Dynamic free volume

A similar theoretical basis for such a relationship can be deduced from the work of Cohen and Turnbull [94], which introduces the concept of a dynamic free volume with a low activation energy that is created and into which solute ions move to and fro. The model is valid when solute movement... 

Volume of vessel (free volume V) Shape of vessel (area and aspect ratio) Type of dust cloud distribution (ISO method/pneumatic-loading method) Dust explosihility characteristics Maximum explosion overpressure P ax Maximum explosion constant K ax Minimum ignition temperature MIT Type of explosion suppressant and its suppression efficiency Type of HRD suppressors number and free volume of HRD suppressors and the outlet diameter and valve opening time Suppressant charge and propelling agent pressure Fittings elbow and/or stub pipe and type of nozzle Type of explosion detector(s) dynamic or threshold pressure, UV or IR radiation, effective system activation overpressure Hardware deployment location of HRD suppressor(s) on vessel... 

Any polymer contains some inner free space free volume distributed in a dynamic manner between its molecular chains (see Section 23.2). When it is exposed to a fluid (liquid or gas) the physical possibility exists for fluid absorption by the polymer, if the fluid molecules or atoms are small enough to fit into local regions of this distributed space during kinetic movements. As this happens, subsequent kinetic chain motion must allow for the newly absorbed fluid molecules and, hence, the polymer s overall volume will adjust accordingly this action will coincide with the formation of more free space around these fluid molecules—so the polymer will swell a little. This process will be continued until an equilibrium is reached ( equilibrium swelling ), by which time the extent of swelling can be considerable. The amount of fluid taken up and the rate at which this happens are both important, and are discussed in this and following sections. 

To account for the variation of the dynamics with pressure, the free volume is allowed to compress with P, but differently than the total compressibility of the material [22]. One consequent problem is that fitting data can lead to the unphysical result that the free volume is less compressible than the occupied volume [42]. The CG model has been modified with an additional parameter to describe t(P) [34,35] however, the resulting expression does not accurately fit data obtained at high pressure [41,43,44]. Beyond describing experimental results, the CG fit parameters yield free volumes that are inconsistent with the unoccupied volume deduced from cell models [41]. More generally, a free-volume approach to dynamics is at odds with the experimental result that relaxation in polymers is to a significant degree a thermally activated process [14,15,45]. 

Generally, the values of the scaling exponent are smaller for polymers than for molecular liquids, for which 3.2 < y < 8.5. A larger y, or steeper repulsive potential, implies greater influence of jamming on the dynamics. The smaller exponent found for polymers in comparison with small-molecule liquids means that volume effects are weaker for polymers, which is ironic given their central role in the historical development of free-volume models. The reason why y is smaller... 

Studies of the effect of permeant s size on the translational diffusion in membranes suggest that a free-volume model is appropriate for the description of diffusion processes in the bilayers [93]. The dynamic motion of the chains of the membrane lipids and proteins may result in the formation of transient pockets of free volume or cavities into which a permeant molecule can enter. Diffusion occurs when a permeant jumps from a donor to an acceptor cavity. Results from recent molecular dynamics simulations suggest that the free volume transport mechanism is more likely to be operative in the core of the bilayer [84]. In the more ordered region of the bilayer, a kink shift diffusion mechanism is more likely to occur [84,94]. Kinks may be pictured as dynamic structural defects representing small, mobile free volumes in the hydrocarbon phase of the membrane, i.e., conformational kink g tg ) isomers of the hydrocarbon chains resulting from thermal motion [52] (Fig. 8). Small molecules can enter the small free volumes of the kinks and migrate across the membrane together with the kinks. 

Steady-state behavior and lifetime dynamics can be expected to be different because molecular rotors normally exhibit multiexponential decay dynamics, and the quantum yield that determines steady-state intensity reflects the average decay. Vogel and Rettig [73] found decay dynamics of triphenylamine molecular rotors that fitted a double-exponential model and explained the two different decay times by contributions from Stokes diffusion and free volume diffusion where the orientational relaxation rate kOI is determined by two Arrhenius-type terms ... 

With further understanding how molecular rotors interact with their environment and with application-specific chemical modifications, a more widespread use of molecular rotors in biological and chemical studies can be expected. Ratiometric dyes and lifetime imaging will enable accurate viscosity measurements in cells where concentration gradients exist. The examination of polymerization dynamics benefits from the use of molecular rotors because of their real-time response rates. Presently, the reaction may force the reporters into specific areas of the polymer matrix, for example, into water pockets, but targeted molecular rotors that integrate with the matrix could prevent this behavior. With their relationship to free volume, the field of fluid dynamics can benefit from molecular rotors, because the applicability of viscosity models (DSE, Gierer-Wirtz, free volume, and WLF models) can be elucidated. Lastly, an important field of development is the surface-immobilization of molecular rotors, which promises new solid-state sensors for microviscosity [145]. 

Several material properties exhibit a distinct change over the range of Tg. These properties can be classified into three major categories—thermodynamic quantities (i.e., enthalpy, heat capacity, volume, and thermal expansion coefficient), molecular dynamics quantities (i.e., rotational and translational mobility), and physicochemical properties (i.e., viscosity, viscoelastic proprieties, dielectric constant). Figure 34 schematically illustrates changes in selected material properties (free volume, thermal expansion coefficient, enthalpy, heat capacity, viscosity, and dielectric constant) as functions of temperature over the range of Tg. A number of analytical methods can be used to monitor these and other property changes and... 

Given this background, we can now provide several examples of how free volumes have been related to thermodynamic and dynamic properties of liquids, and how their measurement has been employed in computer simulations to derive microscopic insights that are otherwise not accessible from experiment. We consider first how to compute thermodynamic values from free volumes and follow that with the relationship of free volumes to dynamics. 

Figure 8 Compressibility factor P/fiksT versus density p = pa3 of the hard-sphere system as calculated from both free-volume information (Eq. [8]) and the collision rate measured in molecular dynamics simulations. The empirically successful Camahan-Starling84 equation of state for the hard-sphere fluid is also shown for comparison. (Adapted from Ref. 71).

Because efficient methods for computing free volumes from molecular simulations were introduced only recently, their connections to the dynamical properties of liquids have yet to be explored systematically. Nonetheless, initial investigations have already allowed scrutiny of some historical notions about these properties. Here, we briefly discuss two of these initial studies. Their results illustrate that some early free-volume based ideas about the origins of dynamics are consistent with simulation data, but those ideas will need significant revision if they are to be applied in a general way. 

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