Diffusion-controlled entry

Smith and Ewart [4] first proposed that the transfer of free radical activity into the interior of a polymer particle takes place by the direct entry of a free radical into a polymer particle. They pointed out that the rate of radical entry into a polymer particle is given by the rate of diffusion of free radicals from an infinite medium of concentration into a particle of diameter dp with zero radical concentration. 

On the other hand, Nomura and Harada [14] proposed a kinetic model for the emulsion polymerization of styrene (St), where they used Eq. 7 to predict the rate of radical entry into both polymer particles and monomer-swollen micelles. In their kinetic model, the ratio of the mass-transfer coefficient for radical entry into a polymer particle kep to that into a micelle kem K lk, 

The concept of radical capture efficiency was further elaborated on by Hansen et al. [15-17]. By applying the theory of mass transfer with simultaneous chemical reactions, they proposed the following expression to represent the net rate of radical absorption by a particle, introducing an absorption efficiency factor F into Eq. 8 

Therefore, F represents a factor that describes the degree to which absorption is lowered compared to irreversible diffusion, and is given by 

A much simpler model for the radical capture (absorption) efficiency F can be derived by introducing the concept of radical desorption from a polymer particle, developed in Section 3.2.1. The probability F for a radical to be captured inside a particle containing n radicals by any chemical reaction (propagation or termination) is given by 

---

Since enzyme catalysis and regulation are dynamic processes, the dynamics of hydrogen bonding is also an important consideration. In relatively weakly hydrogen bonding solvents, such as the first four entries in Table I, the association rate is essentially diffusion controlled, whereas the association rate constants for the last two entries are considerably less than expected for a diffusion-controlled process. This can be understood... 

With an E° value of —0.75 V, entry no. 19 of Table 17, reaction between alkyl halides and alkyllithium compounds, represents a strongly exergonic electron-transfer reaction which is expected to proceed at a diffusion-controlled tate. Experimental rate constants are not available, but such reactions are qualitatively known to be very fast. As we proceed to entry no. 21, two model cases of the nucleophilic displacement mechanism, it can first be noted that the nosylate/[nosylate]- couple is electrochemically reversible the radical anion can be generated cathodically and is easily detected by esr spectroscopy (Maki and Geske, 1961). Hence its E° = —0.61 V is a reasonably accurate value. E° (PhS /PhS-) is known with considerably less accuracy since it refers to an electrochemically irreversible process (Dessy et al., 1966). The calculated rate constant is therefore subject to considerable uncertainty and it cannot at present be decided whether the Marcus theory is compatible with this type of electron-transfer step. In the absence of quantitative experimental data, the same applies to entry no. 22 of Table 17. For the PhS-/BuBr reaction we again suffer from the inaccuracy of E° (PhS /PhS-) what can be concluded is that for an electron-transfer step to be feasible the higher E° value (—0.74 V) should be the preferred one. The reality of an electron-transfer mechanism has certainly been strongly disputed, however (Kornblum, 1975). 

One of the most important parameters in the S-E theory is the rate coefficient for radical entry. When a water-soluble initiator such as potassium persulfate (KPS) is used in emulsion polymerization, the initiating free radicals are generated entirely in the aqueous phase. Since the polymerization proceeds exclusively inside the polymer particles, the free radical activity must be transferred from the aqueous phase into the interiors of the polymer particles, which are the major loci of polymerization. Radical entry is defined as the transfer of free radical activity from the aqueous phase into the interiors of the polymer particles, whatever the mechanism is. It is beheved that the radical entry event consists of several chemical and physical steps. In order for an initiator-derived radical to enter a particle, it must first become hydrophobic by the addition of several monomer units in the aqueous phase. The hydrophobic ohgomer radical produced in this way arrives at the surface of a polymer particle by molecular diffusion. It can then diffuse (enter) into the polymer particle, or its radical activity can be transferred into the polymer particle via a propagation reaction at its penetrated active site with monomer in the particle surface layer, while it stays adsorbed on the particle surface. A number of entry models have been proposed (1) the surfactant displacement model (2) the colhsional model (3) the diffusion-controlled model (4) the colloidal entry model, and (5) the propagation-controlled model. The dependence of each entry model on particle diameter is shown in Table 1 [12]. 

However, some of these models have been refuted, and two major entry models are currently widely accepted. One is the diffusion-controlled model, which assumes that the diffusion of radicals from the bulk phase to the surface... 

Two major entry models - the diffusion-controlled and propagation-controlled models - are widely used at present. However, Liotta et al. [28] claim that the collision entry is more probable. They developed a dynamic competitive growth model to understand the particle growth process and used it to simulate the growth of two monodisperse polystyrene populations (bidisperse system) at 50 °C. Validation of the model with on-line density and on-line particle diameter measurements demonstrated that radical entry into polymer particles is more likely to occur by a collision mechanism than by either a propagation or diffusion mechanism. 

