Radial Distribution of a Dense Lennard-Jones Fluid The Journal of Physical Chemistry, Vol. 83, No. 23, 1979 3033

An Explicit Equation for the Radial Distribution Function of a Dense Lennard-Jones Fluid

Saul Goldman

Guelph- Waterloo Centre for Graduate Work in Chemistry, Guelph Campus, Depatfment of Chemistry, University of Guelph, Guelph, Ontario, Canada N1G 2 W 1 (Received June 12, 1979)

Publication costs assisted by the Natural Sciences and Engineering Research Council of Canada

An equation is given for the radial distribution function (hereafter the rdf) of a dense Lennard-Jones fluid. The equation explicitly expresses the rdf as a continuous function of interparticle distance, temperature, and density. The parameters in the equation were obtained by fitting a three-variable function to molecular dynamics data on the rdf of a Lennard-Jones fluid. For the range of state variables of 0.50 5 T* < 5.0 and 0.35 5 p* 5 1.10 the equation provides rdfs with an average root-mean-squared deviation of 0.034. The error is about half this value in the low density-high temperature regions studied, and it is about double this value in the high density-low temperature regime. The equation predicts the internal energy and the pressure of a Len- nard-Jones fluid with an uncertainty that is comparable to that obtained directly from molecular dynamics results.

Introduction There now exists a fairly extensive body of molecular

dynamics data on the radial distribution function for dense Lennard-Jones 6-12 fluids.' These rd f s are needed in order to use the recently developed liquid state perturba- tion theories wherein the reference state is taken to be a Lennard-Jones fluida2 These theories are important be- cause they can be applied to a wide variety of liquids and solutions that are chemically interesting. For example, these theories have been used to investigate the roles of multipolar forces and nonspherical molecular shapes in phase separations, their influence on excess functions, and related phenomena. Unfortunately, the rdfs from molec- ular dynamics are available only in the form of tables which are inconvenient to use. Consequently, it was con- sidered worthwhile to fit the existing molecular dynamics data on the rdf to an explicit closed-form equation which expresses the rdf as a continuous function of distance, temperature, and density.

The Equation I t turns out to be a fairly difficult problem to write an

equation for the distance dependence of the rdf which would both adequately account for the existing data and also not contain too large a number of parameters. It is perhaps not superfluous to explain why it is important to keep the number of parameters to a minimum. First, it is obviously desirable that the final equation be as simple and as easy to use as possible, and this involves keeping the number of parameters down. Second, if too large a number of parameters is introduced to account for the distance dependence of the rdf, then the noise in the data being fitted causes these parameters to have a high vari- ance, and this in turn introduces a high uncertainty into the functional form of the density and temperature de- pendence of these parameters.

Previous work on the distance dependence of experi- mental rdf data was done by Franchetti3s4 and by Brostow and S o c h a n ~ k i . ~ Franchetti's equation for the rdf (and extensions of it) proves to be inadequate in that it repre- sents the important first peak to be too flat and too broad. Brostow and Sochanski's equation provides a much better fit to experimental data, and the equation adopted here for the distance dependence of the rdf of a Lennard-Jones fluid is similar in spirit to the Brostow-Sochanski equation. We represent the distance dependence of the rdf by a sum of three terms:

0022-3654/79/2083-3033$0 1 .OO/O

IW* - Bl(p*,T*)) + B ~ ( P * , T * ) ) ~ I ) gb(r*,p*,T*) = C(p*,T*)e-ou(r*) r* I Bg(p*,T*)

gb(r*,p*,T*) = expHr* - B5(p*,T*))12/BG(p*,T*)) Bg(p*,T*) 5 r 5 B5(p*,T*)

r* > Bh*,T*)

gb(r*,p*,T*) = 1.0 r* > B5(p*,T*) gc(r*,p*,T*) = 0 r* 5 [B5(p*,T*) - 0.25B&~*,T*)]

g,(r* ,P * 7 T*) = e-B8(p'J'*)r* sin [2a(r* - B6(p*,T*) +

In equation 1 r*, p*, and T* are the reduced distance, particle number density, and absolute temperature, re- spectively,G g(r*,p*,T*) is the rdf, Bl(p*,T*) to Bg(p*,T*) and C(p*,T*) are parameters that depend on p* and T*,

