A-relaxation process

Ultrasonic absorption is used in the investigation of fast reactions in solution. If a system is at equilibrium and the equilibrium is disturbed in a very short time (of the order of 10"seconds) then it takes a finite time for the system to recover its equilibrium condition. This is called a relaxation process. When a system in solution is caused to relax using ultrasonics, the relaxation lime of the equilibrium can be related to the attenuation of the sound wave. Relaxation times of 10" to 10 seconds have been measured using this method and the rates of formation of many mono-, di-and tripositive metal complexes with a range of anions have been determined. 

Sack R. A. Relaxation process and inertial effects. 2. Free rotation in space. Proc. Phys. Soc. (London) B70, 414-26 (1957). 

Kinetic schemes involving sequential and coupled reactions, where the reactions are either first-order or pseudo-first order, lead to expressions for concentration changes with time that can be modeled as a sum of exponential functions where each of the exponential functions has a specific relaxation time. More complex equations have to be derived for bimolecular reactions where the concentrations of reactants are similar.19,20 However, the rate law is always related to the association and dissociation processes, and these processes cannot be uncoupled when measuring a relaxation process. 

Now we can visualise that each of these models represents a relaxation process in the system. However, it is likely that some spacings, or sizes, and interactions occur more often than others, so that each process is not unique. We could represent the m models by a series of unique models where each model occurs / , times. There would be n such models where n is less than m ... 

The iFi terms are the fluorescence lifetimes of fractional contributions a, and the xRJ indicate decay constants due to solvent relaxation (or other excited-state processes) of fractional contribution Pj. The negative sign is indicative of a relaxation process (red shift). Usually, the relaxation process is approximated to a single relaxation time x R by assuming an initial excited state and a final fully relaxed state (see, e.g., Ref. 128). A steady-state fluo-... 

A very important electrochemical phenomenon, which is not well understood, is the so-called memory effect. This means that the charging/discharging response of a conducting polymer film depends on the history of previous electrochemical events. Thus, the first voltammetric cycle obtained after the electroactive film has been held in its neutral state differs markedly in shape and peak position from subsequent ones [126]. Obviously, the waiting time in the neutral state of the system is the main factor determining the extent of a relaxation process. During this waiting time, which extends over several decades of time (1-10 s), the polymer slowly relaxes into an equilibrium state. (Fig. 13) After relaxation, the first oxidation wave of the polymer appears at more... 

A schematic representation of a forcing function (upper curve) and a relaxation process (lower curve). The system, originally at equilibrium, responds to changes in rate constants and equilibrium constants imposed by the forcing function. For a more complete account of forcing functions, see references 1 and 2 below. Based on a diagram from IlgenfritzF... 

Figure 17 shows the temperature dependence of the lateral forces measured at the scanning rate of 10 nms for a high-density PMMA brush (a = 0.8 chains nm ) and an equivalent spin-coated film. The a-relaxation process was clearly observed accompanying a small peak, which was assigned to a surface /1-process. They commented that the surface molecular motion of the brush layer possibly differs from that of the spin-coated film but that it was rather difficult to conclude this because of shghtly scattered data. 

The properties of the F band are, unfortunately, not known, making assignment of the time constants to a specific physical process difficult. In general, the growth time X, about 1 ps, appears to be independent of cluster size since the (S02) clusters produce similar values. The F band decay, x2, is altered substantially in the (S02)n clusters where the decay is slowed with increasing cluster size from about 13 ps to 65 ps for the n = 1 to 5 size range. The decay is attributed to a relaxation process that is slowed by interactions between the excited-state species and the surrounding cluster molecules. However, the exact nature of this relaxation process has not yet been determined. 

It is very important, in the theory of quantum relaxation processes, to understand how an atomic or molecular excited state is prepared, and to know under what circumstances it is meaningful to consider the time development of such a compound state. It is obvious, but nevertheless important to say, that an atomic or molecular system in a stationary state cannot be induced to make transitions to other states by small terms in the molecular Hamiltonian. A stationary state will undergo transition to other stationary states only by coupling with the radiation field, so that all time-dependent transitions between stationary states are radiative in nature. However, if the system is prepared in a nonstationary state of the total Hamiltonian, nonradiative transitions will occur. Thus, for example, in the theory of molecular predissociation4 it is not justified to prepare the physical system in a pure Born-Oppenheimer bound state and to force transitions to the manifold of continuum dissociative states. If, on the other hand, the excitation process produces the system in a mixed state consisting of a superposition of eigenstates of the total Hamiltonian, a relaxation process will take place. Provided that the absorption line shape is Lorentzian, the relaxation process will follow an exponential decay. 

How does a relaxation process take place in a microscopic system To answer this question we amplify two comments made in the preceding section In the first place a representation of the (time-independent) eigenstates of the physical system can be displayed as a superposition of the zero-order states which correspond to the dense and to the sparse parts of the zero-order spectrum of states. In all cases considered we shall focus attention on the properties of the total system. Often, however, we find it convenient to examine the time evolution of one of the zero-order levels in the sparse set of states of one of the component subsystems of the total system. It is suggestive to then think of the remaining subsystem, with its dense manifold of states, as a reservoir. In fact we shall treat all states... 

