Colloids dynamic equations

Even a simple mathematical model for transport on colloids in an aquifer must include dynamic equations for the dissolved phase and for the colloids. The latter equation describes the migration, immobilization, and detachment of the colloids. More sophisticated models include dynamic equations for sorption and desorption of the chemical onto colloids and the stationary solid phase. 

There are two trends in the existing hydrodynamic theories of free films, which differ in the way the (colloidal) interaction forces are taken into account. We mention first the method of Felderhof, who developed a systematic and consistent electrohydrodynamic theory for a nonviscous liquid. His theory was extended to include viscous behavior by Sche and Fijnaut. In these theories the interaction forces are included in the momentum equations. The other theoretical approach considers hydro-dynamic equations without the interaction forces. The influence of these interactions are considered only in the normal stress boundaiy conditions. 

Gelbard, F., and Seinfeld J. H. (1979) The General Dynamic Equation for aerosols—theory and application to aerosol formation and growth, J. Colloid Interface Sci. 68, 363-382. 

Pilinis, C., and Seinfeld, J. H. (1987) Asymptotic solution of the aerosol general dynamic equation for small coagulation, J. Colloid Interface Sci. 115, 472-479. 

These processes can be described by a set of nonlinear, coiqiled electrokinetic and convective diffusion equations for ion densities, in combination widi Q Navier-Stokes equations for the mass current, indicating that colloidal dynamics. 1 are nonlinear ( 71-72). 

Second, the dynamic equations for polymer motion and for colloid motion are qualitatively the same, namely they are generalized Langevin (e.g., Mori-Zwanzig) equations, including direct and hydrodynamic forces on each colloid particle or polymer segment, hydrodynamic drag forces, and random thermal forces due to solvent motion, all leading to coupled diffusive motion. 

Because the forces and the dynamic equations of motion are fundamentally the same, it should not be surprising that the dynamic behaviors of polymers and colloids have substantial similarities. Of course, polymer chains and colloidal spheres do differ in shape, flexibility, and porosity, so the dynamic properties of colloidal spheres and polymers should not be expected to be quantitatively identical. 

Furthermore, the dynamic equations differ in one substantial respect, namely that colloidal spheres are free to move with respect to each other so long as they do not interpenetrate, but the segments ( beads ) of a single polymer chain are obliged to remain attached to each other for all time. Sphere motion at very large concentrations encounters jamming, in which many spheres all get in each other s way, but polymer chains at far smaller concentrations are said to encounter topological obstacles, similar to those found with a poorly-wound ball of yam. 

A second catch is the noise. If one observes the movements of a colloidal particle, the Brownian motion will be evident. There may be a constant drift in the dynamics, but the movement will be irregular. Likewise, if one observes a phase-separating liquid mixture on the mesoscale, the concentration levels would not be steady, but fluctuating. The thermodynamic mean-field model neglects all fluctuations, but they can be restored in the dynamical equations, similar to added noise in particle Brownian dynamics models. The result is a set of stochastic diffusion equations, with an additional random noise source tj [20]. In principle, the value and spectrum of the noise is dictated by a fluctuation dissipation theorem, but usually one simply takes a white noise source. 

We now have all the elements needed to define a self-consistent system of equations to describe the full dynamic properties of a colloidal dispersion in the absence of hydrodynamic interactions. In this section we summarize the relevant equations for both, mono-disperse and multicomponent suspensions, and review some illustrative applications. The general results for A (t), F(k, t), and F k, t) in Equations 1.20,1.23, and 1.24, complemented by either one of the Vineyard-like approximations in Equations 1.25 and 1.26, and with the closure relation in Equation 1.27, constitute the full self-consistent GLE theory of colloid dynamics for monodisperse systems. Besides the unknown dynamic properties, it involves the properties SQi), t), and t), assumed to be deter-... 

To illustrate these ideas let us summarize the general system of equations that constitute the SCGLE theory. In principle, these are the exact results for A (f), F k, f), and t) in Equations 1.20,1.23, and 1.24, complemented with the simplified Vineyard approximation in Equation 1.37 and the simplified interpolating closure in Equation 1.38. This set of equations define the SCGLE theory of colloid dynamics. Its full solution also yields the value of the long-time self-diffusion coefficient which is the order parameter appropriate to detect the glass transition from the fluid side. This is, however, not the only method to detect dynamic arrest transition, as we now explain. 

Yeomans-Reyna, L. and Medina-Noyola, M. 2001. Self-consistent generalized Langevin equation for colloid dynamics. Phys. Rev. E 64 066114. 

Yeomans-Reyna, L., Chavez-Rojo, M. A., Ramfrez-Gonzalez, P. E., Juarez-Maldonado, R., Chavez-Paez, M., and Medina-Noyola, M. 2007. Dynamic arrest within the self-consistent generalized Langevin equation of colloid dynamics. Phys. Rev. E 76 041504. 

