Critical droplet

A homogeneous metastable phase is always stable with respect to the fonnation of infinitesimal droplets, provided the surface tension a is positive. Between this extreme and the other thennodynamic equilibrium state, which is inhomogeneous and consists of two coexisting phases, a critical size droplet state exists, which is in unstable equilibrium. In the classical theory, one makes the capillarity approxunation the critical droplet is assumed homogeneous up to the boundary separating it from the metastable background and is assumed to be the same as the new phase in the bulk. Then the work of fonnation W R) of such a droplet of arbitrary radius R is the sum of the... 

The nucleation time r, i. e.. the time needed to form a critical droplet of deconfined quark matter, can be calculated for different values of the stellar central pressure Pc which enters in the expression of the energy barrier in Eq. (5). The nucleation time can be plotted as a function of the gravitational mass... 

Here Va and are the true velocities at the entrance, of gas and liquid, respectively, and do is the critical droplet diameter. The value of the Wee depends on the degree of shock at the entrance section e.g., for smooth liquid injection, 22 was used, and for tee entrances, 13 to 16. Collier and Hewitt (C6) also measured entrainment in air-water mixtures, and have extended the same correlation to much wider ranges, using We — 13 in the case of jet injection with the results shown in Fig. 9. Anderson et al. (A5), during mass-transfer studies in a water-air-ammonia system, found en-... 

If the wetted perimeter of the droplet at the time of disengagement is only 5% of vO (a reasonable assumption experimental evidence is lacking), the critical droplet size. D, would be reduced to 0.0047 ft (1.400 Sim). In general, it is reasonable to assume that droplet disengagement sizes fall between 0.02 and 0.08 in. (500 - 2.000 sim). 

Fourth, if the liquid-junction potential or IT of TBA+ or FeCp-X+ governs directly the droplet-size effect, the critical droplet radius, below which kobs is leveled off, should depend on the TBA+ concentrations and FeCp-X. However, the critical droplet radius is 5 /an, independent of Ao and FeCp-X (X = H or DCM). Furthermore, in the case of FeCp-DCM, the droplet-size effect on /cobs analogous to the data in Figure 14b can be observed even in the absence of TBA+TPB" and TBA+C1 , with the critical droplet radius being 5 /an. 

The critical droplet radius of kobs is independent of Ao and FeCp-X (X = H or DCM). Furthermore, the ET reaction proceeds efficiently even in... 

As mentioned earlier, experiments indicate that spontaneous condensation is not significant until fairly high supersaturations are achieved. For example, supersaturations of slightly less than 5 are necessary with water vapor in particle free air for the formation of a visible fog by adiabatic expansion of moist air at 0°C. This supersaturation implies a critical droplet diameter of about 0.0015 xm and a cluster of several hundred molecules. 

Figure 9.9 Dimensionless critical droplet size for breakup (capillary number Ca<- = Crj a/r) as a function of viscosity ratio M of the dispersed to the continuous phase for two-dimensional flows in the four-roll mill. The data sets correspond from bottom to top to ff — 1.0(0), 0.8 (A), 0.6 (0), 0.4 (V), and 0.2 ( ), with a defined in Eq. (9-17). The fluids are those described in Fig. 9-7. The solid lines are the predictions of a small-deformation theory, while the dashed lines are for a large-deformation theory. The closed squares are from Rallison s (1981) numerical solutions (see also Rallison and Acrivos 1978). (From Bentley and Leal 1986, with permission from Cambridge University Press.)...

One of the motivations for undertaking this calculation was the fact that the interfacial widths calculated for the planar interface in Section III D were broad enough that the classical nucleation theory predicted a critical droplet that was almost all interface, calling into question the assumption that it could be treated as a bulk crystal with a sharp surface layer. In this nonclassical theory, the properties of the nucleus at the center are not imposed as in the classical theory the system variationally selects the optimal form. It is... 

