Hard sphere self-diffusion

Figure 9 Properties of the attractive colloidal fluid investigated in Ref. 75 (a) self-diffusivity and (b) average free volume versus strength of the interparticle attraction (c) self-diffusivity versus average free volume for the hard-sphere fluid (open circles) and the attractive colloidal fluid (closed circles). Data compiled from Ref. 75.

These experiments suggest that as the long time self-diffusion coefficient approaches zero the relaxation time becomes infinite, suggesting an elastic structure. In an important study of the diffusion coefficients for a wide range of concentrations, Ottewill and Williams14 showed that it does indeed reduce toward zero as the hard sphere transition is approached. This is shown in Figure 5.6, where the ratio of the long time diffusion coefficient to the diffusion coefficient in the dilute limit is plotted as a function of concentration. 

Mass Diffusivity in Liquid Metais and Ailoys. The hard-sphere model of gases works relatively well for self-diffusion in monatomic liquid metals. Several models based on hard-sphere theory exist for predicting the self-diffusivity in liquid metals. One such model utilizes the hard-sphere packing fraction, PF, to determine D (in cm /s) ... 

According to the kinetic theory of gases, the self-diffusivity of a hard-sphere gas is given by DG = (2/5)(u)L, where (u) is the average velocity and L is the mean free path [4]. Because the mean free path of a confined particle in the liquid is about equal to the diameter of its confining volume, the contribution of the confined particle to the self-diffusivity of the liquid may be written... 

Although there has not been much theoretical work other than a quantitative study by Hynes et al [58], there are some computer simulation studies of the mass dependence of diffusion which provide valuable insight to this problem (see Refs. 96-105). Alder et al. [96, 97] have studied the mass dependence of a solute diffusion at an infinite solute dilution in binary isotopic hard-sphere mixtures. The mass effect and its influence on the concentration dependence of the self-diffusion coefficient in a binary isotopic Lennard-Jones mixture up to solute-solvent mass ratio 5 was studied by Ebbsjo et al. [98]. Later on, Bearman and Jolly [99, 100] studied the mass dependence of diffusion in binary mixtures by varying the solute-solvent mass ratio from 1 to 16, and recently Kerl and Willeke [101] have reported a study for binary and ternary isotopic mixtures. Also, by varying the size of the tagged molecule the mass dependence of diffusion for a binary Lennard-Jones mixture has been studied by Ould-Kaddour and Barrat by performing MD simulations [102]. There have also been some experimental studies of mass diffusion [106-109]. 

The hard sphere diameters were then used to calculate the theoretical Enskog coefficients at each density and temperature. The results are shown in Figure 3 as plots of the ratio of the experimental to calculated coefficients vs. the packing fraction, along with the molecular dynamics results (24) for comparison. The agreement between the calculated ratios and the molecular dynamics results is excellent at the intermediate densities, especially for those ratios calculated with diameters determined from PVT data. Discrepancies at the intermediate densities can be easily accounted for by errors in measured diffusion coefficients and calculated diameters. The corrected Enskog theory of hard spheres gives an accurate description of the self-diffusion in dense supercritical ethylene. 

Our previous study (J 6) of self diffusion in compressed supercritical water compared the experimental results to the predictions of the dilute polar gas model of Monchick and Mason (39). The model, using a Stockmayer potential for the evaluation of the collision integrals and a temperature dependent hard sphere diameters, gave a good description of the temperature and pressure dependence of the diffusion. Unfortunately, a similar detailed analysis of the self diffusion of supercritical toluene is prevented by the lack of density data at supercritical conditions. Viscosities of toluene from 320°C to 470°C at constant volumes corresponding to densities from p/pQ - 0.5 to 1.8 have been reported ( 4 ). However, without PVT data, we cannot calculate the corresponding values of the pressure. 

Most free-volume models for diffusion in polymers follow the phenomenological basis set in (55) where the self-diffusion of an ideal liquid of hard spheres ( molecules ) has been analysed. These molecules are confined - for most of the time - in a cage formed by their immediate neighbours. A local fluctuation in density may open a hole within a cage, large enough to permit a considerable displacement of the sphere contained by it. This displacement gives rise to diffusion only if another sphere jumps into the hole before the first sphere returns to its initial position. Diffusion occurs not as a result of an activation process in the ordinary sense but rather as a result of the redistribution of the free-volume within the liquid of hard spheres. 

The model of diffusion of hard spheres is applicable to interpret self-diffusion in liquids which behave according to the van der Waals physical interaction model (56). This might be the case for simple dense fluids at high temperature, T Tg, but it is an oversimplified model for the real diffusion of small organic penetrants in polymers. The functional relationships derived in the model of hard-spheres have been reinterpreted over course of the time, leading to a series of more sophisticated free-volume diffusion models. Some of these models are presented briefly below. 

Liu and Ruckenstein [Ind. Eng. Chem. Res. 36, 3937 (1997)] studied self-diffusion for both liquids and gases. They proposed a semiem-pirical equation, based on hard-sphere theory, to estimate self-diffusivities. They extended it to Lennard-Jones fluids. The necessary energy parameter is estimated from viscosity data, but the molecular collision diameter is estimated from diffusion data. They compared their estimates to 26 pairs, with a total of 1822 data points, and achieved a relative deviation of 7.3 percent. 

Anderko and Lencka find. Eng. Chem. Res. 37, 2878 (1998)] These authors present an analysis of self-diffusion in multicomponent aqueous electrolyte systems. Their model includes contributions of long-range (Coulombic) and short-range (hard-sphere) interactions. Their mixing rule was based on equations of nonequilibrium thermodynamics. The model accurately predicts self-diffusivities of ions and gases in aqueous solutions from dilute to about 30 mol/kg water. It makes it possible to take single-solute data and extend them to multicomponent mixtures. 