= (hT)-' where k is the Boltzmann constant, and u(r*) is the Lennard-Jones 6-12 pair potential. In eq 1 the g, term gives the large asymmetrical first peak in the rdf, the g, term gives the damped oscillatory behavior at large r*, and the gb term smoothly connects the main first peak to the oscillating tail. This equation differs from the one used in ref 5 in two respects. First, our g, term ensures that the rdf will behave correctly as r* approaches zer0.I Second, the damping function in our g, term contains only one parameter, rather than three, as is the case in ref 5.

It was found in the course of the calculations that the density and temperature dependence of the constants B1- ( p * , r ) to Bg(p*,T*) could be adequately represented by polynomials in p* and T*. The functional forms chosen for these parameters were

0 . 2 5 ~ p * , ~ * ) ) /B,(~*,T*)I

3 4 3

k = l j = 2 k=l Bg(p*,T*) = p - 2 Alk(g)p*k-l + 7 W - 2 2 A , (9) *k-1

i k p

(3) The quantity C(p*,T*) in the gb term of eq 1 was adjusted

@ 1979 American Chemical Society

3034

so as to ensure a smooth fit a t r* = Bg(p*,T*) of the func- tion

c(,, * ,T * )e -Bu( r * )

The Journal of Physical Chemistry, Vol. 83, No. 23, 1979

to the function

expHr* - Bs(p*,T*)I2/Bs(p*,T*)) Therefore, C(p*,T*) is not independent of the other pa- rameters and is given by

rdf

4.0

Goldman

-

C(p*,T*) = exp -{B9(p*,T*)-l2 - Bg(p*,T*)-6) - ( ;*

Fitting Procedure The data fitted were in the form of 87 tables which listed

rdf values as a function of r*.9 Each table, which contained between 87 and 147 points, referred to a particular pair of values of p* and T*. The maximum ranges of r*, p* , and T" covered by the tables were 0.775 C r* < 4.475,0.35 C p* C 1.20, and 0.50 C T* C 6.0.'O

There was insufficient rdf data a t low r*'s (Le,, for 0.80 5 r* 5 0.90) to determine values of Bg(p*,T*) from the rdf data alone. In other words, because of the small number of data points in the range 0.80 5 r* 5 0.90, the root- mea-squared deviation in the rdf fits was very insensitive to the value of Bg(p*,T*). Consequently the first step in these calculations was to fit eq 1 to the rdf data, with C(p*,T*) in the first gb term, and Bg(p*,T*) in the second gb term, set equal to zero. Least-squares estimates of the parameters Bl(p*,P) to & ( p * , P ) were obtained (for each pair of values of p* and P) by using a nonlinear regression algorithm based on the "Marquardt compromise procedure".ll A weight function of r*-l was used in the fit so as to give greater relative weight to the rdf s a t small r*. This was done partly because the experimental rdf s for small r* are somewhat more accurate than those for large r*'s,12 and also because the accuracy of any thermo- dynamic functional to be obtained from these rdfs (e.g., energy or pressure) will be much more sensitive to the rdf s at small r* than at large r*.

