Free Volume Approximation and Equation of State for the fcc Phase of Polydisperse Hard Spheres
Tretiakov Konstantin V. *, Wojciechowski Krzysztof W. **
Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17/19, 60-179 Poznań, Poland
* E-mail: kvt@ifmpan.poznan.pl
** E-mail: kww@ifmpan.poznan.pl
Received:
(Received: 24 November 2012; revised: 7 January 2013; accepted: 14 January 2013; published online: 21 March 2013)
DOI: 10.12921/cmst.2013.19.01.59-64
OAI: oai:lib.psnc.pl:434
Abstract:
Monte Carlo simulations of the fcc phase of polydisperse hard spheres near close packing are reported. An experimental equation of state (EoS) is determined numerically in the NpT ensemble with variable shape of the periodic box. The close packing volume extrapolated from the obtained data shows a good agreement with earlier experiments performed by other methods. A new theoretical EoS, based on the free volume approximation, is proposed. The modified EoS fits experimental data for polydisperse hard spheres better than the approximation used before.
Key words:
colloids, equation of state, hard spheres, Monte Carlo simulations, polydispersity
References:
[1] W. W. Wood and J. D. Jacobson, “Preliminary results from a recalculation of the Monte Carlo equation of state of hard spheres”, J. Chem. Phys., 27, 1207, 1957.
[2] B. J. Alder and T. E. Wainwright, “Phase transition for hard sphere system”, J. Chem. Phys., 27, 1208, 1957.
[3] F. H. Stillinger, R. L. Kornegay, and E. A. DiMarzio, “Systematic approach to explanation of rigid disk phase transition”, J. Chem. Phys., 40, 1564, 1964.
[4] F. H. Stillinger and Z. W. Salsburg, “Elasticity in rigiddisk and sphere crystals”, J. Chem. Phys., 46, 3962, 1967.
[5] W. G. Hoover and F. H. Ree, “Melting transition and communal entropy for hard spheres”, J. Chem. Phys., 49, 3609, 1968.
[6] D. Frenkel and A. J. C. Ladd, “Elastic constants of hard-sphere crystals”, Phys. Rev. Lett., 59, 1169, 1987.
[7] K. J. Runge and G. V. Chester, “Monte-Carlo determination of the elastic constants of the hard-sphere solid”, Phys. Rev. A, 36, 4852, 1987.
[8] L. V. Woodcock, “Entropy difference between the facecentred cubic and hexagonal close-packed crystal structures”, Nature, 385, 141, 1997.
[9] P. G. Bolhuis, D. Frenkel, S.-Ch. Mau, and D. A. Huse, “Entropy difference between crystal phases”, Nature, 388, 235, 1997.
[10] L. V. Woodcock, “Entropy difference between crystal phases – Reply”, Nature, 388, 236, 1997.
[11] K. V. Tretiakov and K. W. Wojciechowski, “Efficient Monte Carlo simulations using a shuffled nested Weyl sequence random number generator”, Phys. Rev. E, 60, 7626–7628, 1999.
[12] O. Farago and Y. Kantor, “Fluctuation formalism for elastic constants in hard-spheres-and-tethers systems”, Phys. Rev. E, 61, 2478, 2000.
[13] K. V. Tretiakov and K. W. Wojciechowski, “Poisson’s ratio of the fcc hard sphere crystal at high densities”, J. Chem. Phys., 123, 074509, 2005.
[14] P. Salgi and R. Rajagopalan, “Polydispersity in colloids – implications to static structure and scattering”, Advances in Colloid and Interface Science, 43, 169–288, 1993.
[15] P. N. Pusey and W. van Megen, “Phase-behavior of concentrated suspensions of nearly hard colloidal spheres”, Nature, 320, 340, 1986.
[16] P. N. Pusey, “Colloidal suspensions”, In J. P. Hansen, D. Levesque, and J. Zinn-Justin, editors, Liquids, Freezing and the Glass Transition Elsevier, 1991.
[17] M. E. Leunissen, Ch. G. Christova, A.-P. Hynninen, C. P. Royall, A. I. Campbell, M. Dijkstra A. Imhof, R. van Roij, and A. van Blaaderen, “Ionic colloidal crystals of oppositely charged particles”, Nature, 437, 235, 2005.
[18] V. W. A. de Villeneuve, R. P. A. Dullens, D. G. A. L. Aarts, E. Groeneveld, J. H. Scherff, W. K. Kegel, and H. N. W. Lekkerkerker, “Colloidal hard-sphere crystal growth frustrated by large spherical impurities”, Science, 309, 1231, 2005.
[19] E. Dickinson and R. Parker, “Polydispersity and the fluid-crystalline phase-transition”, J. Phys. Lett., 46, L229, 1985.
[20] P. G. Bolhuis and D. A. Kofke, “Monte carlo study of freezing of polydisperse hard spheres”, Phys. Rev. E, 54, 634, 1996.
[21] P. G. Bolhuis and D. A. Kofke, “Freezing of polydisperse hard spheres”, Phys. Rev. E, 59, 618, 1999.
