Abstract
Application of integral equation theory to complex fluids is reviewed, with particular emphasis to the effects of polydispersity and anisotropy on their structural and thermodynamic properties. Both analytical and numerical solutions of integral equations are discussed within the context of a set of minimal potential models that have been widely used in the literature. While other popular theoretical tools, such as numerical simulations and density functional theory, are superior for quantitative and accurate predictions, we argue that integral equation theory still provides, as in simple fluids, an invaluable technique that is able to capture the main essential features of a complex system, at a much lower computational cost. In addition, it can provide a detailed description of the angular dependence in arbitrary frame, unlike numerical simulations where this information is frequently hampered by insufficient statistics. Applications to colloidal mixtures, globular proteins and patchy colloids are discussed, within a unified framework.
Similar content being viewed by others
References
Baldini, G., Beretta, S., Chirico, G., Franz, H., Maccioni, E., Mariani, P., Spinozzi, F. 1999. Salt induced association of β-lactoglobulin studied by salt light and X-ray scattering. Macromolecules 32, 6128–6138.
Baxter, R.J. 1978. Percus-Yevick equation for hard spheres with surface adhesion. J Chem Phys 49, 2770–2774.
Blum, L. 1975. Mean spherical model for asymmetric electrolytes. I. Method of solution. Mol Phys 30, 1529–1535.
Blum, L., Høye, J.S. 1978. Solution of the Ornstein-Zernike equation with Yukawa closure for a mixture. J Stat Phys 19, 317–324.
Blum, L., Stell, G. 1979. Polydisperse systems. I. Scattering function for polydisperse fluids of hard or permeable spheres. J Chem Phys 71, 42–46.
Carsughi, F., Giacometti, A., Gazzillo, D. 2000. Small Angle Scattering data analysis for dense polydisperse systems: The FLAC program. Comput Phys Commun 133, 66–75.
Ciccariello, S., Gazzillo, D. 1982. A new form of the correction to the Coulombic contribution in the McMillan-Mayer ion-ion potential. Chem Phys Letts 93, 31–34.
Ciccariello, S., Gazzillo, D. 1983. Thermodynamical behaviour and radial distribution functions of dilute systems of charged rigid spheres. Mol Phys 48, 1369–1381.
Ciccariello, S., Gazzillo, D. 1985. A reasonable candidate for the McMillan-Mayer ion-ion potential of strong 1-1 electrolytes. J Chem Soc, Faraday Trans 2,81, 1163–1177.
Ciccariello, S., Gazzillo, D., Dejak, C. 1982. “Effective” dipolar forces in electrolytic solutions of structure-breaking salts. J Phys Chem 86, 2986–2994.
Debye, P., Hückel, E. 1923. Zur Theorie der Elektrolyten. I. Gefrierpunktserniedrigung und verwandte Erscheinungen (On the Theory of Electrolytes. I. Freezing Point Depression and Related Phenomena). Physicalische Zeit 24, 185–206.
El Mendoub, E.B., Wax, J.F., Charpentier, I., Jakse, N. 2008. Integral equation study of the square-well fluid for varying attraction range. Mol Phys 106, 2667–2675.
Fantoni, R., Gazzillo, D., Giacometti, A. 2005a. Stability boundaries, percolation threshold, and two-phase coexistence for polydisperse fluids of adhesive colloidal particles. J Chem Phys 122, 034901-1–15.
Fantoni, R., Gazzillo, D., Giacometti, A. 2005b. Thermodynamic instabilities of a binary mixture of sticky hard spheres. Phys Rev E 72, 011503-1–15.
Fantoni, R., Gazzillo, D., Giacometti, A., Sollich, P. 2006. Phase behaviour of weakly polydisperse sticky hard spheres. Perturbation theory for the Percus-Yevick solution. J Chem Phys 125, 164504-1–16.
Fantoni, R., Gazzillo, D., Giacometti, A., Miller, M.A., Pastore, G. 2007. Patchy sticky hard spheres: Analytical study and Monte Carlo simulations. J Chem Phys 127, 234507-1–16.
Gazzillo, D. 2010. Dipolar sticky hard spheres within the Percus-Yevick approximation plus orientational linearization. J Chem Phys 133, 034511-1–13.
Gazzillo, D. 2011. Fluids of hard spheres with dipolarlike patch interactions and effect of adding an isotropic adhesion. Mol Phys 109, 55–64.