The next step was to obtain values for Bg(p*,T*). While, as mentioned previously, the rms deviation of the rdf fits was insensitive to the value of Bg(p*,T*), the internal en- ergy and the pressure of a Lennard-Jones fluid were strongly sensitive to the values of B9(p*,T*). Therefore, the values of Bg(p*,r*) were obtained by using eq 1 for the rdf with the above values of Bl(p*,T*) to &(p*,T*) in the energy and pressure equations for a Lennard-Jones fluidI3

u*(~*,T*) = 8xp*J"(r*-lo - r*-4 )g(r*,p*,T*) dr* (5)

P*(p*,T*) =

p*T* - 1 6 ~ p * ~ (r*-4 - 2r*-10)g(r*,p*,T*) dr* ( 6 )

Values of U*(p*,T*) and P*(p* ,T ) were available from the molecular dynamics simulations that provided our exper- imental rdfhg A simple algorithm was written to numer- ically solve (at each pair of p*,T*) each of eq 5 and 6 for Bg(p*,P). The arithmetic means of the two solutions were taken to be our fitted values of B9(p*,T*).

The final step was to fit each of the parameters Bl(p*,T*) to Bg(p*,T*) to explicit functions of p* and T*. To do this the "Marquardt compromise procedure" was again applied, this time to eq 2 and 3. The values of Bl(p*,T*) to B9(p*,T*), determined as described above,

Lo

I

r'

Figure 1. Calculated and experimental radial distribution functions. The points are molecular dynamics data taken from ref 9. The curves were obtained from eq 1-4 and the constants in Table I. Solid curve and solid points are for p" = 0.35, T' = 1.1983; dashed curve and open circles are for p" = 1.10, T" = 1.1666.

were used in the left-hand sides of eq 2 and 3 so that the constants Ajk(i) were obtained as the least-squares solutions of eq 2 and 3. In this final fit, all the points (i.e., all the B,(p*,T*) a t each pair of p*,T*) were weighted equally. The constants Alk(I) that resulted from this last fit are entered in Table I. Results

Equation 1, together with eq 2-4 and the entries in Table I, constitutes our equation for the rdf of a dense Lennard-Jones fluid. In this section we illustrate the quality of the fits provided by this equation.

Figure 1 demonstrates the fit a t a high and low density a t (approximately) the same temperature, and Figure 2 demonstrates this for a high and a low temperature a t a fixed density. The systematic positive deviations at r* N

3 are characteristic of our high density results, and are partly a consequence of the weight function r*-l which places relatively greater weight on the points a t small r*. These deviations a t large r* will have little consequence in most applications which involve integrals over products of the rdf (or its derivative) and the potential (or its de- rivative), since these products contribute only very slightly to the integrals a t large r*.

The root-mean-squared deviations for our rdf fits are given in Table 11. The range of these deviations is 0.01- 0.08; the overall average for all the entries is 0.034. I t is seen from Table I1 that the fits are best for the low den- sity-high temperature regime and progressively deteriorate as the high density-low temperature regime is approached. The average root-mean-squared deviation of 0.034 is con- sidered to be of the same magnitude as the expected ac- curacy (as opposed to the precision) of the molecular dy- namics results.

It was considered worthwhile to determine how well our equation predicted rdf s for state conditions that fell within the range used here, but that were not used for our fit. TO do this, our equation was used to obtain rdfs for each of 25 state conditions for which Verlet reported molecular

Radial Distribution of a Dense Lennard-Jones Fluid

4.0

The Journal of Physical Chemistry, Vol. 83, No. 23, 1979 3035

-

w PI

rd' I

0

Figure 2. The curves and points have the same meaning as in Figure 1. Solid curve and solid points are for p * = 1.0, T* = 4.8545: dashed curve and open circles are for p" = 1.0, T' = 0.8113.

- U *

7.0

6.0

5.0

4.0

3.0

2.0

1 .o

0 I I I I

1.0 2.0 3.0 4.0 5.0 6.0

T*

Figure 3. Calculated and experimental values for the internal energy of a LennardJones fluid. The points are from the molecular dynamics results given in Table 2 of ref 9. The curves were obtained with eq 5 together with eq 1-4 and the constants in Table I which were used for gin the integrand of eq 5. The integrations were done numerically by Gauss quadrature.

dynamics rdf's for a Lennard-Jones fluid.14 The root- mean-squared deviations between the predicted and the reported rdrs are given in Table 111. The overall average of the entries in Table I11 turns out to be 0.034, so that, on average, the scatter in our predicted rdf s is the same as that for our fitted rdf's.

