Radial distribution function peaks and coordination numbers in liquids and in amorphous solids

CHEMICAL PHYSICS

Volume 49, number 2

LETTERS

15

RADIAL DISTRIBUTION FUNCTION PEARS AND COORDINATION IN LIQUIDS AND IN AMORPHOUS SOLIDS *

July 1977

NUMBERS

Witold BROSTOW Center for Advanced

Studies.

National

Po@teclmic

Institute.

A.P. 14-740. Me_.rico 14. D.F.

Received 22 March 1977

The moleculnr collision diameter I?,, location RgmlxJ of the maximum of the principtl pcrtk of the bmnry radial disfunction g(R), and location Rumin of the princrpal minimum of the bitwy interaction porentiol II(R) rtrc intcrThe number of nearest neighbors z in liquids and in amorphous solids is contained berelated by R, < R~-,,,,I G R,,inItribution

tween the values of four and eleven.

The structure of liquid and amorphous solid phases is conveniently represented by the binary radial distribution function g(R). Normalized to unity for large distances,g(R) represents the probability of finding another molecule at a distance R from a given molecule. A related function is 4nR2g(R), used for similar purposes. Interactions between a pair of molecules are usually represented by the binary interaction potential zl(R). Relations involving both g(R) and zl (R) can be used to calculate principal equilibrium properties of any phase such as the potentiai energy or the macroscopic pressure; cf. eq. (1) in ref. [l] _ For crystals and dilute gases there exist other convenient procedures for arriving at equilibrium properties. The procedure via g(R) plus u(R) is mainly used for liquids and solutions, dense gases and amorphous solids, and these are the phases that we shall consider here. In particular, we shall deal with three essential parameters related to g(R) and u(R) curves: RgmaxI, the location of the maximum of the main peak of g(R); RtcminI. the location of the minimum of the main potential well of u(R); the coordination number zi, the number of neighbors in jth shell around a given molecule. For simplicity, we shall consider a one component system. Extension to functions g,,,, ’ CR,,,, v) and *A

preliminary report of this work W.LSpresented at the 44th Annual Con_gress of the Association Canadienne-Franl;aisc pour 1’Avancement des Sciences, held at the Universitf de Shcrbrooke on 12-14 May, 1976.

LC,,_,,’ (Rj,,, n), where), and_!” are components in a multicomponent mixture, is straightforward. There exists a relation [2] between g(R) and u(R): cu g(R) = exp!--dR)lkTl = exp[-z~(R)/kT]_v(R)

I:

gJR hi _

(1)

p = N/V is the number

density, N the number of molecules in a system of volume Z’, k the Boltzmann constant and T the thermodynamic temperature. We have

go(R)=

1,

(2)

while g,(R) for i 2 2 are sums of multiple integrals over products of so-called Mayer functions, with each Mayer function depending on zi (R)_ Computer times necessary to calculate g&R) from 11(R) increase rapidly with i, and soon become prohibitively long; most of the calculations are made for i = 1,2,3 or 4. Thus, obtaining g(R) dependence from known u(R) in the above marm is possible only in principle. The inverse procedure of obtaining zl(R) from experimentally established g(R)

was tried also [3] and some useful information obtainel For quantitative predictions, however, the present accuracy of diffractometric measurements ought to be improved further. Moreover, there are as yet no relations between the experimental scattering data and g,(R for i > 2. 285

CHEMICAL PHYSICS LE’lTERS

VoIun~ 49, number 2

Under the circumstances, most of the work on g(R) and u(R) curves is still focused on a small number of characteristic parameters: Rsmaxr, g(Rgmad), R uminI> ZI(R,,~,~), R, defined by -

u(R,)=O.

(3)

more “popular” substances such as Ar one can make fairly long lists [3 ] of Rtcminland LI(RuminI)

different from the RuminI and LL(RuminI) are gene&y data used to find RgnldsI and g(RgmaxI) (calculations

in ref. [3] represent a rare exception from this point of view)_ It is, therefore, surprising that there have been as yet no attempts to interrelate the positions of R gn1axIand Rmcni* I was able to find only one mention of this problem: Egelstaff [5] says that “for a dense liquid the principal peak of g(R) occurs at a position close to the principal minimum of Ll(R)“. We shall try to interrelate ZgmaxI and Rlcminl using the function y (R)_ Differentiating eq. (I) with respect to R we find, that an extremum of g(R) occurs either at n(R) = 00, or when = kT[d lnv(R~e,,)fdR]

Also from eq. (I), at an extremum d In g(R ,,,,)/dR

(4)

of 24(R)

= d In Y (R,,,k)laR

Let us find out, whether extrema

-

-

6)

we can have both principal

at the same value of R, call it R* ; that is

R gmaxl =R urnin ZR&-.

(6)

Given (6), eqs. (4) and (5) are valid simultaneously; sequently du(R=)/dR = dy(R+dR

= 0 _

From eqs. (1) and (2) a solution y=i,

p=o.

con-

(7)

-

and (3) =.W$).

