Landscape equivalent of the shoving model

It is shown that the shoving model expression for the average relaxation time of viscous liquids follows largely from a classical"landscape"estimation of barrier heights from curvature at energy minima. The activation energy involves both instantaneous bulk and shear moduli, but the bulk modulus contributes less than 8% to the temperature dependence of the activation energy. This reflects the fact that the physics of the two models are closely related.

It is shown that the shoving model expression for the average relaxation time of viscous liquids follows largely from a classical "landscape" estimation of barrier heights from curvature at energy minima. The activation energy involves both instantaneous bulk and shear moduli, but the bulk modulus contributes less than 8% to the temperature dependence of the activation energy. This reflects the fact that the physics of the two models are closely related. The physics of highly viscous liquids approaching the calorimetric glass transition continue to attract attention [1,2,3,4,5,6,7,8,9]. A major mystery surrounding these liquids is their non-Arrhenius behavior. If τ is the average relaxation time and τ 0 the microscopic time (of order 10 −13 s), the temperature-dependent activation energy ∆E(T ) is defined [10,11,12] by Only few viscous liquids show Arrhenius temperature dependence of the average relaxation time, i.e., have constant ∆E(T ). Most liquids have an activation energy which increases upon cooling.
The "shoving" model [13] starts from the standard picture of a viscous liquid: At high viscosity almost all molecular motion goes into vibrations around potential energy minima. Only rarely do rearrangements take place which move molecules from one to another minimum. This view was formulated already by Kauzmann in his famous 1948-review [14], and it was the starting point of Goldstein's "potential energy picture" [15] which was recently confirmed by computer simulations [16]. On the short time scale of the barrier transition -expected to last just a few picoseconds -the surrounding liquid behaves as a solid with bulk and shear moduli equal to the instantaneous (i.e., high frequency) bulk and shear moduli. Just as in free volume theories the shoving model assumes that molecular rearrangements only take place when a thermal fluctuation leads to extra space being created locally. One may think of the surroundings as being shoved aside, although this cause-effect reasoning violates time-reversal symmetry. If there is spherical symmetry, the surroundings are subject to a pure shear displacement [7,13,17]. Thus the work done is proportional to the instantaneous shear modulus G ∞ and one finds [13] (where V c is by assumption temperature independent) This expression [18] fits data for the non-Arrhenius be-havior of several molecular liquids [13]; in combination with the Tool-Narayanaswami formalism the shoving model has been applied also to structural (i.e., nonlinear) relaxations [19]. It is the short-time shear modulus which appears in Eq. (2) because the transition itself is fast (compare, e.g., the analogous appearance of shorttime friction constant in the Grote-Hynes theory [20]). The basic assumptions of the shoving model may be summarized as follows [7]: • The main contribution to the activation energy is elastic energy.
• The elastic energy is located in the surroundings of the rearranging molecules.
• The elastic energy is shear energy.
The purpose of this note is to give an alternative justification of Eq. (2) and discuss the interrelation between the two approaches. First, we review a classical argument estimating the height of the barrier between two potential energy minima from the curvature around the minima. This argument is used, e.g., in the Marcus theory for electron transfer reactions [21,22]; in 1987 Hall and Wolynes applied this reasoning in their theory of free energy barriers in glasses [23,24,25]. Consider first a one-dimensional situation with two minima the distance 2a apart (Fig.  1). The thin curve is the potential estimated by second order expansions around the minima. In this approximation the barrier height is determined by the curvature of the potential at the minimum: If the potential here is U (x) = (Λ/2)x 2 , the barrier is given by ∆E = (Λ/2)a 2 . Statistical mechanics implies x 2 = k B T /Λ. Therefore, comparing different situations with differing potentials but same distance between minima, the vibrational meansquare x-fluctuation determines the barrier as follows: The basic assumptions are that the minima are at fixed distance and that the parabola approximation gives a good estimate of the barrier height. Actually, the constant of proportionality in Eq. (3) is of no importance and the true barrier may be consistently smaller than estimated -Eq. (3) applies as long as the barrier scales with that estimated by the parabola approximation. How does the above reasoning apply to the multidimensional configuration space where the hills and valleys of the energy landscape live? At low temperatures a viscous liquid thermally "populates" only deep minima in configuration space. A transition between two deep minima most likely consists of a whole sequence of transitions between intermediate shallow minima [3]. Nevertheless, there is one or more bottlenecks in this sequence. Our basic assumption is that at different temperatures the bottleneck transitions are the same type of local rearrangements and thus with (virtually) same distance between the two minima in configuration space. They