Relaxation controlled diffusion

Reaction scheme, defined, 9 Reactions back, 26 branching, 189 chain, 181-182, 187-189 competition, 105. 106 concurrent, 58-64 consecutive, 70, 130 diffusion-controlled, 199-202 elementary, 2, 4, 5, 12, 55 exchange, kinetics of, 55-58, 176 induced, 102 opposing, 49-55 oscillating, 190-192 parallel, 58-64, 129 product-catalyzed, 36-37 reversible, 46-55 termination, 182 trapping, 2, 102, 126 Reactivity, 112 Reactivity pattern, 106 Reactivity-selectivity principle, 238 Relaxation kinetics, 52, 257 -260 Relaxation time, 257 Reorganization energy, 241 Reversible reactions, 46-55 concentration-jump technique for, 52-55... 

Exchange of counter-ions (and solvent) between the polymer and the solution in order to keep the electroneutrality in the film. In a compacted or stressed film, these kinetics are under conformational relaxation control while the structure relaxes. After the initial relaxation, the polymer swells, and conformational changes continue under counter-ion diffusion control in the gel film from the solution. 

After polarization to more anodic potentials than E the subsequent polymeric oxidation is not yet controlled by the conformational relaxa-tion-nucleation, and a uniform and flat oxidation front, under diffusion control, advances from the polymer/solution interface to the polymer/metal interface by polarization at potentials more anodic than o-A polarization to any more cathodic potential than Es promotes a closing and compaction of the polymeric structure in such a magnitude that extra energy is now required to open the structure (AHe is the energy needed to relax 1 mol of segments), before the oxidation can be completed by penetration of counter-ions from the solution the electrochemical reaction starts under conformational relaxation control. So AHC is the energy required to compact 1 mol of the polymeric structure by cathodic polarization. Taking... 

The overall charge (Qt) consumed to oxidize the film by a potential step from Ec to E has two components the charge consumed to relax the compact structure, which will be called the relaxation charge (Qr), and the charge consumed under diffusion control to complete the oxidation, called the later diffusion charge (Qd)- The following equation is obeyed ... 

When a polymer relaxes at a constant anodic potential, the relaxation and partial opening of the polymeric structure involve a partial oxidation of the polymer. Once relaxed, the oxidation and swelling of the relaxed polymer goes on until total oxidation is reached this is controlled by the diffusion of the counter-ions through the film from the solution. This hypothesis seems to be confirmed by the current decay after the chronoam-perometric maximum is reached. We will focus now on the diffusion control. 

An infinitesimal fraction of the polymer will be considered, consisting of all the segments that are relaxed at the same time (f). As result of the flow given by Eq. (32), the increment of charge stored under diffusion control [dQd(t) in this infinitesimal portion of the polymer at a given time t>f, will be given by... 

Equations (37) and (38), along with Eqs. (29) and (30), define the electrochemical oxidation process of a conducting polymer film controlled by conformational relaxation and diffusion processes in the polymeric structure. It must be remarked that if the initial potential is more anodic than Es, then the term depending on the cathodic overpotential vanishes and the oxidation process becomes only diffusion controlled. So the most usual oxidation processes studied in conducting polymers, which are controlled by diffusion of counter-ions in the polymer, can be considered as a particular case of a more general model of oxidation under conformational relaxation control. The addition of relaxation and diffusion components provides a complete description of the shapes of chronocoulograms and chronoamperograms in any experimental condition ... 

These equations describe the full oxidation of a conducting polymer Submitted to a potential step under electrochemically stimulated confer-mational relaxation control as a function of electrochemical and structural variables. The initial term of /(f) includes the evolution of the current consumed to relax the structure. The second term indicates an interdependence between counter-ion diffusion and conformational changes, which are responsible for the overall oxidation and swelling of the polymer under diffusion control. 

The charge consumed by oxidation swelling under diffusion control, once the structure is relaxed, depends on the anodic potentials applied at each moment. The process can be quantified by Fick s law ... 

As in chronoamperograms, the fraction of the overall oxidation charge involved in relaxation processes is quite small in the absence of any external stress. The share of the overall current at every potential between electrochemical processes occurring under relaxation control and those driven by swelling-diffusion control can be observed in Fig. 66. I(r) has... 