As a final check, we used our equation to predict the internal energy and pressure of a Lennard-Jones fluid over

3036 The Journal of Physical Chemistry, Vol. 83, No. 23, 1979 Goldman

TABLE 11: Root-Mean-Squared Deviations for the Fitted Vs. the Experimental rdf’sa

P * T* rmsd P * T* rmsd P * T* rmsd

P * T* rmsd P * T* rmsd P * T* rmsd P * T* rmsd

0.35 0.35 0.35 0.35 3.0 2.5 2.0 1.7 0.03 0.03 0.03 0.03

0.45 0.60 0.60 0.60 1.20 3.06 2.40 1.97 0.02 0.01 0.02 0.02

0.90 0.90 0.90 0.95 1.03 0.81 0.60 4.69 0.03 0.04 0.04 0.02

1.00 1.00 1.00 1.00 4.85 3.71 2.75 2.20 0.02 0.02 0.02 0.03

1.02 1.02 1.04 1.04 0.82 0.59 4.50 1.50 0.05 0.06 0.02 0.03

1.05 1.06 1.06 1.06 0.61 4.73 1.50 1.19 0.06 0.02 0.03 0.04

1.10 1.10 1.10 1.10 3.96 2.68 2.20 1.81 0.02 0.03 0.03 0.03

--___-__ 0.35 0.35 0.45 1.45 1.20 2.95 0.02 0.02 0.03

0.60 0.60 0.60 1.73 1.50 1.24 0.02 0.02 0.03

0.95 0.95 0.95 3.59 2.71 2.21 0.02 0.02 0.02

1.00 1.00 1.00 1.73 1.27 1.04 0.03 0.03 0.04 1.04 1.05 1.05 1.19 4.80 3.72 0.04 0.02 0.02

1.06 1.06 1.06 1.02 0.79 0.61 0.05 0.06 0.06

1.10 1.10 1.10 1.17 1.02 0.61 0.05 0.05 0.07

-__ 0.45 2.59 0.02

0.90 4.75 0.02

0.95 1 .71 0.03

1.00 0.81 0.05

1.05 2.62 0.03

1.08 4.63 0.03

0.45 1.97 0.02

0.90 3.76 0.02

0.95 1.20 0.03

1.00 0.56 0.07

1.05 2.16 0.03

1.08 1.43 0.04

--__ 0.45 1.70 0.02

0.90 2.21 0.02

0.95 1 .oo 0.04

1.02 1.46 0.04 1.05 1.73 0.03

1.08 1.19 0.04

_______ 0.45 1.48 0.02

0.90 1.64 0.02

0.95 0.95 0.80 0.61 0.04 0.05

1.02 1.02 1.23 0.97 0.03 0.04

1.05 1.05 1.19 1.01 0.04 0.05

1.08 1.08 1.04 0.82 0.05 0.05

0.90 1.19 0.03

0.95 0.49 0.08

1.05 0.86 0.05

1.08 0.60 0.05

a The fitted rdf’s were obtained with eq 1-4 and the constants in Table I. The experimental rdf’s refer to molecular dy- namics data taken from ref 9.

