Pair potential colloidal

The pair potential of colloidal particles, i.e. the potential energy of interaction between a pair of colloidal particles as a function of separation distance, is calculated from the linear superposition of the individual energy curves. When this was done using the attractive potential calculated from London dispersion forces, Fa, and electrostatic repulsion, Ve, the theory was called the DLVO Theory (from Derjaguin, Landau, Verwey and Overbeek). Here we will use the term to include other potentials, such as those arising from depletion interactions, Kd, and steric repulsion, Vs, and so we may write the total potential energy of interaction as... 

There are other approximations in the literature that may be more appropriate for particular conditions.12,27,29,35 In many colloidal systems the value of <5 is 2 nm < <5 < 10 nm and until h < 2S the pair potential will be dominated by the other types of interparticle interactions, i.e. it is of short range, albeit very steep. 

In order to utilise our colloids as near hard spheres in terms of the thermodynamics we need to account for the presence of the medium and the species it contains. If the ions and molecules intervening between a pair of colloidal particles are small relative to the colloidal species we can treat the medium as a continuum. The role of the molecules and ions can be allowed for by the use of pair potentials between particles. These can be determined so as to include the role of the solution species as an energy of interaction with distance. The limit of the medium forms the boundary of the system and so determines its volume. We can consider the thermodynamic properties of the colloidal system as those in excess of the solvent. The pressure exerted by the colloidal species is now that in excess of the solvent, and is the osmotic pressure II of the colloid. These ideas form the basis of pseudo one-component thermodynamics. This allows us to calculate an elastic rheological property. Let us consider some important thermodynamic quantities for the system. We may apply the first law of thermodynamics to the system. The work done in an osmotic pressure and volume experiment on the colloidal system is related to the excess heat adsorbed d Q and the internal energy change d E ... 

We have introduced a statistical mechanical approach, illustrating how the material properties and rheology play a role at the microscopic level. Our main reason for doing this is to determine the microstructure and calculate the macroscopic rheological properties. We can now evaluate the coordination number z from Equation (5.30) for our colloid pair potential in Figure 5.9. The variation of z with volume fraction is shown in Figure 5.10. 

An analogy may be drawn between the phase behavior of weakly attractive monodisperse dispersions and that of conventional molecular systems provided coalescence and Ostwald ripening do not occur. The similarity arises from the common form of the pair potential, whose dominant feature in both cases is the presence of a shallow minimum. The equilibrium statistical mechanics of such systems have been extensively explored. As previously explained, the primary difficulty in predicting equilibrium phase behavior lies in the many-body interactions intrinsic to any condensed phase. Fortunately, the synthesis of several methods (integral equation approaches, perturbation theories, virial expansions, and computer simulations) now provides accurate predictions of thermodynamic properties and phase behavior of dense molecular fluids or colloidal fluids [1]. 

Theories or computer simulations used to calculate the potential of mean force W(r) are typically based on numerous simplifying assumptions and approximations (de Kruif, 1999 Bratko et al., 2002 Prausnitz, 2003 de Kruif and Tuinier, 2005 Home et al., 2007 Jonsson et al., 2007). Therefore they can provide only a qualitative or, at best, semi-quantitative description of the potential of mean force. Such calculations are nevertheless useful because they can serve as a guide for trends in the factors determining the interactions of both biopolymers and colloidal particles. Thus, an increase in the absolute value of the calculated negative depth of W(r) may be attributed to a predominant type of molecular feature favouring aggregation or self-association. To assist with such a theoretical analysis, expressions for some of the mean force potentials will be presented here in the discussion of specific kinds of interactions occurring between pairs of colloidal particles covered by biopolymers in food colloids. 

In general terms, the interactions between the colloidal particles with surfaces covered by adsorbed biopolymer layers can be described qualitatively and quantitatively using the appropriate expression for the potential of mean force W(r). Extending the formalism of equation (3.2), at least four separate contributions to W(r) can contribute to the total free energy of interaction between a pair of colloidal particles in the aqueous dispersion medium (a biopolymer solution) (Snowden et al, 1991 Dickinson, 1992 Israelachvili, 1992 Vincent, 1999 de Kruif, 1999 Praus-nitz, 2003) ... 

