Single-phase boundaries

When a solid-solid reaction starts, a layer of product is formed between the two reactant phases and a single phase boundary converts into two different phase boundaries as... 

Figure 1. Single-phase boundaries for the component oxides of the (Ce,Mo,Te)0 system between 400 and 600° C and regions of formation of non-crystalline reaction products.

Fig. 3 Analyses of bulk contrast scattering intensities as a function of surfactant volume fraction. The scattering intensities were measured at points close to lamellar and single phase boundary where the average mean curvature is nearly zero, (a) Raw data after background correction, (b)-(d) three representative curves comparing the theory and the experimental data. Solid lines are the fits using modified Berk theory...

Table 1 Fitted parameters and calculated interfacial curvatures along the lamellar-single phase boundary, and the other at a fixed salinity 0.49 wt%) two horizontal lines in the phase diagram (one close to ...

Fig. 5 Fitted parameters a, b, and c. (A) As a function of surfactant volume fraction (along the points close to lamellar and single phase boundary), (B) as a function of salinity at a fixed surfactant volume fraction, = 0.20...

Fig. 6 Average Gaussian curvature and average square-mean curvature. (A) As a function of surfactant volume fraction (along the points close to lamellar and single phase boundary), (B) as a function of salinity at a fixed surfactant volume fraction, = 0.20. Solid circles are the calculated average Gaussian curvature using Eq. (30) and solid boxes are the calculated square-mean cuvatures using Eq. (31). In (A), the solid line is a fit with a phenomenological expression, Eq. (28) and the dash line is a fit for the square mean curvature with an equation similar to Eq. (28)...

Turbulent motions in single-phase boundary layers, pipe flows, and jets are not completely random but have coherent or ordered structure [42-45], According to the four-quadrant classification method, the turbulent motions are grouped into four distinct categories, as illustrated in Fig. 2.24. In this figure, ejection denotes a higher momentum fluid motion directed outward, outward interaction denotes a lower momentum fluid motion directed outward, sweep denotes a lower momentum... 

In order to describe any electrochemical cell a convention is required for writing down the cells, such as the concentration cell described above. This convention should establish clearly where the boundaries between the different phases exist and, also, what the overall cell reaction is. It is now standard to use vertical lines to delineate phase boundaries, such as those between a solid and a liquid or between two innniscible liquids. The junction between two miscible liquids, which might be maintained by the use of a porous glass frit, is represented by a single vertical dashed line, j, and two dashed lines, jj, are used to indicate two liquid phases... 

An alloy is cooled from a temperature at which it has a single-phase structure (a) to a temperature at which the equilibrium structure is two-phase (a -i- ji). During cooling, small precipitates of the P phase nucleate heterogeneously at a grain boundaries. The nuclei are lens-shaped as shown below. 

The phase diagram for a binary alloy (Fig. A1.13) shows single-phase fields (e.g. liquid) and two-phase fields (e.g. liquid plus A). The fields are separated by phase boundaries. When a phase boundary is crossed, a phase change starts, or finishes, or both. 

Here we review the properties of the model in the mean field theory [328] of the system with the quantum APR Hamiltonian (41). This consists of considering a single quantum rotator in the mean field of its six nearest neighbors and finding a self-consistent condition for the order parameter. Solving the latter condition, the phase boundary and also the order of the transition can be obtained. The mean-field approximation is similar in spirit to that used in Refs. 340,341 for the case of 3D rotators. 

Since the catalyst is concentrated and operates in the ionic phase, and also probably at the phase boundary, reaction volumes in the biphasic technology are much lower than in the conventional single-phase Dimersol process, in which the catalyst concentration in the reactor is low. As an example, the Difasol reactor volume can be up to 40 times lower than that classically used in the homogeneous process. 

In galvanic cells it is only possible to determine the potential difference as a voltage between two half-cells, but not the absolute potential of the single electrode. To measure the potential difference it has to be ensured that an electrochemical equilibrium exists at the phase boundaries, e.g., at the electrode/electrolyte interface. At the least it is required that there is no flux of current in the external and internal circuits. 

Current use of statistical thermodynamics implies that the adsorption system can be effectively separated into the gas phase and the adsorbed phase, which means that the partition function of motions normal to the surface can be represented with sufficient accuracy by that of oscillators confined to the surface. This becomes less valid, the shorter is the mean adsorption time of adatoms, i.e. the higher is the desorption temperature. Thus, near the end of the desorption experiment, especially with high heating rates, another treatment of equilibria should be used, dealing with the whole system as a single phase, the adsorbent being a boundary. This is the approach of the gas-surface virial expansion of adsorption isotherms (51, 53) or of some more general treatment of this kind. 

It seems probable that a fruitful approach to a simplified, general description of gas-liquid-particle operation can be based upon the film (or boundary-resistance) theory of transport processes in combination with theories of backmixing or axial diffusion. Most previously described models of gas-liquid-particle operation are of this type, and practically all experimental data reported in the literature are correlated in terms of such conventional chemical engineering concepts. In view of the so far rather limited success of more advanced concepts (such as those based on turbulence theory) for even the description of single-phase and two-phase chemical engineering systems, it appears unlikely that they should, in the near future, become of great practical importance in the description of the considerably more complex three-phase systems that are the subject of the present review. 

Analyses of rate measurements for the decomposition of a large number of basic halides of Cd, Cu and Zn did not always identify obedience to a single kinetic expression [623—625], though in many instances a satisfactory fit to the first-order equation was found. Observations for the pyrolysis of lead salts were interpreted as indications of diffusion control. More recent work [625] has been concerned with the double salts jcM(OH)2 yMeCl2 where M is Cd or Cu and Me is Ca, Cd, Co, Cu, Mg, Mn, Ni or Zn. In the M = Cd series, with the single exception of the zinc salt, reaction was dehydroxylation with concomitant metathesis and the first-order equation was obeyed. Copper (=M) salts underwent a similar change but kinetic characteristics were more diverse and examples of obedience to the first order, the phase boundary and the Avrami—Erofe ev equations [eqns. (7) and (6)] were found for salts containing the various cations (=Me). 

The theoretical treatment which has been developed in Sections 10.2-10.4 relates to mass transfer within a single phase in which no discontinuities exist. In many important applications of mass transfer, however, material is transferred across a phase boundary. Thus, in distillation a vapour and liquid are brought into contact in the fractionating column and the more volatile material is transferred from the liquid to the vapour while the less volatile constituent is transferred in the opposite direction this is an example of equimolecular counterdiffusion. In gas absorption, the soluble gas diffuses to the surface, dissolves in the liquid, and then passes into the bulk of the liquid, and the carrier gas is not transferred. In both of these examples, one phase is a liquid and the other a gas. In liquid -liquid extraction however, a solute is transferred from one liquid solvent to another across a phase boundary, and in the dissolution of a crystal the solute is transferred from a solid to a liquid. 

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