Coordination number in liquid argon

Physica 80A (1975) 513-522 © North-Holland Publishing Co.

C O O R D I N A T I O N N U M B E R IN L I Q U I D A R G O N W. BROSTOW and Y. SICOTTE

D~partement de Chimie, Universit~ de Montrdal, J~Iontr6al, Qudbec H3C 3V1, Canada Received 26 February 1975

Structure of liquids is discussed in terms of Voronoi figures and Delaunay polyhedra. Equations of the respective models and precise experimental data recently available for argon lead to calculated values of the coordination number z and the geometric parameter #. The relation betweent z and # is thus found empirically. Further, calculated values of coordination numbers are compared with those coming from thermal-neutron and X-ray scattering data. A selection among the existing methods of obtaining z from the radial distribution function g(R) is made. The method chosen enables one to find z [g(R)] corresponding to the Voronoi structure.

1. Introduction C o o r d i n a t i o n n u m b e r s z are c o n v e n i e n t for characterizing structures o f dense fluid phases, b o t h f r o m theoretical a n d f r o m e x p e r i m e n t a l points o f view. They also serve in p r e d i c t i n g m a c r o s c o p i c p r o p e r t i e s o f industrial i m p o r t a n c e , as e.g. in the study o f G o t o h 1) o f the surface tension o f liquids. The p r o b l e m o f e v a l u a t i n g z f r o m the b i n a r y r a d i a l d i s t r i b u t i o n function g(R) has been c o n s i d e r e d by Pings 2) a n d by M i k o l a j a n d Pingsa); they have discussed in detail no less t h a n f o u r distinct m e t h ods o f calculating z f r o m a given g(R) curve a n d they have m a d e some n u m e r i c a l c o m p a r i s o n s for liquid argon. It t u r n e d out, t h a t different m e t h o d s p r o d u c e d different n u m e r i c a l values. M i k o l a j a n d Pings have n o t given preference to any o f the m e t h o d s ; they only insisted t h a t in each case the m e t h o d used s h o u l d be clearly defined. L a r s s o n , D a h l b o r g a n d Sk61d 4) in their review o f scattering o f slow n e u t r o n s by liquids find also t h a t the n u m b e r o f nearest n e i g h b o r s o b t a i n e d d e p e n d s on extrap o l a t i o n with increasing distance R f r o m the first p e a k in 47~RZg (R). S c h m i d t a n d T o m p s o n 5) reviewing X - r a y scattering by fluids also discuss difficulties in determining: c o o r d i n a t i o n n u m b e r s ; they stress t h a t n o t only ambiguities in assigning areas u n d e r curves b u t also sensitivity o f the f o r m o f g(R) to e x p e r i m e n t a l errors affects values o f z o b t a i n e d . 513

514

W. BROSTOW AND Y. SICOTTE

In a model of the liquid state based on spacial distribution of molecules 6) coordination numbers represent input parameters, along with geometric parameters/~ and with intermolecular potentials u(R). Quantities such as the configurational energy U ~ and volume Vcan be then calculated for a given temperature T. In vie~ of the problem outlined above, we could have hardly followed the usual route of calculating macroscopic properties from a set of microscopic parameters, the latter in our case including the coordination number. We have, therefore, inverted the use of equations and calculated parameters z and # from known macroscopic equilibrium properties. The results obtained are compared with z values resulting from experimental g(R) curves. A choice between different methods of calculating coordination numbers t'rom neutron or X-ray scattering data is then made. Moreover, a relation between coordination number and the geometric parameter ,u is found empirically; such a relation has been sought in vain in the existing mathematics of the subject.

2. Basic relations

In the model of liquids discussed in ref. 6 the geometry of the system is exactly defined. Neighboring molecules are joined by lines, and the links formed represent impenetrable barriers for other molecules. The entire space is thus divided into Delaunay polyhedra: molecules and links become respectively vertices and edges of polyhedra. Taking each link and producing a face perpendicular to it and passing through the point equidistant from terminal molecules results in dividing space into an alternative assembly o f V o r o n o i polyhedra. There exists a dual relationship between the two assembliesV'S). No limitations are imposed upon the structure, so that links outgoing from a given atom may be of any length and the coordination number may vary t¥0m one molecule to another. Thus, when calculating macroscopic properties, summation over atoms with different coordination numbers has to be made6). In the simplest version of the theory, however, an average value oF coordination number z for the entire phase may be used6'9). The configurational energy of the system is then N*

