Critical cluster volume

The critical cluster volume reduces significantly with time due to an increase in segmental orientation for clusters oriented along the flow direction and approaches a steady state. For clusters oriented perpendicularly, the critical volume increases in time over the critical volume in the initial system indicating less favourable conditions for nucleation in the range of disoriented clusters. The reduction of critical volume for highly oriented clusters is associated with reduction in free energy barrier of nucleation. 

Promotion of a number of sub critical clusters to critical nuclei without growth, caused by reduction of critical cluster volume in time, is athermal nucleation. The concept of athermal nucleation, introduced by Fisher and Turnbull [43], consists in changing thermodynamic criterion of cluster stability. General expression of athermal nucleation in the systems with time-dependent thermodynamic parameters was derived in [21,45]. Angular distribution of athermal nucleation in the transient system is proportional to the distribution of critical clusters and the time derivative of the critical cluster volume... 

Fig. 4.11. Time evolution of the reduced critical cluster volume, g (6, t)/g 6. t = 0), vs. orientation angle, 0, between the initial state and the steady-state calculated for uniaxial elongational flow with fixed elongation rate, esr = 1. Af = 100...

The flux of athermal nucleation is positive when associated with a reduction of the critical cluster volume in time due to transient elastic potential of chain deformation and orientation of the chain segments. Fast changes in the critical cluster size may result in athermal nucleation. 

And the free energy of the critical cluster is still Equation 4-4. If the cluster is not spherical (e.g., the cluster could be a cube, or some specific crystalline shape), then the specific relations between i and cluster volume and surface area are necessary to derive the critical cluster size. 

When the supersaturation ratio S becomes greater than unit, the small liquid droplets (i.e. molecular clusters) commence to appear. Almost all the droplets are immediately destroyed due to evaporation and only small fraction of the droplets (critical clusters) with radii greater than a critical radius r have a chance to survive and grow by accretion of vapor molecules (monomers) onto their surface. It is assumed that macroscopic thermodynamics is applied to the critical clusters that are considered as liquid droplets containing the large number of monomers, that is nx>>i. The number of the critical clusters formed per unit time per unit volume is the nucleation rate J so that the number density of dust grains is Nd = JJdt. Expressions for calculation of the nucleation rate and other quantities can be found in the review paper by Draine (1981). 

The classical model is an obvious approximation since the interface may be significantly diffuse and occupy a substantial fraction of the cluster volume. A clean separation between bulk and interfacial energies therefore becomes problematic. The classical model for the critical nucleus may be expected to be a reasonable... 

The percolation probability has different values based on the classical theory site or bond percolation for different structures, as shown in Table 12.2. This critical percolation volume fraction, <, is calculated from the percolation threshold and the space filling factor. The volume fraction for site percolation for various structures is essentially the same as follows. In three dimensions, the site percolation threshold occurs at —16% volume. Near the percolation threshold the average cluster size diverges as does the spanning length of clusters. 

Here Fcrit = - Z Acrit,j is the volume of the critical cluster. Replacing Vent with... 

According to eqs. (4.26a and b), the energy of formation of the critical cluster is also given by its volume Fcrit. This equation holds for both homgeneous and heterogeneous nucleation. If, according to Kaischew [4.4, 4.5], and FJ jj are... 

The linear dimensions of the critical cluster are defined by eq. (4.27) as a function of the overpotential. As discussed already, the presence of the substrate affects the thickness of the cluster via hj only. The larger the adhesion energy p, the smaller is the thickness and, hence, the volume of the critical cluster. If different... 

Here T is the temperature, p is the pressure, a is the surface tension, A is the surface area, V is the volume, 5 is the entropy density, Pi are the particle densities, and pi the chemical potentials of the different components, R is the radius of the critical cluster referred to the surface of tension, the index a specfies the parameters of the cluster while p refers to the ambient phase. The equilibirum conditions coincide with Gibbs expressions for phase coexistence at planar interfaces (R oo) or when, as required in Gibbs classical approach, the surface tension is considered as a function of only one of the sets of intensive variables of the coexisting phases, either of those of the ambient or of those of the cluster phase. In such limiting cases, Gibbs equilibrium conditions... 

In a gas at temperatures sufficient lower than its critical temperature, volume per molecule V/N is much larger than the molecular volume even in the saturated vapor the cluster integrals are all positive and independent of the volume V. 

The model predicts a critical ionic-insulator-to-conductor transition, at which point the interconnection between the clusters is established and above which conductivity is proportional to void volume. Experimentally, the critical void volume is found at a 10% void-volume fraction. 

Figure 11. (a) The conduction gap observed in small clusters of gold, palladium, cadmium, and silver as a function of cluster volume, (b) Normalized slope of the /-K curves (the conductance) as a fraction of cluster volume for the four metals studied. Above a critical volume of ca 4nm the slope becomes size-independent. This is possibly a direct indication of a size-induced-metal-insulator transition, as indicated. Taken from Vinod et a/. ... 

The rate of the primary homogeneous nucleation 5o,hom can be derived by multiplying an impact coefficient 5 with the total surface of all clusters present in a given volume V. The impact coefficient is the number of molecules which are hitting the surface based on a unit of time and surface. The total surface of all critical clusters is given by the number of clusters in a volume V and the surface of a cluster. The rate of homogenous nucleation is... 

Most percolation studies rely on Monte Carlo simulations, whose properties of interest are critical cluster size, bond densities (i.e. number of bonds that lead to network per site), particle volume in the matrix, particle orientation and network geometry at the threshold. In most cases simulations treat the percolation in a statistical manner which means that the particles are randomly distributed in the matrix and network pathways are formed simply by increasing the particle volume fraction in the composite. Although such an approach has merit and provides valuable insight on the network formation, it is far away from reality, especially when polymers are concerned. Addition of particles in a polymer matrix is mainly performed via solution or melt mixing, which means that both particles and polymer chains are in motion and interact with each other. More advanced theoretical approaches do take into consideration the thermodynamic interactiOTis between the composite constituents (particle-particle, particle-polymer and... 

A steep increase in conductivity at intermediate water concentrations can be explained by percolation transition, and every ME mixture will exhibit a specific critical water volume ratio/concentration at which percolation occurs. The increased conductivity leading up to the is caused by an increased number of (still) individual water droplets. The conductivity measured around and above the (j) is due to dynamic droplet clusters or transient water channels, and microscopically droplets do not exist anymore at this stage. The critical water concentration needed to induce percolation usually ranges from 0.1 to 0.26 depending on the ME components, specifically the type of cosurfactant, and the temperature. 

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