Gazzillo, D., Giacometti, A. 2000. Structure factors for the simplest solvable model of polydisperse colloidal fluids with surface adhesion. J Chem Phys 113, 9837–9848.
Gazzillo, D., Giacometti, A. 2002. Polydisperse fluid mixtures of adhesive colloidal particles. Physica A 304, 202–210.
Gazzillo, D., Giacometti, A. 2003a. Pathologies in the sticky limit of hard sphere Yukawa models for colloidal fluids: A possible correction. Mol Phys 101, 2171–2179.
Gazzillo, D., Giacometti, A. 2003b. Structure factor for polydisperse fluids of particles with surface adhesion. J Appl Cryst 36, 832–835.
Gazzillo, D., Giacometti, A. 2004. Analytic solutions for Baxter’s model of sticky hard sphere fluids within closures different from the Percus-Yevick approximation. J Chem Phys 120, 4742–4754.
Gazzillo, D., Giacometti, A., Carsughi, F. 1997. Scattering functions for multicomponent mixtures of charged hard spheres, including the polydisperse limit. Analytic expressions in the Mean Spherical Approximation. J Chem Phys 107, 10141–10153.
Gazzillo, D., Giacometti, A., Della Valle, R.G., Venuti, E., Carsughi, F. 1999a. A scaling approximation for structure factors in the integral equation theory of polydisperse nonionic colloidal fluids. J Chem Phys 111, 7636–7645.
Gazzillo, D., Giacometti, A., Carsughi, F. 1999b. Corresponding-states approach to small-angle scattering from polydisperse ionic colloidal fluids. Phys Rev E 60, 6722–6733.
Gazzillo, D., Giacometti, A., Fantoni, R., Sollich, P. 2006a. Multicomponent adhesive hard sphere models and short-ranged attractive interactions in colloidal or micellar solutions. Phys Rev E 74, 051407-1–14.
Gazzillo, D., Giacometti, A., Fantoni, R. 2006b. Phase behaviour of polydisperse sticky-hard-sphere fluids: Analytical solutions and perturbation theory. Mol Phys 104, 3451–3459.
Gazzillo, D., Fantoni, R., Giacometti, A. 2008. Fluids of spherical molecules with dipolarlike nonuniform adhesion: An analyitically solvable anisotropic model. Phys Rev E 78, 0212010-1–20.
Gazzillo, D., Fantoni, R., Giacometti, A. 2009. Local orientational ordering in fluids of spherical molecules with dipolarlike anisotropic adhesion. Phys Rev E 80, 061207-1–7.
Giacometti, A., Gazzillo, D., Pastore, G., Kanti Das, T. 2005. Numerical study of a binary Yukawa model in regimes characteristic of globular proteins in solutions. Phys Rev E 71, 031108-1–10.
Giacometti, A., Lado, F., Largo, J., Pastore, G., Sciortino, F. 2009a. Phase diagram and structural properties for one-patch particles. J Chem Phys 131, 174114-1–14.
Giacometti, A., Pastore, G., Lado, F. 2009b. Liquidvapor coexistence in square-well fluids: A RHNC study. Mol Phys 107, 555–562.
Giacometti, A., Lado, F., Largo, J., Pastore, G., Sciortino, F. 2010. Effect of patch size and number within a simple model of patchy particles. J Chem Phys 132, 174110-1–15.
Ginoza, M. 1990. Simple MSA solution and thermodynamic theory in a hard-sphere Yukawa system. Mol Phys 71, 145–156.
Ginoza, M., Yasutomi, M. 1997. Analytical model of the equation of state of the hard sphere Yukawa polydisperse fluid: Interaction polydispersity effect. Mol Phys 91, 59–63.
Ginoza, M., Yasutomi, M. 1998a. Analytical model of the static structure factor of a colloidal dispersion: interaction polydispersity effect. Mol Phys 93, 399–404.
Ginoza, M., Yasutomi, M. 1998b. Analytical structure factors for colloidal fluids with size and interaction polydispersities. Phys Rev E 58, 3329–3333.
Ginoza, M., Yasutomi, M. 1998c. Analytical model for the thermodynamic properties of a colloidal dispersion with size and’ charge’ polidispersities. Mol Phys 95, 163–167.