U c = ½NZ J" ~(R) u(R) dR,

(I)

0

where N is the number of molecules, taken in the calculations to be the Avogadro number NA ; ~0(R) dR is the probability that a link length is between R and R + dR, the distribution given by

R2 Q

~t'(R) = - -

exp

(

zu(R) 2kT

l~PR 3) kT

,

(2)

COORDINATION NUMBER IN LIQUID ARGON

515

where

Q=fR~exp( zu(R)2~ #PR3"]~/ dR, R*

(3)

0

k is the Boltzmann constant, T thermodynamic temperature, P macroscopic pressure. The geometric parameter # is the proportionality factor in the volume equation: R*

t.... N/~ j" R3~ (R) dR.

(a)

0

As discussed in ref. 6, it would be advantageous to be able to predict # as/t(z). Formulating such a relation belongs to the theory of packing and covering, based essentially on geometry and topology. The theory in question, however, supplies us with proofs that a three-dimensional space may be completely divided into a set of Delaunay or, alternatively, Voronoi figuresT), but goes very little beyond that. To define the upper integration limit R* in eqs. (1), (3) and (4), let us introduce some terminology. We start From the location of a chosen central molecule (intermolecular distance R = 0) and we move in the positive direction along the R axis. We encounter first the border of the Voronoi figure of the central molecule, to be labelled R~m~,. We then reach the border of the Delaunay figure, to be labelled Rlm.~x; our symbols are related to the respective maxima and minima of the radial distribution function. Further, we arrive at the border of the Voronoi figure of the second molecule. Consistently with our model, this is the upper integration limit: continuing to label characteristic points in the same way, we thus have Ru ..... = R*.

3. Calculations As the most extensive and accurate experimental studies have been made for argon, we have chosen it as the object of calculations. To define the liquid-state range, we have accepted the triple-point temperature as Tt = 83.81 K. This is the value chosen by Pollack1°), while Chen, Aziz and Lim a t), after a careful analysis of literature data and their own vapor-pressure results in the vicinity of Tt arrive at 83.806 + 0.001 K. As the critical point we take Tc = 150.70 K. Grigor and Steele 12) have obtained from measurements 150.6 ___ 0.1 K, Pings and Teague 13) from refractive-index measurements 150.704 __ 0.015 K, the latter value reiterated by Teague and Pingsl~). Analysis of the equilibrium vapor-pressure data for argon has been made by Garside and Smith ~5) and also by Verbeke16). Using no less than 180 experimental points of 9 different authors, Verbeke has found considerable scattering of data

P-V-T

516

W. BROSTOW AND Y. SICOTTE

from different sources. In each of the two papers 15'16 a number of different P ( T ) functions was tested. We have eventually decided to use the 7-parameter formula of Garside and Smith. For liquid density or molar volume Vthere was also a choice of recent data and equations 1T-19). Comparing data for T = 100.00 K, we have found V/cm 3 tool-1 = 30.65 from the corresponding-states formula of Agraval and Thodos tT). Gladun Is) gives an experimental value of 30.40 for 100.23 K, while using the Goldman and Scrase ~9) formula one obtains 30.45. Gladun, who has not represcnted his results by an equation, stresses good agreement of his data with those of Goldman and Scrase. In our calculations we have thus used the equation given in ref. 19. In view of our approach, reliability of the configurational energy U c values is important. U c was calculated by Barker, Fisher and Watts 2°) after a careful analysis of available experimental data. Similar analysis has been more recently made by Streett21), based on the ensemble of literature P V T data and on sound-velocity data o f T h o e n , Vangeel and Van Dae122), and of Streett and CostaminoZ3). The values of Streett 2~) agree within about l °,o with those of Barker et al.2°). We, ourselves, have also made calculations of U c following the procedure described in detail by McGlashan 2~) and using the same data that he has used : the results obtained are generally in agreement with the two sets mentioned. Further, there exist U ° values compiled by RowlinsonZS); while his values for 87.29 K and 9 0 K are slightly higher than those in ref. 20, Rowlinson's value for T, falls exactly on extrapolation of the U ~ (T) curve drawn through the points of Barker et al. [the latter do not give U~(T0]. We have described the data tabulated in ref. 20 plus the triple-point value of Rowlinson by a formula: UC/N;,k -- 11 984.9 - 479.012 T + 6.65010 T 2

- 0.0403425 T 3 + 0.000091 048 T ~.

