Scale, time

Thus for TVp 30 a relaxation time th w 10 s results, 10 times more than the time-step of Rigby and Roe. This number shows that even at high 

Of course, the situation is much worse for melts of long chains in a mutually entangled state, where reptation theory predicts a relaxation time proportional to the third power of N, 

Restrict the study to the small length-scale behavior but abandon the quest for good thermal equilibrium. Of course, in such an approach it is no longer sensible to ask how properties depend on Np, because properties on the length scale of Rg certainly need not be very reliable. This approach is followed in studying the behavior of amorphous polymers in the solid state and their mechanical response (Chapter 5 and Refs 37, 42, 50-58) and certainly is appropriate for studying solid polymers in the crystalline state.The latter problem is not our main concern here, as we concentrate on flexible polymers in the random coil state (solution, melt, amorphous solid). Of course, the preparation of the simulation sample is then very questionable. Cooling from the melt produces rapidly quenched samples which may differ in many physical properties from their slowly cooled experimental counterparts—at least 

Restrict the study to long wavelength properties of the problem by abandoning chemical detail and introduction of a coarse-grained model. It is this approach which will be emphasized in this chapter and many following ones (Chapters 2-4, 6, 7, 9). It is clear that this coarse-graining approach makes sense only for universal properties, and what universal means depends on the problem under study. For example, for dilute polymer solutions one expects that the mean-square gyration radius of a chain varies with Np as for 

Np — 00, where the exponent u for good solvents is universal, i.e., independent of the chemical nature of both polymer and solvent, while the prefactor Cp clearly is not. In a coarse-grained model of a polymer chain we can integrate n chemical monomers into one effective subunit (cf. section 1.3) and thus consider an equivalent chain of N = Np/n segments. While we still expect = CN, the explicit 

The moisture diffusion coefficient D is of order 10 mm /s, which for composites and polymers is about 6 and 7 orders of magnitudes smaller, respectively, than the temperature diffusion coefficient. The above processes can thus be decoupled. 

Capillary motion occurs at about 6 orders of magnitude faster than diffusion. Both processes can be decoupled, with consequent ramifications for diffusion and damage. 

Wicking along the fiber/matrix interphase occurs about 10 times faster than diffusion within the polymeric phase. The two phenomena are weakly coupled. 

Diffusion and polymeric creep occur on approximately similar time-scales. Except for the limiting cases of very thin or very thick plate thicknesses these phenomena are inherently coupled. 

Clark DL (1983) Moisture absorption in hybrid composites. Texas A M University Report MM 4665-83-16, December 

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In many cases faults will only restrict fluid flow, or they may be open i.e. non-sealing. Despite considerable efforts to predict the probability of fault sealing potential, a reliable method to do so has not yet emerged. Fault seal modelling is further complicated by the fact that some faults may leak fluids or pressures at a very small rate, thus effectively acting as seal on a production time scale of only a couple of years. As a result, the simulation of reservoir behaviour in densely faulted fields is difficult and predictions should be regarded as crude approximations only. 

The magnesium ion is made available by migrating pore waters. If the process is continuous on a geologic time scale more and more Mg + is introduced to the system and the porosity reduces again. The rock has been over-dolomitised. 

However, it is easily shown that if the mother wavelet is located in the frequency domain "around"/o (fig 8), then the wavelet a.b(t) is located around f(/a. That is to say, by the mean of the formal identification f = fata it is possible to interpret a time-scale representation as a time-frequency representation [4]. 

In principle, nucleation should occur for any supersaturation given enough time. The critical supersaturation ratio is often defined in terms of the condition needed to observe nucleation on a convenient time scale. As illustrated in Table IX-1, the nucleation rate changes so rapidly with degree of supersaturation that, fortunately, even a few powers of 10 error in the preexponential term make little difference. There has been some controversy surrounding the preexponential term and some detailed analyses are available [33-35]. 

The existence of the polyad number as a bottleneck to energy flow on short time scales is potentially important for efforts to control molecnlar reactivity rising advanced laser techniqnes, discussed below in section Al.2.20. Efforts at control seek to intervene in the molecnlar dynamics to prevent the effects of widespread vibrational energy flow, the presence of which is one of the key assumptions of Rice-Ramsperger-Kassel-Marcns (RRKM) and other theories of reaction dynamics [6]. 

