Crossover Critical Behaviour

1 Crossover from Ising-like to Mean-Field Critical Behaviour 

The general theory of critical phenomena presented in Section 10.2 concerns the asymptotic (Ising-like) thermodynamic critical behaviour. It is possible to extend the theory to include a crossover from fluctuation-induced Ising-like critical behaviour to classical mean-field critical behaviour. For this purpose it is more convenient to replace the potential in Section 10.2 with another potential O with characteristic variables q i and h2 - 

From eq 10.20, we note that for the lattice gas is to be identified with the critical part of the Helmholtz-energy density. From eq 10.11, we see that in the classical mean-field approximation ci has an asymptotic Landau expansion of the form  

In eq 10.82, u 0.472 is a universal coupling constant, u a scaled system-dependent coupling constant, and A a dimensionless cutoff wave number related to the microscopic (molecular) length scale = vq /tcA for the lattice gas Vo is the volume of the unit cell and for fluids vq is the molecular volume. As a measure of a distance from the critical point we consider a parameter k, related to the inverse correlation length, which in the classical limit is proportional to the square root of the inverse susceptibility x and which has the form  

For the rescaling functions 9, and Jf we have adopted approximants originally proposed by Chen et u/. and referred to as crossover model II by Tang et al  

---

The theory of crossover critical behaviour, presented in Section 10.5.1, not only accounts for crossover from asymptotic Ising-like critical behaviour to classical mean field critical behaviour away from (in the limit w O), but also... 

In a blend of immiscible homopolymers, macrophase separation is favoured on decreasing the temperature in a blend with an upper critical solution temperature (UCST) or on increasing the temperature in a blend with a lower critical solution temperature (LCST). Addition of a block copolymer leads to competition between this macrophase separation and microphase separation of the copolymer. From a practical viewpoint, addition of a block copolymer can be used to suppress phase separation or to compatibilize the homopolymers. Indeed, this is one of the main applications of block copolymers. The compatibilization results from the reduction of interfacial tension that accompanies the segregation of block copolymers to the interface. From a more fundamental viewpoint, the competing effects of macrophase and microphase separation lead to a rich critical phenomenology. In addition to the ordinary critical points of macrophase separation, tricritical points exist where critical lines for the ternary system meet. A Lifshitz point is defined along the line of critical transitions, at the crossover between regimes of macrophase separation and microphase separation. This critical behaviour is discussed in more depth in Chapter 6. 

We will discuss if the size exponents (i/ or ve) are affected by the lattice (configurational) fluctuations if r (p) is different from i/ or if i/ is different from i/J for p < 1. This question arises naturally from the application of the Harris criterion [8] to the n-vector model in the n —> 0 limit [2], when the partition function graphs are all SAWs. A naive application of the criterion to the SAWs suggested [6] a possible disorder induced crossover in the critical behaviour of SAW statistics for any amount of disorder (p < 1). A modified... 

We shall proceed as follows. In Section 10.2 we provide the general theoretical frame work for the asymptotic thermodynamic critical behaviour of Ising-like systems. In Section 10.3 we consider the application of the theory to one-component fluids near the vapour-liquid critical point. In Section 10.4 we discuss the extension of the theory to binary fluid mixtures. In Section 10.5 we address the problem of nonasymptotic crossover behaviour of fluids. Concluding remarks are presented in Section 10.6. 

In the simplest case the vapour-liquid critical points of the two components may be connected by a continuous locus of vapour-liquid critical points of mixtures at various concentrations with or without additional liquid-liquid immiscibility. In other cases the locus of vapour-liquid critical points is interrupted and starting from one of the components may wonder off to higher temperatures or may crossover to an upper or a lower consolute point. The principle of isomorphic critical behaviour asserts that the thermodynamic behaviour associated with the critical behaviour in mixtures can still be described by the scaling-law expression of eq 10.1 in terms of two independent scaling fields, hi and hj, and a dependent scaling field hi. The different types of critical phenomena observed experimentally are caused by different relationship of these scaling fields with the actual physical fields. ... 

