Radical distribution function

Zimm, B.H. 1948. The scattering of light and the radical distribution function of high polymer solutions. J. Chem. Phys., 16 1093. 

Jawalkar et al. found that the miscibility between PVA and CS polymers is attributed to hydrogen bond formation between groups of CS (-CH2OH or -NH2) and hydrogen atom of PVA by MD simulation of radical distribution functions for atoms involved in interaction. Figure 6.6 shows the radial distribution function as a function of r for these atoms in the miscible blend ofPVA/CS (1 1) [25]. 

This free radical distribution function, determined via a Monte Carlo technique, was then incorporated into the two-phase emulsion polymerization kinetics model developed by Nelson and Sundberg [44, 45] to predict the monomer conversion versus time data available in the literature. Thus, the only difference between the kinetic model characterized by the following major governing equations [47] and that of Nelson and Sundberg is the method used for calculation of the average free radical population in each polymer phase. 

Chem and Poehlein [52] developed a kinetic model based on the nonuniform free radical distribution function to predict the grafting efficiency of the emulsion emulsion polymerization of styrene in the presence of polybutadiene seed latex particles. The predominant grafting reaction appears to be the attack of growing polystyrene chains on the allyl hydrogen atoms of... 

Sonochemistry is also proving to have important applications with polymeric materials. Substantial work has been accomplished in the sonochemical initiation of polymerisation and in the modification of polymers after synthesis (3,5). The use of sonolysis to create radicals which function as radical initiators has been well explored. Similarly the use of sonochemicaHy prepared radicals and other reactive species to modify the surface properties of polymers is being developed, particularly by G. Price. Other effects of ultrasound on long chain polymers tend to be mechanical cleavage, which produces relatively uniform size distributions of shorter chain lengths. 

When deriving this expression for the average composition distribution, authors of paper [74] entirely neglected its instantaneous constituent, having taken (as is customary in the quantitative theory of radical copolymerization [3,84]) the Dirac delta-function < ( -X) as the instantaneous composition distribution. Its averaging over conversions, denoted hereinafter by angular brackets, leads to formula (Eq. 101). Note, this formula describes the composition distribution only provided copolymer composition falls in the interval between X(0) and X(p). Otherwise, this distribution function vanishes at all values of composition lying outside the above-mentioned interval. 

An interesting question then arises as to why the dynamics of proton transfer for the benzophenone-i V, /V-dimethylaniline contact radical IP falls within the nonadiabatic regime while that for the napthol photoacids-carboxylic base pairs in water falls in the adiabatic regime given that both systems are intermolecular. For the benzophenone-A, A-dimethylaniline contact radical IP, the presumed structure of the complex is that of a 7t-stacked system that constrains the distance between the two heavy atoms involved in the proton transfer, C and O, to a distance of 3.3A (Scheme 2.10) [20]. Conversely, for the napthol photoacids-carboxylic base pairs no such constraints are imposed so that there can be close approach of the two heavy atoms. The distance associated with the crossover between nonadiabatic and adiabatic proton transfer has yet to be clearly defined and will be system specific. However, from model calculations, distances in excess of 2.5 A appear to lead to the realm of nonadiabatic proton transfer. Thus, a factor determining whether a bimolecular proton-transfer process falls within the adiabatic or nonadiabatic regimes lies in the rate expression Eq. (6) where 4>(R), the distribution function for molecular species with distance, and k(R), the rate constant as a function of distance, determine the mode of transfer. 

Mathematically these are radically different functions. Du Di, and D3 are all double exponential decays, but their preexponential factors deviate radically and the lifetimes differ noticeably. The ratio of preexponentials for the fast and slow components vary by a factor of 16 D has comparable amplitudes, while D2has a ratio of short to long of 4, and D3 has a ratio of short to long of 1/4. D4 is a sum of three exponentials. All five functions vary from a peak of about 104 to 25, and all four functions, if overlaid, are virtually indistinguishable. To amplify these differences, we assume that the Gaussiandistribution, Da, is the correct decay function and then show the deviations of the other functions from Do. These results are shown in Figure 4.10. The double exponential D fits the distribution decay essentially perfectly. Even Dj and Ds are a very crediblefit. >4 matches Do so well that the differences are invisible on this scale, and it is not even plotted. 