It follows that there are two kinds of processes required for an arbitrary initial state to relax to an equilibrium state the diagonal elements must redistribute to a Boltzmaim distribution and the off-diagonal elements must decay to zero. The first of these processes is called population decay in two-level systems this time scale is called Ty The second of these processes is called dephasmg, or coherence decay in two-level systems there is a single time scale for this process called T. There is a well-known relationship in two level systems, valid for weak system-bath coupling, that... 

Femtosecond lasers represent the state-of-the-art in laser teclmology. These lasers can have pulse widths of the order of 100 fm s. This is the same time scale as many processes that occur on surfaces, such as desorption or diffusion. Thus, femtosecond lasers can be used to directly measure surface dynamics tlirough teclmiques such as two-photon photoemission [85]. Femtochemistry occurs when the laser imparts energy over an extremely short time period so as to directly induce a surface chemical reaction [86]. 

Thus the average velocity decays exponentially to zero on a time scale detennined by the friction coefficient and the mass of the particle. This average behaviour is not very interesting, because it corresponds to tlie average of a quantity that may take values in all directions, due to the noise and friction, and so the decay of the average value tells us little about the details of the motion of the Brownian particle. A more interesting... 

Due to the conservation law, the diffiision field 5 j/ relaxes in a time much shorter than tlie time taken by significant interface motion. If the domain size is R(x), the difhision field relaxes over a time scale R Flowever a typical interface velocity is shown below to be R. Thus in time Tq, interfaces move a distanc of about one, much smaller compared to R. This implies that the difhision field 6vj is essentially always in equilibrium with tlie interfaces and, thus, obeys Laplace s equation... 

General first-order kinetics also play an important role for the so-called local eigenvalue analysis of more complicated reaction mechanisms, which are usually described by nonlinear systems of differential equations. Linearization leads to effective general first-order kinetics whose analysis reveals infomiation on the time scales of chemical reactions, species in steady states (quasi-stationarity), or partial equilibria (quasi-equilibrium) [M, and ]. 

This is no longer the case when (iii) motion along the reaction patir occurs on a time scale comparable to other relaxation times of the solute or the solvent, i.e. the system is partially non-relaxed. In this situation dynamic effects have to be taken into account explicitly, such as solvent-assisted intramolecular vibrational energy redistribution (IVR) in the solute, solvent-induced electronic surface hopping, dephasing, solute-solvent energy transfer, dynamic caging, rotational relaxation, or solvent dielectric and momentum relaxation. 

For very fast reactions, as they are accessible to investigation by pico- and femtosecond laser spectroscopy, the separation of time scales into slow motion along the reaction path and fast relaxation of other degrees of freedom in most cases is no longer possible and it is necessary to consider dynamical models, which are not the topic of this section. But often the temperature, solvent or pressure dependence of reaction rate... 

The dependence of k on viscosity becomes even more puzzling when the time scale of motion along the reaction coordinate becomes comparable to that of solvent dipole reorientation around the changing charge distribution... 

From stochastic molecnlar dynamics calcnlations on the same system, in the viscosity regime covered by the experiment, it appears that intra- and intennolecnlar energy flow occur on comparable time scales, which leads to the conclnsion that cyclohexane isomerization in liquid CS2 is an activated process [99]. Classical molecnlar dynamics calcnlations [104] also reprodnce the observed non-monotonic viscosity dependence of ic. Furthennore, they also yield a solvent contribntion to the free energy of activation for tlie isomerization reaction which in liquid CS, increases by abont 0.4 kJ moC when the solvent density is increased from 1.3 to 1.5 g cm T Tims the molecnlar dynamics calcnlations support the conclnsion that the high-pressure limit of this unimolecular reaction is not attained in liquid solntion at ambient pressure. It has to be remembered, though, that the analysis of the measnred isomerization rates depends critically on the estimated valne of... 

Harris A L, Berg M and Harris C B 1986 Studies of chemical reactivity in the condensed phase. I. The dynamics of iodine photodissociation and recombination on a picosecond time scale and comparison to theories for chemical reactions in solution J. Chem. Phys. 84 788... 

Spectroscopic detemiination of the HE rotational distribution is another story. In both the chemical laser and infrared chemiluminescence experiments, rotational relaxation due to collisions is faster or at least comparable to the time scale of the measurements, so that accurate detemiination of the nascent rotational distribution was not feasible. However, Nesbitt [40, 41] has recently carried out direct infrared absorption experiments on the HE product under single-collision conditions, thereby obtaining a fiill vibration-rotation distribution for the nascent products. 

As a rule, in diemial unimolecular reaction systems at modest temperatures, is well separated from the other eigenvalues, and thus the time scales for incubation and relaxation are well separated from the steady-... 

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