Nonclassical Critical Behavior Research Articles

The relaxor enigma — charge disorder and random fields in ferroelectrics

Substitutional charge disorder giving rise to quenched electric random-fields (RFs) is probably at the origin of the peculiar behavior of relaxor ferroelectrics, which are primarily characterized by their strong frequency dispersion of the dielectric response and by an apparent lack of macroscopic symmetry breaking at the phase transition. Spatial fluctuations of the RFs correlate the dipolar fluctuations and give rise to polar nanoregions in the paraelectric regime as has been evidenced by piezoresponse force microscopy (PFM) at the nanoscale. The dimension of the order parameter decides upon whether the ferroelectric phase transition is destroyed (e.g. in cubic PbMg1/3Nb2/3O3, PMN) or modified towards RF Ising model behavior (e.g. in tetragonal Sr1−x BaxNb2O6, SBN, x ≈ 0.4). Frustrated interaction between the polar nanoregions in cubic relaxors gives rise to cluster glass states as evidenced by strong pressure dependence, typical dipolar slowing-down and theoretically treated within a spherical random bond-RF model. On the other hand, freezing into a domain state takes place in uniaxial relaxors. While at T c non-classical critical behavior with critical exponents ρ ≈ 1.8, β ≈ 0.1 and α ≈ 0 is encountered in accordance with the RF Ising model, below T c ≈ 350 K RF pinning of the walls of frozen-in nanodomains gives rise to non-Debye dielectric response. It is relaxation- and creep-like at radio and very low frequencies, respectively.

Crossover scaling from classical to nonclassical critical behavior

We study the crossover between classical and nonclassical critical behaviors. The critical crossover limit is driven by the Ginzburg number G. The corresponding scaling functions are universal with respect to any possible microscopic mechanism which can vary G, such as changing the range or the strength of the interactions. The critical crossover describes the unique flow from the unstable Gaussian to the stable nonclassical fixed point. The scaling functions are related to the continuum renormalization-group functions. We show these features explicitly in the large-$N$ limit of the $\mathrm{O}(N) {\ensuremath{\varphi}}^{4}$ model. We also show that the effective susceptibility exponent is nonmonotonic in the low-temperature phase of the three-dimensional Ising model.

Turbidity, light scattering, and coexistence curve data for the ionic binary mixture triethyl n-hexyl ammonium triethyl n-hexyl borate in diphenyl ether

We report turbidity, light scattering, and coexistence curve data for a solution of triethyl n-hexyl ammonium triethyl n-hexyl borate in diphenylether. We recently reported that the present sample shows much higher turbidity than that of K. S. Zhang, M. E. Briggs, R. W. Gammon, and J. M. H. Levelt Sengers [J. Chem. Phys. 109, 4533 (1998)] for an earlier sample. An analysis of the data shows that nonclassical critical behavior is favored in the reduced temperature range from 10−5 to 10−2. At fixed reduced temperature, the correlation length is about twice as large as that of the previous sample. The correlation length amplitude calculated from the fit is 1.4 nm±0.1 nm. A detailed data analysis points out the limitations of turbidity measurements far away from the critical point. The intensity of scattered light was measured at 90°. Multiple scattering is relevant in the wider vicinity of the critical point and was corrected for by a Monte Carlo simulation method. An Ising-type exponent for the correlation length was obtained: ν=0.641±0.003, and the amplitude of the correlation length ξ0=1.34 nm±0.01 nm agrees with that of the turbidity experiment. Mean-field behavior can be ruled out. The refractive indices of coexisting phases were measured in the reduced temperature range from t=10−4 to 0.04. These measurements disagree with results reported by R. R. Singh and K. S. Pitzer [J. Chem. Phys. 92, 6775 (1990)]. The present data lead to an exponent β=0.34±0.01, close to the Ising value. The coexistence curve is much narrower than that of Singh and Pitzer. Crossover could not be detected in any of the experiments. Two-scale-factor universality could be confirmed for this and another ionic system within the experimental uncertainty.

Critical scaling laws and a classical equation of state

In this paper we present a method which modifies a classical equation of state by incorporating the nonclassical critical behavior. As an example we have applied our procedure to the Carnahan-Starling-DeSantis (CSD) equation of state. The resulting equation reproduces the universal scaling behavior near the critical point and reduces to the universal ideal-gas behavior at low densities. We show that the renormalized CSD equation yields an improved and consistent representation of both mechanical and caloric thermodynamic properties. In addition, the suppression of the critical temperature due to the critical fluctuations is clearly demonstrated.