The bimolecular termination reaction in free-radical polymerization is a typical example of a diffusion controlled reaction, and is chain-length-depen-dent [282-288]. When pseudobulk kinetics appUes, the MWD formed can be approximated by that resulting from bulk polymerization, and it can be solved numerically [289-291]. As in the other extreme case where no polymer particle contains more than one radical, the so-caUed zero-one system, the bimolecular termination reactions occur immediately after the entrance of second radical, so unique features of chain-length-dependence cannot be found. Assuming that the average time interval between radical entries is the same for all particles and that the weight contribution from ohgomeric chains formed... 

At low monomer concentration (that is, at high monomer conversion), entry, propagation, and termination become diffusion-controlled processes. 

The entry processes, and hence the rate coefficients, and are assumed to be diffusion controlled... 

The rate constant for the reverse of Reaction 1 is 1.8 X 107 liter/mole sec. (5). This value is somewhat less than would be expected for a diffusion controlled reaction. If the pre-exponential factor is near the 1010 liter/mole sec. considered normal for an activation-controlled reaction of an ion with a neutral molecule, the Arrhenius energy of activation would be about 3.8 kcal./mole in reasonable agreement with the value of 4.5 kcal./mole based on the AHr entry in Table II. Since the transition state for Reaction 1 almost certainly has negative charge more dispersed than in the neighborhood of an hydroxide ion, the pre-exponential term for the reverse reaction may even be somewhat more positive than the normal 1010 liter/mole sec., and the enthalpy of activation would then be larger also. Even if the correct enthalpy of activation is less than the value quoted in Table II, the difference could hardly be more than 2 kcal./mole. 

This equation, as a double reciprocal plot, is similar to Eq. (5) but is applied to probe mobility, not quencher dynamics. The same values for the dissociation rate constants (Table 17) were obtained when employing these two different methods with quenchers in different phases, suggesting that the underlying assumptions for the derivation of Eqs. (27) and (28) were reasonable. The entry rate constants were diffusion controlled, and the exit rate constants varied by a factor of 3. In analogous fashion to the polycyclic aromatic hydrocarbon probes, the exit rate constants were faster for the more polar ketones p-methoxyacetophenone and acetophenone compared to isobutyro-phenone or propiophenone [193,194],... 

Triangular Pulse With the triangular pulse method, each side of a membrane is initially held at a constant potential so that permeation occurs in a normal manner [109]. After a steady state is achieved, an anodic or cathodic triangular pulse is applied to the entry side and the change in the oxidation current is measured at the output side. The duration of the pulse is typically 0.01 to 0.03 s. Analytical solutions for the current have been obtained for pure diffusion control and for entry-limited diffusion control. An anodic current peak is obtained in response to the triangular pulse, and the time corresponding to the half-peak width is characteristic of the type of kinetic control. 

The above controversy regarding the physical mechanism of radical entry is reflected in the theoretical expressions which were developed for describing the rate of radical entry by various workers. Hansen and Ugelstad [27,41] summarized the various dependencies of the rate of radical capture on the dimensions of the particles or micelles as follows. Depending on what is the rate-determining step in the absorption process, radical capture rate may be proportional to either the radius of the micelles or particles (for diffusion control in the water phase, when the water solubility of the radicals is low and/or the particles are large), their surface area (for diffusion control in the monomer/polymer phase where the diffusion constant in this phase is low), or their volume (when the particles... 

For conventional surfactants with a long alkyl chain (m > 16) and dimeric (gemini) surfactants with m>8, the entry of a surfactant in a micelle is slower than for a diffusion-controlled process. The surfactant residence time and the micelle lifetime can become long with respect to the values found for conventional surfactants. 

Figure 12.5 shows the characteristic signal temperature plots obtained with these systems. It can be seen that a plateau is reached for types (a) and (b). At this point the output of the device is independent of temperature, indicating that the reaction is diffusion-controlled at a point remote from the sensor, that is, the sinter in type (a) and the chimney entry in type (b). No such plateau exists for the type (c) system. Operation of the detector under diffusion-controlled conditions limits the response time to several seconds, but offers several advantages ... 

Wragg, A. A., Tagg, D. J., and Patrick, M. A., 1980, Diffusion controlled current distributions near cell entries and comers, J. Appl. Electrochem., 10 43-47. 

The main technique used to look at exchange processes in equilibrium systems employs labeled surfactants, particularly with ESR spectroscopy. Fox s ESR study [92] of a paramagnetic surfactant in micellar solution was the first of its kind, and yielded a solution-micelle monomer exchange rate of 10 s" at room tanperature for 2,2,6,6-tetramethylpiperidine-oxidedodecyldimethylammonium bromide. These techniques, along with time-resolved luminescence quenching, have shown that the entry of surfactant molecules into micelles is near-diffusion controlled, whereas loss from micelles is rate limiting, and hence kinetically controlled [93]. A decade later (1981), Bolt and Turro [94] were able to find the separate exit and reentry rate constants for 10-(4-bromo-l-naphthoyl)decyltrimethylammonium bromide as 3.2 x 10 s and 5.7 x 10 mok s", respectively. 

---