Phases transitions

A phase is a specific state of matter that is uniform throughout in composition cuid physiccd state. The hquid and vapor states of water cffe two of its phcises. The term phcise is more specific than state of matter because a substance may exist in more than one solid form, each one of which is a solid phase. There are at least 

12 forms of ice. No substance has more than one gaseous phase, so gas phase and gaseous state are effectively synonyms. The only substance that exists in more than one liquid phase is helium, although some evidence suggests that water might also have two liquid phases. 

The conversion of one phase of a substance to another phase is called a phase transition. Thus, vaporization (liquid — gas) is a phase transition, as is a transition between solid phases (such as aragonite — calcite in geological processes). With a few exceptions, phase transitions are accompanied by a change of enthalpy, for the rearrangement of atoms or molecules usually requires or releases energy. 

Definition A phase transition is an event which entails a discontinuous (sudden) change of at least one property of a material. 

Generally, a phase transition is triggered by an external stress which most commonly is a change in temperature or pressure. Properties that can change discontinuously include volume, density, specific heat, elasticity, compressibility, viscosity, color, electric conductivity, magnetism and solubility. As a rule, albeit not always, phase transitions involve structural changes. Therefore, a phase transition in the solid state normally involves a change from one to another modification. 

If a modification is unstable at every temperature and every pressure, then its conversion into another modification is irreversible such phase transitions are called monotropic. Enantiotropic phase transitions are reversible they proceed under equilibrium conditions (AG = 0). The following considerations are valid for enantiotropic phase transitions that are induced by a variation of temperature or pressure. 

If one of these quantities experiences a discontinuous change, i.e. if AS 0 or AV 0, then the phase transition is called a first-order transition according to Ehrenfest. It is accompanied by the exchange of conversion enthalpy AH = TAS with the surroundings. 

First-order phase transitions exhibit hysteresis, i.e. the transition takes place some time after the temperature or pressure change giving rise to it. How fast the transformation proceeds also depends on the formation or presence of sites of nucleation. The phase transition can proceed at an extremely slow rate. For this reason many thermodynamically unstable modifications are well known and can be studied in conditions under which they should already have been transformed. 

One-component, two-phase systems are discussed in the first part of this chapter. The major part of the chapter deals with two-component systems with emphasis on the colligative properties of solutions and on the determination of the excess chemical potentials of the components in the solution. In the last part of the chapter three-component systems are discussed briefly. 

The common characteristics of phase transitions are that the Gibbs energy is continuous. Although the conditions of equilibrium and the continuity of the Gibbs energy demand that the chemical potential must be the same in the two phases at a transition point, the molar entropies and the molar volumes are not. If, then, we have two such phases in equilibrium, we have a set of two Gibbs-Duhem equations, the solution of which gives the Clapeyron equation (Eq. (5.73)) 

The primes distinguish the separate phases. This equation may also be written as 

The entropy, enthalpy, and volume quantities appearing in Equations (10.1) and (10.2) are molar quantities, but care must be taken that such quantities refer to the same mass of material. This requirement is readily proved. Consider one phase, the primed phase, to contain ( ) moles of component based on a specified chemical formula, and assume that no chemical reaction takes place in this phase. Assume that the double-primed phase contains (h,) moles of component but that a reaction, possibly polymerization or decomposition, takes place in this phase. The chemical reaction may be expressed as = i vfii = 0, where the sum is taken over all reacting species. The two Gibbs-Duhem equations may then be written as 

If a is the fraction of the original number of moles of component 1 which react, the number of moles of species 1 at equilibrium is (n ) (l — a) and the number of moles of each new species is a(ni) vi/v1. Then Equation (10.4) may be written as 

An enormous number of phase transitions are known to occur in common solid compounds. For example, silver nitrate undergoes a displacive phase transition from an orthorhombic form to a hexagonal form at a temperature of approximately 162°C that has a enthalpy of 1.85 kj/mol. In many cases, the nature of these transitions are known, but in other cases there is some uncertainty. Moreover, there is frequently disagreement among the values reported for the transition temperatures and enthalpies. Even fewer phase transitions have been studied from the standpoint of kinetics, although it is known that a large number of these transformations follow an Avrami rate law. There is another complicating feature of phase transitions that we will now consider. 

