Monomer diffusion equation

Eq. (5.223) coincides with the monomer diffusion equation proposed by Evans et al. [149] if the rate constant Rb in [149] is replaced by c°[rc ,(c + a c, )]. However, the obtained result is not restricted to the interpretation of the coefficients only, which have been used before. Eq. (5.224) does not coincide with the corresponding diffusion equation in [149] even if we replace Rb by this expression. Unlike the equations derived in the preceding works, the system (5.223) and (5.224) takes into account the polydispersity of micelles and the two-step nature of the micellisation. Actually, the release or incorporation of monomers in the second step of disintegration or formation of micelles is determined not only by their transition from the micellar to the premicellar region and their subsequent disintegration (as characterised by the parameter J) but also by the alteration of the size distribution of micelles. The latter change... 

The second major difference found in vapor-liquid extraction of polymeric solutions is related to the low values of the diffusion coefficients and the strong dependence of these coefficients on the concentration of solvent or monomer in a polymeric solution or melt. Figure 2, which illustrates how the diffusion coefficient can vary with concentration for a polymeric solution, shows a variation of more than three orders of magnitude in the diffusion coefficient when the concentration varies from about 10% to less than 1%. From a mathematical viewpoint the dependence of the diffusion coefficient on concentration can introduce complications in solving the diffusion equations to obtain concentration profiles, particularly when this dependence is nonlinear. On a physiced basis, the low diffiisivities result in low mass-transfer rates, which means larger extraction equipment. 

As mentioned earlier, k may be reduced by an electrostatic interaction, whose magnitude may be given the Fuchs symbol W1, but also because of reversibility (30). Because the radical, upon collision with a particle, may add monomer units and/or terminate with a radical already present in the particle, Hansen and Ugelstad applied the theory of Dankwerts for diffusion with reaction to determine the overall capture rate. Under these conditions the diffusion equation of Fick must be modified to read ... 

Equation (58) indicates that an increase in initiatior concentration will not enhance the rate of polymerization. It can be used for estimating the molecular mass of the polymer assuming, of course, the absence of transfer. The ratio N/q corresponds to the mean time of polymer growth and molecular mass is equal to the product of the number of additions per unit time and the length of the active life time of the radical, kpN/e. An increase in [I] also means a higher value of q, and thus a shortening of the chains. As in Phase II, the polymerized monomer in the particles is supplemented by monomer diffusion from the droplets across the aqueous phase a stationary state is rapidly established with constant monomer concentration in the particle. The rate of polymerization is then independent of conversion (see, for example the conversion curves in Fig. 7). We assume that the Smith-Ewart theory does not hold for those polymerizations where the mentioned dependence is not linear [132], The valdity of the Smith-Ewart theory is limited by many other factors. 

It is well-known that the adsorption kinetics from non-micellar solutions can be described mathematically by a corresponding boundary problem for the diffusion equation [104, 105]. In the case of micellar solutions the diffusion equation for monomers must contain terms taking into account the influence of micelles. The single diffusion equation of monomers must be replaced by a system of two equations (for monomers and micelles). At last, it is necessary to introduce an additional boundary condition, which takes into account that micelles are not surface active. This are all alterations in the formulation of the mathematical problem. However, it will be shown below that the new problem is essentially more complex and can be solved analytically only for very particular situations and after introduction of additional simplifications. 

The first attempt to take into account the two-step kinetic theory of micellisation was made by Fainerman [147]. With that end in view two pairs of diffusion equations (for micelles and monomers) were written down for two situations eorresponding to the fast and slow proeesses. Approximate solutions of the boundary problems for these equations were used subsequently in the course of analysis of experimental data on the adsorption kinetics from micellar solutions [77, 85, 87, 88]. However, as it has been shown by Dushkin et al. [137], this approaeh is equivalent to the PFOR model for the slow proeess and probably eannot be applied to the description of the adsorption kinetics for the fast process. 

A more rigorous approach to the description of the colloid surfactant diffusion to the interfaee was proposed by Noskov [133]. The reduced diffusion equations for micelles and monomers, which take into account the multistep nature of micellisation and the polydispersity of micelles, were derived for time intervals corresponding to the fast and slow processes using the method applied initially by Aniansson and Wall to uniform systems. Analogous equations have been derived later by Johner and Joanny [135] and also by Dushkin et al. [137]. Recently Dushkin has studied also the adsorption kinetics in the framework of a simplified model of quasi-monodisperse micelles. In this case the assumption of the existence of two kinds of micelles permits to study the main features of the surface tension relaxation in real micellar solution [138]. The main steps of the derivation of surfactant diffusion equations in micellar solutions are presented below [133, 134]. 

The equation of monomer diffusion originates from (5.153) in a similar way... 

The Eqs. (5.210) and (5.211) describe the diffusion of monomers and micelles as before. The physical meaning of Eq. (5.212) is not so obvious. It can be regarded as the equation of the local balance of surfactants in micelles. If the initial distribution of micelles is homogeneous and only the monomer concentration is perturbed, the first relaxation proeess can lead to the dependence of the aggregation numbers on space coordinates and time even in absence of a concentration gradient of the total number of micelles. Surfactants can be transferred not only as a result of the monomer diffusion but due to the diffusion of aggregates of different aggregation numbers. This effect is described by equation (5.212). 

