Phase transition points

It is in order to ask, whether basically melting occurs in a certain temperature range, or whether the range arises due to inaccurate thermometric measurement. 

We focus now on a substance, consisting of a single component. The thermodynamic criterion for a phase transition, or in other words that two phases (0 and ( ) may coexist, is that the chemical potentials of both phases in question are equal. We express the chemical potential as a function of temperature and pressure, = /u(r, p). The chemical potential as an intensive function may not depend on the size of the system. Since we are dealing with a single component substance. 

The free enthalpy as condition of equilibrium dG = 0 arises naturally, if the system is in contact with a thermal reservoir and a pressure reservoir. The constraint dn -I-dn = 0 arises from the conservation of mass and the assumption that chemical reactions are not allowed. Near equilibrium, a plot of the chemical potentials looks like that in Fig. 6.11. 

The data are taken from steam tables [14, p. 447]. Traditionally, in the steam tables, the specific quantities are given rather than the molar quantities. Moreover, the entropy is normalized not to absolute zero, but to 0 °C. The full curves are the chemical potentials for the liquid phase at fixed pressure as a function of the temperature. The curves are accessible by measurement only at temperatures below the boiling point. The dashed curves represent the chemical potentials for the gaseous phase at fixed pressure as a function of the temperature. These data are accessible by measurement only at temperatures above the boiling point. Both types of chemical potential are extrapolated in the region that is not accessible to measurement. 

The experimental accessible points appear to make a kink at the phase transition. Thus, the chemical potentials of the two phases are running in a way that they can cut only in a point. Therefore, notably for a single substance, the condition of equal chemical potential is fulfilled not in a temperature range, but at a single temperature. Therefore, the term transition point is justified. At a temperature below the intersect, jji fi and in 0 to get dG 0. Therefore, the phase 0 is stable in this region. The situation is reverse at a temperature beyond the intersect. An infinitesimal deviation of the temperature from the temperature at equal chemical potential will cause the less stable phase completely to disappear. 

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Finally, the use of the constant pressure minimization algorithm allows searching for phenomena that can be considered as precursors of pressure-induced transitions. For example, the predicted behaviour of the anatase cell constants as a function of pressure shows that the a(P) and c(P) plots are only linear for P<4 GPa, the value that is close to both the theoretical and experimental transition pressures. At higher pressures the a constant starts to grow under compression, indicating inherent structural instability. In the case of ratile there is a different precursor effect, nami y at 11 GPa the distances between the titanium atom and the two different oxygens, axial and equatorial, become equal. Once again, the pressure corresponds closely to the phase transition point. 

Campbell say that they found by differential thermal analysis that the phase transition point is 224°, the mp is 575°, and that decompn occurs at 628—30° (Ref 2) CA Registry No 13454-84-7... 

Fig. 14. Temperature dependences of the layer spacing in the smectic Ad and Cd phases for terminally polar polyphilic compounds F4H11OCB (squares), FsHnOCB (circles), FgHioOCB (triangles) and FioHnOCB (diamonds). The arrows indicate the smectic Ad - smectic Cd phase transition points (Ostrovskii et al. [45])...

In order to investigate the phase transition in the monolayer state, the temperature dependence of the Jt-A isotherm was measured at pH 2. The molecular area at 20 mN rn 1, which is the pressure for the LB transfer of the polymerized monolayer, is plotted as a function of temperature (Figure 2.6). Thermal expansion obviously changes at around 45 °C, indicating that the polymerized monolayer forms a disordered phase above this temperature. The observed temperature (45 °C) can be regarded as the phase transition point from the crystalline phase to the liquid crystalline phase of the polymerized organosilane monolayer. 

Cholesterol s presence in liposome membranes has the effect of decreasing or even abolishing (at high cholesterol concentrations) the phase transition from the gel state to the fluid or liquid crystal state that occurs with increasing temperature. It also can modulate the permeability and fluidity of the associated membrane—increasing both parameters at temperatures below the phase transition point and decreasing both above the phase transition temperature. Most liposomal recipes include cholesterol as an integral component in membrane construction. 

Enthalpy can be measured by liquid chromatography where enthalpy is a slope of the relationship between In k and the inverse value of the absolute temperature. A schematic diagram is shown in Figure 6.7. The slope depends upon the solutes being retained by the same liquid chromatographic mechanism. An example is given in Table 6.4. The results, measured on an octadecyl-bonded vinyl alcohol copolymer gel, did not show a simple linear relationship. This is due to a conformation change of the octadecyl-bonded vinyl alcohol copolymer gel stationary phase material, which has a phase transition point at about 33 °C. 