From a thermodynamic standpoint, we know that at the temperature at which the two phases are in equilibrium, the free energy, G, is the same for both phases. Therefore, 

As a result, there will be a continuous change in G as the transition of one phase into another takes place. However, for some phase transitions (known as first-order transitions), it is found that there is a discontinuity in the first derivative of G with respect to pressure or temperature. It can be shown that the partial derivative of G with pressure is the equal to volume, and the derivative with respect to temperature is equal to entropy. Therefore, we can express these relationships as follows  

For the first case, as the sample is heated, there will be a change in the volume of the sample that can be followed by a technique known as dilatometry. For changes in entropy, use can be made of the fact that AG = 0, and from Eq. (8.51) we find that 

Although there are other ways, one of the most convenient and rapid ways to measure AH is by differential scanning calorimetry. When the temperature is reached at which a phase transition occurs, heat is absorbed, so more heat must flow to the sample in order to keep the temperature equal to that of the reference. This produces a peak in the endothermic direction. If the transition is readily reversible, cooling the sample will result in heat being liberated as the sample is transformed into the original phase, and a peak in the exothermic direction will be observed. The area of the peak is proportional to the enthalpy change for transformation of the sample into the new phase. Before the sample is completely transformed into the new phase, the fraction transformed at a specific temperature can be determined by comparing the partial peak area up to that temperature to the total area. That fraction, a, determined as a function of temperature can be used as the variable for kinetic analysis of the transformation. 

Although the first theoretical treatment of the surface phase transitions appeared within the frames of the TPC modef a few years after the first experimental evidences of this phenomenon, problems closely related to phase transitions tantalized the relevant research area for more than two decades. These are the interrelated issues of the polarization catastrophe, the equivalence or not of the electrical variables as well as the equivalence or not of the various statistical mechanical treatments. Due to their significance, these issues are discussed separately in the section below. Here, we focus our attention on the types and properties of the phase transitions predicted by the models for electrosorption. 

The molecular models may predict at least two types of phase transitions Transitions leading to (i) two immiscible surface solutions, and (ii) surface precipitation. 

This is the conventional type of phase transitions predicted by both the maaoscopic and molecnlar (two-dimensional) models, and they are 

the critical properties depend upon the adsorption processes that take place on the electrode surface. For example, if the adsorption is described by the equilibrium Eq. (23) with m = 1 and assume for simplicity that A(j) = A(j), then the critical values of and 0 are Af =2 and0e = O.5. 

Therefore, the adsorbed layer undergoes a phase separation transition when A takes values greater than 2. The typical feature of this type of transition is the appearance of transition loops at the plots of 6 V.T.. A(j) and v.r.. A(j . These loops may be replaced by vertical steps if we use the generalized ensemble A introduced or make 

Phase diagrams have been summarized recently by Gordon and will not be discussed here. Janz and co-workers have made an extensive study of the solid-solid and solid-liquid phase transitions of 33 quaternary ammonium salts by differential thermal analysis. The results show changes of alkyl chain conformation in the cations at or below the melting point, especially for the longer chains. Thus in the use of simple models without internal degrees of freedom one must be wary of where these degrees of freedom will manifest themselves in the observable properties. 

The tetraalkylammonium iodides have also been examined by Levkov et For this series with four identical alkyl chains on the cation they show that the total entropy of up to three premelting transitions is constant for the chains with odd numbers of carbon atoms at about 8 eu, while that for even numbers increases with chain length, starting at 11 eu for the ethyl 

Salt d5(total), eu/mole Salt zl5(total), eu/mole 

The melting of the alkali acetates has also been studied volumetrically and spectroscopically by Hazelwood et The sodium salt expands upon melting by 3.9%, which compares well with sodium nitrate, but the potassium salt actually contracts about 1% upon melting. The volumetric data indicate the possibility of transitions in the solid similar to the quaternary ammonium salts. 