The kinetics data of the geminate ion recombination in irradiated liquid hydrocarbons obtained by the subpicosecond pulse radiolysis was analyzed by Monte Carlo simulation based on the diffusion in an electric field [77,81,82], The simulation data were convoluted by the response function and fitted to the experimental data. By transforming the time-dependent behavior of cation radicals to the distribution function of cation radical-electron distance, the time-dependent distribution was obtained. Subsequently, the relationship between the space resolution and the space distribution of ionic species was discussed. The space distribution of reactive intermediates produced by radiation is very important for advanced science and technology using ionizing radiation such as nanolithography and nanotechnology [77,82]. 

Equation 2 indicates that if the distance r between the cation radical and the electron becomes less than the reaction radius R, they recombine. The typical shape of/(r,ro)> the initial distribution function, is expressed by the following exponential or Gaussian. 

Figure 13 (a) Comparison of the simulation curve with the analytical solution given by Hong and Noolandi [11,13,83]. The initial distribution function and the Onsager length (r ) were taken as a delta function S(r—ro) of = 7.5 nm and = 30 nm, respectively. The dots and the solid line indicate the analytical solution and the simulation curve, respectively, (b) Time-dependent distribution function that was obtained from the simulation of (a), r indicates the distance between the cation radical and the electron. 

Bamford and Tompa (93) considered the effects of branching on MWD in batch polymerizations, using Laplace Transforms to obtain analytical solutions in terms of modified Bessel functions of the first kind for a reaction scheme restricted to termination by disproportionation and mono-radicals. They also used another procedure which was to set up equations for the moments of the distribution that could be solved numerically the MWD was approximated as a sum of a number of Laguerre functions, the coefficients of which could be obtained from the moments. In some cases as many as 10 moments had to be computed in order to obtain a satisfactory representation of the MWD. The assumption that the distribution function decreases exponentially for large DP is built into this method this would not be true of the Beasley distribution (7.3), for instance. 

Computations of the EPR line shape made in ref. 71 with the help of eqn. (18) for the model distribution functions f(R) that are frequently used in radiation chemistry, have shown that the shape of the wings of the EPR lines is far more sensitive to changes in the distribution functions of radical pairs over the distances than to changes in such a conventionally used parameter as the line width between the points of the maximum slope, A//p. Thus, to estimate the distances of tunneling and their variations in the course of a reaction it is necessary to analyze the shape of the wings of the EPR lines. 

Photodissociation dynamics studies in a nozzle-cooled beam have been reported along with an elegant analysis of the data (146). Only CN radicals in the v" = 0 level are reported because of the difficulty of detecting small amounts of excited radicals in the upper vibrational level. The results that were obtained by fitting the observed rotational distributions with Boltzmann rotational distribution functions are summarized in Table 3. 

It can be established by the following reasoning. If n = %, each particle contains at most one free radical. Growing chains in the latex particles can thus either grow or be terminated instantaneously by entrant free radicals. These mutually exclusive kinetic events immediately prescribe the Flory most probable distribution function for the growing chains (12) this is an exponential distribution function with a polydispersity index of 2.00 (13). 

Fig. 7.10. EPR spectra (a) of the (=28Si-0-) 9Si radicals in the bulk (solid line [38]) and on the surface (dashed line) of silica and distribution function (b) for aiso(29Sia) values for bulk radicals (solid line - experiment [38], dashed line - calculations, see text for details).

Figure 6 Results of the analyses for the classical free radical distribution with 0 and 1% noise in the AC function.

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