The diffusion equations of micelles and monomers obtained in the preceding sections allow us to formulate a mathematical problem of surfactant diffusion to the interface. Investigation of the adsorption kinetics is reduced then to the solution of this problem. It is noteworthy that the diffusion equations (5.210) - (5.211), (5.223), (5.224), (5.226), (5.228) and the results given in the preceding sections on the relaxation kinetics of the concentration perturbations in the... 

If the relationship between ci and Cm is known, the set of Eqs. (5.229), (5.230) can be reduced to a single diffusion equation. Joos and Van Hunsel assumed that the diffusion proceeds essentially slower than the second (slow) step of micellisation so that there is equilibrium between micelles and monomers at any moment [84]... 

If the PFOR model is used, an analytical solution of the surfactant diffusion problem in a micellar solution can be obtained. For this model only the monomer diffusion has to be considered, which is described by the equation [78, 83, 85, 87-89,93,137,138,147]... 

Eq. (5.248) and its modification for a deformed surface [154], together with the corresponding equations for AT [152] and Eq. (5.243) are the only analytical results obtained as solution of the boundary value problem for the diffusion equations of micelles and monomers. An approximate relation for Ay can be also obtained without integration of the diffusion equations with the help of the penetration theory [155], In this case the derivative on the right hand side of Eq. (5.237) is replaced by the ratio of finite differences... 

Peskin et al [1993] have proposed the Brownian ratchet theory to describe the active force production. The main component of that theory was the interaction between a rigid protein and a diffusing object in front of it. If the object undergoes a Brownian motion, and the fiber undergoes polymerization, there are rates at which the polymer can push the object and overcome the external resistance. The problem was formulated in terms of a system of reaction-diffusion equations for the probabilities of the polymer to have certain number of monomers. Two limiting cases, fast diffusion and fast polymerization, were treated analytically that resulted in explicit force/velocity relationships. This theory was subsequently extended to elastic objects and to the transient attachment of the filament to the object. The correspondence of these models to recent experimental data is discussed in the article by Mogilner and Oster [2003]. 

Then, the concentration dependence of the self-diffusion coefficient can be explained as follows. In dilute solutions (but still much higher than the cmc where contribution from the monomer diffusion can be neglected), the self-diffusion coefficient is dominated by the first term of Equation 7.20, Dm- As the concentration increases. Dm decreases rapidly due to the entanglement of worm-like micelles, whereas the second term increases due to the decrease of ((d) depends on the concentration only slightly [13, 18, 21)), leading to the minimum in the plot of Ds vs. c. 

V acrylate the bulk monomer viscosity of the acrylate. Equation 4.15 is an approximation as the diffusion coefficient of a diffusing entity does not only depend upon the viscosity of the medium, but also on the charateristics of the dififusant itself At 50°C, the monomer diffusion coeficient of MMA is relatively well determined and has been found to be equal to 4.11TO m s [36]. The viscosities of MMA, MA, EA and BA, that are needed to estimate a diffusion coefficient for these monomers on the basis of equation 4.15, were calculated according to [37] ... 

Notice that Eq. (46) does not contain a polymerization reaction term. Because the multigrain model assumes that polymerization takes place at the surface of the catalyst fragment embedded within the primary particle, the reaction term appears as one of the two required boundary conditions [Eq. (48)]. Equation (48) states that the rate of monomer diffusion at the surface of the catalyst fragment, Rc, equals the rate of monomer consumption due to polymerization at rate of R, and Eq. (49) imposes the condition that the concentration at the surface of the primary particle equals the equilibrium concentration of monomer absorbed onto the polymer phase,... 

KAPRAL - If the diffusion equation contains monomer interactions and a diffusion tensor that incorporates hydrodynamic interactions, the calculation of the stationary density is not simple. How are the extrema in this possibility density determined in order to apply your method ... 

In Equation 3.26, T is the equilibrium surface excess, C the bulk concentration, t the time, and D the surfactant monomer diffusion coefficient. Eastoe et al. have measured the time dependence of the DST and the relaxation time %2 for solutions of many surfactants nonionic, dimeric, and zwitterionic. In all instances the fitting of the data to Equation 3.26 with the experimentally determined value of %2 was poor. The authors concluded that the micelle dissociation may have an effect on the measured DST only if the concentration of monomeric surfactant in the subsurface diffusion layer is limiting or when the micelle lifetimes are extremely long. No surfactant for which this last condition is fulfilled was evidenced by the authors. They also concluded that the rapid dissociation of monomers from micelles present in the subsurface was not likely to limit the surfactant adsorption and thus the DST. 

Limitations on neutron beam time mean that only selected surfactants can be investigated by OFC-NR. However, parametric and molecular structure studies have been possible with the laboratory-based method maximum bubble pressure tensiometry (MBP). This method has been shown to be reliable for C > 1 mM.2 Details of the data analysis methods and limitations of this approach have been covered in the literature. Briefly, the monomer diffusion coefficient below the cmc, D, can be measured independently by pulsed-field gradient spin-echo NMR measurements. Next, y(t) is determined by MBP and converted to F(0 with the aid of an equilibrium equation of state determined from a combination of equilibrium surface tensiometry and neutron reflection. The values of r(f) are then fitted to a diffusion-controlled adsorption model with an effective diffusion coefficient which is sensitive to the dominant adsorption mechanism 1 for... 

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