In view of these findings, it would be desirable to perform a systematic study of finite-size effects. In fact, it is well known that continuous (second-order) phase transitions become shifted and rounded owing to the finite size of the samples used in numerical 

Since one has x = Z- VarlQ), it follows that the variance of the order parameter actually vanishes in the thermodynamic limit. Owing to this evidence we conclude that for the range of parameters used, rather than undergoing a true phase transition, the Ag/Au(100) system actually exhibits a crossover between the DOS and the DDS. A possible reason for the observation of a crossover instead of a true DPT could be the absence of symmetry between the adsorption and desorption processes. To test this possibility we measured the relaxation times for Ag-covered and uncovered surfaces as a function of the applied overpotential (see Fig. 20). 

Two relaxation times are measured for each process, namely, (1) and for the desorption processes up to 0Ag = 1/2 and 0Ag = 0, respectively, and (2) and for the adsorption processes up to 0Ag = 1/2 and 0Ag = U respectively. The results obtained, plotted in Fig. 20, show that for low overpotentials the relaxation times corresponding to both processes are different, quantitatively confirming the asymmetry between them. Also, for large overpotentials the relaxation times tend to be almost the same, suggesting that in this limit the asymmetry may be irrelevant. However, 

To further characterize the crossover between different states, we take advantage of the well-defined peak exhibited by x and Var (I Q ) (see Figs. 17 19). So, the crossover period (t ) and the corresponding crossover chemical potential (/rp ) are identified with the location of the above-mentioned peak. The results obtained are displayed in Fig. 21, which shows a logarithmic dependence of cross Qjj cross jjj g ate diagram of Fig. 21, the full line shows the border between DOSs obtained for intermediate values of both CToss cross DDSs that are found for large enough values 

Grand ganonical Monte Carlo simulations using realistic interatomic potentials were performed for a significant number of metallic systems, allowing us to draw a number of interesting conclusions. One of the novel features of the work is the exploration of the electrochemical phenomena of UPD and OPD in terms of lattice models that consider the many-body interactions typical of metallic systems. Thus, without the need to assume a particular type of interaction potential between the particles, phase transition phenomena in metallic monolayers could be studied. These studies comprised the formation 

In the chapter opener, we discussed the change of solid carbon dioxide (dry ice) directly to a gas. Such a change of a substance from one state to another is called a chaise of state or phase transition. In the next sections, we will look at the different kinds of phase transitions and the conditions under which they occur. (A phase is a homogeneous portion of a system—that is, a given state of either a substance or a solution.) 

In general, each of the three states of a substance can change into eitha- of the otha-states by undergoing phase transition. 

Melting is the change of a solid to the liquid state (melting is also referred to as fusion). For example. 

Freezing is the change of a liquid to the solid state. The freezing of liquid water to ice is a common example. 

The casting of a metal object involves the melting of the metal, then its freezing in a mold. 

Simulation of free-surface and interfacial flows is a topic with many practical applications, e.g., the formation of droplet clouds or sprays from liquid jets, 

Porous medium (cubes represent solid phase) 

Simulation methods for problems with free surfaces governed by Navi-er Stokes equations were reviewed by Scardovelli and Zaleski (1999). The specific problems of these simulations are the location of the interface and the choice of the spatial discretization  

Let us introduce the VOF method in more detail. The phase volume function fi(r) is defined by/) = 1 in phase 1 and f x — 0 in phase 2 cf. Eq. (1). A discrete analog of the phase volume function fx is the scalar field tpi known as the volume fraction (or area fraction in spatially 2D cases). It represents the fraction of the volume of the voxel with size h and the coordinate of its center r filled with the phase 1, 

The central problem of the VOF method is the reconstruction and representation of the interface. For example, in two spatial dimensions, the interface is considered to be a continuous, piecewise smooth line the problem of its reconstruction is that of finding an approximation to the section of the interface in each cut cell by knowing only the volume fraction 4 in that cell and in the neighboring ones. 

