Transitions second-order

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. 

Other examples of order-disorder second-order transitions are found in the alloys CuPd and Fe Al. Flowever, not all ordered alloys pass tlirough second-order transitions frequently the partially ordered structure changes to a disordered structure at a first-order transition. 

A related phenomenon with electric dipoles is ferroelectricity where there is long-range ordermg (nonzero values of the polarization P even at zero electric field E) below a second-order transition at a kind of critical temperature. 

Here the coefficients G2, G, and so on, are frinctions ofp and T, presumably expandable in Taylor series around p p and T- T. However, it is frequently overlooked that the derivation is accompanied by the connnent that since. . . the second-order transition point must be some singular point of tlie themiodynamic potential, there is every reason to suppose that such an expansion camiot be carried out up to temis of arbitrary order , but that tliere are grounds to suppose that its singularity is of higher order than that of the temis of the expansion used . The theory developed below was based on this assumption. 

There are many other examples of second-order transitions involving critical phenomena. Only a few can be mentioned here. 

Syimnetrical tricritical points are predicted for fluid mixtures of sulfur or living polymers m certain solvents. Scott (1965) in a mean-field treatment [38] of sulfiir solutions found that a second-order transition Ime (the critical... 

Scott R L 1953 Second-order transitions and critical phenomena J. Chem. Phys. 21 209-11... 

Figure B3.6.3. Sketch of the coarse-grained description of a binary blend in contact with a wall, (a) Composition profile at the wall, (b) Effective interaction g(l) between the interface and the wall. The different potentials correspond to complete wettmg, a first-order wetting transition and the non-wet state (from above to below). In case of a second-order transition there is no double-well structure close to the transition, but g(l) exhibits a single minimum which moves to larger distances as the wetting transition temperature is approached from below, (c) Temperature dependence of the thickness / of the enriclnnent layer at the wall. The jump of the layer thickness indicates a first-order wetting transition. In the case of a conthuious transition the layer thickness would diverge continuously upon approaching from below.

There is no discontinuity in volume, among other variables, at the Curie point, but there is a change in temperature coefficient of V, as evidenced by a change in slope. To understand why this is called a second-order transition, we begin by recalling the definitions of some basic physical properties of matter ... 

Since V experiences a change of slope at the second-order transition, that is, (3V/3T)p and (3V/3p)j have different values on each side of the transition, it is a and /3 that show the discontinuities at the second-order transition rather than V itself. The term second order comes about since the quantities showing the discontinuity, a and /3 among others, may be written as second derivatives of G. That is, we apply Eq. (4.3) to Eqs. (4.43) and (4.44) to obtain... 

Figures 4.14a and b are the analogs of Figs. 4.3a and b they schematicaUy describe second- and first-order transitions, respectively. It is the discontinuity in these second-order properties that characterizes a second-order transition.

This expression describes the variation of the pressure-temperature coordinates of a first-order transition in terms of the changes in S and V which occur there. The Clapeyron equation cannot be applied to a second-order transition (subscript 2), because ASj and AVj are zero and their ratio is undefined for the second-order case. However, we may apply L Hopital s rule to both the numerator and denominator of the right-hand side of Eq. (4.47) to establish the limiting value of dp/dTj. In this procedure we may differentiate either with respect to p. 

Since Eqs. (4.48) and (4.49) both describe the same limit, they must be equal at the second-order transition point ... 

For a second-order transition Eq. (4.53) is the analog of Eq. (4.5), which is useful for first-order transitions. Equation (4.53) and the first- and second-order terminology are due to Ehrenfest. 

If a Tg value corresponding to a true second-order transition exists, it is a value lower than those based on short-term observations. The latter should then be regarded as approximations to the true value. 

Use these data to evaluate Tg, assuming that the latter is a true second-order transition. Compare your results with the values in Table 4.4 and comment on the agreement or lack thereof. 

The melting point of commercial Teflon PEA is 305°C, ie, between those of PTEE and EEP. Second-order transitions are at —100, —30, and 90°C, as determined by a torsion pendulum (21). The crystallinity of the virgin resin is 65—75%. Specific gravity and crystallinity increase as the cooling rate is reduced. An ice-quenched sample with 48% crystallinity has a specific gravity of 2.123, whereas the press-cooled sample has a crystallinity of 58% and a specific gravity of 2.157. 

Stabilization of the Cellular State. The increase in surface area corresponding to the formation of many ceUs in the plastic phase is accompanied by an increase in the free energy of the system hence the foamed state is inherently unstable. Methods of stabilizing this foamed state can be classified as chemical, eg, the polymerization of a fluid resin into a three-dimensional thermoset polymer, or physical, eg, the cooling of an expanded thermoplastic polymer to a temperature below its second-order transition temperature or its crystalline melting point to prevent polymer flow. 

Poly(vinylchloride). Cellular poly(vinyl chloride) is prepared by many methods (108), some of which utili2e decompression processes. In all reported processes the stabili2ation process used for thermoplastics is to cool the cellular state to a temperature below its second-order transition temperature before the resia can flow and cause coUapse of the foam. 

Poly(vinyl chloride) has a good resistance to hydrocarbons but some plasticisers, particularly the less polar ones such as dibutyl sebacate, are extracted by materials such as iso-octane. The polymer is also resistant to most aqueous solutions, including those of alkalis and dilute mineral acids. Below the second order transition temperature, poly(vinyl chloride) compounds are reasonably good electrical insulators over a wide range of frequencies but above the second order transition temperature their value as an insulator is limited to low-frequency applications. The more plasticiser present, the lower the volume resistivity. 

Now, assume that we are getting closer to the critical point of our transition, i.e., to the point of the second-order transition. In the case of a uniform system the critical region can be described by the divergent correlation length of statistical fluctuations [138]... 

All the above scaling relations have one common origin in the behavior of the correlation length of statistical fluctuations, in a finite system [140,141]. Namely, the most specific feature of the second-order transition is the divergence of at the transition point, as is described by Eq. (22). In the finite system, the development of long-wavelength fluctuations is suppressed by the system size limitation can be, at the most, of the same order as L. Taking this into account, we find from Eqs. (22) and (26) that... 

Another interesting version of the MM model considers a variable excluded-volume interaction between same species particles [92]. In the absence of interactions the system is mapped on the standard MM model which has a first-order IPT between A- and B-saturated phases. On increasing the strength of the interaction the first-order transition line, observed for weak interactions, terminates at a tricritical point where two second-order transitions meet. These transitions, which separate the A-saturated, reactive, and B-saturated phases, belong to the same universality class as directed percolation, as follows from the value of critical exponents calculated by means of time-dependent Monte Carlo simulations and series expansions [92]. 

C. A. Voigt, R. M. Zilf. Epidemic analysis of the second-order transition in the Zilf Gulari Barshad surface-reaction model. Phys Rev. E 56 R6241-R6244, 1997. 

In fact, the mean-field estimate for H, given by equation 3.64, predicts a second-order transition. While abrupt changes in entropy characterize sudden transitions between regions of periodic and chaotic rules, smooth changes in entropy as A is increased instead suggest that the rule path sometimes passes through a region of... 

---