[22] S. E. Phan, W. B. Russel, J. X. Zhu, and P. M. Chaikin, “Effects of polydispersity on hard sphere crystals”, J. Chem. Phys., 108, 9789, 1998.
[23] T. Nogawa, N. Ito, and H. Watanabe, “Dynamical study of a polydisperse hard-sphere system”, Phys. Rev. E, 82, 021201, 2010.
[24] M. Yang and H. Ma, “Solid-solid transition of the sizepolydisperse hard sphere system”, J. Chem. Phys., 130, 031103, 2009.
[25] J. L. Barrat and J. P. Hansen, “On the stability of polydisperse colloidal crystals”, J. Phys., 47, 1547, 1986.
[26] R. McRae and A. D. J. Haymet, “Freezing of polydisperse hard-spheres”, J. Chem. Phys., 88, 1114, 1988.
[27] P. N. Pusey, “The effect of polydispersity on the the crystallization of hard spherical colloids”, J. Phys., 48, 709, 1987.
[28] R. P. Sear, “Phase separation and crystallisation of polydisperse hard spheres”, Europhys. Lett., 44, 531, 1998.
[29] P. Bartlett and P. B. Warren, “Reentrant melting in polydispersed hard spheres”, Phys. Rev. Lett., 82, 1979, 1999.
[30] P. Bartlett, “Freezing in polydisperse colloidal suspensions”, J. Phys.: Cond. Matt., 12, A275, 2000.
[31] H. Xu and M. Baus, “Free-volume theory of the freezing of polydisperse hard-sphere mixtures: Initial preparation, fractionation, and terminal polydispersity”, J. Chem. Phys., 118, 5045, 2003.
[32] C. C. Huang and H. Xu, “Polydisperse hard sphere mixtures: equations of state and the melting transition”, Mol. Phys., 102, 967, 2004.
[33] P. Bartlett, “Thermodynamic properties of polydisperse hard spheres”, Mol. Phys., 97, 685, 1999.
[34] M. Yang and H. Ma, “Elasticity of a polydisperse hardsphere csrystal”, Phys. Rev. E, 78, 011404, 2008.
[35] P. N. Pusey, W. van Megen, P. Barlett, B. J. Ackerson, J. G. Rarity, and S. M. Underwood, “Structure of crystals of hard colloidal spheres”, Phys. Rev. Lett., 63, 2753, 1989.
[36] J. Mattsson, H. M. Wyss, A. Fernandez-Nieves, K. Miyazaki, Zh. Hu, D. R. Reichman, and D. A. Weitz, “Soft colloids make strong glasses”, Nature, 462, 83, 2009.
[37] M. Yang and H. Ma, “Effect of polydispersity on the relative stability of hard-sphere crystals”, J. Chem. Phys., 128, 134510, 2008.
[38] M. Fasolo and P. Sollich, “Equilibrium phase behavior of polydisperse hard spheres”, Phys. Rev. Lett., 91, 068301, 2003.
Monte Carlo simulations of the fcc phase of polydisperse hard spheres near close packing are reported. An experimental equation of state (EoS) is determined numerically in the NpT ensemble with variable shape of the periodic box. The close packing volume extrapolated from the obtained data shows a good agreement with earlier experiments performed by other methods. A new theoretical EoS, based on the free volume approximation, is proposed. The modified EoS fits experimental data for polydisperse hard spheres better than the approximation used before.
Key words:
colloids, equation of state, hard spheres, Monte Carlo simulations, polydispersity
References:
[1] W. W. Wood and J. D. Jacobson, “Preliminary results from a recalculation of the Monte Carlo equation of state of hard spheres”, J. Chem. Phys., 27, 1207, 1957.
[2] B. J. Alder and T. E. Wainwright, “Phase transition for hard sphere system”, J. Chem. Phys., 27, 1208, 1957.
[3] F. H. Stillinger, R. L. Kornegay, and E. A. DiMarzio, “Systematic approach to explanation of rigid disk phase transition”, J. Chem. Phys., 40, 1564, 1964.
[4] F. H. Stillinger and Z. W. Salsburg, “Elasticity in rigiddisk and sphere crystals”, J. Chem. Phys., 46, 3962, 1967.
[5] W. G. Hoover and F. H. Ree, “Melting transition and communal entropy for hard spheres”, J. Chem. Phys., 49, 3609, 1968.
[6] D. Frenkel and A. J. C. Ladd, “Elastic constants of hard-sphere crystals”, Phys. Rev. Lett., 59, 1169, 1987.
[7] K. J. Runge and G. V. Chester, “Monte-Carlo determination of the elastic constants of the hard-sphere solid”, Phys. Rev. A, 36, 4852, 1987.
[8] L. V. Woodcock, “Entropy difference between the facecentred cubic and hexagonal close-packed crystal structures”, Nature, 385, 141, 1997.
[9] P. G. Bolhuis, D. Frenkel, S.-Ch. Mau, and D. A. Huse, “Entropy difference between crystal phases”, Nature, 388, 235, 1997.
[10] L. V. Woodcock, “Entropy difference between crystal phases – Reply”, Nature, 388, 236, 1997.