J-F. Wilson, USP 3223185(1965) CA 64, 6153(1966) [The instability of AN (used in expls and fertilizers) in storage results from change in water content or changes in vol at the phase transition points. The improvement in stability is achieved by coating the AN particles with a compn consisting of a major amt of a clay (such as attapulgite, kaolin or diatomaceous earth), and a minor amt of an oil-sol alk-earth metal salt of petroleum sulfonic acids (sucb as Ca petroleum sulfonate)]... 

For the gels, which swell under the applied mechanical force, the dependence of deformation on the stress is characterized by some specific features. In particular, at some critical values of the applied force, the deformation can increase in a jump-like manner by hundreds of percent (Fig. 25). For the samples which are very close to the phase transition point even a very small increase in tension causes noticeable changes in the degree of swelling and in the dimensions of the gels. 

It is useful to check whether this kind of relations is valid for other systems like ferromagnetics and ferroelectrics too. Here the order parameters are the magnetization M and the polarization P, respectively. At high temperatures (T > Tc), and zero external field these values are M = 0 (paramagnetic phase) and P = 0 (paraelectric phase) respectively. At lower temperatures close to the phase transition point, however, spontaneous magnetization and polarization arise following both the algebraic law M, P oc (Tc - Tf. 

An important step in developing the mean-field concept was done by Landau [8, 10]. Without discussing the relation between such fundamental quantities as disorder-order transitions and symmetry lowering, we just want to note here that his theory is based on thermodynamics and the derivation of the temperature dependence of the order parameter via the thermodynamic potential minimization (e.g., the free energy A(r),T)) which is a function of the order parameter. It is assumed that the function A(rj,T) is analytical in the parameter 77 and thus near the phase transition point could be expanded into the series in 77 usually it is a polynomial expansion with temperature-dependent coefficients. Despite the fact that such a thermodynamical approach differs from the original molecular field theory, they are quite similar conceptually. In particular, the r.h.s. of the equation of state for the pressure of gases or liquids and the external field in ferromagnetics, respectively, have the same polynomial form. 

Figure 9.2 shows the correlation functions for the reactive state just below the phase transition point (Yco = 0.56 < yz). The steady state is reached in a very short time. The particle composition on the surface is a mixture of A and B particles with many empty sites in between. 

We have introduced in this Section a stochastic model for the A+ 2B2 —> 0 reaction which is equivalent to the ZGB-model [2] and thus remedies a deficiency of a previously presented model [13]. In this model we obtain for the case of no diffusion for the phase transition points y = 0.395 and y2 = 0.565, which are in good or fair agreement with the results of the ZGB-model (y = 0.395 and yi = 0.525). In the model [13] where the reaction occurs only if A particle jumps to active site occupied by a B particle, we obtain y — 0.27 and 7/2 = 0.65 (for D — 10). Because the reaction occurs only due to diffusion, we cannot directly compare this model with the ZGB-model in which no diffusion exists. But the value of t/2 is in agreement with computer simulations of the extended ZGB-model including diffusion (t/2 = 0.65 for a high diffusion rate) [3]. The value of y should not be influenced by the additional aspect of A-diffusion because too few A particles... 

We have studied the system (9.1.39) to (9.1.41) by means of the Monte Carlo method on a disordered surface where the active sites form a percolation cluster built at the percolation threshold and also above this threshold [25]. Finite clusters of active sites were removed from the surface to study only the effect of the ramification of the infinite cluster. The phase transition points show strong dependence on the fraction of active sites and on the... 

For S > Sc we obtain an infinite cluster for which in principle a reactive state exists. We use this fact to define the percolation threshold in a kinetic way for the particular reaction at hand as the transition point from the reactive (Rco2 > 0) to the non-reactive (Rco2 — 0) state. As we have shown above, this transition happens in such a way that the kinetic phase transition points of 2/1 and are approaching each other if S —> Sc [25]. At S = Sc the... 

In curves 4 the system behaviour is shown under the conditions of curves 3 but now including the effect of A-desorption, equation (9.1.42). In this case we obtain a reactive state. We observe a phase transition of the first order at y — 0.268. For Yco < U the lattice is completely occupied by particles B. A phase transition point y2 does not exist but we obtain a smooth transition over a wide range of kco Due to the desorption the state poisoned by A does not exist. 