(9)

R,

(?)

= RgInaxI*

from eqs. (4) and (9) we obtain dtc(R,

(??)

= RgmdxI)/dfZ = 0.

The interaction potential drops rapidly with increasing distance at and around R = R, and there is no inflection of U(R) curve. Thus, relations incorrect, and necessarily &s <&nLK:I

(?) and (??) are

00)

*

From (6)-(8) and (10) we obtain the displacement limits of RgmZ.I as R0 < RgmaxI

G RrmGnI -

(11)

in a way, the result (11) confirms qualitative notions popular among experimental diffractometrists, including the opinion of Egelstaff [S] quoted above. A very convenient structural parameter is the coordination number zi. It forms the basis of several theories of condensed phases, including lattice, cell, hole and free volume theories. In spite of the popularity, or perhaps because of it, the coordination number is surrounded by misunderstandings and confusion. To begin with,

there are at least four methods

of cakula-

ting zi from g(R) curves. Mikolaj and Pings [7] considered these methods in detail. They have not given preference to any of the methods, but urged that in each case the method used should be clearly defined. The same problem was considered in ref. [8] and the following equation: RzminG+ 1)

of (7) is

Zj=p

(8)

Thus, (6) is possible in the limit of vanishing density. This suggests a procedure of finding RrrminIfrom a series of RgmavI values by determining the limit R grnaxl(p+ 0). When the density is non-zero and increases, it is only natural for the distance fZW_u(r to 286

Ar limit of (1)

Posing tentatively

parameters proposed by various authors. Each pair of parameters characterizing the u(R) curve is usually bases on a number of thermodynamic properties, on direct experimental determination by molecular beams, and more recently also on quantum-mechanical calculations [4] _ In any case, the data used to evaluate

d&$,&ldR

decrease; this has been found experimentally for by Mikolaj and Pings [6] _ As a possible opposite ofRlfnlaxl let us try the other characteristic point u(R) curve, the collision diameter R,. From eqs.

&J)

For

15 July 1977

s Rzmin( 11

4nR?-g(R) dR

(12)

was proposed; the subscript zmin refers to a minimum of the function 4nR2g(R); the peak located between R~rnin(i) and R.zrnin(i+ 1) corresponds to jth coordination shell. As discussed in ref. 181, eq. (12) represents a generalization of “method D” of MikoIaj and Pings and of at least one other method of computing coordination numbers.

Volume 49, number 2

CHEMICAL PHYSICS LETTERS

Given that z] can be defined in purely e x p e r i m e n t a l terms, it is o n l y natural to ask, w h a t are the p e r m i t t e d values o f the n u m b e r o f nearest neighbors z 1 o f molecules in liquids and in a m o r p h o u s solids ? Eisenstein and Gingrich [9] c o n c l u d e d f r o m their X-ray scattering data for liquid Ar that 6 ~< z I ~< b e t w e e n 10.2 and 10.9 depending o n the procedure.

(777)

While m a n y measurements were made since [ 9 ] , the validity o f relation (???) for liquids is s o m e h o w persistently still accepted. But let us take the simplest three-dimensional m o d e l o f liquids, which is that o f hard spheres. The m i n i m u m n u m b e r o f nearest neighbors is four, since each sphere must rest o n three others and will s u p p o r t at least o n e * ; this is, necessarily, the limiting value o f z I at the liquid--vapor critical point. Further, in spite o f m a n y a t t e m p t s t o push in the thirteenth sphere s o m e h o w , the m a x i m u m number o f spheres o f equal size touching a given one is twelve [ 1 0 ] . Phases with z I = 12 are crystals, either hexagonal or face-centered cubic. In a liquid, or in an a m o r p h o u s solid with no crystalline regions, the maxim u m c o o r d i n a t i o n n u m b e r has to be lesser b y at least one. F o r m a t i o n o f crystalline regions in Bernal's physical m o d e l o f the liquid state [11 ] is w o r t h reminding in this c o n t e x t . Expefimefits show that z I in liquids increases along with decreasing t e m p e r a t u r e ; this implies z I = 11 at the triplet p o i n t . In conclusion, instead o f relation (???) we have 4 < z 1 <--A1 .

(13)

It is instructive to c o n f r o n t relation (13) with the e x p e r i m e n t a l data o t h e r than those in ref. [ 9 ] . There is a general consensus, that the m o s t accurate experiment-derived g(R) values are n e u t r o n scattering data o f YarneU et al. [12] for liquid Ar at 85.0 K. Using the tabulated values o f g ( R ) in ref. [ 1 2 ] , Rzminll = 0.503 n m was f o u n d by i n t e r p o l a t i o n [ 8 ] . Since then, however, a precise analytical formula for g(R) was established [ 1] , and the parameters representing the data o f Yarnell et al. f o u n d . Using n o w the g(R) formula f r o m ref. [1 ] and a c o m p u t e r f u n c t i o n minization p r o c e d u r e (ZXPOWL), one obtains a m u c h m o r e * Similarly, we have in two dimensions the corresponding value zl = 3 and in one dimension z I = 2.

