Time scales of dynamical processes

Time scales of dynamic processes in lipid systems are wide and range from picoseconds to hours or even months. Examples are discussed below. Here, we just refer to rotational motions as the fastest dynamical events and cell death as the slowest event. What is relevant to stress here is the lack of any specific time scale. 

NMR time scale. The time scale of dynamic processes that can be observed with an NMR spectrometer. 

Figure 2.13 Dynamic NMR experiments and time scale of dynamic processes associated with proteins. Reprinted from Ref. [71], Copyright 2005 Eisevier.

Methods to determine co-crystal solubility are based on thermodynamic and kinetic approaches. Thermodynamic equilibrium experiments provide a measure of co-crystal solubilization processes, while kinetic studies provide insight about the time scales of dynamic processes and concentration fluctuations during co-crystal dissolution. From equilibrium measurements, we can extract information about the origin of co-crystal solution phase behavior and fine tune solution processes by controlling thermodynamic solubility. We can use the knowledge gained from equilibrium studies to design kinetic studies and separate the thermodynamic and kinetic contributions to the co-crystal dissolution and transformation behavior. 

William Russel May I follow up on that and sharpen the issue a bit In the complex fluids that we have talked about, three types of nonequilibrium phenomena are important. First, phase transitions may have dynamics on the time scale of the process, as mentioned by Matt Tirrell. Second, a fluid may be at equilibrium at rest but is displaced from equilibrium by flow, which is the origin of non-Newtonian behavior in polymeric and colloidal fluids. And third, the resting state itself may be far from equilibrium, as for a glass or a gel. At present, computer simulations can address all three, but only partially. Statistical mechanical or kinetic theories have something to say about the first two, but the dynamics and the structure and transport properties of the nonequilibrium states remain poorly understood, except for the polymeric fluids. 

ADAS is centred on generalized collisional-radiative (GCR) theory. The theory recognizes relaxation time-scales of atomic processes and how these relate to plasma relaxation times, metastable states, secondary collisions etc. Attention to these aspects - rigorously specified in generalized collisional-radiative theory - allow an atomic description suitable for modeling and analyzing spectral emission from most static and dynamic plasmas in the fusion and astrophysical domains [3,4]. 

The partition of the total system into a spin part and a lattice part is, in principle, fuzzy. Other nuclear or electronic spins can either be included in the spin space or into the lattice. In a perturbation treatment, it depends on the interaction strength and on the time-scales of different processes. If nuclear spins are interacting over long periods (e.g. within a small molecule), the different spins cannot be considered separately. However, electron spins can in many cases be put in the lattice space since the difference in time scales for the electron spin and nuclear spin dynamics is prohibitive an effective coupling. 

The extension of Gillespie s algorithm to spatially distributed systems is straightforward. A lattice is used to represent binding sites of adsorbates, which correspond to local minima of the potential energy surface. The discrete nature of KMC coupled with possible separation of time scales of various processes could render KMC inefficient. The work of Bortz et al. on the n-fold or continuous time MC CTMC) method can lead to computational speedup of the KMC method, which, however, has been underutilized most probably because of its difficult implementation. This method classifies all atoms in a finite number of classes according to their transition probability. Probabilities are computed a priori and each event is successful, in contrast to the Metropolis method (and other null event algorithms) whose fraction of unsuccessful (null) events increases drastically at low temperatures and for stiff problems. In conjunction with efficient search within a class and dynamic variation of atom coordi-nates, " the CPU time can be practically independent of lattice size. After each event, the time is incremented by a continuous amount. 

Boero et al. (1998) used Car-Parrinello molecular dynamics to study the polymerization of ethylene at titanium sites in MgCl2-supported Ziegler-Natta catalysts. Their objectives were to evaluate the reaction mechanism, in addition to determining the free energy profile of the polymerization process. Obviously, the characteristic time scale of this process is much greater than the picosecond time scale directly accessible by the simulation. Thus, it is not possible to observe the polymerization process via a straightforward Car-Parrinello simulation. 

Even highly dynamic natural water systems may be at equilibrium with respect to certain processes this depends on the time scale of the process. Hence, there always exist in natural waters regions or environments that are locally at equilibrium, even though gradients exist throughout the system as a whole. 

At steady state, yss = Ku. Linear dynamic models such as Eq. (18.3) provide a theoretical means to determine the time scale of the process. For a step change in u (= M), the solution to Eq. (18.3) can be found analytically. 

We start our exposition with basic dynamical features of polymers (often probed experimentally) and focus on mechanical relaxation. Here we encounter one of the most familiar properties of polymers, namely viscoelasticity. In general, polymers do not behave like solids or liquids instead, they take an intermediate position, by which elastic or plastic behavior depends on the time-scale of the process under observation. 

Each of the above processes occurs numerous times during emulsification and the time scale of each process is very short, typically a microsecond. This shows that emulsification is a dynamic process, and that events occurring in the microsecond domain could be very important [9, 10]. 

The fact that we choose the same factor g for both space and time is related to the typical scaling of wave-like phenomena, where the time scale of a process is linearly proportional to the corresponding length scale. However, hydrodynamics also includes diffusion of momentum, where the time scale is proportional to the square of the length scale. These processes occur on a much longer time scale, and to capture the slow dynamics we introdnce a second clock that is even more coarsegrained,... 

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]. 

Many of the fiindamental physical and chemical processes at surfaces and interfaces occur on extremely fast time scales. For example, atomic and molecular motions take place on time scales as short as 100 fs, while surface electronic states may have lifetimes as short as 10 fs. With the dramatic recent advances in laser tecluiology, however, such time scales have become increasingly accessible. Surface nonlinear optics provides an attractive approach to capture such events directly in the time domain. Some examples of application of the method include probing the dynamics of melting on the time scale of phonon vibrations [82], photoisomerization of molecules [88], molecular dynamics of adsorbates [89, 90], interfacial solvent dynamics [91], transient band-flattening in semiconductors [92] and laser-induced desorption [93]. A review article discussing such time-resolved studies in metals can be found in... 

The SHG and SFG teclmiques are also suitable for studying dynamical processes occurring on slower time scales. Indeed, many valuable studies of adsorption, desorption, difhision and other surface processes have been perfomied on time scales of milliseconds to seconds. 

STM has not as yet proved to be easily applicable to the area of ultrafast surface phenomena. Nevertheless, some success has been achieved in the direct observation of dynamic processes with a larger timescale. Kitamura et al [23], using a high-temperature STM to scan single lines repeatedly and to display the results as a time-ver.sn.s-position pseudoimage, were able to follow the difflision of atomic-scale vacancies on a heated Si(OOl) surface in real time. They were able to show that vacancy diffusion proceeds exclusively in one dimension, along the dimer row. 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

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