(5)

The coefficients in eq. (5) have been found using a computer by a least-squares procedure. Barker et al. do not give values of configurational energy for T > 140 K. Eq. (5) gives at the critical point U~/NAk (150.70 K) = - 2 8 7 . Rowlinson's respective value for T = 150.86 K (what he presumes to be the critical point) is - 289, an entirely satisfactory agreement. Thus, our formula (5) covers the entire liquid-state range. Another choice to be made is that of the interatomic potential u(R). Studying the equation of state of crystals Madan 26) has used 3 different potentials. He has found that results depend considerably on the choice of the potential. There is little doubt that the same statement would apply to the problem at hand. Direct experimental determination of potential parameters by molecular beams produces interval of values within a contour rather than a single value. Consequently, as discussed by Brooks, Kalos and Grosser27), choosing between potentials on the basis of such data is very difficult. We have decided to use two potential functions : that

COORDINATION NUMBER IN LIQUID ARGON

51 7

of Bae 28'a9) and of Barker et al.2°). The Bae potential with two adjustable parameters gives results comparable to three-parameter formulas28); the minimum for Ar is at 3.50 A29). The Barker et al. potential 2°) represents a superposition of two earlier rather complicated equations; it has minimum at 3.76 A. Most potentials proposed for argon, as discussed in another paper3°), have minima located between these two values. Finally, we have found the upper integration limit R* from experimental g(R) data. Yarnell and his colleagues 31) have determined the radial distribution function of Ar at 85 K by slow neutron scattering; these are presumably the most accurate such data presently available. F r o m their tabulated values of g(R) we have found that the second minimum of the function 47~R2Ng (R)/V is a t R l l m i n = R* = 5.03 A. It has been observed by Mikolaj and Pings 32) that the extrema of this function move slightly with the temperature. They have studied a number of Ar states for various temperatures and densities. A m o n g the measurements described in ref. 32 however, there is only one (the run no. 31) corresponding to a liquid in equilibrium with its vapor. As Mikolaj and Pings give positions of maxima only. we have calculated for their run no. 31 R* = 0.5 (Ri~ox + RH~,,x) = 5.64 ~ at 148.14 K. There was little point in taking other data into consideration, as earlier data are considered to be distinctly less accurate. Moreover, e.g., the results of Henshaw 33) are for 84 K, i.e., almost the same temperature as these of Yarnell et al. Given the value of R* from Yarnell data for a temperature close to the triple point, and R* from Mikolaj and Pings data for a temperature close to the critical point, we have simply assumed linear dependence R*(T) along the entire existence line of liquid Ar saturated with its vapor. The parameters so obtained plus a pair of trial values of z and/z were substituted in eqs. (2) and (3). The lower integration limit was taken as 3.180 A. Then, using a value of U c from eq. (5), a new value of z was calculated from eq. (1). This was followed by obtaining a new value of # from eq. (4) and from the G o l d m a n and Scrase volume formula19). The second computer cycle started with the second pair of z and # values substituted into eq. (2). After several such cycles (three to six, usually) values of/z and z which ceased to change within five significant figures were accepted as final.

4. Results and discussion

The results obtained are collected in table ~I. B F W refers to the potential of Barker, Fisher and Watts2°). Our choice of linear dependence of the upper integration limit on temperature was, as explained in the previous section, due to accessibility of two sufficiently accurate values only. To find out, to what extent the assumed R*(T) might affect the results, we have also made calculations taking other values of R*. At T = 85 K, for example, using the BFW potential, we have found

518

W. BROSTOW AND Y. SICOTTE TABLE 1 Properties of liquid Ar in equilibrium with its vapor between the triple point and the critical point

T (K)

P V -- U~ (Jcm-3) (cn.lamol i) Nak

R* (/~)

Potential

II

Bae BFW Bae BFW Bae BFW Bac BFW Bae BFW Bae BFW Bae B FW Bae BFW Bae BF\V Bae BFW Bae BF'W Bae BFW Bae BFW Bae BFW Bae BFW Bae BFW

1.0204 0.8158 1.0240 0.8187 1.0381 0.8299 1.0511 0.8407 1.0630 0.8511 .0869 0.873'0 .1173 0.9044 .1466 0.9339 .1617 0.9491 .1779 0.9660 .2228 1.0151 .2469 .0434 .2685 .0687 .3330 .1376 .4526 .2515 .7266 1.4903

