^{1,a)}and D. M. Heyes

^{1,b)}

### Abstract

The authors investigate the behavior of a model fluid for which the interaction energy between molecules at a separation is of the form , where and are constants and is a large integer. The particular properties they study are the pressure , the mean square force , the elasticshear modulus at infinite frequency , the bulk modulus at infinite frequency , and the potential energy per molecule . They show that if is sufficiently large it is possible to derive the properties of the system in terms of two parameters, the values of the cavity function and of its derivative at the position . As an example they examine in detail the cases with and for three different temperatures and they test the theory by comparison with a computer simulation of the system. They use the simulated pressure and the average mean square force to determine the two parameters and use these values to evaluate other properties; it is found that the theory produces results which agree with computer simulation to within approximately 3%. It is also shown that the model, when the parameter is large, is equivalent to Baxter’s sticky-sphere model with the strength of the adhesion determined by the value of and the temperature. They use Baxter’s solution of the Percus-Yevick equations for the sticky-sphere model to determine the cavity function and from that the values of the same properties. In this second approach there are no free parameters to determine from simulation; all properties are completely determined by the theory. The results obtained agree with computer simulation only to within approximately 6%. This suggests that for this model one needs a better approximation to the cavity function than that provided by the Percus-Yevick solution. Nevertheless, the model looks promising for the study of (typically small) colloidal liquids where the range of attraction is short but finite when compared to its diameter, in contrast to Baxter’s sticky-sphere limit where the attractive interaction range is taken to be infinitely narrow. The continuous function approach developed here enables important physical properties such as the infinite shear modulus to be computed, which are finite in experimental systems but are undefined in the sticky-sphere model.

I. INTRODUCTION AND MODELS

II. DETAILS OF THE CALCULATIONS

III. RESULTS

IV. CONCLUSIONS

### Key Topics

- Colloidal systems
- 11.0
- Cavitation
- 9.0
- Elastic moduli
- 7.0
- Elasticity
- 6.0
- Atomic and molecular interactions
- 5.0

## Figures

A plot of the function , the radial derivative of the Mayer function [see Eq. (1.9)], for , , and .

A plot of the function , the radial derivative of the Mayer function [see Eq. (1.9)], for , , and .

A comparison of the theoretical and simulated cavity functions for in the region where is significant. Note the small range of shown. The curve is the theoretical curve obtained from Baxter’s result for sticky spheres with adhesion chosen according to Eq. (1.14). The curves labeled have been obtained by simulation with steadily decreasing spatial resolution as the index increases. There were 864 particles in the simulation and the time step was 0.0006 reduced units. The apparent peak and rapid fall in the simulated values of the cavity function for below about 1.0 are, we believe, due to the difficulty of obtaining reliable values of this function from simulation when the pair distribution function is very small.

A comparison of the theoretical and simulated cavity functions for in the region where is significant. Note the small range of shown. The curve is the theoretical curve obtained from Baxter’s result for sticky spheres with adhesion chosen according to Eq. (1.14). The curves labeled have been obtained by simulation with steadily decreasing spatial resolution as the index increases. There were 864 particles in the simulation and the time step was 0.0006 reduced units. The apparent peak and rapid fall in the simulated values of the cavity function for below about 1.0 are, we believe, due to the difficulty of obtaining reliable values of this function from simulation when the pair distribution function is very small.

A comparison of the theoretical and simulated cavity functions for . The subscripts and denote the theoretical and simulated cavity functions, respectively. There were 864 particles in the simulation and the time step was 0.0003 reduced units. Similar remarks to those given to the caption of Fig. 2 apply in respect of the simulated values of when .

A comparison of the theoretical and simulated cavity functions for . The subscripts and denote the theoretical and simulated cavity functions, respectively. There were 864 particles in the simulation and the time step was 0.0003 reduced units. Similar remarks to those given to the caption of Fig. 2 apply in respect of the simulated values of when .

A comparison of the theoretical and simulated pair distribution functions for . The subscripts and denote the theoretical and simulated pair distribution functions, respectively.

A comparison of the theoretical and simulated pair distribution functions for . The subscripts and denote the theoretical and simulated pair distribution functions, respectively.

## Tables

This table shows the values of the required auxiliary integrals , , , , and defined in Sec. II for three temperatures .

This table shows the values of the required auxiliary integrals , , , , and defined in Sec. II for three temperatures .

The quantities given in this table are the simulated values for the dimensionless ratio derived from as described in Sec. III. The columns are headed by the property from which they are derived. The data are for and .

The quantities given in this table are the simulated values for the dimensionless ratio derived from as described in Sec. III. The columns are headed by the property from which they are derived. The data are for and .

Values of the interaction energy per molecule in units of for the cases, and 72. The energy is obtained directly from simulation. The energy is obtained from Eq. (2.14) in the text using the simulated values of the cavity function and its derivative at , as given in Table IV. The energy is obtained in the same way as except that Baxter’s values for the cavity function and its derivative are employed.

Values of the interaction energy per molecule in units of for the cases, and 72. The energy is obtained directly from simulation. The energy is obtained from Eq. (2.14) in the text using the simulated values of the cavity function and its derivative at , as given in Table IV. The energy is obtained in the same way as except that Baxter’s values for the cavity function and its derivative are employed.

The table shows values of the cavity function and of its first derivative at when the value of is 144 or 72; the subscript refers to the values calculated from the simulated results for Z and ; refers to the values obtained from Baxter’s solution of the PY equations for the sticky-sphere model using our values of the stickiness .

The table shows values of the cavity function and of its first derivative at when the value of is 144 or 72; the subscript refers to the values calculated from the simulated results for Z and ; refers to the values obtained from Baxter’s solution of the PY equations for the sticky-sphere model using our values of the stickiness .

The column shows the simulated values of the compressibility factor for the two cases, and 72. The other columns are from theory and are based upon the cavity function obtained by Baxter from his solution of the PY equations for his sticky-sphere model; the strength of the stickiness is as determined by Eq. (1.14) from the potential. has been obtained from Eq. (2.10) and depends only upon the value of this cavity function and its derivative at the radial distance . For and we used Baxter’s formulas for the virial pressure and compressibility pressure, respectively, both of which depend upon the value of the cavity function throughout the range.

The column shows the simulated values of the compressibility factor for the two cases, and 72. The other columns are from theory and are based upon the cavity function obtained by Baxter from his solution of the PY equations for his sticky-sphere model; the strength of the stickiness is as determined by Eq. (1.14) from the potential. has been obtained from Eq. (2.10) and depends only upon the value of this cavity function and its derivative at the radial distance . For and we used Baxter’s formulas for the virial pressure and compressibility pressure, respectively, both of which depend upon the value of the cavity function throughout the range.

This table shows values of for three temperatures. is the simulated value; is the theoretical value obtained using the approximation of Eq. (2.11) with Baxter’s value for the cavity function at .

This table shows values of for three temperatures. is the simulated value; is the theoretical value obtained using the approximation of Eq. (2.11) with Baxter’s value for the cavity function at .

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