Phase transition critical point

Long-range fluctuations in matter close to a critical point (phase transition) can lead to long-range forces, which are called critical Casimir forces. 

General considerations regarding the character of the transition of a substance from a metal to a dielectric state lead to the conclusion that such a transition occurs as a normal phase transition even up to high temperatures. For mercury and other low-boiling metals the critical point of transition from a liquid to a gaseous state probably corresponds to a lower temperature. One should expect the existence in some region of two separate (at different pressures and temperatures) transitions, from a metallic to a nonmetallic state, and from a liquid to a gaseous state, i.e., the existence of a liquid nonmetallic phase which transforms into a metal with increased pressure, and into a gas with decreased pressure. 

First, recall the close analogies that exist between pure-component critical points and those of binary-mixtures. Unlike simple phase changes, which represent transitions between stable and metastable behavior, all critical points represent transitions between stable and unstable behavior. For pure components, the transition is driven by mechanical instabilities, and at vapor-liquid critical points pure fluids have... 

In all these cases, if the nontrivial fixed point is stable, then it represents a critical phase with characteristic exponents, while if it is unstable it represents a critical type phase transition with its own characteristic exponents. The reunion behaviour in Sec. 4.2.3 and the reunion exponents[13] are examples of nontrivial exponents at a stable fixed point. The unstable fixed point will be associated with a diverging length scale with an exponent = l/ e as in Eq. (32). 

An interesting type of quantum phase transition are boundary transitions where only the degrees of freedom of a subsystem become critical while the bulk remains imcritical. The simplest case is the so-called impurity quantum phase transitions where the free energy contribution of the impurity (or, in general, a zero-dimensional subsystem) becomes singular at the quantum critical point. Such transitions occur in anisotropic Kondo systems, quantum dots, and in spin systems coupled to dissipative baths as examples. Impurity quantum phase transitions require the thermodynamic limit in the bulk (bath) system but are completely independent from possible phase transitions of the bath. A recent review of impurity quantum phase transitions can be found in Ref. 42. 

First indirect experimental observations of the critical Casimir force were made by Chan and Garcia [152]. They measured the thickness of He films on a copper substrate and detected a thinning of the films close to the critical point of transition to superfluidity, indicating an attractive critical Casimir force. For a He/ He mixture close to the tricritical point, the same authors found a repulsive critical Casimir force, which caused film thickening on the copper substrate [153] (for a later, refined theoretical analysis, see Ref [154]). The tricritical point is the point in the phase diagram where the superfluidity transition line terminates at the top coexistence line of He/He. 

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. 

Figure A2.5.1. Schematic phase diagram (pressure p versus temperature 7) for a typical one-component substance. The full lines mark the transitions from one phase to another (g, gas liquid s, solid). The liquid-gas line (the vapour pressure curve) ends at a critical point (c). The dotted line is a constant pressure line. The dashed lines represent metastable extensions of the stable phases.

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.

Domb C and Lebowitz J (eds) 1984 Phase Transitions and Critical Phenomena vol 9 (London, New York Academic) oh 1. Lawrie I D and Sarbach S Theory of tricritical points oh 2. Knobler C M and Scott R L Multicritical points in fluid mixtures. Experimental studies. 

SmA phases, and SmA and SmC phases, meet tlie line of discontinuous transitions between tire N and SmC phase. The latter transition is first order due to fluctuations of SmC order, which are continuously degenerate, being concentrated on two rings in reciprocal space ratlier tlian two points in tire case of tire N-SmA transition [18,19 and 20], Because tire NAC point corresponds to the meeting of lines of continuous and discontinuous transitions it is an example of a Lifshitz point (a precise definition of tliis critical point is provided in [18,19 and 20]). The NAC point and associated transitions between tire tliree phases are described by tire Chen-Lubensky model [97], which is able to account for tire topology of tire experimental phase diagram. In tire vicinity of tire NAC point, universal behaviour is predicted and observed experimentally [20]. 

Fig. 3. PF diagram for a pure fluid (not to scale) point c is the gas—liquid critical state, is the constant pressure at which phase transition occurs at...

Helium Purification and Liquefaction. HeHum, which is the lowest-boiling gas, has only 1 degree K difference between its normal boiling point (4.2 K) and its critical temperature (5.2 K), and has no classical triple point (26,27). It exhibits a phase transition at its lambda line (miming from 2.18 K at 5.03 kPa (0.73 psia) to 1.76 K at 3.01 MPa (437 psia)) below which it exhibits superfluid properties (27). 

On the other hand, the formation of the high pressure phase is preceded by the passage of the first plastic wave. Its shock front is a surface on which point, linear and two-dimensional defects, which become crystallization centers at super-critical pressures, are produced in abundance. Apparently, the phase transitions in shock waves are always similar in type to martensite transitions. The rapid transition of one type of lattice into another is facilitated by nondilTusion martensite rearrangements they are based on the cooperative motion of many atoms to small distances. ... 

FIG. 4 Qualitative phase diagram close to a first-order irreversible phase transition. The solid line shows the dependence of the coverage of A species ( a) on the partial pressure (Ta). Just at the critical point F2a one has a discontinuity in (dashed line) which indicates coexistence between a reactive state with no large A clusters and an A rich phase (hkely a large A cluster). The dotted fine shows a metastability loop where Fas and F s are the upper and lower spinodal points, respectively. Between F2A and Fas the reactive state is unstable and is displaced by the A rich phase. In contrast, between F s and F2A the reactive state displaces the A rich phase. 

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