[11] K. V. Tretiakov and K. W. Wojciechowski, “Efficient Monte Carlo simulations using a shuffled nested Weyl sequence random number generator”, Phys. Rev. E, 60, 7626–7628, 1999.
[12] O. Farago and Y. Kantor, “Fluctuation formalism for elastic constants in hard-spheres-and-tethers systems”, Phys. Rev. E, 61, 2478, 2000.
[13] K. V. Tretiakov and K. W. Wojciechowski, “Poisson’s ratio of the fcc hard sphere crystal at high densities”, J. Chem. Phys., 123, 074509, 2005.
[14] P. Salgi and R. Rajagopalan, “Polydispersity in colloids – implications to static structure and scattering”, Advances in Colloid and Interface Science, 43, 169–288, 1993.
[15] P. N. Pusey and W. van Megen, “Phase-behavior of concentrated suspensions of nearly hard colloidal spheres”, Nature, 320, 340, 1986.
[16] P. N. Pusey, “Colloidal suspensions”, In J. P. Hansen, D. Levesque, and J. Zinn-Justin, editors, Liquids, Freezing and the Glass Transition Elsevier, 1991.
[17] M. E. Leunissen, Ch. G. Christova, A.-P. Hynninen, C. P. Royall, A. I. Campbell, M. Dijkstra A. Imhof, R. van Roij, and A. van Blaaderen, “Ionic colloidal crystals of oppositely charged particles”, Nature, 437, 235, 2005.
[18] V. W. A. de Villeneuve, R. P. A. Dullens, D. G. A. L. Aarts, E. Groeneveld, J. H. Scherff, W. K. Kegel, and H. N. W. Lekkerkerker, “Colloidal hard-sphere crystal growth frustrated by large spherical impurities”, Science, 309, 1231, 2005.
[19] E. Dickinson and R. Parker, “Polydispersity and the fluid-crystalline phase-transition”, J. Phys. Lett., 46, L229, 1985.
[20] P. G. Bolhuis and D. A. Kofke, “Monte carlo study of freezing of polydisperse hard spheres”, Phys. Rev. E, 54, 634, 1996.
[21] P. G. Bolhuis and D. A. Kofke, “Freezing of polydisperse hard spheres”, Phys. Rev. E, 59, 618, 1999.
[22] S. E. Phan, W. B. Russel, J. X. Zhu, and P. M. Chaikin, “Effects of polydispersity on hard sphere crystals”, J. Chem. Phys., 108, 9789, 1998.
[23] T. Nogawa, N. Ito, and H. Watanabe, “Dynamical study of a polydisperse hard-sphere system”, Phys. Rev. E, 82, 021201, 2010.
[24] M. Yang and H. Ma, “Solid-solid transition of the sizepolydisperse hard sphere system”, J. Chem. Phys., 130, 031103, 2009.
[25] J. L. Barrat and J. P. Hansen, “On the stability of polydisperse colloidal crystals”, J. Phys., 47, 1547, 1986.
[26] R. McRae and A. D. J. Haymet, “Freezing of polydisperse hard-spheres”, J. Chem. Phys., 88, 1114, 1988.
[27] P. N. Pusey, “The effect of polydispersity on the the crystallization of hard spherical colloids”, J. Phys., 48, 709, 1987.
[28] R. P. Sear, “Phase separation and crystallisation of polydisperse hard spheres”, Europhys. Lett., 44, 531, 1998.
[29] P. Bartlett and P. B. Warren, “Reentrant melting in polydispersed hard spheres”, Phys. Rev. Lett., 82, 1979, 1999.
[30] P. Bartlett, “Freezing in polydisperse colloidal suspensions”, J. Phys.: Cond. Matt., 12, A275, 2000.
[31] H. Xu and M. Baus, “Free-volume theory of the freezing of polydisperse hard-sphere mixtures: Initial preparation, fractionation, and terminal polydispersity”, J. Chem. Phys., 118, 5045, 2003.
[32] C. C. Huang and H. Xu, “Polydisperse hard sphere mixtures: equations of state and the melting transition”, Mol. Phys., 102, 967, 2004.
[33] P. Bartlett, “Thermodynamic properties of polydisperse hard spheres”, Mol. Phys., 97, 685, 1999.
[34] M. Yang and H. Ma, “Elasticity of a polydisperse hardsphere csrystal”, Phys. Rev. E, 78, 011404, 2008.
[35] P. N. Pusey, W. van Megen, P. Barlett, B. J. Ackerson, J. G. Rarity, and S. M. Underwood, “Structure of crystals of hard colloidal spheres”, Phys. Rev. Lett., 63, 2753, 1989.
[36] J. Mattsson, H. M. Wyss, A. Fernandez-Nieves, K. Miyazaki, Zh. Hu, D. R. Reichman, and D. A. Weitz, “Soft colloids make strong glasses”, Nature, 462, 83, 2009.
[37] M. Yang and H. Ma, “Effect of polydispersity on the relative stability of hard-sphere crystals”, J. Chem. Phys., 128, 134510, 2008.
[38] M. Fasolo and P. Sollich, “Equilibrium phase behavior of polydisperse hard spheres”, Phys. Rev. Lett., 91, 068301, 2003.