Bratko, D., Striolo, A., Wu, J.Z., Blanch, J.M., Prausnitz, J.M. (2002). Orientation-averaged pair potentials between dipolar proteins or colloids. Journal of Physical Chemistry B, 106, 2714-2720. 

FIG. 13.4 Stereo pairs of colloidal dispersions generated using computer simulations, (a) Polystyrene latex particles at a volume fraction of 0.13 with a surface potential of 50 mV. The 1 1 electrolyte concentration is 10 7 mol/cm3. The structure shown is near crystallization. (The solid-black and solid-gray particles are in the back and in the front, respectively, in the three-dimensional view.) (b) A small increase in the surface potential changes the structure to face-centered cubic crystals. (Redrawn with permission from Hunter 1989.)... 

Below we take into account the non-linear terms in the kinetic equations containing functionals J (coupling spatial correlations of similar and dissimilar particles) but neglect the perturbation of the pair potentials assuming that il(r, t) = l3U(r). This is justified in the diluted systems and for the moderate particle interaction which holds for low reactant densities and loose aggregates of similar particles. However, potentials of mean force have to be taken into account for strongly interacting particles (defects) and under particle accumulation when colloid formation often takes place [67],... 

The other kind of systems largely studied, consists of polymethylmethacrylate (PMMA) or silica spherical particles, suspended in organic solvents [23,24]. In these solvents Q 0 and uy(r) 0. The particles are coated by a layer of polymer adsorbed on their surface. This layer of polymer, usually of the order of 10-50 A, provides an entropic bumper that keeps the particles far from the van der Waals minimum, and therefore, from aggregating. Thus, for practical purposes uw(r) can be ignored. In this case the systems are said to be sterically stabilized and they are properly considered as suspensions of colloidal particles with hard-sphere interaction [the pair potential is of the form given by Eq. (5)]. 

FIG. 16 Effective pair potential between the colloidal particles in the systems of Fig. 15. In (a) and (b) are shown the cases with n = 0.023 and n = 0.48, respectively. The lines are the results of deconvoluting the radial distribution function using the Ornstein-Zernike equation and three different closure relations HNC, MSA and PY. The closed circles represent the potential of the mean force, which coincides with u(r) at low concentrations. Adapted from Carbajal-Tinoco et al. [42]. 

Our consideration can easily be extended on a variety of colloidal systems. For example, the behavior of a charged particle near a (charged) wall can be described in a similar way, only with a DLYO-like potential instead of a Coulombic-type pair potential used throughout the paper (see [32], for example). 

Preliminary inspection of Equation 2.16 reveals that the Gibbs pair potential leads to repulsion at small interplate separations and attraction at large distances. Because it is U°n and not 17 F that is the appropriate pair potential for describing the effective interaction between the mth and nth macroions in solution under isobaric conditions, the different analytic properties of and U, have profound implications for colloid science. 

A potential function can also be used to describe the force between a pair of colloidal particles. The electromagnetic forces that contribute to W r) can be grouped into several categories, namely excluded volume (or steric), van der Waals, electrostatic, hydrogen bonding, and hydrophobic--------------------------------------------------------------... 

The theory of the kinetic stability of lyophobic colloids is based on the projjerties of the pair potential between charged colloidal particles, which for spherical particles is characterized by an attractive potential due to disjjersion forces of the form of Eq. (6.19), to be discussed in the next section, and a long-range repulsive screened electrostatic potential of the form... 

When two double layers overlap a repulsive pair potential develops which leads to a repulsive pressure. Dispersed like-charged colloids hence repel each other... 

A direct link between theoretical and experimental work on depletion-induced phase separation of a colloidal dispersion due to non-adsorbing polymers was made by De Hek and Vrij [56, 109]. They mixed sterically stabilized silica dispersions with polystyrene in cyclohexane and measured the limiting polymer concentration (phase separation threshold). Commonly, one uses the binodal or spinodal as experimental phase boundary. A binodal denotes the condition (compositions, temperature) at which two or more distinct phases coexist, see Chap. 3. A tie-line connects two binodal points. A spinodal corresponds to the boundary of absolute instability of a system to decomposition. At or beyond the spinodal boundary infinitesimally small fluctuations in composition will lead to phase separation. De Hek and Vrij [56] used the pair potential (1.21) to estimate the stability of colloidal spheres in a polymer solution by calculating the second osmotic virial coefhcient B2 ... 

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