Properties of the glass instability treated within a mode coupling theory

The anomalies at the liquid glass transition discussed recently by Bengtzelius et al. within a mode coupling theory are demonstrated to be due to an isolated eigenvalue of a certain stability matrix to approach unity at the critical point. Within this scenario it is shown how to derive the asymptotic results for the correlations functions analytically up to the determination of two eigenvectors and the evaluation of some wave vector integrals. As a result it is found that the Debye-Waller factor, the Lamb-Mössbauer factor, the localization length for a tagged particle, and the elastic moduli approach their asymptotic limit at the glass instability point with critical exponent one half. The critical dynamics for the coherent and incoherent scattering functions and for the transversal currents is given by a single wave vector independend scaling function. A formula for the critical exponent parameter is obtained and the scaling equation is shown to agree with the one discussed earlier for a schematic model.Dedicated to B. Mühlschlegel on the occasion of his 60th birthday     

Exactly solvable toy model that mimics the mode coupling theory of supercooled liquid and glass transition

A toy model is proposed which incorporates the reversible mode coupling mechanism responsible for ergodic-nonergodic transition of the mode coupling theory (MCT) of structural glass transition with trivial Hamiltonian. The model can be analyzed without relying on uncontrolled approximations inevitable in the current MCT. The strength of hopping processes can be easily tuned and the ideal glass transition is reproduced only in a certain range of the strength. On the basis of the analyses of our model, we discuss about a sharp ergodic-nonergodic transition and its smearing out by "hopping."     

On the applicability of mode coupling theory to a ϕ 4-model with first order phase transition

A ? ⁴-model with symmetric double-well-like on-site potential and anharmonic, infinite range interactions is investigated. This model exhibits a first order phase transition at a temperature T _c. The time-dependent displacement correlation function is studied in the framework of the mode coupling theory (MCT). Depending on the choice of slow modes, MCT makes qualitatively different predictions which are compared with MD-results. These numerical results suggest that only the order parameter mode {ie1-1} should be considered as slow. In that case it is shown that MCT yields a dynamical transition in the supercooled high-temperature phase {ie1-2} at a temperature T* which coincides with the spinodal temperature T _s (T _s = 0 for our model) where the metastable supercooled phase becomes instable.     

Water confined in MCM-41: a mode coupling theory analysis

In this paper we analyze molecular dynamics simulation results on supercooled water in a MCM-41 pore in order to test the mode coupling theory. A layer analysis must be performed for water in the pore in order to exclude the contribution of water bound to the strongly hydrophilic surface. Upon supercooling a range of temperatures is reached where the liquid follows the mode coupling theory. From the power law behavior of the relaxation times extracted from the Kohlrausch-William-Watts fit to the self-intermediate scattering function, we obtain the crossover temperature T(C) and the γ exponent of the theory. The time-temperature superposition principle is also satisfied. A fit to the von Schweidler law yields a coefficient b from which all the other parameters of the theory have been calculated. In particular, we obtained the same value of γ as extracted from the power law fit to the relaxation times, in agreement with the requirements of the theory. For very low temperatures, the mode coupling theory no longer holds as hopping processes intervene and water turns its behavior to that of a strong liquid.     

Colloidal glass transition: beyond mode-coupling theory

A new theory for the dynamics of concentrated colloidal suspensions and the colloidal glass transition is proposed. The starting point is the memory function representation of the density correlation function. The memory function can be expressed in terms of a time-dependent pair-density correlation function. An exact, formal equation of motion for this function is derived and a factorization approximation is applied to its evolution operator. In this way a closed set of equations for the density correlation function and the memory function is obtained. The theory predicts an ergodicity breaking transition similar to that predicted by mode-coupling theory, but at a higher density.     

The synergetic theory of the glass transition in liquids

The glass transition is treated as a spontaneous emergence of the shear components of strain and stress elastic fields upon cooling a liquid at a rate exceeding the critical value. The stationary elastic strains and stresses and the effective relaxation time are determined within the adiabatic approximation. It is shown that the glass transition process occurs through the mechanism of a first-order kinetic transition with allowance made for the strain dependence of the shear modulus. The critical cooling rate turns out to be proportional to the thermal diffusivity and unrelaxed shear modulus and inversely proportional to the temperature derivative of the relaxed shear modulus and the square of the heat conductivity length of the sample.     