A relaxation process will occur when a compound state of the system with large amplitude of a sparse subsystem component evolves so that the continuum component grows with time. We then say that the dynamic component of this state s wave function decays with time. Familiar examples of such relaxation processes are the a decay of nuclei, the radiative decay of atoms, atomic and molecular autoionization processes, and molecular predissociation. In all these cases a compound state of the physical system decays into a true continuum or into a quasicontinuum, the choice of the description of the dissipative subsystem depending solely on what boundary conditions are applied at large distances from the atom or molecule. The general theory of quantum mechanics leads to the conclusion that there is a set of features common to all compound states of a wide class of systems. For example, the shapes of many resonances are nearly the same, and the rates of decay of many different kinds of metastable states are of the same functional form. 

It is meaningful to consider a decay process on this limited time scale, which in turn is determined by the properties of the system under consideration. The introduction of the recurrence time defines, appropriately, the notion of irreversibility for the decay process. In particular, we see that if the recurrence time is sufficiently long (p sufficiently large) it is meaningful to consider a relaxation process even in a system with only a quasicontinuum of levels. 

By marrying molecular dynamics to transition state theory, these questionable assumptions can be dispensed with, and one can simulate a relaxation process involving bottlenecks rigorously, assuming only 1) classical mechanics, and 2) local equilibrium within the reactant and product zones separately. For simplicity we will first treat a situation in which there is only one bottleneck, whose location is known. Later, we will consider processes involving many bottlenecks, and will discuss computer-assisted heuristic methods for finding bottlenecks when their locations are not known a priori. 

First of all, we define the transition rates for our stochastic model using an ansatz of Kawasaki [39, 40]. In the following we use the abbreviation X for an initial state (07 for mono- and oion for bimolecular steps), Y for a final state (ct[ for mono- and a[a n for bimolecular steps) and Z for the states of the neighbourhood ( cr f 1 for mono- and a -1 a -1 for bimolecular steps). If we study the system in which the neighbourhood is fixed we observe a relaxation process in a very small area. We introduce the normalized probability W(X) and the corresponding rates 8.(X —tY Z). For this (reversible) process we write down the following Markovian master equation... 

Such a short spin-equilibrium relaxation time raises the question of whether discrete spin state isomers exist. Their existence is affirmed by two observations. One is the persistence of electronic spectral bands typical of the low-spin 2E state over a wide temperature range in solid samples (98). The other is the observation of EPR signals characteristic of the 2E state in both solids and solutions between 4 and 293 K (98,139). At very low temperatures EPR signals of both spin states can be observed simultaneously (98). At low temperatures hyperfine splitting into eight lines is observed from coupling with the 1 = 7/2 Co nucleus. As the temperature is raised the spectral features broaden and the hyperfine resolution is lost. This implies a relaxation process on the EPR time scale of 1010 sec-1, or a relaxation time of the order 0.1 nsec, consistent with the upper limit set by the ultrasonic experiments. 

The processes observed in the depolarized Rayleigh spectrum correspond to internal modes of motion. Thus, they may have relaxation times which substantially exceed those obtained from the longitudinal or bulk relaxation alone. Nevertheless they are a part of the a relaxation process as it is normally observed in the creep compliance. All processes with the same shift factors make up the full a relaxation. In liquids with substantial depolarized Rayleigh scattering the slowly relaxing part of the W scattering is also dominated by the orientation fluctuations associated with the internal modes of motion. Each internal mode contributes some intensity, but it is believed that fairly short wavelength modes dominate the scattered intensity. 

Large perturbation from a given stale of fluctuation leads to a relaxation process toward a state of equilibrium. The relaxation time, for instance, measures the deviation from quasistationary equilibrium of a process carried out at a finite rate. 

Quantitative comparison of dipole relaxation characteristics of polymers listed in Table 10, shows that at temperatures Tr (i.e. lower than Tg) a relaxation process,... 

Table 18. Values of correlation parameter g, relaxation time tJ.p and activation energy EJ p of a relaxation process of dipole polarization for some comb-like poly(methacrylates) in toluene solutions...

The effect of the side chain bulkiness has been further studied on a series of chloro derivatives of poly(ethyl methacrylate)(PEMA). Though poly(2-chloroethyl methacrylate) exhibits69 a pronounced peak at Ty = 117 K, poly(2,2,2-trichloroethyl methacrylate), poly(2,2,2-trichloro-l-methoxyethyl methacrylate), and poly(2,2,2-trichloro-l-ethoxyethyl methacrylate) do not show (Fig. 6) any low-temperature loss maximum above the liquid nitrogen temperature157. However, these three polymers probably display a relaxation process below 77 K as indicated by the decrease in the loss modulus with rising temperature up to 100 K. Their relaxation behavior seems to be similar to that of PEMA rather than of poly(2-chloroethyl methacrylate) which is difficult to explain. 

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