occur, however, as temperature changes in different surroundings, so the minima involved are different and have differing x 2 . The relevant mean-square x-fluctuation is to be taken in the direction between the minima. The final assumption needed is that this quantity is typical for the minima, i.e., it is equal to the average over all directions. Consequently, the relevant mean-square x-fluctuation may be evaluated simply as the vibrational mean-square fluctuation averaged over all directions. We estimate this quantity by its ensemble average over all minima, x 2 . Note that, since the configuration space average x 2 is equal to the single atom mean square displacement in three dimensions, this quantity is experimentally acces-sible in, e.g., neutron scattering where it enters into the Debye-Waller factor exp(−Q 2 x 2 ) (where Q is scattering vector) [26]. -Recently, Starr, Sastry, Douglas, and Glotzer arrived at Eq. (3) from the free volume perspective [27]. Their work includes a direct numerical confirmation for a glass-forming polymer melt by calculating the free volume v f and subsequently showing, with v f as "mediator," that the temperature dependencies of x 2 and τ are consistent with Eq. (3). The next step is to relate x 2 to the instantaneous bulk and shear moduli. On short time scales a viscous liquid behaves like a solid [14,15,16,28]. In particular, it has well-defined vibrational eigenstates. We assume that the entire phonon spectrum scales with the long wavelength limit of the phonon dispersion relation. For a one-dimensional solid there is only one elastic constant, C. The above Λ is proportional to C and consequently Unfortunately, it is not possible to test Eq. (4) directly because there are no measurements of the instantaneous isothermal bulk modulus. There are several ways to quantify variations in ∆E(T ), a liquid's "fragility" [30].
The standard approach utilizes the quantity m introduced by Plazek, Ngai, Böhmer, and Angell [31]: m = d log 10 (τ )/d(T g /T )| T =Tg where T g is the calorimetric glass transition temperature defined by τ (T g ) = 10 3 s. Simple Arrhenius behavior corresponds to m = 16; most glass-forming liquids have fragilities between 50 and 150. As an alternative Tarjus, Kivelson, and coworkers proposed to measure the degree of non-Arrhenius behavior at any given temperature by the normalized activation energy: ∆E(T )/∆E(T →∞) [12]. It is not obvious a priori, however, that scaling to the high-temperature limit is physically relevant. Inspired by the Grüneisen parameter [26] and Granato's recent work on interstitialcy relations for the viscosity [32] we suggest that a useful unbiased measure of how much activation energy changes with temperature is its logarithmic derivative, d ln ∆E(T )/d ln T . Because activation energy increases as T decreases, it is convenient to change sign. We thus define the temperature index I ∆E of the activation energy by It is straightforward to show that m = 16 1 + I ∆E (T g ) . We proceed to express I ∆E in terms of temperature indices of instantaneous moduli (same definition). First, note that the temperature index I of a sum, f 1 + f 2 , is a convex combination of the temperature indices I j of f j : Substituting Eq. (7) into Eq. (6) leads to where It is straightforward to show [33] that one always has α < 0.08 .
Thus more than 92% of the temperature index of the activation energy derives from the instantaneous shear modulus. The minute influence of the bulk modulus comes about because of three factors: • There are two transverse phonon degrees of freedom, but only one longitudinal.
• Longitudinal phonons are associated with a larger elastic constant than transverse phonons and thus give less than one third contribution to the activation energy (Eq. (4)).
• G ∞ affects also the longitudinal phonons.
We conclude that simplifying [34], but not unreasonable, assumptions in the landscape approach lead to a prediction for the non-Arrhenius behavior which in practice is going to be hard to distinguish from that of the shoving model. The really interesting question is: What is the relation between the physics of the two approaches? At first sight they seem quite different. There are, however, similarities leading us to conclude that the physics are actually closely related: The first shoving model assumption (main contribution to the activation energy is elastic energy) is equivalent to the landscape assumption of a parabolic potential with a curvature which determines the activation energy. The second shoving model assumption (elastic energy is in the surroundings of the reorienting molecules) is consistent with the landscape assumption that the vibrational mean-square fluctuation is typical, because vibrational eigenstates in an isotropic solid involve all atoms. And the final shoving model assumption (elastic energy is mainly shear energy) is consistent with Eq. (10). -Finally, we would like to remind that the original shoving model derivation of Eq.
(2) assumed spherical symmetry [13]. In more realistic scenarios there must be some volume change in the surroundings of the reorienting molecules and thus some contribution to the activation energy from the instantaneous bulk modulus. While it is difficult to give absolute bounds on the magnitude of the bulk contribution, it is noteworthy that Granato -supported by many others (see Ref. [35] and its references) -finds that for defect creation in a crystal the work is overwhelmingly that of a shear deformation. This work was supported by the Danish Natural Science Research Council.