The Propagation Rate of Steps

If the potential of an atomically smooth (stepless) singular face is changed, e.g., to a value below the reversible potential, the enhanced deposition rate would increase the adatom concentration above its equilibrium value. Cad, until the opposite reaction /a,ad, increasing with Cad, takes the balance with /c,ad- Then, with /c.ad = a,ad, from Eq. (8) it follows that 

Here q on is the amount of electricity needed for the completion of one monatomic layer. The factor 2 in Eq. (12) accounts for the two equivalent fluxes from both sides of the step. 

A fraction of the number of atoms incorporated into the step edge at a cathodic overpotential is supplied by direct transfer from the solution. The rate of this direct transfer reaction is determined by the frequencies of deposition and dissolution, Eqs. (1) and (2), applied to the step sites st, i.e., sites adjacent to the step edge. We can define cathodic, anodic, and exchange current densities for these sites by analogy to the adsorption sites [see Eq. (7)]. Then, with v t = DT / mon, the partial propagation rate Vr r due to direct transfer will be given by 

If the exchange current densities /o,ad and /o,st are of the same order of magnitude, the ratio As/5 predetermines a much higher contribution from surface diffusion. 

If two steps are closer than twice the mean displacement distance of adatoms, jco A, the regions of depleted adatom concentration overlap and the surface diffusion decreases in rate with respect to that obtained from Eq. (12). 

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In general, the propagation rate of steps proved to be anisotropic at lower (room) temperatures, producing polygonized growth steps. At lower pH values and higher temperatures, the growth rate becomes isotropic. 

This equation can also be written in terms of the propagation rates of the different types of addition steps which generate the sequences ... 

C2H ] appears in step 3 and this term will drop out from the overall rate because it is not a propagation step, and although it produces C3H this step is not essential for the continuance of the chain. Rate of propagation rate of step 3. This will leave a two-term equation, which can be approximated by the long chains approximation, since rate of initiation [Pg.404]

With eq. (2.47), the propagation rate of a step advancing by direct transfer only (/ = dt) is... 

From the initial linear slope of the current transients, the propagation rate of monatomic steps can easily be determined as a function of overvoltage. Fig. 5.13 shows that this function is linear in the range 0 mV < <6 mV, giving a propagation rate constant k -v/ tj = 2.2 cm s V ... 

The assumption i = Uoo for / = Ic and u = 0 for / < /c is an oversimplification. It is obvious that the time lapse Tc and hence the period of rotation should be in reality larger because the step adjacent to the new one propagates with a speed smaller than Doo. The calculation is complicated because the propagation rates of all the adjacent steps are also dependent on / and hence the propagation rate of the spiral is given by a system of differential equations. A numerical calculation on this system of equations was given by Budevski et who found that irrespective of the form of the spiral, the period of rotation of a polygonized spiral is... 

Here o stands for the propagation rate of a linear step and rJiAfi) is the radius of the circular 2D critical nucleus at the apphed supersaturation A p.. 

In any application of a copolymer the rate of formation of the product, its molecular weight, and the uniformity of its composition during manufacture are also important considerations. While the composition of a copolymer depends only on the relative rates of the various propagation steps, the rate of formation and the molecular weight depend on the initiation and termination rates as well. We shall not discuss these points in any detail, but merely indicate that the situation parallels the presentation of these items for homopolymers as given in Chap. 6. The following can be shown ... 

Autoca.ta.Iysis. The oxidation rate at the start of aging is usually low and increases with time. Radicals, produced by the homolytic decomposition of hydroperoxides and peroxides (eqs. 2—4) accumulated during the propagation and termination steps, initiate new oxidative chain reactions, thereby increasing the oxidation rate. 

The result of the steady-state condition is that the overall rate of initiation must equal the total rate of termination. The application of the steady-state approximation and the resulting equality of the initiation and termination rates permits formulation of a rate law for the reaction mechanism above. The overall stoichiometry of a free-radical chain reaction is independent of the initiating and termination steps because the reactants are consumed and products formed almost entirely in the propagation steps. 

Acrylamide polymerization by radiation proceeds via free radical addition mechanism [37,38,40,45,50]. This involves three major processes, namely, initiation, propagation, and termination. Apart from the many subprocesses involved in each step at the stationary state the rates of formation and destruction of radicals are equal. The overall rate of polymerization (/ p) is so expressed by Chapiro [51] as ... 

The propagation rate constant did not depend on the monomer concentration which corresponds to the first-order propagation step. The activation energy of the propagation calculated according to the variation of Kp with temperature was found to be 6.5 0.5 kcal/mole. 

Also, the rates of the propagation steps are equal to one another (see Problem 8-4). This observation is no surprise The rates of all the steps are the same in any ordinary reaction sequence to which the steady-state approximation applies, since each is governed by the same rate-controlling step. The form of the rate law for chain reactions is greatly influenced by the initiation and termination reactions. But the chemistry that converts reactant to product, and is presumably the matter of greatest importance, resides in the propagation reactions. Sensitivity to trace impurities, deliberate or adventitious, is one signal that a chain mechanism is operative. 

The explanation for autoacceleration is as follows. As polymerisation proceeds there is an increase in the viscosity of the reaction mixture which reduces the mobility of the reacting species. Growing polymer molecules are more affected by this than either the molecules of monomer or the fragments arising from decomposition of the initiator. Hence termination reactions slow down and eventually stop, while initiation and propagation reactions still continue. Such a decrease in the rate of the termination steps thus leads to the observed increase in the overall rate of polymerisation. 

The chemical mechanisms of transition metal catalyses are complex. The dominant kinetic steps are propagation and chain transfer. There is no termination step for the polymer chains, but the catalytic sites can be activated and deactivated. The expected form for the propagation rate is... 

If the chains are long, the composition of the copolymer and the arrangement oi units along the chain are determined almost entirely by the relative rates of the various chain propagation reactions. On the other hand, the rate of polymerization depends not only on the rates of these propagation steps but also on the rates of the termination reactions. Copolymer composition has received far more attention than has the rate of copolymerization. The present section will be confined to consideration of the composition of copolymers formed by a free radical mechanism. 

The product of an isomerization polymerization is thus determined by the relative rates of the propagation and isomerization steps i.e., it is kinetically determined. If isomerization is much faster than propagation, the homopolymer of B is obtained competitive rates will lead to A-B copolymers. 

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