83.81

0.072

28.25

707

5.017

85.00

0.083

28.40

707

5.030

90,00

O. 139

29.02

696

5.083

95.00

0.222

29.70

677

5.137

100.00

0.337

30.45

653

5.190

110.00

0.692

32.18

606

5.297

120.00

1.259

34.40

567

5.404

127.00

1.815

36.41

54[

5.479

130,00

2.100

37.45

528

5,511

133.00

2.414

38.64

512

5.543

140.00

3.279

42.38

458

5.618

143.00

3.712

44.72

423

5.650

145.00

4.024

46.79

395

5.672

148.15

4.558

51.93

341

5.705

150.00

4.899

58.54

302

5.725

150.70

5.034

70.47

287

5.733

z -

10.727 a n d / x -- 0.8139 f o r R* = 5.500 ]k. A s s u m i n g R* -

in z = 10.737 a n d / z

Zcalc '

8.777 10.715 8.789 10.729 8.720 t0.638 8.565 10.441 8.359 10.178 7.976 9.677 7.969 9.294 7.520 9.046 7.424 8.915 7.297 8.741 6.815 8.[00 6.473 7.653 6.191 7.283 5.611 6.536 5.!59 5.962 4.950 5.711

-exp.

11.4

6.9

6.0

6, 150 fi,, r e s u h e d

= 0.81 18. C o m p a r i n g t h e s e d a t a a m o n g t h e m s e l v e s a n d a l s o

w i t h t h e r e s p e c t i v e v a l u e s in t a b l e 1, o n e f i n d s o u t t h a t t h e p a r t i c u l a r c h o i c e s o f R* m a d e h a v e n o t a f f e c t e d t h e r e s u l t s in a n y s i g n i f i c a n t w a y . T h e s a m e c o n c l u s i o n is r e a c h e d v a r y i n g R * a t o t h e r t e m p e r a t u r e s , a n d a l s o w h e n u s i n g t h e Bae p o t e n t i a l . I n s p e c t i n g t h e r e s u l t s , c o n s i d e r first t h e p r o b l e m o f / ~ ( z ) d e p e n d e n c e . O n e f i n d s t h a t / , i n c r e a s e s w i t h d e c r e a s i n g z; this is, q u a l i t a t i v e l y , i n d e p e n d e n t o f t h e p o t e n -

COORDINATION NUMBER IN LIQUID ARGON

519

tial used. To explain such a behavior, consider for simplicity a two-dimensional system. Both Voronoi and Delaunay figures now become polygons, and tetrahedra turn into triangles, while ,u becomes proportional to the surface area. A Yoronoi polygon of a given molecule contains z triangles. Label nearest neighbors of the central molecule consecutively as 1, 2, 3, etc., and label in the same way corresponding triangles in the Voronoi polygon of the central molecule. Remove now one of the neighbors, say 2. Surface areas of triangles 1 and 3 increase now in such a way, that they absorb completely among them the former triangle 2. Moreover, due to perpendicularity of Voronoi borders with respect to intermolecular links, triangles 1 and 3 gain also some additional area beyond the former triangle 2. The net result is thus an increase in the surface area of the Voronoi polygon. Thus decrease in the coordination number is bound to increase the geometric parameter/~. The same kind of argument holds for a three-dimensional system also, what explains the/z(z) behavior found in table I. While we thus understand qualitatively the interrelation between the geometric parameter and the coordination number, a quantitative formula is clearly desirable. As discussed in section 2, the theory of packing and covering appears far from supplying us with such a result. We have, therefore, represented the/z values in the table by a series in descending powers of the coordination number. There was a choice between two sets of #(z) values, depending on the u(R) potential chosen. We have decided on the set obtained using the B F W potential, and this for at least two reasons : a) it is from the same paper 2°) that values of configurational energy for our eq. (5) were taken; b) Barker and his colleagues claim that their interaction potential °'gives excellent agreement with thermodynamic properties of solid, liquid and gaseous argon". We have described the z and ,u values found with the BFW potential by the formula

u = 0.7576

2.4142 34.706 - + - Z

(6)

Z2

Numerical values of the coefficients in eq. (6) have been found by a computer leastsquares procedure. Return now to the problem of determining the average coordination number. Among the methods discussed by Pings 2) and Mikolaj and Pings3), the first three bear no relation to the geometric scheme outlined in section 2. Accordingly, we are not going to consider them; they are in any case treated in detail by Mikolaj and Pings. We are going to deal with the approach labeled by Mikolaj and Pings "method D". We write 112

:: = .[ 4~,RZ ( N / V ) g(R) d R Rt

(7)

520

W. BROSTOW A N D Y. SICOTTE

and we consider three sets of integration limits: R1 = 0, R1 = R i .... R1 = Rimi.,

Rz = R . .... = R*, l,

R 2 = R i .... q- A , R,

= R~i+l~,o,°.

