Diffusion coefficients of a monomer

As it was mentioned earlier, in Chapter 3, the chain termination rate constant depends on the translation and segmental diffusion coefficients of a monomer and a macroradical, and can be rewritten by the equation ... 

L of a periodic system, in agreement with previous studies [68,91,112]. Simulations for various system sizes for polymers of lengths Nm = 10, 20, and 40 allow an extrapolation to infinite system size, which yields Do/y/k Ta /mRi 1.7 x 10 , in good agreement with the diffusion coefficient of a monomer in the same solvent. The values of are about 30% larger than the finite-system-size values presented in Fig. 10. Similarly the diffusion coefficient for a polymer chain with excluded vol-lune interactions displays the dependence Dh l/Rn [73]. 

The dynamical properties of polymer molecules in solution have been investigated using MPC dynamics [75-77]. Polymer transport properties are strongly influenced by hydrodynamic interactions. These effects manifest themselves in both the center-of-mass diffusion coefficients and the dynamic structure factors of polymer molecules in solution. For example, if hydrodynamic interactions are neglected, the diffusion coefficient scales with the number of monomers as D Dq /Nb, where Do is the diffusion coefficient of a polymer bead and N), is the number of beads in the polymer. If hydrodynamic interactions are included, the diffusion coefficient adopts a Stokes-Einstein formD kltT/cnr NlJ2, where c is a factor that depends on the polymer chain model. This scaling has been confirmed in MPC simulations of the polymer dynamics [75]. 

In contrast, the Zimm model considers the motion of beads (or monomers) to be hydrodynamically coupled with other monomers. Both the polymer and the solvent molecules within the pervaded volume of the chain move together in dilute solutions. The diffusion coefficient of a chain in the Zimm model is of the same form as the Stokes-Einstein relation [Eq. (8.9)] for diffusion of a colloidal particle in a liquid ... 

Consider a molecule made out of two /-arm stars with Kuhn segments per arm with junction points connected by a central linear strand of Abb Kuhn monomers. This molecule is called a pom-pom polymer. If/= 1, this molecule is linear, while the H-polymer corresponds to /=2. Estimate the /-dependence of relaxation time and diffusion coefficient of a melt of monodisperse pom-pom polymers for /> 1. Consider only single-chain modes and assume that the coordination number of an entanglement network is z. 

Here Dmna is the diffusion coefficient of a monomeric radical, p expresses how the diffusion coefficient scales with degree of polymerization, and a is the root-mean-square end-to-end distance per square root of the number of monomer units in a polymer chain. 

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] ... 

NMR spectroscopic measurements provide support for these conclusions. Specifically, the diffusion coefficient of a mixture of 40 and 43 was first measured at low monomer concentration (total monomer concentration =1.4 mM). Under these conditions, a relatively large diffusion coefficient is obtained (cf. Fig. 12.30e). On the other hand, at a lOx higher monomer concentration (total mraiomer concentration = 14 mM), a diffusion coefficient of roughly 0.33 of that recorded at low concentration is obtained (cf. Fig. 12.30e). Such a finding is consistent with the formation of larger aggregates ([40 43] ) in solution, which are less free to diffuse in solution. 

Fig. 6.2-2. The diffusion coefficient of a dimerizing solute. As a solute dimerizes, its average diffusion coefficient changes from that of the monomer to that of the dimer. The concentration Cj at which this occurs is roughly the reciprocal of the association constant K.

Reference 115 gives the diffusion coefficient of DTAB (dodecyltrimethylammo-nium bromide) as 1.07 x 10" cm /sec. Estimate the micelle radius (use the Einstein equation relating diffusion coefficient and friction factor and the Stokes equation for the friction factor of a sphere) and compare with the value given in the reference. Estimate also the number of monomer units in the micelle. Assume 25°C. 

The permeability tests for alkali metal ions in the aqueous solution were also conducted. When an aqueous salt solution moves to cell 2 through the membrane from cell 1, the apparent diffusion coefficient of the salt D can be deduced from a relationship among the cell volumes Vj and V2, the solution concentration cx and c2, the thickness of membrane, and time t6 . In Table 12, permeabilities of potassium chloride and sodium chloride through the 67 membrane prepared by the casting polymerization technique from the monomer solution in THF or DMSO are compared with each other and with that the permeability through Visking dialyzer tubing. The... 

It is appropriate to differentiate between polymerizations occuring at temperatures above and below the glass transition point(Tg) of the polymer being produced. For polymerizations below Tg the diffusion coefficients of even small monomer molecules can fall appreciably and as a consequence even relatively slow reactions involving monomer molecules can become diffusion controlled complicating the mechanism of polymerization even further. For polymerizations above Tg one can reasonably assume that reactions involving small molecules are not diffusion controlled, except perhaps for extremely fast reactions such as those involving termination of small radicals. 

The diffusion coefficients of this system were determined for disordered micelles and bcc spheres [47]. They were found to be retarded as compared to the disordered state. This retardation is consistent with a hindered diffusion process, D Do exp(- AxN ), with D0 being the diffusion coefficient in the absence of any interactions (i.e. for y -> 0), and A is a prefactor of order unity. Hence, the diffusion barrier increases with the enthalpic penalty xNa, where N represents the number of monomers in the foreign block. In the simplest description of hindered diffusion, the prefactor A remains constant. This model describes the experimental data poorly as A was found to increase with xNa [47]. 

Each submolecule will experience a frictional drag with the solvent represented by the frictional coefficient /0. This drag is related to the frictional coefficient of the monomer unit (0- If there are x monomer units per link then the frictional coefficient of a link is x(0- If we aPply a step strain to the polymer chain it will deform and its entropy will fall. In order to attain its equilibrium conformation and maximum entropy the chain will rearrange itself by diffusion. The instantaneous elastic response can be thought of as being due to an entropic spring . The drag on each submolecule can be treated in terms of the motion of the N+ 1 ends of the submolecules. We can think of these as beads linked... 

Another unique attribute of polymerizations of multifunctional monomers is the dominance of reaction diffusion as a termination mechanism [134,136, 143-146]. Reaction diffusion involves the mobility of radicals by propagation through unreacted functional groups. This termination mechanism is physically different from translation and segmental diffusion termination mechanisms which involve the diffusion of polymer macroradicals and chain segments to bring radicals within a reaction zone before terminating. Whereas normal termination mechanisms are related to the diffusion coefficient of the polymer, reaction diffusion must be considered differently. In essence, reaction diffusion is... 

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