Stiffness instability in short-range critical wetting.

Recent theoretical work has shown that an interface separating two fluid phases suffers changes in its (bare) effective stiffness, \ensuremath{\Sigma}\ifmmode \tilde{}\else \~{}\fi{}(l)=\ensuremath{\Sigma}${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathrm{\ensuremath{\infty}}}$+\ensuremath{\Delta}\ensuremath{\Sigma}\ifmmode \tilde{}\else \~{}\fi{}(l), when located at a distance l from a planar wall: terms varying as ${\mathit{l}}^{\mathit{k}}$${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathit{j}\mathrm{\ensuremath{\kappa}}\mathit{l}}$ appear in \ensuremath{\Delta}\ensuremath{\Sigma}\ifmmode \tilde{}\else \~{}\fi{} (where 0\ensuremath{\le}k\ensuremath{\le}j=1,2, . . . and \ensuremath{\kappa} is the inverse bulk correlation length in the fluid wetting the wall). This may induce first-order wetting transitions when critical wetting had been expected. This general behavior of \ensuremath{\Delta}\ensuremath{\Sigma}\ifmmode \tilde{}\else \~{}\fi{}(l) is confirmed using an integral/adsorption constraint to determine l, in place of the original crossing constraint. The exact linearized functional renormalization-group technique is used to analyze the full wetting-phase diagram as a function of T, of \ensuremath{\omega}=${\mathit{k}}_{\mathit{B}}$${\mathit{T}}_{\mathit{c}\mathit{W}}$${\mathrm{\ensuremath{\kappa}}}^{2}$/4\ensuremath{\pi}\ensuremath{\Sigma}\ifmmode \tilde{}\else \~{}\fi{}(${\mathit{T}}_{\mathit{c}\mathit{W}}$), and of q, the amplitude of the -${\mathit{le}}^{\mathrm{\ensuremath{-}}2\mathrm{\ensuremath{\kappa}}\mathit{l}}$ term in \ensuremath{\Delta}\ensuremath{\Sigma}\ifmmode \tilde{}\else \~{}\fi{}. For dimensions d>3, any positive q (as generally expected) yields first-order wetting. The same is true for d=3 provided \ensuremath{\omega}1/2; but when \ensuremath{\omega}>1/2 nonclassical critical behavior is still found for small q〈${\mathit{q}}_{\mathit{t}}$(\ensuremath{\omega})〉0. Detailed expressions are obtained for 〈l〉, ${\ensuremath{\xi}}_{\mathrm{\ensuremath{\parallel}}}$, etc., in the various critical and first-order regions. Numerical estimates show that previous Ising-model simulations probably encountered weakly first-order wetting transitions which might explain discrepancies with earlier renormalization-group predictions.

NMR evidence for universal nonclassical critical behavior of incommensurately modulated crystals.

Nonclassical universal critical behavior is revealed in several one dimensionally modulated incommensurate (IC) crystals by means of quadrupolar perturbed NMR. In the IC phase the temperature dependence of the NMR line shape leads to the same critical order parameter exponent \ensuremath{\beta} for all systems. Above the phase transition a nonclassical exponent was found for the critical contributon to the spin-lattice relaxation. All results agree with the theoretical predictions of the 3d XY model.

Chemical reaction driven phase transitions and critical points

Two simple examples of model (mean-field) equations of state for phase equilibrium in chemically reactive systems are examined for ‘‘unexpected’’ phase equilibria. They are, essentially, exactly soluble and give classical critical behavior. One of these leads to a lower critical solution temperature that results in closed-loop coexistence curves similar to those seen in hydrogen-bonding mixtures. The second leads to less familiar, but interesting phase diagrams that exhibit a phenomenon analogous to critical azeotropy. The same phenomena occur in two examples of lattice gas models the partition functions of which can be mapped exactly to that of the Ising model thus resulting in nonclassical critical behavior. These models demonstrate how a chemical reaction can provide a mechanism leading to interesting phase equilibria and critical phenomena.

Comprehensive theory of simple fluids, critical point included.