(7a) (7b) (7c)

By, say, eq. (7a) we shall mean eq. (7) with the integration limits specified by (a). Eq. (7a) represents the method D of Mikolaj and Pings. They have limited their treatment to the first coordination shell. Eq. (7b) is used by Kaplow and his colleagues 34'35) for calculating coordination numbers in liquids and also amorphous and crystalline solids. It is applicable for any ith coordination shell. Finally, eq. (7c) corresponds exactly to the geometrical model as discussed in section 2: integration limits correspond to borders of the Voronoi figure of any ith neighbor of the central molecule. Now eq. (7a) represents the special case of eq. (7c) with f - - 1 ; it should be remembered, that the interval 0 <= R < RI,,,~, does not contribute anything to the integral (7). Eq. (7b) of Kaplow et al. may be considered also as a special case of (7c), namely with Ri ..... - R~.... = R ~ + 1~.... - R i .... = [ . The fact that eq. (7c) corresponds exactly to representing a phase of matter as an assembly of Voronoi polyhedra (also called Wigner-Seitz cells) is clearly independent of any experimental information. But we are now going to consider available, admittedly scarce, experimental evidence. Taking g ( R ) values for 85 K tabulated by Yarnell et al. 31) we have performed integration according to eq. (7c). The result is given in the last column of table I. The difference between the experimental coordination number and the one obtained using the Bae potential is rather large. By contrast, the BFW potential produces a coordination number fairly close to the experimental value. We have mentioned above reasons for ascribing more weight to the potential of Barker et al. Other available experimental z values are those calculated by Mikolaj and Pings3), using their own X-ray data32). As already mentioned, among states studied by Mikolaj and Pings there is only one located on the liquid-vapor coexistence line on the liquid side. There is, however, one more point (run no. 22), at 143 K, located fairly close to the saturation line, and we have included the corresponding result in table I also. Both values of z of Mikolaj and Pings quoted here result, of course, from their method D ; their methods A, B and C give values consistently lower. Comparing the results of Mikolaj and Pings with calculations, we find that each experimental value is between the two alternative calculated coordination numbers. This only stresses the importance of having an accurate intermolecular potential u ( R ) . Coordination numbers are essential for many other approaches to the liquid state, apart of the geometric model characterized in section 2; the quasilattice theory of Guggenheim3°,) cell theories 37-39) and the theory of simple liquids of Franchetti 4°) at least ought to be mentioned here. Analysis of experimental methods of

COORDINATION NUMBER IN LIQUID ARGON

52l

determining z in refs. 2 and 3 has caused some repercussions and also some extrapolations. For instance, Hildebrand et al. 4~) have somehow deduced from the work of Mikolaj and Pings 3) that the coordination number is "misleading as a parameter". We would like to stress that there is no ambiguity in calculating the coordination number from experimentally measurable quantities according to eq. (7c); the only problem involved is that of accuracy of determinations, in order to be able to well locate the function minima. As for our procedure of inverting eq. (i) so as to calculate coordination numbers, we find that the results obtained are comparable to experimental ones. Thus, when experimental data for performing the integration (7c) are lacking, eq. (1) can be used to find the coordination number. The problem with eq. (1) is also one of accessibility of reliable input data, the intermolecular potential included. Finally, we would like to comment on the values of coordination numbers found by our calculations. From the early data of Eisenstein and Gingrich 42) it follows that the coordination number of argon changes from 12 in the solid close to the triple point to 2 in vapor around the critical point. Our data show that liquid atoms with coordination number as high as eleven exist at the triple point, while coordinations as low as five have to exist at the critical point.

Acknowledgements Discussions of various aspects of this work, with Dr. E. Braun of the Institute of Physics, the National University, Mexico City, Dr. U. Dahlborg of the Institute of Reactor Physics, Royal Institute of Technology, Stockholm, Professor C.J.Pings of California Institute of Technology, Pasadena, and Professor N.S. Snider of Queen's University, Kingston, Ontario are gratefully acknowledged.

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