Effective Critical Exponents

In Eq. (15) the second term reflects the gain in entropy when a chain breaks so that the two new ends can explore a volume Entropy is increased because the excluded volume repulsion on scales less than is reduced by breaking the chain this effect is accounted for by the additional exponent 9 = y — )/v where 7 > 1 is a standard critical exponent, the value of 7 being larger in 2 dimensions than in 3 dimensions 72 = 43/32 1.34, 73j 1.17. In MFA 7 = 1, = 0, and Eq. (15) simplifies to Eq. (9), where correlations, brought about by mutual avoidance of chains, i.e., excluded volume, are ignored. 

The experimental data corresponding to one labeled arm in stars of f=12 (good solvent) [68] shows, as expected, Kratky plot ordinates that increase monotonously with q. However, the plateau is only obtained with an apparent critical exponent of 2/3 (i.e., greater than the theoretical value, v-3/5). This seems an indication of the arm stretching effect, though the scaling and RG theoretical predictions describe this effect only in terms of a pre-exponential factor [11,42]. 

Close to the gel point, in the range AX/X <0.1, the static modulus cannot be measured. Strong relaxation effects are present even at the lowest frequency which could be used, to be consistent with the kinetics (one period of oscillation = 67s). Beyond this range for AX/X >0.1, G (0,015Hz) corresponds to the static relaxed modulus. A critical exponent for the relaxed modulus can be determined by using the equation X - X... 

If there is no interaction between similar reactants (traps) B, they are distributed according to the Poisson relation, Ab (r, t) = 1. Besides, since the reaction kinetics is linear in donor concentrations, the only quantity of interest is the survival probability of a single particle A migrating through traps B and therefore the correlation function XA(r,t) does not affect the kinetics under study. Hence the description of the fluctuation spectrum of a system through the joint densities A (r, ), which was so important for understanding the A4-B — 0 reaction kinetics, appears now to be incomplete. The fluctuation effects we are interested in are weaker here, thus affecting the critical exponent but not the exponential kinetics itself. It will be shown below that adequate treatment of these weak fluctuation effects requires a careful analysis of many-particle correlations. 

Fig. 6.21. The critical exponent a(t) as a functions of time for d = 1. Curves 1 and 2 correspond to the initial concentrations n(0) = 1 0.1 0.01, respectively. Full curves show the asymmetric case, Da = 0 broken curves are symmetric case, Da = Dr. Dotted lines show neglect of the many particle effects.

As it is seen in these figures, the higher n(0), the faster the asymptotics is achieved. For the immobile reactant A and d = 1, a(t) systematically exceeds that for the equal mobilities which leads to faster concentration decay in time. The results for d = 2 and 3 are qualitatively similar. Their comparison with the one-dimensional case demonstrates that the concentration decay is now much faster since the critical exponents strive for a = 3/4 and a = 1/2 for the symmetric and asymmetric cases, respectively, which differ greatly from the classical value of a = 1. Respectively, the gap between symmetric and asymmetric decay kinetics grows much faster than in the d = 1 case. Therefore, the conclusion could be drawn that the effect of the relative particle mobility is pronounced better and thus could be observed easier in t ree-dimensional computer simulations rather than in one-dimensional ones, in contrast to what was intuitively expected in [33]. 

Lastly, we would like to mention here results of the two kinds of large-scale computer simulations of diffusion-controlled bimolecular reactions [33, 48], In the former paper [48] reactions were simulated using random walks on a d-dimensional (1 to 4) hypercubic lattice with the imposed periodic boundary conditions. In the particular case of the A + B - 0 reaction, D = Dq and nA(0) = nB(0), the critical exponents 0.26 0.01 0.50 0.02 and 0.89 0.02 were obtained for d = 1 to 3 respectively. The theoretical value of a = 0.75 expected for d = 3 was not achieved due to cluster size effects. The result for d = 4, a = 1.02 0.02, confirms that this is a marginal dimension. However, in the case of the A + B — B reaction with DB = 0, the asymptotic longtime behaviour, equation (2.1.106), was not achieved at all - even at very long reaction times of 105 Monte Carlo steps, which were sufficient for all other kinds of bimolecular reactions simulated. It was concluded that in practice this theoretically derived asymptotics is hardly accessible. 

Fig. 6.33. The critical exponent, equation (4.1.68), as a function of time for asymmetric (a) and symmetric (b) cases. The initial particle concentrations n(0) 1.0 (full curves), 0.1 (dashed curves). Dotted lines are obtained neglecting the many-particle effects. Parameter...

The origin of this unusual behaviour is partly clarified from Fig. 6.34(a) where the relevant curves 2 demonstrate the same kind of the non-monotonous behaviour as the critical exponents above. Since, according to its definition, equation (4.1.19), the reaction rate is a functional of the joint correlation function, this non-monotonicity of curve 2 arises due to the spatial re-arrangements in defect structure. It is confirmed by the correlation functions shown in Fig. 6.34(a). The distribution of BB pairs is quasi-stationary, XB(r,t) X°(r) = exp[(re/r)3], which describes their dynamic aggregation. (The only curve is plotted for XB in Fig. 6.35(a) for t = 102 (the dotted line) since for other time values XB changes not more than by 10 per cent.) This quasi-steady spatial particle distribution is formed quite rapidly already at t 10° it reaches the maximum value of XB(r, t) 103. The effect of the statistical aggregation practically is not observed here, probably, due to the diffusion separation of mobile B particles. 

Chapters 13 and 14 use thermodynamics to describe and predict phase equilibria. Chapter 13 limits the discussion to pure substances. Distinctions are made between first-order and continuous phase transitions, and examples are given of different types of continuous transitions, including the (liquid + gas) critical phase transition, order-disorder transitions involving position disorder, rotational disorder, and magnetic effects the helium normal-superfluid transition and conductor-superconductor transitions. Modem theories of phase transitions are described that show the parallel properties of the different types of continuous transitions, and demonstrate how these properties can be described with a general set of critical exponents. This discussion is an attempt to present to chemists the exciting advances made in the area of theories of phase transitions that is often relegated to physics tests. 

Applying superposition approximations to the Ising model, one finds an evidence for the phase transition existence but the critical parameter to is systematically underestimated (To is overestimated respectively). Errors in calculation of to are greater for low dimensions d. Therefore, the superposition approximation is effective, first of all, for the qualitative description of the phase transition in a spin system. In the vicinity of phase transition a number of critical exponents a, /3,7,..., could be introduced, which characterize the critical point, like oc f-for . M oc (i-io), or xt oc i—io for the magnetic permeability. Superposition approximations give only classical values of the critical exponents a = ao, 0 = f o, j — jo, ., obtained earlier in the classical molecular field theory [13, 14], say fio = 1/2, 7o = 1, whereas exact magnitudes of the critical exponents depend on the space dimension d. To describe the intermediate order in a spin system in terms of the superposition approximation, an additional correlation length is introduced, 0 = which does not coincide with the true In the phase... 

In the linear approximation the critical exponent a t) for d = 2 and 3 approaches monotonously its asymptotic value a = ao = 1- In contrast, incorporation of the fluctuation effects leads to the emergence of the maximum a(t), better pronounced for d = 2 the position of a maximum a t)... 

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