Smoluchowski-like equation

The subscripts of H indicate variables with respect to which differentiation has to be made in Eq. (87) (0 for no differentiation) before the dummy variables are set equal to 1/jq and 1/k2 as in Eqs. (89) and (90). The required derivatives are obtained from the Smoluchowski-like equation (88) by... 

Modeling with the Smoluchowski-like equation generalized to take into account FSSE is not limited to the simple RAf polymerization. A kinetics approach similar to that described in this section have been used to study crosslinking reactions of epoxy resins with components introduced into the system at different times [17]. Kinetic equations analogous to Eq. (101) have been derived [48] for an RA2 + R B2 system as well as for systems containing 3-functional monomers having functional groups of intrinsically different reactivities [49]. 

It is not very difficult to extend formally the treatment presented in Sect. 8, namely the Smoluchowski-like equation (Eq. 88), to model, besides the substitution effect, the ability of functional groups to react intramolecularly. For the simplest case of RA3 homopolymerization, a crude method [63] is to code the molecules with four indices two of which count the units with two or three reacted functional groups that are engaged in cycles. The Smoluchowski-like equation reads then... 

Smoluchowski s equation, like the fragmentation equation, can be written in terms of the scaling distribution. Furthermore, general forms may be determined for the tails of the scaling distribution—limits of small mass, xls(t) < 1, and large mass, x/s(t) > 1. The details can be found in van Dongen and Ernst (1988). 

In order to establish the validity condition of a diffusion like equation for the probability of escape of a particle over a potential barrier, the solution of the modified Fokker-Planck equation is compared to the solution of the modified Smoluchowski equation. Since the main contribution to the determination of the escape probability comes from the neighborhood of the maximum in the potential energy (x = x J, the potential energy function was approximated by a parabolic function and the original Fokker-Planck equation was approximated at the vicinity of xmax by (Chandrasekhar, 1943) ... 

At the above optimal concentration, the rate of coagulation of the aerosol happens to be rather small. According to Smoluchowski s equation, the particle number concentration decreases like... 

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

Note that the particle diffusion term is ignored, just like particle dispersion due to SGS motions (this was found justified in a separate simulation). The shape of the sink term in the right-hand term of this equation is due to Von Smoluchowski (1917) while the local value of the agglomeration kernel /i0 is assumed to depend on the local 3-D shear rate according to a proposition due to Mumtaz et al. (1997). 

Equation 6.18 is graphed in Fig. 6.6 for the cases q = 1, 2, 3. The number density of primary particles, pj(t), decreases monotonically with time as these particles are consumed in the formation of floccules. The number densities of the floccules, on the other hand, rise from zero to a maximum at t = (q - l)/2KDp0, and then decline. This mathematical behavior reflects creation of a floccule of given size from smaller floccules, followed by a period of dominance, and finally consumption to form yet larger particle units as time passes. Both experimental data and computer simulations, like that whose visualization appears in Fig. 6.1, are in excellent qualitative agreement with Eq. 6.18 when they are used to calculate the pq(t).13,14 Thus the von Smoluchowski rate law with a uniform rate coefficient appears to capture the essential features of diffusion-controlled flocculation processes. 

A molecular theory for the effect of flow on the orientation of ideal monodisperse rigid rod-like polymers has been developed by Doi (1980) and by Hess (1976). In this theory, a Smoluchowski equation is derived for the probability (u) that a rod-like molecule is oriented parallel to a unit vector u ... 

Doi first proposed the generalized dynamic equations for the concentrated solution of rod-like polymers. Such constitutive equations can be derived from the molecular theory developed by Doi and Edwards (1986). The basis for the molecular theory is the Smoluchowski equation or Fokker-Planck equation in thermodynamics with the mean field approximation of molecular interaction. 

The Smoluchowski equation is used to relate average flow velocity (Vav) to electric field strength E in electroosmotic flows. Under the conditions of a thin double layer or a large channel height (i.e., a plug-like or constant velocity profile), it can be expressed as follows ... 

Like in the case of electrophoresis, the Smoluchowski equation is only valid for particles with thin double layers and negligible surface conductance (low zeta potentials). The theory was later generalized to arbitrary Ka values by Booth [43] for low zeta potentials, and was developed for arbitrary by Stigter [44], Considering the fact that rather concentrated suspensions are often used in sedimentation potential determinations, theories have also been elaborated to include these situations [45-47]. 

One of the ways to get an analytical solution of the Stnoluchowski equation is to use some transformations, which reduce the Smoluchowski equation into Pick s equation [34,35], The interesting fact is that for sufficiently large V , the solution of the Smoluchowski equation behaves like a travelling wave , i.e. it starts to behave like a solution of the following equation ... 

The excess of the volume charge in the diffuse layer causes the origin of the electric potential liquid solution. It is dependent on the distance y from the Helmholtz layer (Figure 8). The potential (f> is conventionally set zero at a big distance from the wall. The value of the potential in the diffuse layer in the closest vicinity to the Helmholtz layer (y = 0) is called the zeta potential, 4> 0) = When longitudinal driving electric field E is applied, the velocity fEOF of the plug-like EOF is related to the zeta potential by the Helmholtz-Smoluchowski equation ... 

Finally, we would like to elaborate the proposed protocol of the high-friction map, eqn (13.17). Its construction is based purely on the thermodynamic consideration, eqn (13.15), validated by the central limit theorem. Therefore it may offer a general rule to obtain the Smoluchowski limit to any phase-space dynamics under study. The protocol proposed in this chapter is based on the fact that the map is universal at formal level and is therefore obtainable with thermodynamic consideration. It means the Smoluchowski dynamics can be taken care of by the related Fokker-Planck equation, upon the universal map being carried out. It is worth pointing out that the resultant diffusion operator in eqn (13.18) clearly originates from only the Hamiltonian part of the... 

The reptation model, like the Rouse model, supposes that the friction involved in dragging the chain through its tube is proportional to the chain length, 5 = N i, Equation (33.33). The diffusion constant Dtubo for the chain moving through the tube is given by the Einstein-Smoluchowski relation, Equa-... 

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