Critical behaviour

WIson K G 1971 Renormalization group and critical phenomena. II. Phase space cell analysis of critical behaviour P/rys. Rev. B 4 3184-205... 

The nematic to smectic A phase transition has attracted a great deal of theoretical and experimental interest because it is tire simplest example of a phase transition characterized by tire development of translational order [88]. Experiments indicate tliat tire transition can be first order or, more usually, continuous, depending on tire range of stability of tire nematic phase. In addition, tire critical behaviour tliat results from a continuous transition is fascinating and allows a test of predictions of tire renonnalization group tlieory in an accessible experimental system. In fact, this transition is analogous to tire transition from a nonnal conductor to a superconductor [89], but is more readily studied in tire liquid crystal system. 

Obviously 9 =0 corresponds to the SmA phase. This transition is analogous to the nonnal-superfluid transition in liquid helium and the critical behaviour is described by the AT model. Further details can be found elsewhere [18, 19 and 20]. 

E. V. Aibano. Critical behaviour of a forest fire model with immune trees. J Phys A (Math Gen) 27 L881-L886, 1994. 

K. Binder. Critical behaviour at surfaces. In C. Domb, J. L. Lebowitz, eds. Phase Transitions and Critical Phenomena, Vol. 8. London Academic Press, 1983, pp. 1-144. 

T. Riste, E.J. Samuelson, K. Otnes, and J. Feder, Critical behaviour of SrTiOs near the... 

Wilding, N. B., Simulation studies of fluid critical behaviour, J. Phys. Condens. Matter... 

The transition from ideal elastic to plastic behaviour is described by the change in relaxation time as shown by the stress relaxation in Fig. 66. The immediate or plastic decrease of the stress after an initial stress cr0 is described by a relaxation time equal to zero, whereas a pure elastic response corresponds with an infinite relaxation time. The relaxation time becomes suddenly very short as the shear stress increases to a value equal to ry. Thus, in an experiment at a constant stress rate, all transitions occur almost immediately at the shear yield stress. This critical behaviour closely resembles the ideal plastic behaviour. This can be expected for a polymer well below the glass transition temperature where the mobility of the chains is low. At a high temperature the transition is a... 

To explore the implications of this model for critical behaviour it is noted that Kc = f Kc) so that [/(/<,..)] is infinite. Secondly, suppose that K is near Kc, such that approximately... 

The rheology of the sol-gel transition was undertaken with special care in order to avoid gel disruption. A critical behaviour for the shear modulus with respect to the helix amount, is noticed. A simple relation between the rheological parameters and the degree of helix formation is pointed out in a limited range of helix amounts (X<15%). These experiments will continue on the fully matured gels. 

In contrast to critical behaviour, where the NMR relaxation rate shows a max-imiun (or a corresponding Ti minimiun) at Tc, thermally activated slowing down provides a Ti minimiun for lTc = 1, i.e. at the border between the fast motion and the slow motion regimes. Since according to Eq. 11 In(rc) is proportional to T Ti is usually plotted in logarithmically versus T", as for example shown in Fig. 11a. The slopes above and below the minimum are proportional to the activation energy E, . In Fig. 11b a typical tempera-... 

The preceding discussion should indicate how the chemically interesting phenomenon of spin cross-over becomes quite a complex one to understand in the solid state. It must be pointed out that in spite of the extensive studies, there are many aspects of spin-state transitions (e.g. critical behaviour, soft modes) that are yet to be explored. 

These mixing rules were applied to perform critical-point calculations and the critical behaviour of some highly non-ideal systems [47, 48],... 

Several codimension-two bifurcations have already been mentioned. Although they occur in restricted subspaces of parameter space and would therefore be difficult to locate experimentally, their usefulness lies in their role as centres for critical behaviour. Emanating from each local codimen-sion-two point will be two or more of the above codimension-one bifurcation curves. Their usefulness in studying dynamics is akin to that of the triple point in thermodynamic phase equilibria in which boundaries between three different phases come together at a point in a two-parameter diagram. Because some of these codimension-two points have been studied and classified analytically, finding one can provide clues about what other codimension-one bifurcation curves to expect near by and thus aids in the continuation of all of the bifurcation curves in the excitation diagram. 

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. 

Canter, K.F. and Roellig, L.O. (1970). Critical behaviour of positrons in low temperature gaseous helium. Phys. Rev. Lett. 25 328-330. 

GMP contains ten principles that introduce employees to critical behaviours established by FDA and industry leaders to maintain good manufacturing practices in plants. 

In the rest of this chapter, we will discuss briefly the theoretical ideas and the models employed for the study of failure of disordered solids, and other dynamical systems. In particular, we give a very brief summary of the percolation theory and the models (both lattice and continuum). The various lattice statistical exponents and the (fractal) dimensions are introduced here. We then give brief introduction to the concept of stress concentration around a sharp edge of a void or impurity cluster in a stressed solid. The concept is then extended to derive the extreme statistics of failure of randomly disordered solids. Here, we also discuss the competition between the percolation and the extreme statistics in determining the breakdown statistics of disordered solids. Finally, we discuss the self-organised criticality and some models showing such critical behaviour. 

Cuvelier, G., Peigney-Noiuy, C., and Launay, B. 1990. Viscoelastic properties of physical gels critical behaviour at the gel point, in Gums and Stabilisers for the Food Industry 5, eds. G. O. Phillips, D. J. Wedlock and P. A. Williams, pp. 549-552, IRL Press, Oxford, UK. 

J. I. Siepmann, S. Karaborni, and B. Smit, Nature, 365, 330 (1993). Simulating the Critical Behaviour of Complex Fluids. 

Salje EKH (1994) Phase transitions and vibrational spectroscopy in feldspars. In Feldspars and Their Reactions. Parson 1 (ed) Kluwer, Dordrecht, The Netherlands, p 103-160 Salje EKH, Bismayer U (1981) Critical behaviour of optical modes in ferroelastic Pb3(P04)2- Pb3(As04)2, Phase Transitions 2 15-30... 

Below the gel point, the system is self-similar on length scales smaller than the correlation length with a power law distribution of molar masses with Fisher exponent r = 5/2 [Eq. (6.78)]. Each branched molecule is a self-similar fractal with fractal dimension 27 = 4 for ideal branched mole-cules in the mean-field theory. The lower limit of this critical behaviour is the average distance between branch points (= ). There are very few... 

Anisotropy in the Critical Behaviour of TTF-TCNQ and TSeF-TCNQ 469... 

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