Length and Timescales

The phenomenological approach does not preclude a consideration of the molecular origins of the characteristic timescales within the material. It is these timescales that determine whether the observation you make is one which sees the material as elastic, viscous or viscoelastic. There are great differences between timescales and length scales for atomic, molecular and macromolecular materials. When an instantaneous deformation is applied to a body the particles forming the body are displaced from their normal positions. They diffuse from these positions with time and gradually dissipate the stress. The diffusion coefficient relates the distance diffused to the timescale characteristic of this motion. The form of the diffusion coefficient depends on the extent of ordering within the material. 

For an ideal gas the diffusion coefficient is related to the mean free path of the gas molecule, 2, which represents the mean distance between collisions for that molecule. 

For a molecule at RTP this is of the order of a few hundred molecular diameters. In our ideal gas there is a distribution of velocities of the molecules about a mean value c. The mean free path defines a length scale in gases. As the density of the gas is increased and the mean free path approaches the molecular dimensions, a short-range molecular order develops and the material condenses to a liquid. The diffusional length scale is now much shorter range as a molecule encounters its 

In contrast to a gas, the short-range order in the fluid demands that significant structural relaxation must have occurred when the molecule has diffused a distance equivalent to the distance to the surrounding shell of nearest neighbours. This is of the order of a molecular radius. Since the diffusive process is described by the square of the mean distance moved by a molecule in a time t then 

This expression represents the structural relaxation time of a liquid so that if a strain is applied to the material it will relax the stress with a time characterised by t. This prompts the question What is the form of the stress when a strain is applied To answer this question we must consider linear viscoelasticity in detail. 

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An alternative, and interesting, possibility is to introduce a phase-separating blend as the fluid component (a schematic of this system is shown in Fig. 11). The phase-separating A-B polymer blend will evolve, and phase separate, at its own length- and timescales. However, to minimize the interface between the A and B domains of the polymer blend it may be desirable for the length-scale of phase separation to conform to the wavelength of undulation growth found in... 

Now consider the next larger length and timescales or , and x or xr. When L , r and t x, xr, transport is ballistic in nature and local thermodynamic equilibrium cannot be defined. This transport is nonlocal in space. One has to resort to time-averaged statistical particle transport equations. On the other hand, if L , , and t x, xr, then approximations of local thermodynamic equilibrium can be assumed over space although time-dependent terms cannot be averaged. The nonlocality is in time but not in space. When both L , r and t x, xr, statistical transport equations in full form should be used and no spatial or temporal averages can be made. Finally, when both L , , and t x, xr, local thermodynamic equilibrium can be applied over space and time leading to macroscopic transport laws such as the Fourier law of heat conduction. 

Local Volume Averaging. The principle of volume averaging and the requirement of existence of the local thermal equilibrium between the fluid and solid phases was discussed in the section entitled Conduction Heat Transfer. In addition to the diffusion time and length scale requirements for the existence of the local thermal equilibrium, the residence timescales (the time it takes for a fluid particle to cover the length scales d, i, and L) must be included in the length and timescale requirements. 

The imderlying model in dissipative particle dynamics is usually developed in such a way that the mass, length and timescales are all unity. This is similar to the use of reduced units for the Lennard-Jones potential (Section 4.10.5). A particular advantage of such an approach is that a single simulation may often be able to explain the behaviour of many different systems. With a mass of 1 the force acting on a particle is equal to its acceleration. In DPD there are three forces on each bead [Groot and Warren 1997] ... 

The theory and measurement techniques pertaining to the canonical Brownian motion of a particle in a fluid have been outlined above. We now describe two interesting areas of current research that probes deeper - motion at short length and timescales and motion when the step lengths and waiting times between steps are part of a broad-tail distribution. 

Ballistic to Brownian Transition What Happens at Confined Length and Timescales... 

Illustration of the different length and timescales relevant for the simulation of nanocomposites. A selection of representative methods employed at each scale is shown. MC, Monte Carlo MD, molecular dynamics NEMD, nonequilibrium molecular dynamics. (Reprinted from Praprotnik, M. et aL, Annu. Rev. Phys. Chem., 59,545,2008.)... 

In addition to providing a microscopic tool for observing the outcomes of physicochemical processes in extraordinary detail, molecular dynamics simulations can, in principle, provide a valuable technique for obtaining thermodynamic variables and rate constants via integration over selected portions of the molecular dynamics trajectory. Several techniques have heen recently employed that allow this kind of analysis, even with the present hmitations regarding length and timescales, such as time-accelerated molecular dynamics [228, 229]. 

Despite these tremendous capabilities, whieh are only just being now applied to many of the outstanding problems in corrosion science, there remain several key ehaUenges there is still some way to go for the computational science eommunity to be able to represent the kinds of length and timescales assoeiated with macro- and mesoscopic corrosion processes. Some of these challenges include ... 

The ultimate goal is a reversible bottom-up, top-down approach, based on first principles QM, to characterize properties of materials and processes at a hierarchy of length and timescales. This will improve our ability to design, analyze, and interpret experimental results, perform model-based prediction of phenomena, and to control precisely the multi-scale nature of material systems for multiple applications. Such an approach is now enabling us to study problems once thought to be intractable, including reactive turbulent flows, composite material instabilities,... 

However, rather than waiting for future developments, scientists and engineers have applied themselves to inventing a multitude of clever ways to negotiate the many different length and timescales that are manifest in the properties of polymers. This article is an attempt to portray a cross section of the methods that have been developed for this multitude of purposes, as well as to describe some of the outstanding results that have been obtained to date. 

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