We have studied above a model for the surface reaction A + 5B2 -> 0 on a disordered surface. For the case when the density of active sites S is smaller than the kinetically defined percolation threshold So, a system has no reactive state, the production rate is zero and all sites are covered by A or B particles. This is quite understandable because the active sites form finite clusters which can be completely covered by one-kind species. Due to the natural boundaries of the clusters of active sites and the irreversible character of the studied system (no desorption) the system cannot escape from this case. If one allows desorption of the A particles a reactive state arises, it exists also for the case S > Sq. Here an infinite cluster of active sites exists from which a reactive state of the system can be obtained. If S approaches So from above we observe a smooth change of the values of the phase-transition points which approach each other. At S = So the phase transition points coincide (y 1 = t/2) and no reactive state occurs. This condition defines kinetically the percolation threshold for the present reaction (which is found to be 0.63). The difference with the percolation threshold of Sc = 0.59275 is attributed to the reduced adsorption probability of the B2 particles on percolation clusters compared to the square lattice arising from the two site requirement for adsorption, to balance this effect more compact clusters are needed which means So exceeds Sc. The correlation functions reveal the strong correlations in the reactive state as well as segregation effects. 

In this Section we focus our attention on the development of the formalism for complex reactions with application to the formation of NH3. The results obtained (phase transition points and densities of particles on the surface) are in good agreement with the Monte Carlo and cellular automata simulations. The stochastic model can be easily extended to other reaction systems and is therefore an elegant alternative to the above-mentioned methods. 

If we enlarge the diffusion rate of the H atoms up to D = 10 as shown in Fig. 9.15, the value of the phase transition point y and the coverages of N, NH and NH2 are almost unaffected by this change, only the concentration of H atoms drops more rapidly for j/n > 2/i- This behaviour shows increased reactivity of H atoms which are now more mobile. The coverages of the other particles depend only on the concentration of the activated sites of the surface but not on the reactivity of the H atoms. 

The results obtained from the CA model for S = 1/8 are in quite good agreement with the results obtained from the stochastic ansatz for D = 1 and D = 10 the value of the phase transition point y is found to be y 0.4 in both models. In the stochastic model the density of adsorbed N atoms is smaller compared to the CA model because of the blocking effect of activated sites which arises from the different reaction mechanism. 

We have also performed calculations for higher diffusion rates (D = 100) and for the triangular lattice (coordination number z — 6). The qualitative behaviour is in complete agreement with the calculation presented here. For the case S = 1 the increase of the diffusion rate or change of the lattice structure leads to a very small shift of the phase transition point y to higher values of j/n- This trend is clear because the reactivity of the H atoms is increased by the larger mobility. For S < 1 nearly no effect is observed which means that the system s behaviour is mainly dominated by the number of activated sites. The correlation of the adsorbed particles are rather small as expected for S < 1. 

Let us study now a stochastic model for the particular a+ib2 -> 0 reaction with energetic interactions between the particles. The system includes adsorption, desorption, reaction and diffusion steps which depend on energetic interactions. The temporal evolution of the system is described by master equations using the Markovian behaviour of the system. We study the system behaviour at different values for the energetic parameters and at varying diffusion and desorption rates. The location and the character of the phase transition points will be discussed in detail. 

It has been established that the volume phase transition of gels is an universal phenomenon [17]. Dynamic light scattering studies indicate that the dynamic fluctuations of the density correlation diverges in the vicinity of the volume phase transition point of the gel. It has also been shown that the time scale of the density fluctuations become slow in the vicinity of the volume phase transition... 

In spite of the constant density of the gel, the friction of the poly(N-isopropylacrylamide) gel reversibly decreases by three orders of magnitude and appears to diminish as the gel approaches a certain temperature. This phenomenon should be universal and may be observed in any gel under optimal experimental conditions of the solvent composition and the temperature because the unique parameter describing the friction is the correlation length which tends to diverge in the vicinity of the volume phase transition point of gels. The exponent v for the correlation length obtained from the frictional experiment is far from the theoretical value. It will, therefore, be important to study a poly(N-isopropylacrylamide) gel prepared at the critical isochore where the frictional property of gel may be governed by the critical density fluctuations of the gel. 

The quasi-chemical approximation gives only qualitative results and appears to be particularly inaccurate at temperatures below the "order-disorder phase transition points of T = 0.567 EAA at 0 = 1/2. 

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