This relative importance of relaxation and diffusion has been quantified with the Deborah number, De [119,130-132], De is defined as the ratio of a characteristic relaxation time A. to a characteristic diffusion time 0 (0 = L2/D, where D is the diffusion coefficient over the characteristic length L) De = X/Q. Thus rubbers will have values of De less than 1 and glasses will have values of De greater than 1. If the value of De is either much greater or much less than 1, swelling kinetics can usually be correlated by Fick s law with the appropriate initial and boundary conditions. Such transport is variously referred to as diffusion-controlled, Fickian, or case I sorption. In the case of rubbery polymers well above Tg (De < c 1), substantial swelling may occur and... 

A normal proton transfer was defined by Eigen as one whose rate in the thermodynamically favourable direction was diffusion-controlled (Eigen, 1964). By use of relaxation techniques Eigen was able to show that many proton transfers involving oxygen and nitrogen acids and bases were in this category. If the reactions (5) of an acid (HA) with a series of bases (B-) shows normal proton-transfer behaviour, the rate coefficients in the forward... 

Perlmutter-Hayman and Shinar (15, 16) have studied by temperature-jump the reactions of bases with different acid-base indicators having intramolecular hydrogen bonds. With Tropaeolin 0, direct attack of the base on the hydrogen bridge predominates according to their interpretation, whereas, for Alizarin Yellow G, the observed relaxation is ascribed chiefly to diffusion controlled reaction between the base and that part of the indicator present in the open form. Thus, data exist that lead one to doubt the generality of statement number 5. 

Is a primary constraint the central problem in any analysis of ionization mechanisms is the kinetic study of the Interconversion processes between the different species for such a kinetic investigation to be complete all the elementary processes should be analyzed for their energetic and dynamic properties. Since the elementary steps in ionic association-dissociation processes are usually very fast - to the limit of diffusion- controlled reactlons-their kinetic investigation became only feasible with the advent of fast reaction techniques, mainly chemical relaxation spectrometric techniques. 

Relaxation time may be too slow transient runs into diffusion control... 

Knibbe, D. E. Diffusion-controlled stress relaxation of swollen rubber-like networks, Rotterdam University Press 1968. 

In Section 4.7, we discussed the relaxation process of SE s in a closed system where the number of lattice sites is conserved (see Eqn. (4.137)). A set of coupled differential equations was established, the kinetic parameters (v(x,iq,x )) of which describe the rate at which particles (iq) change from sublattice x to x. We will discuss rate parameters in closed systems in Section 5.3.3 where we deal with diffusion controlled homogeneous point defect reactions, a type of reaction which is well known in chemical kinetics. 

Our neglect of the discreteness of the crystal lattice does not introduce noticeable discrepancies. Continuum theory, based on the original concept by Smoluchowski, is quite acceptable for the description of diffusion controlled relaxation of irregular structure elements in crystals. 

When the point defect relaxation is diffusion controlled, we can use Eqn. (5.89) to determine k. After setting rAB = aAX (= unit cell dimension), it is found that at even moderate temperatures (= 100°C), x is on the order of a millisecond or less. This r is many orders of magnitude shorter than relaxation times for nonstoichiometric compounds where the point defect pairs equilibrate at external surfaces (Section 5.3.2). In other words, intrinsic defects equilibrate much faster than extrinsic defects if, during the defect equilibration, the number of lattice sites is conserved. 

Figure 10-10. Representation of the chemical potential of A during the heterogeneous solid state reaction A+B = AB. a) Diffusion control, b) interface control at b2, c) rate control by rearrangement (relaxation) of A in B in zone A (B), d) simultaneous diffusion and interface control (bj).

Proceeding systematically, diffusion controlled a-fi transformations of binary A-B systems should be discussed next when a and / are phases with extended ranges of homogeneity. Again, defect relaxations at the moving boundary and in the adjacent bulk phases are essential for their understanding (see, for example, [F. J. J. van Loo (1990)]). The morphological aspects of this reaction type are dealt within the next chapter. 

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