) enclosed areas in fig. 3 are regions where the superposition approximation underestimates (overestimates) the true distribu80

18 May 1984

i

z c

a Z c

004 t

I

Superposilmn Approximolion

oool -I 0

-05

Cosme

Fig.

3. The “distribution

0.0 ot

the

Subtended

05

1.0

Anqle

OF cosines” for the Lennard-Jones

liquid at density p* = 0.85 and tempcraturc 7” = 0.70. Also shown is the superposition approstmation to the same quantity. The shaded enclosed arcas indicated the regions where the superposition approximation underestimates the true distribution. whereas the unshaded enclosed areas indicate the regions where the superposition approsimation overestimates the true distribution.

tion.

In addition,

the

approximation

flattens

out

the

important structure at large angles. Overall the distribution predicted by the superposition approximation is in general shifted to higher angles compared to the true distribution, and the use of the superposition ap prosimstion in theoretical formulae would lead to erroneous predictions of the static [ 11 and dynamic [ 161 properties of the liquid near the triple point. The orientational order in the fee solid is well known at zero temperature and using the present measure is shown in Jig. I. For the fist time, however. we have measured the broadening of the distribution of cosines as the temperature increases. The measured distribution at the temperature T* = 0.50 is displayed in fig. 4, again for the Lennard-Jones system at the density p*= 0.85. Also shown in fig. 4 is the superposition approximation for the same distriiution. It has long been known that the superposition approximation fails for solids: the surprising feature of this more detailed orientational information is that it is accurate for small subtended angles. and only becomes qualitatively wrong as the subtended angle increases to 180”. The superposition approximation completely

Volume

107. number

CHEMICAL

1

PHYSlCS

006~

LETTERS

1s xly

tion 3 to localized interface to allow

regions. a clear “picture”

is generated

quickly,

adjustment

in fact

quickly

of tile thermodynamic

198-1

of the cnougia conditions

complete melting or crystallization and ruining the simulation. More improtantly. if the average value of !2(0) is X2, (0) in the bulk liquid and 52, (0) in the bulk solid. and we defiie a normalized to prevent

order

local Q&T)=

parameter [R(e;f)

-

n,(e)p[s2,(e)

- q(e)]

.

(5)

the correlation function (fi(f3:r) fi(U:r’)) provides a description oiorientational correlations within the interface. This and other applications are being investigated.

Cosme

of

the

Sublended

Angle

Acknowledgement Fig.

4. Same as ft. 3, but for ihe Lennard-Jones = 0.85 and T' = 0 .50 .

misses the high probabilities of finding tending angles of I 20° and 180°.

solid at p* =

neighbourssub-

It isa pleasure to thank Professor H. SllillerKrumbhaar and Dr. C. Lehnlan for helpful dIscussions and hospitality St the Kernforschunesanlage Jiilich. West Germany. where part of rhis calculation was performed.

4. Discussion References

There is clearly much to be gained from studying the “distribution of cosines” in a liquid as a function of the temperature and density. but this information is not widely available. There is particular interest in following the “sharpening” of the peaks as a liquid is supercooled and perhaps forced into a “glassy” state. Among many possible applications we discuss one of particular value. In equilibrium simulations of the solid-liquid interface [ I7,1 S] there are many problems, including

size depcndencc.

long relaxation

times

solid) and stability of the interface. It is even difficult to defiie the exact location of the “interface”. especially if the simulation is started slightly aw3y from equilibrium and the interface is advancing rapidly to freeze or melt the entire system. From figs. 3 and 4 it is clear that the distribution of cosines evaluated at the angle lSO” provides a natural order parameter for deciding whether a given region of particles is “liquid-like” or “solid-like”. Of course, the full distribution (particularly at large angles) provides more detailed information. By restricting the average over particles in sec-

(comparable

with

defect

relaxation

times

in the

X.D.J. Haymel. S.A. Rkz and \V.G. \hdden. J. Chrm. Phys. 75 119Sl) 3696; \VJ. SlcX:eil,\V.C. Madden. A.DJ. Hsgmcf and S.X. Rice, J. Chem. Ply. 76 (19S3) 3SS. D.R. Nelson. Phys. Rev. Letrcrs 50 (1963) 9S2. PJ. Strinhxdt. D.R. Nelson and .\I. Ronchrrti. Phys. Rev. Letters-17 (1981) 1297: Phls. Rev. 61s (1983) 761. L\.DJ. Hsymet. Phys. Rev. B27 (19s;) 1723. C.D. Scott and D.L. Wtdrr. h’xurr 201 (1963) 3s;: J.D. Bernal. Rot. Roy. Sot. A160 (196-1) 199. C.S. Hsu and A. Rahmzm. J. Chem. Ph)s. 70 (1979, 5234: 71 (1979) 4974. R.D. Mountain. Phys. Rev. 17-6 (19s’) X59; erratum A27 (19S3) 2767: RD. Mountain and PK. Basu. J. Chem. Phys. 75 (19631 7318. V.A. Poluchin. \I.\!. Drugurov, \‘.F. L’chov zmd S..-\. Vatolin. J. P!~ys. (Pxis) 41 (1980) CS-X1. J. Hafncr, J. Phys. Fl?- (1982) L205. BJ. Beme. cd., Srstisticll mechanics. Pan .A (Plenum Press, Sew I’ork. 1977). D.A. young 3nd BJ. Alder. J. Chrm. Phys. 60 (197-l) 1154. \V. Brostow, JP. Dusswlr and B.L. Fo\. J. Comput. Phys. 19 (1976) Sl. J.L. Finney, Proc. Roy. Sot. A319 (1970) 379.

Vo!umc 107, number 1

CHEMICAL PHYSICS LETTERS

[ 141 R. Collins, in: Phase transitions and critical phenomena, Vol. 2. eds. C. Domb and M.S. Green (Academic Press, New York, 1972).. [IS] S. Gupta, J.M. Haile and W.A. Steele, Chem. Phys. 72 (1982) 425. [16! J. Bosse, E. Leutheusser and S. Tip, Phys. Rev. A27 (1983) 1696.

82

18 May 1981

1171 L.F. RuU and S. Tosvaerd, J. Chem. Phys. 78 (1983) 3273; J .Q. Browhton and G.H. Gibner. J. Chem. Phys. 79 (I 983) 5095. [18] D.W. Oxtoby and A.D.J. Haymet. J. Chem. Phys. 76 (1982) 6262.