Clusters critical

The last step, the formation of metal, takes place if in the reducing atmosphere (MX excess) a critical cluster size is reached (e.g., by heating the solution to more than 100°C), thus forming a giant cluster whose metal core is nearly identical to the bulk material and which can disproportionate to metal and Mm species without any further significant change of its bondings. 

Consider the formation of a two-dimensional nucleus and show that the Gibbs energy of a critical cluster is inversely proportional to rj. For this purpose introduce a boundary energy which is proportional to the perimeter of the cluster. 

Consider the formation of hemispherical nuclei of mercury on a graphite electrode. The intefacial tension of mercury with aqueous solutions is about 426 mN m-1. From Eq. (10.16) calculate the critical cluster sizes for 7 = —10, —100, —200 mV. Take z — 1 and ignore the interaction energy of the base of the hemisphere with the substrate. 

The interaction between ions or molecules leads to the formation of a critical cluster or nucleus. 

Figure 7.1. Free energy of formation of a cluster as a function of size N (a cluster of N atoms) the size of the critical cluster (nucleus).

The spontaneous growth of clusters is possible after the maximum in AG is reached. This critical cluster is the nucleus of the new phase and is characterized by... 

Figure 4-1 Extra Gibbs free energy of clusters as a function of cluster radius. The critical cluster size is when the extra free energy reached the maximum. A5m c = 56 J/K/mol, Vc = 46 cm /mol, a = 0.3 J/m, Te = melting temperature = 1600 K, and system temperature = 1500 K. AGc mX ASm ( (r—Tg) = —5600 J/mol. The radius of the critical cluster is r = 2aVg/(AGm c) = (2) (0.3) (46 x 10 )/5600m = 4.93 nm. The Gibbs free energy of the critical cluster relative to the melt is AG = (16/3)7tCT /(AGm c/l g)2 = 3.05 x lO i J.

Because AGg m < 0 for crystallization to occur, r is greater than zero. The free energy of the critical cluster (relative to the melt) is... 

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. 

Nucleation rate based on the classical nucleation theory The nucleation rate is the steady-state production of critical clusters, which equals the rate at which critical clusters are produced (actually the production rate of clusters with critical number of molecules plus 1). The growth rate of a cluster can be obtained from the transition state theory, in which the growth rate is proportional to the concentration of the activated complex that can attach to the cluster. This process requires activation energy. Using this approach, Becker and Coring (1935) obtained the following equation for the nucleation rate ... 

The mechanisms of droplet (or liquid germs) formation from a supersaturated vapour phase is still the subject of many investigations. After giving a brief account of the classical theory [64], which, as shown above, provides a simple method for estimating the energy barrier to overcome before effective nucleation is started, and permits the estimation of the critical cluster size, a complementary approach will be presented. 

Other computer simulations were made to test the classical theory. Recently, Ford and Vehkamaki, through a Monte-Carlo simulation, have identified fhe critical clusters (clusters of such a size that growth and decay probabilities become equal) [66]. The size and internal energy of the critical cluster, for different values of temperature and chemical potential, were used, together with nucleation theorems [66,67], to predict the behaviour of the nucleation rate as a function of these parameters. The plots for (i) the critical size as a function of chemical potential, (ii) the nucleation rate as a function of chemical potential and (iii) the nucleation rate as a function of temperature, suitably fit the predictions of classical theory [66]. 

The nucleation theorems have been used to obtain information about critical clusters from experimental data, see Ford IJ (1997) Phys Rev E 56 5615 Ford IJ (1996) J Chem Phys 105 8324. These theorems state that, from known size and internal energy of the critical cluster, the nucleation rate can be deduced as a function of the temperature and the supersaturation. 

As seal from Eq. (7.251), the value of TV depends upon several parameters of the system, e.g., the edge surface energy, . It also depends on the overpotential T, and one can see that the size of the critical cluster decreases with an increase in hjl. For 2D nucleation on quasi-perfect silver single crystals, the number of atoms in the minimum nucleus size at which AG begins to decrease with an increase in A varies from 25 to 67 atoms as t varies from -10 to -6 mV. 

All approaches are based either on the thermodynamical description of the gas-solid phase transition by classical nucleation theory or on a detailed discussion of the relevant chemical reactions leading finally to critical clusters (e.g. review by Gail, Sedlmayr, 1987d). We will refrain from a presentation of these various approaches but only list the basic molecules from which the primary condensates are likely to be formed ... 

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). 

Equation 19.17 may be interpreted in a simple way. If the equilibrium concentration of critical clusters of size Afc were present, and if every critical cluster that grew beyond size Mc continued to grow without decaying back to a smaller size, the nucleation rate would be equal to J = (3CNexp[-AQc/(kT)]. However, the actual concentration of clusters of size Mc is smaller than the equilibrium concentration, and many supercritical clusters decay back to smaller sizes. The actual nucleation rate is therefore smaller and is given by Eq. 19.17, where the first term (Z) corrects for these effects. This dimensionless term is often called the Zeldovich factor and has a magnitude typically near 10-1. 

The spontaneous growth of clusters is possible after the maximum in AG is reached. This critical cluster is the nucleus of the new phase and is characterized by equal probability for growth and dissolution. The growth of clusters before a maximum is reached, and when average AG is increasing, is due to statistical energy... 

Clusters grow until a critical cluster size (nucleus) is reached that is energetically favorable to sustain growth... 

The rate at which critical sized clusters are formed is very sensitive to the height of the free energy barrier (AG), or equivalent to the extent of penetration into the metastable region. As the critical cluster size becomes smaller, so does the free energy barrier that must be overcome to form the critical cluster. With increasing... 

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