Radius, critical

Given the expression IX-20 for the interfacial tension dependence on radius, derive the form of the dependence of the critical radius on the radius, given that the tension is proportional to (a) 1/r and (b) 1/r. ... 

Otherwise it will evaporate as above. The two regimes are separated by a critical radius R... 

In the LS analysis, an assembly of drops is considered. Growth proceeds by evaporation from drops withi < R and condensation onto drops R > R. The supersaturation e changes in time, so that e (x) becomes a sort of mean field due to all the other droplets and also implies a time-dependent critical radius. R (x) = a/[/"(l)e(x)]. One of the starting equations in the LS analysis is equation (A3.3.87) withi (x). 

FIG. 22-33 A practical isomotive field geometry, showing eo, the critical radius characterizing the isomotive electrodes. Electrode 3 is at ground potential, while electrodes 1 and 2 are at Vj = V. and V2 = V = —V. respectively. The inner faces of electrodes 1 and 2 follow = o [sin (36/2)]" , while electrode 3 forms an angle of 120 about the midline. 

Figure 1.9 The balance of endothermic surface energy and the exothermic formation of the stable condensed phase during nucleation from the vapour phase. The critical radius, above which the nuclei become stable, is where the resultant Gibbs energy change has zero slope...

When the nucleus is a liquid, the angle 6 is called tire wetting angle. It can be seen that the critical radius in heterogeneous nucleation is given by the same equation as tlrat for homogeneous nucleation, but the radius now refers... 

In the total particle size distribution, some particles of small diameter decrease in radius, and those in the larger diameter range increase in radius during Ostwald ripening. There will therefore be a radius at which particles neither decrease nor grow in size and if [Pg.210]

Figure 10.2 The time-temperature-nucleation curve showing the balance between the rate of nucleation and the critical radius which produces a maximum rate...

This result has been plotted out in Fig. 7.1. It shows that there is a maximum value for Wjr corresponding to a critical radius r. For r < r (dV /dr) is positive, whereas for r > r it is negative. This means that if a random fluctuation produces a nucleus of size r < r it will be unstable the system can do free work if the nucleus loses atoms and r decreases. The opposite is true when a fluctuation gives a nucleus with r > r. Then, free work is done when the nucleus gains atoms, and it will tend to grow. To summarise, if random fluctuations in the liquid give crystals with r > r then stable nuclei will form, and solidification can begin. 

If we compare eqns (7.11) and (7.3) we see that the expressions for the critical radius are identical for both homogeneous and heterogeneous nucleation. But the expressions for the volume of the critical nucleus are not volume is... 

ITence, show that the critical radius is given by r- = 2y,p/ AG., ... 

The nucleation rate is, in fact, critically dependent on temperature, as Fig. 8.3 shows. To see why, let us look at the heterogeneous nucleation of b.c.c. crystals at grain boundaries. We have already looked at grain boundary nucleation in Problems 7.2 and 7.3. Problem 7.2 showed that the critical radius for grain boundary nucleation is given by... 

Grain boundary nucleation will not occur in iron unless it is cooled below perhaps 910°C. At 910°C the critical radius is... 

Nucleation of solids from liquids critical radius for homogeneous and heterogeneous nucleation... 

Here R is a critical radius for nucleation [20,56] on a faceted surface... 

By putting the right-hand side in Eqs. (52) and (53) equal to zero, one receives the equilibrium value of local radius of curvature R (or, Rt), which is nothing but the Wulff construction. For an anisotropic step tension 7(0), there is a local critical radius defined as... 

The elution of such gels is an example not of size exclusion but rather of hydrodynamic fractionation (HDF). However, it must be remembered that merely being able to physically fit an insoluble material through the column interstices is not the only criterion for whether the GPC/HDF analysis of an insoluble material will be successful. A well-designed HDF packing and eluant combination will often elute up to the estimated radius in Eq. (5), but adsorption can drastically limit this upper analysis radius. For example, work in our laboratory using an 8-mm-bead-diameter Polymer Laboratories aqueous GPC column for HDF found that that column could not elute 204 nM pSty particles, even though Eq. (5) estimates a critical radius of —1.5 jam. 

The shear-stress distribution is uneven in a capillary. Since an interfacial slippage takes place only at a point where the shear stress exceeds a critical value, a critical radius r,- can be defined as ... 

Thus, for a given fluid (Ry constant), the critical radius rt is determined entirely by the pressure drop and becomes progressively larger as the pressure drop is reduced. Flow-ceases when the shear stress at the wall R falls to a value equal to the yield stress Ry. 

Figure 15. Electrocapillary energy for the formation of a breakthrough pore in a thin surface oxide film on metals as a function of pore radius.7 AE E - Epzc, where Epzc is the potential-of-zero charge of the film-free metal. Al is the activation banier for the formation of a breakthrough pore and r is its critical radius. M, metal OX, oxide film EL, electrolyte solution, h a 2 x I O 9 m, am = 0.41 J m-2, a = 0.01 J m-2, ACj= 1 F m"2. a, AE=0.89 V b, AE=0.9 V c,A = 1.0 V. (From N. Sato, J. Electmckem. Soc. 129,255,1982, Fig. 2. Reproduced by permission of The Electrochemical Society, Inc.)...

From the conceptual diagram in Fig. 15, it is obvious that if the radius of the nucleus exceeds the critical radius r the nucleus will grow into a macroscopically ruptured small pore. The passive film is more or less defective and the size of the defect will fluctuate from moment to moment. It is therefore reasonable to assume a certain probability that pore nuclei larger than the critical radius are formed in the film. 

The critical radius at Tg is a multiple of Droplets of size N > N are thermodynamically unstable and will break up into smaller droplets, in contrast to that prescribed by F N), if used naively beyond size N. This is because N = 0 and N = N represent thermodynamically equivalent states of the liquid in which every packing typical of the temperature T is accessible to the liquid on the experimental time scale, as already mentioned. In view of this symmetry between points N = 0 and N, it may seem somewhat odd that the F N) profile is not symmetric about. Droplet size N, as a one-dimensional order parameter, is not a complete description. The profile F N) is a projection onto a single coordinate of a transition that must be described by order parameters—the... 

Figure 7. Experimental data (symbols) for TNB s viscosity [78] superimposed on the results of the fitting procedure (line) from Lubchenko and Wolynes [47] are shown. Ta is diown by a tickmark. (TNB = trinaphthyl benzene). The temperature Ter signifies a crossover from activated to collisional viscosity, dominant at the lower and higher temperatures, respectively (see text). The temperature is varied between the boiling point and the glass transition. The right-hand side panel depicts the temperature dependence of the length scales of cooperative motions in the liquid. The thick solid and dashed lines are the critical radius and the cooperativity length respectively. Taken from Ref. [47] with permission.

It can be seen in A that the value of r is remarkably constant over a wide range of temperatures, but that it starts to approach infinity at some critical temperature, T. In contrast, the number of nuclei produced in B is maximum at some particular temperature. This shows us that cdthough the critical radius does not change significantly over a wide range of temperature, the numbers of homogeneous nuclei produced is very dependent upon temperature. 

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