TABLE 111:

P * T* rmsd P * T* rmsd P * T* rmsd

Root-Mean-Squared Deviations for Predicted rdf’sa

0.880 0.880 0.880 0.850 0.850 0.850 0.850 0.850 0.850 0.850 0.850 1.095 0.936 0.591 2.888 2.202 1.273 1.127 0.880 0.786 0.719 0.658 0.03 0.04 0.04 0.02 0.02 0.03 0.03 0.04 0.04 0.04 0.11 0.824 0.750 0.750 0.750 0.750 0.650 0.650 0.650 0.650 0.650 0.500 0.820 2.845 1.304 1.070 0.827 3.669 1.827 1.584 1.036 0.900 1.360 0.04 0.02 0.02 0.03 0.05 0.02 0.02 0.02 0.04 0.05 0.02

0.450 0.450 0.450 2.934 1.710 1.552 0.02 0.03 0.02

__I__ -

a The calculated rdf’s were obtained by eq 1-4 and the constants in Table I. The data against which comparisons are being made are molecular dynamics results taken from ref 14.

TABLE IV: Root-Mean-Squared Deviations of the Predicted Pressure and Energy of a Lennard-Jones Fluida

P * -- (0.35 - 0.50) (0.50 - 1.06) (1 .06- 1.10)

rmsd ( U * ) 0.022 0.052 0.071 rmsd (P*) 0.056 0.26 0.28

a The calculated values of U* and P* were obtained from eq 5 and 6, respectively. Equations 1-4 and the constants in Table I were used for the rdf in the integrands of eq 5 and 6. The integrations were done numerically by Gauss quadrature. The experimental U*’s and P*’s were molecular dynamics results taken from Table 2 of ref 9. The entries aye averages over the entire range of tempera- tures for which data were available.

a range of state conditions. To do this, our equation was used for g(r*,p*,T*) in the integrals of eq 5 and 6, and the integrations were performed by Gauss quadrature. Some typical results are illustrated in Figures 3 and 4. It is seen from these figures that no systematic errors (in a positive or negative sense) arise in the predicted values of U* or P* with either p* or T*. The magnitudes of the uncer- tainties in these predictions are given in Table IV. In this table the deviations were simply grouped into low, medi- um, and high density ranges, since these root-mean- squared deviations varied only slowly with density, and showed no significant variation with temperature. A com- parison of the entries in Table IV with the scatter diagrams for U* and P* given in Figure 4 of ref 9 indicates that the uncertainties in our predicted values of U* and P* are of

0. - 1 1

0 1.0 2.0 3.0 4.0 5.0 6.0 T‘

Flgure 4. Calculated and experimental values for the pressure of a Lennard-Jones fluid. The points are from molecular dynamics results given in Table 2 of ref 9. The curves were obtained with eq 6 together with eq 1-4 and the constants in Table I which were used for gin the inegrand of eq 6. The integrations were done numerically by Gauss quadrature.

Gas-Phase Study of Me3SiI f HBr G Me,SiBr -I- HI

about the same size as those that come directly out of the molecular dynamics simulations.

Acknowledgment. The author thanks Professor Keith E. Gubbins for preprints of his work and for his helpful correspondence and the Natural Sciences and Engineering Research Council of Canada for financial assistance.

References and Notes (1) The Lennard-Jones 6-12 pair potential has the form

u ( r ) = 4c[ (u / r ) ' * - ( ~ / r ) ~ ]

u ( r ) has a minimum equal to -c at r = 2%; u ( r ) is zero when r

(2) C. G. Gray, K. E. Gubbins, and C. H. Twu, J . Chem. fhys., 69, 182 (1978), and references contained therein.

(3) S. Franchetti, Nuovo Cimento 6, 55, 335 (1968). (4) S. Franchetti, Nuovo Cimento 6, 10, 211 (1972). (5) W. Brostow and J. S. Sochanski, fhys. Rev. A , 13, 882 (1976). (6) The asterisk superscript indicates reduced variable, with c and u used

for the reduction: density = pu3 = Nu3/ V = p " ; temperature = kT/t = T ' ; pressure = fu3/c = f " ; energy = U/Nc = U " ; distance = r / u = r * .

= u.

