RESEARCH INTERESTS
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
TOOL-NARAYANASWAMY-MOYNIHAN
("TNM")
and
NONLINEAR
ADAM-GIBBS ("SCHERER-HODGE")
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."