Critical Phenomena

characteristic behavior of substances in the vicinity of phase transition points. Typical critical phenomena include an increase in compressibility upon approaching the critical point of liquid-vapor equilibrium, an increase in magnetic susceptibility and dielectric constant in the vicinity of the Curie points of ferromagnets and feorroelectrics (Figure 1), an anomaly in heat capacity at the point of transition of helium to the superfluid state (Figure 2), slowing of the mutual diffusion of substances near the critical points of mixtures of stratifying liquids, and an anomaly in the propagation of ultrasound.

Figure 1. Dependence of the magnetic susceptibility X of a ferromagnetic (a single crystal of nickel) in the vicinity of the Curie point T_c on the reduced temperature τ = (T – T_c)/T_c

In a narrower sense, phenomena caused by an increase in fluctuations of the thermodynamic quantities (such as density) in the vicinity of phase transition points are classified as critical.

A significant increase in fluctuations leads to considerable variations in the density of a substance from point to point at the critical point of liquid-vapor equilibrium. The fluctuation nonuniformity of the substance significantly affects its physical

Figure 2. Dependence on temperature T of the specific heat c_p of helium (at constant pressure) near the point of transition to the super-fluid state (2.19°K)

properties. For example, there is a significant increase in dispersion and absorption of radiation. The size of density fluctuations near the liquid-vapor critical point reaches thousands of angstroms and is comparable to a wavelength of light. As a result, the substance becomes completely opaque and most of the incident light is scattered. The substance acquires an opalescent (dull milky white) color, and critical opalescence of the substance is observed.

An increase in fluctuations also leads to dispersion of sound and intense sound absorption (Figure 3), retardation in the establishment of thermal equilibrium (at the critical point, the time for establishment of thermal equilibrium is measured in hours), a change in the nature of Brownian motion, and anomalies in viscosity and heat conduction in pure substances.

Figure 3. Dispersion (1) and absorption (2) of sound in argon near the critical temperature T_c of the liquid-vapor transition: (A) intensity of sound passing through substance, (A₀) initial sound intensity, (v_R) speed of sound

Analogous phenomena are observed in the vicinity of the critical points of binary mixtures; here they result from the development of fluctuations in the concentration of one component in the other. Thus, at the critical point for separation of liquid metals—for example, in Li-Na and Ge-Hg systems—critical scattering of X rays is observed (Figure 4). In the vicinity of the Curie points of ferromagnetics and ferroelectrics, where the fluctuations in magnetization and dielectric polarization increase, there are sharp anomalies in the scattering and polarization of neutron beams passing through them (Figure 5) and in the propagation of sound and a high-frequency electromagnetic field. In ordering of alloys (for example, metal hydrides) and the establishment of long-range orientation order in molecular crystals (as in solid methane, carbon tetrachloride, and ammonium halides), typical critical phenomena are also observed that are related to an increase in fluctuations of the corresponding physical property, such as the ordering of the atomic arrangement in alloys or the average orientation of molecules in the crystal in the vicinity of the phase transition point.

The inherent similarity of the critical phenomena in phase transitions for very different types of objects makes it possible to examine them from a single viewpoint. For example the existence of an identical temperature dependence of a number of physical quantities near second-order phase transitions for all objects has been established. To find this dependence, the physical quantities are expressed in the form of exponential functions of the reduced temperature τ = (T – T_c)/T_c, where T_c is the critical temperature, or other reduced quantities. For example, the compressibility of a gas (∂ V/∂p) _T, the susceptibility of a ferromagnet (∂M/∂H)_p, _T or ferroelectric (∂D/∂E)_p,T and the analogous quantity (∂x/μ)_p,T for mixtures with a liquid-liquid or liquid-vapor equilibrium critical point depend identically on temperature in the vicinity of the critical point and may be expressed by the standard formulas

(1) (∂V/∂p)_T, (∂M/∂H)_p,T, (∂D/∂E)_{p, T}, (∂x/∂μ)_p,T^{∼ τ-γ}

Here V, p, and T are volume, pressure, and temperature; M and D are the magnetization and polarization of the substance; H

Figure 4. Temperature dependence of X-ray scattering by a lithium-sodium liquid metal mixture. A sharp maximum in the number of quanta of scattered X radiation as recorded by a counter per unit time occurs near the critical solubility point of the mixture (301 °C).

and E are the magnetic and electric field strengths; and μ is the chemical potential of a component of the mixture with concentration x. The critical exponent γ may have identical or close values for all systems. Experimentally determined values of γ lie between 1 and 4/3, although the error in the determination of γ is often of the same order as the discrepancy in the experimental results. An analogous temperature dependence of the specific heat c for all these systems has the form

The values of α lie between zero and ∼0.2, and in a number of experiments a was found to be close to 1/8. For the heat capacity of helium at the point of transition to the superfluid state (the λ-point), formula (2) undergoes a change in form: c_p ∼ In τ.