A. Coniglio and W. Klein (1980) Clusters and Ising critical droplets a renormalisation group approach. J. Phys. A 13, pp. 2775-2780... 

The possibility of a direct proof of the question which of the mentioned predictions reflects the situation more correctly was opened by recent molecular dynamics simulations of argon condensation performed by Wedekind et al. The difference of the average temperature of the critical droplets AT = To-T, according to their simulations, is shown in Fig. 4 (upper curves in the Fig.4, taken from Refs. 17-18). It seems to be obvious that the (average) temperature of the critical clusters is larger as compared with the temperature of the surrounding vapor. 

In order to compute the temperature difference between the argon critical droplet and the vapour, we have to express via the entropy... 

Fig. 35. (a) Order parameter profile 0(z) across an interface between two coexisting phases the interface being oriented perpendicular to the z-direclion. (b) The radial order parameter profile for a marginally stable critical droplet in a metastable state which is close to the coexistence curve, (c) Same as (b) but for a state close to the spinodal curve, 0sp. In (a) and (b) the intrinsic thickness of the interface is of the order of the correlation length ifeoex whereas in (c) it is of the order of the critical droplet radius / . From Binder (1984b). 

Obviously, A F p) increases for small p (where the surface term SdPd X fmi dominates), reaches a maximum AF at a critical droplet radius / , and then decreases again due to the negative volume term. In this classical nucleation theory, it is straightforward to obtain the critical droplet radius R from... 

Fig. 40. Schematic description of unstable thermodynamic fluctuations in the two-phase regime of a binary mixture AB at a concentration cb (a) in the unstable regime inside the two branches tp of the spinodal curve and (b) in the metastable regime between the spinodal curve tp and the coexistence curve The local concentration c(r) at a point r = (x. y, z.) in space is schematically plotted against the spatial coordinate x at some time after the quench. In case (a), the concentration variation at three distinct times t, ti, u is indicated. In case (b) a critical droplet is indicated, of diameter 2R , the width of the interfacial regions being the correlation length Note that the concentration profile of the droplet reaches the other branch ini, of the coexistence curve in the droplet center only for weak supersaturations of the mixture, where cb -

The critical droplet size that can exist in the flow at a given thermodynamic mode is determined by the velocity of movement of the mixture of water and petroleum, interfacial tension between the phases and pulsation of flow. 

All the curves in Figure 15.5 pass through a maximum. These maxima occur at the critical droplet diameter, Dpc,... 

Fig. 32.6 Critical droplet size in the dripping mode. Electric field applied to a capillary containing distilled water. Experiment conditions Voltage applied at capillary tip was varied from 0 to 10,000 V flow rate set at 0.07 mL min , ground electrode is set at 10 mm below the emitter tip...

This mean-field theory to calculate the interfacial profile and interfacial free energy can be extended to compute also for a critical droplet the order parameter profile (for a binary mixture near the critical point,... 

Formation of such droplets must then be an activated process whose rate is proportional to exp [—AF /(A 7 ]. We can estimate this rate using equation (4.2.6) for the interfacial energy y, and the result is that the rate of homogeneous nucleation we should expect for polymer systems is vanishingly small. In practice nucleation is usually aided by the presence of other interfaces, for example impurity particles such as dust or the container walls may well be able to nucleate critical droplets with much lower activation energies (heterogeneous nucleation) or indeed with no activation energy at all. We will return to this subject in section 5.3 when we discuss the effects of surfaces on phase separation. 

If water drops are formed over a boihng kettle over which there exists an upward convection current of 0.02 m s , what will be the critical droplet diameter for which the drops will begin to fall ... 

For water, the values for surface tension and density are yig = 72.99 N/m and p = 1,000 kg/m, respectively g is the gravitational constant (9.81 m/s ). Using (4.2), a critical droplet size of 2.7 mm can be obtained. A droplet smaller than jc generally remains stuck when placed on a solid surface because of the contact angle hysteresis. On the other hand, gravity would flatten a droplet of radius r > k. ... 

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