III. Single-Particle Motion—A Stochastic Approach A. Stochastic Formulation of Friction Self-Diffusion in Hard-Sphere Fluids Collisional Friction... 

Consider the difficult case of self-diffusion in a hard-sphere fluid. It is useful to define the caging factor x as < / p(0) >/8/iBMkT and a shielding factor S by = (S/2)(Jm- Then Eq. (3.7) can be rearranged to give the self-diffusion coefficient... 

In Section III B the encounter model formulation of the diffusion coefficient was tested for self-diffusion = m ) for a hard-sphere fluid, and in Section HIE the model was evaluated for the case of a large test particle... 

The results for the encounter model for hard spheres were obtained by closure, as was done for self-diffusion in Section Iff B. The pair friction was modeled either by the screened FPE result,... 

Figure 10. Mass dependence of self-diffusion coefficient of hard-sphere test particle of mass mg in hard-sphere fluid of particles of mass m for Rg = R at density pR, = 0.9 for Rodger-Sceats friction (-----) and Smoluchowski friction (--------). 0—simulation results of Herman and Alder.

To close this Section we comment on two papers that do not fit under any neat heading. The first of these is by Xiao et al,261 who study the final stages of the collapse of an unstable bubble or cavity using MD simulations of an equilibrated Lennard-Jones fluid from which a sphere of molecules has been removed. They find that the temperature inside this bubble can reach up to an equivalent of 6000 K for water. It is at these temperatures that sonolumines-cence is observed experimentally. The mechanism of bubble collapse is found to be oscillatory in time, in agreement with classical hydrodynamics predictions and experimental observation. The second paper, by Lue,262 studies the collision statistics of hard hypersphere fluids by MD in 3, 4 and 5 dimensions. Equations of state, self-diffusion coefficients, shear viscosities and thermal conductivities are determined as functions of density. Exact expressions for the mean-free path in terms of the average collision time and the compressibility factor in terms of collision rate are also derived. Work such as this, abstract as it may appear, may be valuable in the development of microscopic theories of fluid transport as well as provide insight into transport processes in general. 

Taking the hard sphere colloids as a reference state, the mean-square displacement (MSD) in dilute suspensions is associated with the particle self-diffusion whereas at finite volume fractions the onset of interactions marks the alteration of the dynamics. The latter can be probed by the intermediate scattering function C(, t) which measures the spatiotemporal correlations of the thermal volume fraction fluctuations [91]. Figure depicts two representations (lower inset and main plot) of the non-exponential for a nondilute hard sphere colloidal... 

In the presence of size polydispersity, there is an additional incoherent contribution to C(, t) decaying through the self-diffusion coefficient Ds((/)) [42,43,91]. The latter can also be measured for monodisperse hard sphere suspensions at finite concentrations at qR corresponding to the first minimum of S( ), i.e., when the interactions can be ignored [91]. These three different diffusion coefficients exhibit distinctly different dependence on q and 0. From these three transport quantities, Z>cou( ) is absent in monodisperse homopolymers, whereas Ds can hardly be measured in polydisperse homopolymers due to the vanishingly small contrast. 

Fig. 21 Steady state incoherent intermediate scattering functions d> (r) as functions of accumulated strain yt for various shear rates y the data were obtained in a col loidal hard sphere dispersion at packing fraction

The direct calculation of the collective contribution DJDs to the self-diffusion coefficient is complicated by the inadequate temperature dependence of the shear viscosity in ref. [ ]. Indeed, it is easy to verify that the ratio r / r g for the model argon increases with temperature on isochors. From the physical viewpoint, this result is inadequate. It is worth noting that for ( ) < 0.4 the values of r from ref. f ] and those determined on the basis of the Enskog theory for hard spheres diameter of which coincides with the effective diameter... 

The dependence of self-diffusion on density, p, and temperature can be described in all nonpolar liquids with good accuracy by modifications of the hard-sphere model. This model also provides quantitative descriptions for self-diffusion of the normal alkanes and these studies could be extended to polyethylene waxes with an average chain length of 154 carbon atoms. 

In a recent work, the translational motion of 4- -hexyl-4 -cyanobiphenyl (6CB) was studied in the isotropic phase by atomistic molecular dynamics simulation [134], The mean-square displacement showed evidence of sub-diffusive dynamics, with a plateau that became very apparent at the lowest temperatures. A three-time self-intermediate scattering function revealed that this plateau was connected with a homogeneous dynamics that, at longer times, became heterogeneous and finally exponential. These features, which are shared by, for example, a high-density system of hard spheres, support the universal character of the translational dynamics of liquids in their supercooled regime. 

Clarke has examined the thermodynamic equation of state and the specific heat for a Lennard-Jones liquid cooled through 7 at zero pressure. He found that drops with decreasing temperature near where the selfdiffusion becomes very small. Wendt and Abraham have found that the ratio of the values of the radial distribution function at the first peak and first valley shows behavior on cooling much like that observed for the volume of real glasses (Fig. 6), with a clearly defined 7. Stillinger and Weber have studied a Gaussian core model and find a self-diffusion constant that drops essentially to zero at a finite temperature. They also find that the ratio of the first peak to the first valley in the radial distribution function showed behavior similar to that found by Wendt and Abraham" for Lennard-Jones liquids. However, the first such evidence for a nonequilibrium (i.e. kinetic) nature of the transition in a numerical simulation was obtained by Gordon et al., who observed breakaways in the equation of state and the entropy of a hard-sphere fluid similar to those in real materials. 

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