In the vicinity of critical points, the dependence of the specific volume of a gas on pressure, of the magnetic or electric moment of a system on field strength, and of the concentration of a mixture on chemical potential of the components may be expressed similarly (as an exponential expression). At a constant temperature T_c, these relationships may be given as follows:

The experimental values for δ lie between 4 and 5.

A number of other quantities also depend identically on the reduced temperature, including the difference in the specific volumes of a liquid (V_liq) and vapor (V_vap) that are in equilibrium below the critical point, the magnetic or electric moment of a substance in the ferromagnetic or ferroelectric state in the absence of an external field, the difference in the concentrations of two phases (x₁ and x₂) in a stratifying mixture, and the square

Figure 5. Dependence on temperature of the intensity N of a scattered neutron beam passing through a nickel sample. The scattering was measured at angles θ = 10.2' and θ = 37' (for θ = 10.2' the scale is reduced by a factor of 3). The scattering maximum corresponds to the Curie point T_c and the position of the maximum does not de-pend on the scattering angle.

root of the density ρ_s of the superfluid component in helium II:

The values found for β are close to ⅓ (from 5/16 to ⅜). The constants α, β, γ, and δ characterize the behavior of physical quantities in the vicinity of second-order phase transition points and are called critical exponents.

In some samples—for example, in ordinary superconductors and many ferroelectrics—critical phenomena are not found in virtually the entire range of temperatures close to the critical point. On the other hand, the properties of ordinary liquids in a significant temperature range near the critical point or the properties of helium near the λ-point are determined almost entirely by critical effects. Such behavior is related to the nature of the action of intermolecular forces. If these forces decrease sufficiently quickly with increasing distance, then fluctuations play a significant role in such substances, and critical phenomena appear long before the critical point. On the other hand, if the intermolecular forces have a comparatively long action radius, as in the case of Coulomb and dipole-dipole interactions in ferroelectrics, then the average force field established in the substance will be distorted very little by fluctuations, and critical phenomena will be found only extremely close to the Curie points.

Critical phenomena are cooperative effects—that is, effects resulting from the properties of the entire system of particles and not from the individual properties of each particle. The problem of cooperative effects has not yet been entirely solved, and thus there is no exhaustive theory of critical phenomena.

All the current approaches to a theory of critical phenomena are based on the empirical fact of the increase of the nonuniformity of a substance upon approaching the critical point and introduce the concept of the fluctuation correlation radius r_c, which is similar in essence to the average fluctuation dimension. The correlation radius characterizes the distance at which fluctuations affect one another and thus are interdependent and “correlated.” This radius for all samples depends on temperature according to the exponential relationship

(5)r_c^∼ τ^-v

The assumed values for v lie between ½ and ⅔.

Equations (1), (2), and (5) indicate that the values of the corresponding quantities become infinite at points at which τ approaches zero (see Figures 1, 2, and 3). Thus, the correlation radius infinitely increases upon approaching the phase transition point. This means that any part of the system examined at the phase transition point senses the changes occurring in the other parts. On the other hand, far from the transition point, the fluctuations are statistically independent and random changes in the state of a substance at a given point of the sample have no effect on the rest of the substance. The scattering of light by a substance is an obvious example.

In the case of scattering of light by independent fluctuations (Rayleigh scattering), the intensity of the scattered light is inversely proportional to the fourth power of the wavelength and is approximately the same in any direction (Figure 6,a). Scattering by correlated fluctuations —critical scattering—differs in that the intensity of the light is proportional to the square of the wavelength and has a specific directional pattern (Figure 6,b).

Among the theories of critical phenomena, the theory that considers a substance near the phase transition point as a system of fluctuating regions of dimensions of the order of r_c has gained popularity. It is called the theory of scale transformations, or scaling theory. Scaling theory does not permit computation of the critical exponents from the properties of the molecules making up the substance, but it gives the relationship between the exponents, which permits calculation of all the exponents if any two of them are known. The relationship between the critical exponents makes possible the determination of the equation of state and hence calculation of the various thermodynamic quantities from a relatively limited amount of experimental data. An

Figure 6. (a) Directional pattern for scattering of light on independent fluctuations of liquid density; (b) scattering of light by correlated fluctuations (scattering at the critical temperature). The scale in (b) is considerably reduced.

analogous theory relates the critical exponents of kinetic properties, such as viscosity, heat conduction, diffusion coefficient, and sound absorption, as well as other properties that have anomalies at phase transition points, by a few relationships with the exponents of the thermodynamic quantities. This theory is called dynamic scaling theory, in contrast to static scaling, which relates only to the thermodynamic properties of matter.