We present a comprehensive theory of fluids which has the typical accuracy of a good liquid-state theory in the dense regime but in addition has a genuine nonclassical critical behavior. The theory is based on the hierarchical reference theory of fluids decoupled with an approximation inspired by the optimized random-phase approximation. The Lennard-Jones interaction is studied in detail above the critical temperature.

Born–Green hierarchy for continuum percolation

We present a projection operator technique that yields hierarchies of integral equations satisfied exactly by the n-point connectedness functions in a continuum version of the site-bond percolation problem. The n-point connectedness functions carry the same structural information for a percolation problem as the n-point correlation functions do for a thermal problem. Our method extends the Potts model mapping of Fortuin and Kastelyn to the continuum. We use the projection operator technique to produce an integral equation hierarchy for percolation similar to the Born–Green thermal hierarchy. The Kirkwood superposition approximation (SA) is extended to percolation in order to close this hierarchy and yield a nonlinear integral equation for the two-point connectedness function. We discuss the fact that this function, in the SA, is the analytic continuation to negative density of the two-point correlation function in a corresponding thermal problem. The Born–Green–Yvon (BGY) equation for percolation is solved numerically, both by an expansion in powers of the density, and iteratively, using the modified Picard method. We argue, both analytically and numerically, that the BGY equation for percolation, unlike its thermal counterpart, shows nonclassical critical behavior, with η=1 and γ=2.2±0.2. Finally, we develop a sequence of refinements to the superposition approximation that can be used to give increasingly accurate calculations of the two-point connectedness function.

Critical behavior in nonequilibrium phase transitions.

The nonequilibrium phase transitions occurring in a fast-ionic-conductor model and in a reaction-diffusion Ising model are studied by Monte Carlo finite-size scaling to reveal nonclassical critical behavior; our results are compared with those in related models.

Universality classes for the isotropic-to-nematic transition in systems of interacting, semiflexible polymers.

This paper discusses the behavior of simple models of interacting, semiflexible, nonchiral polymers. The Ursell-Mayer expansion is used to systemize calculation of the properties of such systems and avoid problems encountered with the strong, bonding interactions in a direct analysis of this problem. The formation of a nematic phase is discussed within a field theory which is equivalent, order by order in perturbation theory, to the Ursell-Mayer expansion. Symmetry analysis allows two universality classes; one has the usual, tensor order parameter, the other has a vectorlike order parameter. The vectorlike order parameter has no direct physical interpretation (distinguishing the transition from a ferroelectric transition) but must be introduced because of the strong, bonding interactions between monomers on a polymer. The behavior of some physical observables near the critical point in systems with phase transitions controlled by the vectorlike order parameter is discussed, including a brief discussion of nonclassical critical behavior. Evidence is given that transitions with either type of order parameter are realizable in model systems.

A thermodynamic ‘‘paradox’’ associated with nonclassical critical phenomena in mixtures

Nonclassical critical behavior gives rise to an apparent thermodynamic paradox near the critical point in many if not most two-component mixtures. This paradox occurs at both gas–liquid and liquid–liquid critical points. It is resolved by observing that in these mixtures the thermal expansion coefficient is negative sufficiently close to the critical point and diverges to −∞ there. As a consequence, the thermodynamic state is not uniquely determined by the variables P, v, x in the vicinity of the critical point. While for gas–liquid critical points the region in which this occurs is unobservably small, the predicted effect should be observable for liquid–liquid critical points.

Continuous ferroelastic phase transition of a KBr:KCN mixed crystal.

The ferroelastic phase transition of ${(\mathrm{KBr})}_{0.27}$${(\mathrm{KCN})}_{0.73}$ has been studied by x-ray diffraction, ultrasonics, and inelastic neutron scattering. It is the first example of a cubic crystal where the elastic shear constant ${C}_{44}$ softens completely corresponding to the $m=2$ universality class. ${C}_{44}$ and the Bragg intensities show a nonclassical critical behavior.

Random-field critical behavior and the Ginzburg criterion.

The Ginzburg criterion for crossover from classical to nonclassical critical behavior is derived in the presence of random fields. Experiments on dilute antiferromagnets in a field are examined, and the possibility of such a crossover is investigated. We also point out that the (necessarily) symmetric logarithmic divergence observed in birefringence measurements can be accompanied by a discontinuity, and a fitting form which allows for this and other corrections to scaling is proposed.

