Padé approximant for hard sphere ＋ square well and hard sphere ＋ square well ＋ square shoulder model fluids

Abstract

The coupling parameter series expansion (CPSE) is used to calculate coefficientsai (i=1,2,3,…7) of the high temperature series expansion (HTSE) for two hard sphere ＋ square well models and two hard sphere ＋ square well ＋ square shoulder models, the four sets of coefficients are used to construct 28 Padé approximant variants Pn∕d for the four models. Based on the 28 Padé variants, six common thermodynamic functions for the four models are calculated and compared with corresponding results based on MC simulations reported in the literature and the HTSE truncated to seventh order in the present work, respectively. Several conclusions are generalized from the comparisons: (i) the commonly used rule of the minimum value for n−d to choose the best Padé approximation may not select the best one, which depends on the model and relevant parameters values and has to be determined in advance by reference to simulations. (ii) From the whole table of 28 Padé approximants, reliable (throughout the text, by “reliable” we mean the results are at least qualitatively correct) variants always can be singled out, which not only provides an improvement over the raw HTSE truncated at the optimal order but also is free of pole(s) and consequently applicable over reasonably wide phase space of interest; particularly, the improvement can be immense for most but not all of the models considered when the excess heat capacity is being considered. (iii) Among the reliable Pn∕dapproximations picked, some have high values of n+d indicating that the higher-order HTSE coefficients calculated by the CPSE provide important and reliable information for the unknown excess Helmholtz free energy, and the conventional Padé summation indeed provides an effective method to utilize the information. (iv) The Padé summation can be a resort in statistical mechanics perturbation theories of simple potential fluids because of its simplicity in the actual application and possibility in improving the raw HTSE; however, to ensure the use reliably, one comprehensive pre-research should be carried out.