15 July 1977

precise value ofRzminll = 0.4903 nm. This value, used as the upper integration limit in eq. (12), produces z I = I0.9. Since we are at 85.0 K, and the triple point o f Ar is at 83.81 K [ 8 ] , the value o f z 1 n o w o b t a i n e d agrees p e r f e c t l y with the upper limit o f formula (13) (and also with the early low t e m p e r a t u r e data in ref. [9] ). Still, this is only a single value. Extensive calculations o f z 1 , using an equivalent o f our eq. (12) and the diffractometric data for Ne, Ar and Kr f r o m over half a d o z e n sources were made b y van L o e f [ 1 3 ] . He presented his results on a z ! versus P[Ocr diagram, where Pcr is the density at the liquid--vapor critical point. His diagram contains points with z I < 4, but these are all for P/Pcr "<~1, i.e. for gaseous phases. Vml L o e f does not discuss the limits o f z I in his paper, but in a subsequent private c o m m u n i c a t i o n he points out that the value z i = 4 is significant. As for the upper limit o f z I , the van L o e f diagram simply does not contain a n y values o f z I > 11, e x c e p t for z I = 12 for each o f the substances studied at the triple point but in the solid phase. Thus, e x p e r i m e n t a l data for m o n a t o m i c liquids are in c o m p l e t e agreement with our relation (13). The way we have arrived at (13) suggests, that it applies to o t h e r liquids and solutions also. The only possible e x c e p t i o n m a y be a " m i x e d " c o o r d i n a t i o n number in a system o f molecules widely differing in size, since a large sphere m a y conceivably have m o r e than twelve m u c h smaller spheres simultaneously touching it. A m o r p h o u s solids have some peculiar properties, such as lower values o f z I for Ge than in the corresponding liquid [ 1 4 ] . But relation (13) clearly applies to a m o r p h o u s solids also. One more p r o b l e m involving the c o o r d i n a t i o n nunibers ought to be m e n t i o n e d . The n u m b e r o f faces f l o f a V o r o n o i p o l y g o n (a domain, a W i g n e r - S e i t z cell) o f a given molecule is usually larger than z 1, and in general zi --/=3~.However, f l is o f t e n also called the coordination n u m b e r , a n o t h e r source o f confusion. The relation b e t w e e n z I and f l forms the subject o f a n o t h e r paper [ 1 5 ] . I thank Professor Neil Snider o f Chemistry Departm e n t , Queen's University, and Dr. J.S. Sochanski o f Physics Department, The University o f Quebec, for discussions as well as c o m m e n t s o n the manuscript. Similar thanks are due to the late Professor Douglas McEachern o f the Center for Advanced Studies, Mexico City. This work forms a part o f a program sup287

Volume 49, number

2

CXEMICAL

PHYSICS

ported financially by the National Research Council of Canada, Ottawa.

References [l] [2]

[ 3J [4: [S]

288

W. Brostow and J.S. Sochanski, Phys. Rev. A 13 (1976) 882 C.J. Pings, in: physics of simple liquids, eds. N.H.V. Temperley, J.S. Rowlinson and G.S. Rusbbrooke (NorthHolland, Amsterdam, 1968) ch. 10. 15’. Brostow, Chcm. Phys. Letters 35 (1975) 387. A.J. Thakkar and V.H. Smith Jr., Chem. Phys Letters 24 (1974) 1.57. P.A. l$&taff, An introduction to the liquid state (AcJdcntic Press, New York, 1967) legend for fig. 2.1.

LETIERS

15 July 1977

[6] P.G. Mikolaj and C.J. Pings, J. Chem. Phys. 46 (1967) 1401, cq_ (9). [7] P-G. Mikolaj and C.J. Pings, Phys. Chem. Liquids 1 (1968) 93. [8] W. Brostow and Y. Sicotte, Physica A 80 (1975) 513. (91 A. Eisenstein and N.S. Girtgrich, Phys. Rev. 62 (1942) 261. [IO] K. Schtitte and B.L. van der Waerden, &lath. Ann. 125 (1953) 325. [ 1 l] J-D. Bernal, Proc. Roy. Sot. A 280 (1964) 299. [ 121 J.L. Yarnell, M.J. Katz, R.G. Wenzcl and S.H. Koenig. Phys. Rev. A 7 (1973) 2130. [ 131 J.J. van Loef. Physica 62 (1972) 345. [ 141 S.P. Isherwood, B.R. Orton and R. Manaila, J. NonCryst. Solids 8-10 (1972) 691. [ 151 A. Boivin, W. Brostow, J.-P. Dussault and B. Fox, in preparation.