Gray, C.G., Gubbins K.E. 1984. Theory of Molecular Fluids, Vol. I. Clarendon Press, Oxford.
Hansen, J.P., McDonald, I.R. 2006. Theory of Simple Liquids, 3rd Edition. Academic Press, Amsterdam.
Hong, L., Cacciuto, A., Luijten, E., Granick, S. 2008. Cluster of amphiphilic colloidal spheres. Langmuir 24, 621–625.
Høye, J.S., Stell, G. 1977. New self-consistent approximations for ionic and polar fluids. J Chem Phys 67, 524–529.
Jackson, G., Chapman, W.G., Gubbins, K.E. 1988. Phase equilibra of associating fluids: Chain molecules with multiple bonding sites. Mol Phys 65, 1057–1079.
Kahl, G., Schöll-Paschinger, E., Stell, G. 2002. Phase transition and critical behavior of simple fluids and their mixtures. J Phys: Condens Mat 14, 9153–9169.
Kern, N., Frenkel, D. 2003. Fluid-fluid coexistence in colloidal systems with short-ranged strongly directional attractions. J Chem Phys 118, 9882–9889.
Lado, F. 1982a. A local thermodynamic criterion for the Reference-Hypernetted Chain equation. Phys Lett A 89, 196–198.
Lado, F. 1982b. Integral equations for fluids of linear molecules. I. General formulation. Mol Phys 47, 283–298.
Lado, F. 1982c. Integral equations for fluids of linear molecules. II. Hard dumbell solution. Mol Phys 47, 299–312.
Morita, T. 1960. Theory of classical fluids: Hypernetted chain approximation III. Prog Theor Phys 23, 829–845.
Pawar, A.B., Kretzchmar, I. 2010. Fabrication, assembly, and application of patchy particles. Macromol Rapid Commun 31, 150–168.
Pini, D., Stell, G., Wilding, N.B. 1998. A liquid-state theory that remains successful in the critical region. Mol Phys 95, 483–494.
Santos, A., Fantoni, R., Giacometti, A. 2009. Thermodynamic consistency of energy and virial routes: An exact proof within the linearized Debye-Hückel theory. J Chem Phys 131, 181105-1–3.
Schöll-Paschinger, E., Levesque, D., Weiss, J.J., Kahl, G. 2005. Phase diagram of a binary symmetric hardcore Yukawa mixtures. J Chem Phys 122, 024507-1–7.
Sciortino, F., Giacometti, A., Pastore, G. 2009. Phase diagram of Janus particles. Phys Rev Lett 103, 237801-1–4.
Sciortino, F., Giacometti, A., Pastore, G. 2010. A numerical study of one-patch colloidal particles: from square-well to Janus. Phys Chem Chem Phys 12, 11869–11877.
Spinozzi, F., Gazzillo, D., Giacometti, A., Mariani, P., Carsughi, F. 2002. Interaction of proteins in solution from smallangle scattering: a perturbative approach. Biophys J 82, 2165–2175.
Stell, G. 1991. Sticky spheres and related systems. J Stat Phys 63, 1203–1221.
Triolo, R., Blum, L., Floriano, M.A. 1977. Simple electrolytes in the Mean Spherical Approximation. III. A workable model for aqueous solutions. J Chem Phys 67, 5956–5959.
Triolo, R., Blum, L., Floriano, M.A. 1978. Simple electrolytes in the Mean Spherical Approximation. II. Study of a refined model. J Phys Chem 82, 1368–1370.
Vervey, E.J., Overbeek, J.T.G. 1948. Theory of the Stability of Lyophobic Colloids. Elsevier, Amsterdam.
Vrij, A. 1978. Light scattering of a concentrated multicomponent system of hard spheres in the Percus-Yevick approximation. J Chem Phys 69, 1742–1747.
Walther, A., Müller, A.H. 2008. Janus particles. Soft Matter 4, 663–668.
White, R.P., Lipson, J.E.G. 2007. Square-well mixtures: a study of their coexistence using theory and simulations. Mol Phys 105, 1983–1997.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Gazzillo, D., Giacometti, A. Effects of polydispersity and anisotropy in colloidal and protein solutions: An integral equation approach. Interdiscip Sci Comput Life Sci 3, 251–265 (2011). https://doi.org/10.1007/s12539-011-0106-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12539-011-0106-5