If a system is described in terms of the Gibbs function G then a change in G can be written 

Suppose 1 mol of gaseous sulfur dioxide is compressed at a temperature fixed at 30.0°C. The volume is measured at each pressure, and a graph of volume against pressure is constructed (Fig. 10.18). At low pressures, the graph shows the inverse dependence (V 1/F) predicted by the ideal gas law. As the pressure increases, deviations appear because the gas is not ideal. At this temperature, attractive forces dominate therefore, the volume falls below its ideal gas value and approaches 4.74 L (rather than 5.50 L) as the pressure approaches 4.52 atm. 

FIGURE 10.18 As 1 mol SOj is compressed at a constant temperature of 30°C, the volume at first falls somewhat below its ideal gas value. Then, at 4.52 atm, the volume decreases abruptly as the gas condenses to a liquid. At a much higher pressure, a further transition to the solid occurs. 

For pressures up to 4.52 atm, this behavior is quite regular and can be described by the van der Waals equation. At 4.52 atm, something dramatic occurs The volume decreases abruptly by a factor of 100 and remains small as the pressure is increased further. What has happened The gas has been liquefied solely by the application of pressure. If the compression of SO2 is continued, another abrupt (but small) change in volume will occur as the liquid freezes to form a solid. 

Condensed phases also arise when the temperature of a gas is reduced at constant pressure. If steam (water vapor) is cooled at 1 atm pressure, it condenses to liquid water at 100°C and freezes to solid ice at 0°C. Liquids and solids form at low temperatures once the attractive forces between molecules become strong enough to overcome the kinetic energy of random thermal motion. 

FIGURE 10.19 Direct transitions among all three states of matter not only are possible but are observed in everyday life. 

Azide Process Temp (K) AH (cal/g) AH (kcal/mole) AS (cal/mole deg) Reference 

In crystals with molecular ion groups, like N3, the crystal potential function can be modulated via some of the anharmonic terms in the potential. The effect of this could be a decrease in the barrier height for rotation of the molecular ion with increasing temperature, resulting in the flipping of the molecular ion to another equivalent orientation. The entropy change for such a system is given by the relation 

Here the statistical assembly is distributed through W complexions with equal fractions in each. Such a phase transition is an order-disorder or X transition. As a consequence of such a change the single-well potential of an ordered crystal changes to a multiple-well potential of the disordered system. If the transition is approached from the lower temperature side, a small displacement of the ionic sites may also occur, giving rise to a unit cell of higher symmetry. 

Azide Librational mode (cm ) Translational mode (cm ) T(Eg) ratio Metal ion mass ratio 

A phase transition can occur well above room temperature and give rise to a system with long-range disorder and a X-type transition temperature, Also if 

A molecular solid consists of atoms or molecules held togetiier by intermolecular attractive forces. In molecular solids, the attractive forces include hydrogen bonds and dipole-dipole forces (e.g.. Ice [H O (5)]). 

Phase transition or change of state refers to the change of a substance s phase from one state to another. In this section, we will talk about the different phase changes, the associated terminology, and the key ideas related to all these. 

The melting properties of polypropylene are complicated and depend on process conditions and measurement techniques. However, the melting point is usually aroimd 165°C and the glass transition temperature is approximately — 15°C. The peak recrystallization temperature in quiescent melts lies near 110-120°C. 

This relation is called the lever rule in analogy with a lever supported at S, in equilibrium with weights jc and (1 — jc) attached to either end. 