There are several attractive features of such a mesoscopic description. Because the dynamics is simple, it is both easy and efficient to simulate. The equations of motion are easily written and the techniques of nonequilibriun statistical mechanics can be used to derive macroscopic laws and correlation function expressions for the transport properties. Accurate analytical expressions for the transport coefficient can be derived. The mesoscopic description can be combined with full molecular dynamics in order to describe the properties of solute species, such as polymers or colloids, in solution. Because all of the conservation laws are satisfied, hydrodynamic interactions, which play an important role in the dynamical properties of such systems, are automatically taken into account. 

The computer interface system lends itself well to the determination of interfacial tension and contact angles using Equation 3 and the technique described by Pike and Thakkar for Wilhelmy plate type experiments (20). Contact angles for crude oil/brine systems using the dynamic Wilhelmy plate technique have been determined by this technique and all three of the wetting cycles described above have been observed in various crude oil/brine systems (21) (Teeters, D. Wilson, J. F. Andersen, M. A. Thomas, D. C. J. Colloid Interface Sci., 1988, 126, in press). The dynamic Wilhelmy plate device also addresses other aspects of wetting behavior pertinent to petroleum reservoirs. 

The above model assumes that both components are dynamically symmetric, that they have same viscosities and densities, and that the deformations of the phase matrix is much slower than the internal rheological time [164], However, for a large class of systems, such as polymer solutions, colloidal suspension, and so on, these assumptions are not valid. To describe the phase separation in dynamically asymmetric mixtures, the model should treat the motion of each component separately ( two-fluid models [98]). Let Vi (r, t) and v2(r, t) be the velocities of components 1 and 2, respectively. Then, the basic equations for a viscoelastic model are [164—166]... 

In Section 4.7c we outlined the types of effects one can expect in the response of charged dispersions to deformation. In this section, we present some results for the viscosity of charged colloids for which electroviscous effects could be important. As mentioned above, we shall not go into the theoretical details behind the equations since they require a fairly advanced knowledge of fluid dynamics and, in some cases, statistical mechanics. Moreover, some of... 

Note that the above study is performed for a simple system. There exists a large body of literature on the study of diffusion in complex quasi-two-dimensional systems—for example, a collodial suspension. In these systems the diffusion can have a finite value even at long time. Schofield, Marcus, and Rice [17] have recently carried out a mode coupling theory analysis of a quasi-two-dimensional colloids. In this work, equations for the dynamics of the memory functions were derived and solved self-consistently. An important aspect of this work is a detailed calculation of wavenumber- and frequency-dependent viscosity. It was found that the functional form of the dynamics of the suspension is determined principally by the binary collisions, although the mode coupling part has significant effect on the longtime diffusion. 

If the soil suspension were instead an aqueous solution, a scale of activity values for Na+ could be defined in terms of emf data obtained for standard reference solutions of prescribed (Na+), in exactly the same way as the scale of (H) values (the operational pH scale) is defined (arbitrarily) in terms of emf data for standard buffer solutions.39,40 However, the success of this extrathermo-dynamic calibration technique depends entirely on the extent to which E, and B in the standard reference solutions are the same as E, and B in the solution of interest. For the case of a soil suspension, the presence of colloidal material may cause these two parameters to differ very much from what they would be in a reference aqueous solution. If the difference is indeed large, the value of (Na+), m, or any other ionic activity estimated with the help of standard solutions and an equation like Eq. s2.23 would be of no chemical significance. 

The third category of methods that have been used to simulate colloid interactions are generally referred to as Molecular Dynamics (MD) methods. These methods can be considered to be the most exact and computationally intensive, and have been adapted to colloid interactions from the general field of fluid mechanics. MD methods proceed by the numerical approximation of the equations of motion, and thus are deterministic and primarily applicable to systems with ideal or simple geometry. The majority of MD applications involve the interactions between spheres, flat surfaces, cylinders, and combinations of such geometries. 

For colloidal semiconductor systems, Albery et al. observed good agreement between the value of the radial dispersion obtained from dynamic light scattering and the value found from application of the above kinetic analysis to flash photolysis experiments [144], It should be remembered that this disperse kinetics model can only be applied to the decay of heterogeneous species under unimolecular or pseudo-first order conditions and that for colloidal semiconductors it may only be applied to dispersions whose particle radii conform to equation (37), i.e., a log normal distribution. However, other authors [145] have recently refined the model so that assumptions about the particle size distribution may be avoided in the kinetic data analysis. 

Simulation techniques suitable for the description of phenomena at each length-scale are now relatively well established Monte Carlo (MC) and Molecular Dynamics (MD) methods at the molecular length-scale, various mesoscopic simulation methods such as Dissipative Particle Dynamics (Groot and Warren, 1997), Brownian Dynamics, or Lattice Boltzmann in the colloidal domain, Computational Fluid Dynamics at the continuum length-scale, and sequential-modular or equation-based methods at the unit operation/process-systems level. 

A broad range of physical and chemical processes occur as a result of an encounter between pairs of particles. Depending on the physical phenomenon being investigated, the particles may range in size from the atomic scale to colloidal, while the bath density may range from that of a low-pressure gas to a dense liquid. The primary aim of this chapter is to show that a simple model for encounter dynamics can be derived from the Fokker-Planck equation (FPE), which applies, within limits, over the entire range of particle size and bath density. 

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