Equations 10.85 to 10.88 define what has been called a two-term crossover Landau model (CLM). In the classical limit (A/ic 1) the crossover function Y approaches unity and one recovers from eq 10.85 the classical expansion of eq 10.83. In the critical region (A/k 1) the crossover function approaches zero as Y x k/uA) and one recovers from eq 10.85 the power-law expansions specified in Table 10.5 with expressions for the critical amplitudes listed in Table 10.8. The values for the critical-amplitude ratios implied by the crossover Landau model are included in Table 10.3. The nonasymptotic critical behaviour is governed by u and A/cJ or, equivalently by u and by Nq, known... 

The coupling constant u controls the magnitude of the corrections to the asymptotic power-law behaviour as can be seen from Table 10.8. The Ginzburg number Nq is a measure of the value of the temperature variable t, where the crossover from Ising-like to mean-field critical behaviour occurs. 

Another theoretical formulation of crossover from Ising-like to classical mean-field critical behaviour has been derived by Bagnuls and Bervillier. It has been further developed and applied to one-component fluids by Garrabos and coworkers. ... 

The theory described in Section 10.5.1 accounts for the crossover from Ising-like critical behaviour to asymptotic mean-field critical behaviour. To extend the range of applicability one may consider crossover from Ising-like critical behaviour to nonasymptotic mean-field critical behaviour by including higher-order terms in the classical Landau expansion of eq 10.83 ... 

The critical behaviours of higher order satellites near a continuous phase transition are of particular interest, as they provide the opportunity to study the crossover exponents of the respective symmetry breaking perturbations in the spin Hamiltonian. Each order of satellite has an associated order parameter critical exponent given by / = 2 - a -where a is the specific heat exponent, and crossover exponent As an example, if the transition is described by the 3DXY model, then the exponent 2 measures the crossover caused by a perturbation of uniaxial symmetry. For this model, theory predicts that = o P, with... 

Povodyrev et aJ [30] have applied crossover theory to the Flory equation ( section A2.5.4.1) for polymer solutions for various values of N, the number of monomer units in the polymer chain, obtaining the coexistence curve and values of the coefficient p jj-from the slope of that curve. Figure A2.5.27 shows their comparison between classical and crossover values of p j-j for A = 1, which is of course just the simple mixture. As seen in this figure, the crossover to classical behaviour is not complete until far below the critical temperature. 

However, for more complex fluids such as high-polymer solutions and concentrated ionic solutions, where the range of intemiolecular forces is much longer than that for simple fluids and Nq is much smaller, mean-field behaviour is observed much closer to the critical point. Thus the crossover is sharper, and it can also be nonmonotonic. 

De Gennes applied his method to study the chain behaviour similar to that used by Wilson (1971) to study magnetic phenomena. He pointed out that at a certain concentration the behaviour of a polymer chain is analogous to the magnetic critical and tricritical phenomena. De Gennes classifies the concentration c into three categories the dilute solution d, the semi-dilute solution c and the concentrated solution c". The concentration c is equivalent to the critical point where the crossover phenomenon occurs from randomness... 

This limiting behaviour characterizes Kuhnian chains which are continuous limits of chains with excluded volume, just as Brownian chains are continuous limits of chains with independent links. Kuhnian chains (z - oo), like Brownian chains (z = 0), are critical objects which depend only on one length X. Non-zero finite values of z correspond to a crossover domain between two different critical types of behaviour. 

Des Cloizeaux introduced the conception of a critical object in reference to a polymer chain. With = 0 the chain as a critic2d object is the Brownian chain. With z —> oo one will obtain a critical object as the limit of the chain with interacting segments (the Kuhnian chain). Between these two limits there is a crossover region with a finite value of z. All the physical quantities at a given 6 (or 5) increase with a however, in the 2isymptotic limit the universal behaviour and the correctness of the scaling relationships can be expressed. 

There exist also systems with midticritical points in their phase diagrams and with complex crossover behaviour from one set of critical exponents to another when the range of reduced temperature is laige enough. Examples of mixed and diluted systems will be considered in the last section of this review. Here we discuss only some results of the precise experiments demonstrating the unique features of the phase transitions in dipolar uniaxial ferromagnets. 

If a fluid possesses two or more mesoscopic length scales, a competition between these scales may cause crossover between different behaviours, each associated with a particular meso-scale. In this section, we discuss only two examples of such competition near-critical polymer solutions and near-critical finite-size fluids. 

---