When transition from a solid to liquid or from a liquid to vapor takes place, there is a discontinuous change in the entropy. This can clearly be seen (Fig. 7.11) if we plot molar entropy as a function of T, for a fixed p and 

The same is tme for other derivatives of Gm such as = dG/dp)j. The chemical potential changes continuously but its derivative is discontinuous. At the transition temperature, because of the existence of latent heat, the specific 

The characteristic features of second-order phase transitions are shown in Fig. 7.12. In this case the changes in the thermodynamic quantities are not so drastic changes in 5 and Vm are continuous but their derivatives are discontinuous. Similarly, for the chemical potential it is the second derivative that is discontinuous the specific heat does not have a singularity but it has a discontinuity. Thus, depending on the order of the derivatives that are discontinuous, phase transitions are classified as transitions of first and second order. 

However, experiments showed that p was in the range 0.3-0.4, not equal to 0.5. Along the critical isotherm, as the critical pressure Pc is approached from above, the theory predicted 

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S. Chains in the S phase are also oriented normal to the surface, yet the unit cell is rectangular possibly because of restricted rotation. This structure is characterized as the smectic E or herringbone phase. Schofield and Rice [204] applied a lattice density functional theory to describe the second-order rotator (LS)-heiTingbone (S) phase transition. 

The free energy of a monolayer domain in the coexistence region of a phase transition can be described as a balance between the dipolar electrostatic energy and the line tension between the two phases. Following the development of McConnell [168], a monolayer having n circular noninteracting domains of radius R has a free energy... 

P. Meakin, Fractal Aggregates and Their Fractal Measure, in Phase Transitions arul Critical Phenomerui, Vol. 12, C. Domb and J. L. Lebowitz, eds.. Academic, New York, 1988. 

Fuerstenau and Healy [100] and to Gaudin and Fuerstenau [101] that some type of near phase transition can occur in the adsorbed film of surfactant. They proposed, in fact, that surface micelle formation set in, reminiscent of Langmuir s explanation of intermediate type film on liquid substrates (Section IV-6). 

S. A. Safran, Theory of Structure and Phase Transitions in Globular Microemulsions, in Micellar Solutions and Microemulsions, S. H. Chen and R. Rajagopalan, eds.. Springer-Verlag, New York, 1990, Chapter 9. 

Phospholipid molecules form bilayer films or membranes about 5 nm in thickness as illustrated in Fig. XV-10. Vesicles or liposomes are closed bilayer shells in the 100-1000-nm size range formed on sonication of bilayer forming amphiphiles. Vesicles find use as controlled release and delivery vehicles in cosmetic lotions, agrochemicals, and, potentially, drugs. The advances in cryoelec-tron microscopy (see Section VIII-2A) in recent years have aided their characterization [70-72]. Additional light and x-ray scattering measurements reveal bilayer thickness and phase transitions [70, 71]. Differential thermal analysis... 

Stigter and Dill [98] studied phospholipid monolayers at the n-heptane-water interface and were able to treat the second and third virial coefficients (see Eq. XV-1) in terms of electrostatic, including dipole, interactions. At higher film pressures, Pethica and co-workers [99] observed quasi-first-order phase transitions, that is, a much flatter plateau region than shown in Fig. XV-6. 

While the v-a plots for ionized monolayers often show no distinguishing features, it is entirely possible for such to be present and, in fact, for actual phase transitions to be observed. This was the case for films of poly(4-vinylpyri-dinium) bromide at the air-aqueous electrolyte interface [118]. In addition, electrostatic interactions play a large role in the stabilization of solid-supported lipid monolayers [119] as well as in the interactions between bilayers [120]. 

Consider how the change of a system from a thennodynamic state a to a thennodynamic state (3 could decrease the temperature. (The change in state a —> f3 could be a chemical reaction, a phase transition, or just a change of volume, pressure, magnetic field, etc). Initially assume that a and (3 are always in complete internal equilibrium, i.e. neither has been cooled so rapidly that any disorder is frozen in. Then the Nemst heat... 

