First- and Second-Order Phase Transitions

What gives rise to condensation and fusion We know that the hypothetical substance we call an ideal gas does not condense or solidify, and this is because it possesses no intermo-lecular interactions. The molecules of real gases do interact in a significant way, and that is the fundamental basis for condensation and fusion. At long range, there is an attraction between real gas particles. This attraction covers a considerable range in real gas systems. 

There are a number of characteristics of the type of phase transitions we have considered so far. In practice, these characteristics are often used to determine data points for phase equilibrium lines. One of the obvious characteristics is a discontinuity in enthalpy as a function of temperature. Consider a sample of ice at -100°C and a pressure of 1 bar. The enthalpy changes smoothly as the temperature is increased until the system reaches 0°C, the melting temperature. At this point, there is a jump in the enthalpy corresponding to the enthalpy of melting. After all the ice has melted, the temperature can be increased and the enthalpy will be a new but different function of the temperature. Since the heat capacity, Cp, is the derivative of the enthalpy with respect to temperature, Cp as a function of temperature is also discontinuous at the phase transition. Vaporization of liquid water at 100°C and 1 bar leads to a sharp increase in volume. Thus, volume is a discontinuous function at a phase transition. The same holds for entropy. 

Typical behavior of heat capacity, volume, and entropy of a pure substance as a function of temperature given that the system undergoes a first-order phase transition (left) versus a second-order phase transition (right) at the temperature T.  

The chemical potential of two phases at equilibrium must be identical, and so the chemical potential of a substance as a function of temperature is continuous at the phase transition. The discontinuous functions just mentioned, volume, entropy, enthalpy, and heat capacity, are all related to first derivatives of the chemical potential. Therefore, the type of transitions we have considered so far, the ones to which we are most accustomed, is categorized as having a discontinuity in the first derivative of the chemical potenhal with respect to temperature at the phase transition. These are called first-order phase transitions. 

A system consisting of more than one pure substance is one that has some particular number of components. Components are chemically distinct species. The thermodynamic 

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The same equations, albeit with damping and coherent external driving field, were studied by Drummond et al. [104] as a particular case of sub/second-harmonic generation. They proved that below a critical pump intensity, the system can reach a stable state (field of constant amplitude). However, beyond the critical intensity, the steady state is unstable. They predicted the existence of various instabilities as well as both first- and second-order phase transition-like behavior. For certain sets of parameters they found an amplitude self-modula-tion of the second harmonic and of the fundamental field in the cavity as well as new bifurcation solutions. Mandel and Erneux [105] constructed explicitly and analytically new time-periodic solutions and proved their stability in the vicinity of the transition points. 

Overbeck, G. A. Honig, D. Mobius, D. Visualization of first and second order phase transitions m eicosanol monolayers using Brewster angle microscopy. Langmuir 1993, 9, 555. 

Figure 4.14 Bulk phase diagrams fix(T) [see Eq. (1.88)] where G, M, and D refer to one-phase regions of gaseous, mixed liquid, and demixed liquid phases, respectively. Pairs of neighboring phases coexist for state points represented by solid lines where thick and thin lines refer to first- and second-order phase transitions, respectively, (a) sab = 0-40, (b) ab = 0.50, (c) ab = 0.56, (d) ab = 0.70.

Define the first- and second-order phase transitions and give one experimental method to differentiate between them. 

Apart from temperature, hydrostatic pressure is the other intensive thermodynamical parameter that can be modified with high-pressure cells to build (T, P) phase diagrams. By changing the relative distances between the atoms and molecules, the strength of the interactions are modified, thereby modifying the transition temperature or even inducing new phases. The change of the transition temperature as a function of pressure depends whether the transition is continuous (second order) or discontinuous (first order). The Clausius-Clapeyron (dTc/dP) and Ehrenfest (dTc/dP)2 relationships apply to first- and second-order phase transitions, respectively,... 

Amaral, J. S., Silva, N. J. O. Amaral, V. S. (2007). A mean-field scaling method for first- and second-order phase transition ferromagnets and its application in magnetocaloric studies, Appl. Phys. Lett. 91(17) 172503. 

Phase transitions can be classified according to whether they are first or second order, this classification originally being introduced by Ehrenfest. Changes in various thermodynamic properties, as well as an order parameter, (Section 1.6), for first- and second-order phase transitions as a function of temperature are illustrated in Fig 1.4. A first-order transition is defined by discontinuities in first derivatives of chemical potential. Enthalpy, entropy and volume can all be defined by appropriate first derivatives of chemical potential and all change discontinuously at a first-order phase transition. The heat capacity is defined as the derivative of enthalpy with respect to temperature. It is thus infinite for a first-order transition. The physical meaning of this is apparent when the boiling of water is considered. Any heat absorbed... 

This method allows study of first- and second-order phase transitions of pure compounds. The mesogenic sample is enclosed in a rigid but dilatable metallic cell. The sample pressure and temperature are simultaneously recorded. The thermobarograms obtained exhibit a clear change of slope at the phase transition. The measuring apparatus is called a metabolmeter and requires only a very small amount of mesogen. 

For thermal analysis experiments it is important to compare the main distinctiveness of these first- and second-order phase transitions. With a first-order phase transition, each of the thermodynamic potentials can exist on both sides of the boundary, i.e., the functions can be extrapolated into the region of and vice versa, denoting thus a metastable state of the phase (7) above and vice versa for (2) below Teq, From this it follows that the phase can be overheated at T>Teq (less common) and phase 2 undercooled at T[Pg.252]

According to Erenfest s classification, there are first-order and second-order phase transitions. The first-order transitions are discontinuous, and the second-order transitions are continuous. The KDP-type phase transitions for many years were regarded as purely continuous ones. Then it was shown by Kobayashi et al... 