Hierarchical models and chaotic spin glasses

Renormalization-group studies in position space have led to the discovery of hierarchical models which are exactly solvable, exhibiting nonclassical critical behavior at finite temperature. Position-space renormalization-group approximations that had been widely and successfully used are in fact alternatively applicable as exact solutions of hierarchical models, this realizability guaranteeing important physical requirements. For example, a hierarchized version of the Sierpiriski gasket is presented, corresponding to a renormalization-group approximation which has quantitatively yielded the multicritical phase diagrams of submonolayers on graphite. Hierarchical models are now being studied directly as a testing ground for new concepts. For example, with the introduction of frustration, chaotic renormalization-group trajectories were obtained for the first time. Thus, strong and weak correlations are randomly intermingled at successive length scales, and a new microscopic picture and mechanism for a spin glass emerges. An upper critical dimension occurs via a boundary crisis mechanism in cluster-hierarchical variants developed to have well-behaved susceptibilities.

On the density anomaly in sulfur at the polymerization transition

We show that the density anomaly in sulfur near the polymerization transition observed by Borst and co-workers can be explained, at least in part, as a manifestation of nonclassical critical behavior. We combine the lattice solution model recently proposed to study equilibrium polymerization in sulfur with a phenomenological assumption to produce a dramatic improvement over earlier theories. Some discrepancies between the theory and experiment remain.

Critical and tricritical phenomena in ‘‘living’’ polymers

The equilibrium polymerization of ‘‘living polymers’’ such as polytetrahydrofuran and poly-α-methylstyrene can be usefully described by the n=0 limit of the n vector model of magnetism. Nonclassical critical behavior is predicted near the ceiling temperature when the initiator concentration is sufficiently small. The small initiator concentration implies a small magnetic field in the n→0 vector model. The agreement of our theory with the limited experimental data available is good, and appears to confirm nonclassical critical behavior. Solutions of living polymers can be described by an annealed dilute n→0 vector model, and interesting new phase diagrams are predicted that include tricritical points analogous to those found in He3–He4 mixtures at very low temperature and in models of sulfur solutions and dilute ferromagnets. The relationship between the phase diagrams in these systems is elucidated.

Critical behavior in fluids and fluid mixtures

The presence of large fluctuations near a critical point leads to thermodynamic anomalies not present in classical, i.e., analytic equations, such as that of van der Waals. We introduce the concepts of universality, critical exponents, scaling laws, “field” and “density” variables, “strong” and “weak” directions. Experimental evidence for the presence of nonclassical critical behavior in fluids and fluid mixtures is presented. A scaled thermodynamic potential represents the thermodynamic data of steam, ethylene and isobutane accurately. It is valid up to 1.07 T c, and within ±30% from ρ c. Methods for joining it to an analytic equation are discussed. The generalization to a nonclassical description of fluid mixtures is described, and applications are given. The engineer may require the nonclassical description in custody transfer, design of supercritical power cycles and supercritical extraction. The classical approach is used here to explain peculiarities of dilute mixtures recently reported by several experimenters.

Tricritical Points in the Equilibrium Polymerization of Sulfur Solutions.

A dilute $n\ensuremath{\rightarrow}0$ vector model provides a useful description of equilibrium polymerization in a solvent. Such polymerization occurs in liquid sulfur solutions and leads to a lower critical solution point that is analogous to the tricritical point found in $^{3}\mathrm{He}$-$^{4}\mathrm{He}$ mixtures. In the mean-field approximation, the model is identical to an earlier theory of Scott. Nonclassical critical behavior can explain certain discrepancies between Scott's theory and experiment.

Critical points and tricritical points in liquid sulfur solutions

Equilibrium polymerization in a solvent can be described by the n→0 limit of a dilute n-vector model of magnetism in a small magnetic field. In the molecular field approximation the model becomes identical to the earlier theory of Scott for liquid sulfur solutions. The lower critical solution temperature in sulfur solutions is found to be intimately associated with a tricritical point in the magnetic model which accounts for the distinctive shape found by Scott for the high-temperature phase separation curve. The connection with the magnetic model also establishes a relationship between the phase diagrams predicted by Scott for sulfur solutions and those predicted by Blume, Emery, and Griffiths for He3xnHe4 mixtures. Modern theory of critical and tricritical phenomena suggests that incorporation of nonclassical critical behavior in the dilute n→0 vector model may help to resolve certain discrepancies between Scott’s mean field theory and experimental coexistence curves for sulfur solutions.