Basic thermodynamics, statisticai thermodynamics, third-iaw entropies, phase transitions, mixtures and soiutions, eiectrochemicai systems, surfaces, gravitation, eiectrostatic and magnetic fieids. (in some ways the 3rd and 4th editions (1957 and 1960) are preferabie, being iess idiosyncratic.)... 

Fundamentais of thermodynamics. Appiications to phase transitions. Primariiy directed at physicists rather than chemists. Reid C E 1990 Chemical Thermodynamics (New York McGraw-Fliii)... 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

The CS pressures are close to the machine calculations in the fluid phase, and are bracketed by the pressures from the virial and compressibility equations using the PY approximation. Computer simulations show a fluid-solid phase transition tiiat is not reproduced by any of these equations of state. The theory has been extended to mixtures of hard spheres with additive diameters by Lebowitz [35], Lebowitz and Rowlinson [35], and Baxter [36]. 

Stell G 1995 Criticality and phase transitions in ionic fluids J. Stat. Phys. 78 197... 

Lee T D and Yang C N 1952 Statistical theory of equations of state and phase transitions II. Lattice gas and Ising models Phys. Rev. 87 410... 

Goldenfeld L 1992 Lectures in Phase Transitions and Renormalization Group (New York Addison-Wesley) Goodstein D L 1974 States of Maffer(Englewood Cliffs, NJ Prentice-Hall and Dover)... 

Reiss H and Hammerich ADS 1986 Hard spheres scaled particle theory and exact relations on the existence and structure of the fluid/solid phase transition J. Phys. Chem. 90 6252... 

Stanley H E 1971 Introduction to Phase Transitions and Critical Phenomena (Oxford Oxford University Press)... 

Phase transitions at which the entropy and enthalpy are discontinuous are called first-order transitions because it is the first derivatives of the free energy that are disconthuious. (The molar volume V= (d(i/d p) j is also discontinuous.) Phase transitions at which these derivatives are continuous but second derivatives of G... 

The initial classification of phase transitions made by Ehrenfest (1933) was extended and clarified by Pippard [1], who illustrated the distmctions with schematic heat capacity curves. Pippard distinguished different kinds of second- and third-order transitions and examples of some of his second-order transitions will appear in subsequent sections some of his types are unknown experimentally. Theoretical models exist for third-order transitions, but whether tiiese have ever been found is unclear. 

Phase transitions in binary systems, nomially measured at constant pressure and composition, usually do not take place entirely at a single temperature, but rather extend over a finite but nonzero temperature range. Figure A2.5.3 shows a temperature-mole fraction T, x) phase diagram for one of the simplest of such examples, vaporization of an ideal liquid mixture to an ideal gas mixture, all at a fixed pressure, (e.g. 1 atm). Because there is an additional composition variable, the sample path shown in tlie figure is not only at constant pressure, but also at a constant total mole fraction, here chosen to be v = 1/2. 

Figure A2.5.30. Left-hand side Eight hypothetical phase diagrams (A through H) for ternary mixtures of d-and /-enantiomers with an optically inactive third component. Note the syimnetry about a line corresponding to a racemic mixture. Right-hand side Four T, x diagrams ((a) tlirough (d)) for pseudobinary mixtures of a racemic mixture of enantiomers with an optically inactive third component. Reproduced from [37] 1984 Phase Transitions and Critical Phenomena ed C Domb and J Lebowitz, vol 9, eh 2, Knobler C M and Scott R L Multicritical points in fluid mixtures. Experimental studies pp 213-14, (Copyright 1984) by pennission of the publisher Academic Press.

It has long been known from statistical mechanical theory that a Bose-Einstein ideal gas, which at low temperatures would show condensation of molecules into die ground translational state (a condensation in momentum space rather than in position space), should show a third-order phase transition at the temperature at which this condensation starts. Nonnal helium ( He) is a Bose-Einstein substance, but is far from ideal at low temperatures, and the very real forces between molecules make the >L-transition to He II very different from that predicted for a Bose-Einstein gas. 

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