Ian M Hodge


The areas of most interest to me are The Glass Transition and Glassy State Relaxation and Solid Electrolytes.

The Glass Transition and Glassy State Relaxation
Nobel Laureate in solid state physics, P. W. Anderson, has stated (italics added):
(A)  "The spin glass ... requires a whole new version of statistical mechanics. Glass itself remains one of the deepest puzzles of physics". [Physics Today p9 July 1991],
(B) "The deepest and most interesting unsolved problem in solid state physics is probably the theory of the nature of glass and the glass transition". [Science 267 1616 (1995)].

         I am interested in the phenomenology of the kinetics of the glass transition, particularly those aspects that affect enthalpy relaxation and the associated kinetics of DSC scans. My goal is to provide theorists with reliable experimental data for them to explain or interpret. One of the most intriguing properties of the glass transition is the apparent close relationship between its kinetic and thermodynamic aspects, so my interests embrace thermodynamics as well. The thermodynamic aspects of the glass transition are controversial, however, because they are based entirely on extrapolation. My personal view is that if thermodynamics is dismissed as irrelevant, then too large a body of agreement between thermodynamic and kinetic properties must  be dismissed as fortuitous.

Questions that require better answers include:
(1) What is the exact relationship between the kinetics of the glass transition and of glassy state relaxation? It is usually assumed, for very good reason but without quantitative justification, that these processes are the same. The difficulty is that the glass transition is determined by longer time processes and glassy state relaxation by shorter time processes (thus the need to reduce temperature to make their time scale match that of the annealing time), and the issue of whether these are parts of the same distribution is not easily resolved. Stated otherwise, if the short time part of the distribution is incorrect for the glass transition the description of the glass transition will not be compromised, but the description of annealing would be. There is solid evidence that the short time components of the distribution governing the glass transition indeed determine the annealing distribution, because sub-Tg endotherms in annealed glasses with broad distributions for the glass transition are well acounted for [papers (18) and (20) in list of publications].

(2) What is the theoretical basis of the phenomenology? The Nonlinear Adam-Gibbs ("Scherer-Hodge") model is the best extant account, but it is imperfect (vide infra)

(3) How can enthalpy relaxation parameters be determined more accurately? (A preliminary attempt is described in part of reference [41]).

The glass transition has three canonical features ("the three Nons"): Non-Arrhenius Thermal Activation, Nonexponentiality, and Nonlinearity.

Non-Arrhenius Thermal Activation This gives rise to very large effective activation energies for the average relaxation/retardation time. In the supercooled liquid state above the glass transition temperature range, an excellent description is given by the empirical Fulcher equation:

, for which the effective activation energy is . The activation energy is termed "effective" because it often exceeds chemical bond strengths (!). The accepted interpretation of this fact is that many moities must cooperatively rearrange for relaxation to occur.

Nonexponentiality. Usually well described by the "stretched exponential" or "Williams-Watts" (WW) function,  . Nonexponentiality is expected for a cooperative relaxation process, but the details are obscure. 

Nonlinearity. Within the glass transition temperaure range, the Fulcher equation must be generalized to account for non-linearity because when the system falls out of equilibrium. It is indicated by the experimental finding that the relaxation/retardation time for any relaxing property P(t) depends on the value of P - thus can also be expressed as a function of time as P relaxes. Nonlinearity is the focus of my interest. The best extant accounts of  nonlinearity are provided by the "Tool-Narayanaswamy-Moynihan" (TNM) equation

  , and the "Scherer-Hodge" relation


In these equations Tf,p is the fictive  temperature, defined as the temperature at which the nonequilibrium value of property P would be the equiibrium value - thus equilibrium is characterized by . The quantity x is unity for a completely linear process and zero for a totally nonlinear process (for which depends only on Tf, and not on T). I have shown that the parameters in these apparently quite different formalisms are related: and . The kinetic parameter T2 can be identified with the thermodynamic (Kauzmann) temperature TK at which the configurational entropy extraploates to zero. Thus one of the most problematic aspects of  the glass transition kinetics, nonlinearity, can plausibly be related to one of the more controversial issues of glass transition science - its thermodynamic aspects. The two expressions for Tf,p(t), as a function of thermal history T(t) starting at a temperature To at which equilibrium prevails, are



Nonlinearity in both expresiions arises from occurring in the innermost integrand.

Solid Electrolytes
  I am a proponent of the complex electric modulus function M*, defined as the reciprocal of the complex relative permittivity . The average relaxation time for M* defines the time scale for relaxation of the electric field E at constant displacement D, which differs from the average retardation time for that defines the time scale for relaxation of the displacement D at constant electric field E (analogous to the distinction between the frequencies of longitudinal and transverse optical phonons). When combined with the complex resistivity , the electric modulus can be used, inter alia, to obtain detailed information on intergranular impedances that is obscured by traditional complex permittivity and complex conductivity analyses. The complex electric modulus also has the useful property of suppressing high capacitance phenomena, such as electrode polarization and impedances associated with thin layers. Thus M* is particularly suited for the analysis of electrical properties of heterogeneous materials.

  However, M* is controversial and I have co-authored a paper [publication 38] defending it and correcting misleading and incorrect statements in the literature. Essentially, it is difficult to physically distinguish between the electric current and the displacement current for ionic conduuctivity. They can very easily be experimentally separated (quadrature components of the complex conductivity or complex permittivity), but their physical distinction is the issue that is debated.

             The following is distilled from an email debate I have had about the validity of M* (11/20/08). "The displacement current arises from localized ion hopping between adjacent sites - as is well known this gives rise to a Debye dielectric loss. The electric current arises from long range migration of ions, BUT THIS MIGRATION MUST OCCUR BY A SEQUENCE OF THE SAME INDIVIDUAL HOPS THAT PRODUCE THE DISPLACEMENT. CURRENT. The electric and displacement currents can easily be measured separately from the quadrature components of the (equivalent) complex resistivity or complex permittivity , but because of the conundrum just described their physical distinction is not obvious. The position of myself, Moynihan, Ngai, Angell and many others is that there is NO physical distinction - they both arise from ion hopping between sites. This indistinguishability has direct experimental support, namely that fmax for e"(residual) tracks exactly with the limiting low frequency conductivity . They both have identical activation energies for example. Thus the M* formalism is based on the idea that the electric and displacement currents arise from the same process of ionic hopping, and that their indistinguishability is therefore inevitable. There is nothing fundamentally wrong with this position. For example, the Maxwell equation for curl H does not distinguish between the electric and displacement currents. The position of M* advocates is that a separation is fundamentally impossible for ionic conductors.
              The occurrence of the limiting high frequency relativie permittivity, , in the relation beween and the conductivity relaxation time is not at all problematic - polarizability has an obvious influence on ionic mobility . Consider the atomic level version of the Maxwell relaxation time : . If tau is equated to a librational lifetime () and = 10 (a typical value for ionic conductors), then (max) = 9E-14*10/10^-13 = 10 S/m, which is very close to the maximum observed ionic conductivities (beta alumina) of about 3 S/m."