Time and length scales

The smallest length scale for each of energy carriers in solid, that is, phonon, electron, or photon, is its wavelength, A. The wavelength ranges for these carriers are the following  

The scattering of energy carriers results in resistance to energy transport. In the absence of scattering, the conductivity will be infinite. There are several time and length scales associated with the scattering process, which are discussed in the succeeding text. 

Collision time is the duration of collision. In classical physics, collisions are considered instantaneous. However, there is a finite collision time during wave scattering. The collision time is defined as the ratio of wavelength of the carrier and the propagation speed of the carrier. The typical values of collision time (t ) are as follows  

Mean free time, r is the average time between collisions. Generally, t The typical values of mean free time are  

Relaxation time is the statistical time lag value (a nonnegative constant) needed to establish steady-state heat flow conditions in a small elemental volume of material when a temperature gradient is suddenly imposed on the boundary. Chandrasekharaiah (1986) has reported relaxation time for different types of materials. The relaxation time for gases is in the range of 10 s, and the relaxation time for metals is in the range of 10 s. Relaxation time, is associated with local thermodynamic equilibrium. Equilibrium is achieved in 5-20 collisions, Note A system may have different momentum (t ) and energy 

To integrate the equations of motion in a stable and reliable way, it is necessary that the fundamental time step is shorter than the shortest relevant timescale in the problem. The shortest events involving whole atoms are C-H vibrations, and therefore a typical value of the time step is 2fs (10-15s). This means that there are up to one million time steps necessary to reach (real-time) simulation times in the nanosecond range. The ns range is sufficient for conformational transitions of the lipid molecules. It is also sufficient to allow some lateral diffusion of molecules in the box. As an iteration time step is rather expensive, even a supercomputer will need of the order of 106 s (a week) of CPU time to reach the ns domain. 

Of course there are many phenomena that equilibrate on the nanosecond timescale. However, the majority of relevant events take much more time. For example, the ns timescale is much too short to allow for the self-assembly of a set of lipids from a homogeneously distributed state to a lamellar topology. This is the reason why it is necessary to start a simulation as close as possible to the expected equilibrated state. Of course, this is a tricky practice and should be considered as one of the inherent problems of MD. Only recently, this issue was addressed by Marrink [56]. Here the homogeneous state of the lipids was used as the start configuration, and at the end of the simulation an intact bilayer was found. Permeation, transport across a bilayer, and partitioning of molecules from the water to the membrane phase typically take also more time than can be dealt with by MD. We will return to this point below. 

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Although much as been done, much work remains. Improved material models for anisotropic materials, brittle materials, and chemically reacting materials challenge the numerical methods to provide greater accuracy and challenge the computer manufacturers to provide more memory and speed. Phenomena with different time and length scales need to be coupled so shock waves, structural motions, electromagnetic, and thermal effects can be analyzed in a consistent manner. Smarter codes must be developed to adapt the mesh and solution techniques to optimize the accuracy without human intervention. 

The difficulties arise from the enormous variation in time and length scales. They range from microscopic oscillation periods of the order... 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

J. T. Padding and A. A. Louis, Hydrodynamic interactions and Brownian forces in colloidal suspensions coarse-graining over time and length scales, Phys. Rev. E 74, 031402 (2006). 

How can one hope to extract the contributions of the different normal modes from the relaxation behavior of the dynamic structure factor The capability of neutron scattering to directly observe molecular motions on their natural time and length scale enables the determination of the mode contributions to the relaxation of S(Q, t). Different relaxation modes influence the scattering function in different Q-ranges. Since the dynamic structure factor is not simply broken down into a sum or product of more contributions, the Q-dependence is not easy to represent. In order to make the effects more transparent, we consider the maximum possible contribution of a given mode p to the relaxation of the dynamic structure factor. This maximum contribution is reached when the correlator in Eq. (32) has fallen to zero. For simplicity, we retain all the other relaxation modes = 1 for s p. 

The integral-scale turbulence frequency is the inverse of the turbulence integral time scale. The turbulence time and length scales are defined in Chapter 2. 

The principal length and time scales, and the Reynolds numbers that are used to characterize a fully developed turbulent flow are summarized in Table 2.1. Conversion tables between the time and length scales, written in terms of the turbulence Reynolds number Rez, are given in Tables 2.2 and 2.3. 

Pulsed field gradient (PFG)-NMR experiments have been employed in the groups of Zawodzinski and Kreuer to measure the self-diffusivity of water in the membrane as a function of the water content. From QENS, the typical time and length scales of the molecular motions can be evaluated. It was observed that water mobility increases with water content up to almost bulk-like values above T 10, where the water content A = nn o/ nsojH is defined as the ratio of the number of moles of water molecules per moles of acid head groups (-SO3H). In Perrin et al., QENS data for hydrated Nation were analyzed with a Gaussian model for localized translational diffusion. Typical sizes of confining domains and diffusion coefficients, as well as characteristic times for the elementary jump processes, were obtained as functions of A the results were discussed with respect to membrane structure and sorption characteristics. ... 

The other major limitation of membrane simulations is the time and length scale we are able to simulate. We are currently able to reach a microsecond, but tens to hundreds of nanosecond simulations are more common, especially in free energy calculations. The slow diffusion of lipids means we are not able to observe many biologically interesting phenomena using equilibrium simulations. For example, we would not observe pore formation in an unperturbed bilayer system during an equilibrium simulation, and even pore dissipation is at the limits of current computational accessibility. 

As long as the boundary and initial conditions remain unchanged, the band profiles on the reduced time and length scale depend only on the column efficiency. The conventional boundary and initial conditions for all modes of chromatography state that (1) the column is equilibrated with the mobile phase prior to the beginning of the separation (2) the sample is then injected as a rectangular pulse and (3) the separation proceeds as required by the specific mode selected. The amount of sample injected is determined by the volume and the concentration of the feed injected. As long as we avoid serious volume overload, the actual values of these two parameters are immaterial. Only their product, i.e., the amount injected, will influence the band profile. 

For the two flow regimes of River G discussed in Illustrative Example 24.1, calculate (a) the characteristic time and length scale for vertical mixing (b) the characteristic time and length scale for transversal mixing and (c) the dispersion coefficient. 

The coefficient of vertical diffusivity is calculated from Eq. 24-32 and from a evaluated in Illustrative Example 24.2. The characteristic time and length scales for vertical mixing are given by Eqs. 24-33 and 24-34. The following table summarizes the results ... 

Figure 2. Illustration of simulation techniques available at various time and length scales. QC means first principles, quantum chemical methods. MD refers to classical molecular dynamics methods. (Monte Carlo methods are useful in roughly the same range of time and distance.) Methods for connecting QC, MD, and continuum methods are indicated in parentheses.

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