Systematic field theory of the RNA glass transition

We prove that the L?ssig-Wiese (LW) field theory for the freezing transition of the secondary structure of random RNA is renormalizable to all orders in perturbation theory. The proof relies on a formulation of the model in terms of random walks and on the use of the multilocal operator product expansion. Renormalizability allows us to work in the simpler scheme of open polymers, and to obtain the critical exponents at 2-loop order. It also allows us to prove some exact exponent identities, conjectured by LW.     

Critical slowing down exponents of mode coupling theory

A method is provided to compute the exponent parameter λ yielding the dynamic exponents of critical slowing down in mode coupling theory. It is independent from the dynamic approach and based on the formulation of an effective static field theory. Expressions of λ in terms of third order coefficients of the action expansion or, equivalently, in terms of six point cumulants are provided. Applications are reported to a number of mean-field models: with hard and soft variables and both fully connected and dilute interactions. Comparisons with existing results for the Potts glass model, the random orthogonal model, hard and soft-spin Sherrington-Kirkpatrick, and p-spin models are presented.     

Role of nonadjacent mode coupling in the theory of microbending

The error caused by the neglect of coupling between nonadjacent modes in the theory of microbending is calculated by comparison with the exact result. This comparison is made for two-dimensional fibers that have a power-law index profiles and various curvature spectra. For power-law spectra, closed from expressions for the microbending losses are given.     

High-order mode-coupling theory for the colloidal glass transition

A theoretical approach is developed to derive a hierarchy of mode-coupling equations for the dynamics of concentrated colloidal suspensions, which improves the prediction of the colloidal glass transition. Our derivation is based on a matrix formalism for stochastic dynamics and the resulting recursive expressions for irreducible memory functions. The 1st order truncation of the generalized mode-coupling closure recovers mode-coupling theory, whereas its 2nd and 3rd order truncations provide corrections. The predictions of the transition volume fraction and Debye-Waller parameter for the hard-sphere colloidal system improve with the increasing mode-coupling order and compare favorably with experimental measurements.     

Diffusion in a metallic melt at the critical temperature of mode coupling theory

According to mode coupling theory, liquidlike motion becomes frozen at a critical temperature T(c) well above the caloric glass transition temperature T(g). Here, for the first time, we report on radiotracer diffusion in a supercooled Pd43Cu27Ni10P20 alloy from T(g) to the equilibrium melt. Liquidlike motion is seen to set in exactly above T(c) as evidenced by a gradual drop of the effective activation energy. This strongly supports the mode coupling scenario. Isotope effect measurements, which have never been carried out near T(c) in any material, show atomic transport up to the equilibrium melt to be far away from the hydrodynamic regime of uncorrelated binary collisions.     

Absence of a “ferromagnet-spin glass” reentrant phase transition in quasi-two-dimensional ferromagnetic systems with competing exchange interactions

It is shown on the basis of the results of magnetic investigations for the example of the intercalated layered compounds Cr_1/3−x Ni_xTaS₂ that in quasi-two-dimensional ferromagnets with competing exchange interactions there is no reentrant “ferromagnet-spin glass” phase transition all the way down to liquid-helium temperatures. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 2, 155–158 (25 January 1997)     

A statistical-mechanical theory of slow dynamics near the glass transition

A statistical-mechanical theory of slow dynamics near the glass transition in two kinds of glass-forming systems, (M) molecular systems and (S) suspensions of colloids, is presented from a unified point of view based on the Tokuyama-Mori projection operator method. The exact diffusion equations for the coherent- and the incoherent-intermediate scattering functions are first derived, whose memory functions are convolutionless in time and contain the correlation effects due to the hydrodynamic interactions in (S). The analytic expressions of the memory functions are then calculated within the mode-coupling theory (MCT) approximation and are shown to coincide with the conventional ones obtained by MCT. Alternative mode-coupling equations are thus obtained in (M) and (S) separately. Self-diffusion is also discussed. Alternative equations for the mean-square displacement and the non-Gaussian parameter are also derived within MCT approximation. All results in both the systems are compared with those obtained by MCT.