The Journal of Physical Chemistry, Vol. 83, No. 23, 1979 3037

(7) It is a theoretical resultathat as r * - 0, g ( r ' , p * , T " ) - Ke-Bu(''), where K, which is constant with respect to r ' , is greater than 1.0. The equation for the rdf given in ref 5 (eq 2) is incorrect in the limit r * - 0 , since according to this equation g - (a positive constant) as r * - 0.

(8) B. Widom, J . Chem. fhys., 39, 2803 (1963). (9) These data may be obtained from the British Library, Lending Division,

Boston Spa, Yorkshire, U.K., where they are deposited under their supplementary publication scheme. The general reference for these tables is J. J. Nicolas, K. E. Gubbins, W. B. Streett, and D. J. Tildesley, Mol. fhys., 37, 1429 (1979).

(10) While all 87 tables were used in our fitting procedures it was found that the accuracy of the predicted rdf's deteriorated significantly for p" > 1.10 and T' -+ 5.0. Consequently, it is recommended that the equation presented in this work be used only within the range

(1 1) The program used provides least-squared estimates of the parameters for a nonlinear, multivariable function. The program was a modlfied version of SHARE SDA 3094.01 (NLIN) by D. W. Marquardt, taken from the Share Program Library.

(1 2) The cutoff procedure used in molecular dynamics calculations shoukl have a greater adverse affect on the accuracy of the rdf's at large r' than at small r * .

(13) D. A. McQuarrie, "Statistical Mechanics", Harper & Row, New York, 1976.

(14) L. Verlet, Phys. Rev., 165, 201 (1968).

0.35 < p* < 1.10, 0.50 < r * < 5.0.

Gas-Phase Study of the Reaction System Me3SiI -t HBr Me3SiBr 4- HI. Equilibrium and the Bond Dissociation Energy D (Me3Si-I)

Alan M. Doncaster and Robin Walsh"

Department of Chemistry, University of Reading, Whiteknights, Reading RG6 ZAD, England (Received April IO, 1979)

The title reaction has been studied in the temperature range 290-390 "C (but mainly at 321 "C) by a UV spectrophotometric method. The reaction reaches a steady state, and experiments starting from both sides over a range of partial pressures of reactants indicate that equilibrium is reached. At 321 "C the equilibrium constant is 12.2 f 2.4. The third-law enthalpy change, AH"298.15, is -13.4 f 1.2 kJ mol-l. From the literature value for ;Wf"(Me3SiBr) a value of AHf"298.2(Me3SiI(g)) = -218 f 4 kJ mol-' is derived. Additionally, from an estimate of ;Wf"(Me3Si.), the bond dissociation energy D(Me3Si-I) = ca. 322 kJ mol-' (76.9 kcal mol-l) may be deduced. The kinetics of this system are discussed briefly.

Introduction

shown that the principal decomposition pathway in the early stages involves molecular elimination of HI with concomitant formation of an olefina1p2 Studies of these pyrolyses2 and their reverse reactions, the addition of HI to olefin^,^ have been useful in providing both kinetic and

thus opening up the possibility of the reaction step Me2Si=CH2 + HBr - Me3SiBr (2)

Regardless of the question of mechanism, a study of this reaction system offers the additional opportunity to ob- serve the equilibrium process

Investigations of the pyrolyses of alkyl iodides have

Me3SiI + HBr MeRSiBr + HI

Experimental Section Apparatus . A spectrophotometric technique was em-

ployed to study this reaction in situ in a static system. A quartz reactor was placed in an electrically heated metal furnace, and one beam from a Varian Techtron 635 spec- trometer was diverted by means of mirrors to make a double pass through the cell reactor before returning to

Me3SiI e Me2Si=CH2 + HI (1)

The difficulty of this reaction could be either kinetic (a high activation energy) or thermodynamic (an unfavorable equilibrium constant) in origin. We decided to investigate this question and the possible intermediacy of Me2Si=CH2 by the device of adding the alternative trapping agent HBr,

0022-3654/79/2083-3037$01 .OO/O 0 1979 American Chemical Society