Another illustration of such a difference between first-order and second-order phase transitions was provided in Chapter 3. It was shown that the density and the molar volume both undergo discontinuous changes upon melting, while they only change in slope but the coefficient of thermal expansion changes discontinuously at the glass transition. 

There are many more first- versus second-order phase transitions, state functions versus path-dependent functions, and so forth. However interwoven, the subject can be divided roughly into two parts as presented in Figure 3.1. One part concentrates on the heat and work transferred between a system and its surroundings. The other part attends to the relationships between a system s state variables and functions. There are quite a number of these beginning with temperature (I), pressure (p), and volume (V), as introduced in Chapter 1. If the chemist chooses a quantity such as enthalpy (H), there is quite a story to tell about its relation to other system properties such as compressibility, heat capacity, and so on. Suffice to say that the variables and functions form the infrastructure for thermodynamics under the umbrella of physical laws. 

The phase transition of DNP could further be characterized by specific heat measurements for monomer and thermally polymerized single crystals (Fig. 9.46) [98]. These data support the description of the phase transition of the monomer crystals as a tricritical transition. This means it is a borderline case between a first-order and second-order phase transition, with a distribution of transition temperatures. The transition enthalpy was much lower than the corresponding order-disorder transition, in agreement with results obtained by Ber-tault et al. via Raman spectroscopy, which proved the importance of displacive contributions to the DNP phase transition [99]. 

From the above considerations, it should be clear that running an adiabatic scanning calorimeter in the constant heating (or cooling) modes makes it possible to determine latent heats when present and distinguish between first-order and second-order phase transitions. On the basis of Cp = P/t, it is also possible to obtain information on the pretransi-tional heat capacity behavior, provided one is able to collect sufficiently detailed and ac-... 

The Maier-Saupe tlieory was developed to account for ordering in tlie smectic A phase by McMillan [71]. He allowed for tlie coupling of orientational order to tlie translational order, by introducing a translational order parameter which depends on an ensemble average of tlie first haniionic of tlie density modulation noniial to tlie layers as well as / i. This model can account for botli first- and second-order nematic-smectic A phase transitions, as observed experimentally. 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

Observe how in each of these four events, H is zero until, at some critical Ac (which is different for different cases), H abruptly jumps to some higher value and thereafter proceeds relatively smoothly to its final maximum value i max = log2(8) = 3 at A = 7/8. In statistical physics, such abrupt, discontinuous changes in entropy are representative of first-order phase transitions. Interestingly, an examination of a large number of such transition events reveals that there is a small percentage of smooth transitions, which are associated with a second-order phase transition [li90a]. 

Cp is the specific heat at constant pressure, k is the compressibility at constant temperature. The conversion process of a second-order phase transition can extend over a certain temperature range. If it is linked with a change of the structure (which usually is the case), this is a continuous structural change. There is no hysteresis and no metastable phases occur. A transformation that almost proceeds in a second-order manner (very small discontinuity of volume or entropy) is sometimes called weakly first order . 

MnAs exhibits this behavior. It has the NiAs structure at temperatures exceeding 125 °C. When cooled, a second-order phase transition takes place at 125 °C, resulting in the MnP type (cf. Fig. 18.4, p. 218). This is a normal behavior, as shown by many other substances. Unusual, however, is the reappearance of the higher symmetrical NiAs structure at lower temperatures after a second phase transition has taken place at 45 °C. This second transformation is of first order, with a discontinuous volume change AV and with enthalpy of transformation AH. In addition, a reorientation of the electronic spins occurs from a low-spin to a high-spin state. The high-spin structure (< 45°C) is ferromagnetic,... 

Fig. 35 Phase diagrams AB miktoarm-star copolymers for n = 2, n = 3, n = 4 and n = 5. mean-field critical point through which system can transition from disordered state to Lam phase via continuous, second-order phase transition. All other phase transitions are first-order. From [112]. Copyright 2004 American Chemical Society...

It is easily shown that a first-order phase transition is obtained for cases were d < 0, whereas behaviour at the borderline between first- and second-order transitions, tricritical behaviour, is obtained for d = 0. In the latter case the transitional Gibbs energy is... 

The Greek indices a,j3= II, B,G) count colors, the Latin indices i = u,d,s count flavors. The expansion is presented up to the fourth order in the diquark field operators (related to the gap) assuming the second order phase transition, although at zero temperature the transition might be of the first order, cf. [17], iln is the density of the thermodynamic potential of the normal state. The order parameter squared is D = d s 2 = dn 2 + dG 2 + de 2, dR dc dB for the isoscalar phase (IS), and D = 3 g cfl 2,... 

Fig. 29. Phase diagram of the model Eq. (22) for coadsorption of two kinds of atoms in the temperature-coverage space. Circles indicate a second-order phase transition, while crosses indicate first-order transitions. Point A is believed to be a tricritical point and point B a bicritical point. The dashed curve shows the boundary from the Blume-Capel model on a square lattice with a nearest-neighbor coupling equal to 7 in the present model (for - 0 Eq. (22) reduces to this model), only the ordered phase I then occurs. From Lee and Landau. )...

A second-order phase transition is one in which the enthalpy and first derivatives are continuous, but the second derivatives are discontinuous. The Cp versus T curve is often shaped like the Greek letter X. Hence, these transitions are also called -transitions (Figure 2-15b Thompson and Perkins, 1981). The structure change is minor in second-order phase transitions, such as the rotation of bonds and order-disorder of some ions. Examples include melt to glass transition, X-transition in fayalite, and magnetic transitions. Second-order phase transitions often do not require nucleation and are rapid. On some characteristics, these transitions may be viewed as a homogeneous reaction or many simultaneous homogeneous reactions. 

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