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 Systems Biology
Shear thinning in deeply supercooled melts

Edited by Peter G. Wolynes, University of California at San Diego, La Jolla, CA, and approved May 7, 2009 (received for review January 23, 2009)
Abstract
We compute, on a molecular basis, the viscosity of a deeply supercooled liquid at high shear rates. The viscosity is shown to decrease at growing shear rates, owing to an increase in the structural relaxation rate as caused by the shear. The onset of this nonNewtonian behavior is predicted to occur universally at a shear rate significantly lower than the typical structural relaxation rate, by approximately two orders of magnitude. This results from a large size—up to several hundred atoms—of the cooperative rearrangements responsible for mass transport in supercooled liquids and the smallness of individual molecular displacements during the cooperative rearrangements. We predict that the liquid will break down at shear rates such that the viscosity drops by approximately a factor of 30 below its Newtonian value. These phenomena are predicted to be independent of the liquid's fragility. In contrast, the degree of nonexponentiality and violation of the Stokes–Einstein law, which are more prominent in fragile substances, will be suppressed by shear. The present results are in agreement with existing measurements of shear thinning in silicate melts.
NonNewtonian rheological response of complex fluids undergirds many industries and laboratory procedures, yet because of the extraordinarily broad dynamic range exhibited by these systems, securing microscopic insight—either by direct modeling (1–5) or phenomenological treatments (6–10)—has been difficult. This extraordinary breadth comes about for two somewhat distinct reasons: (i) the broad distribution of lengths of moving units—as in polymers or mixtures of granular and fluid matter—which we may call “builtin heterogeneity”; (ii) the kinetic arrests/slowdowns emerging in otherwise homogeneous systems, owing to cooperative effects at high densities. To begin constructing a systematic microscopic description of nonNewtonian response of complex matter, here, we consider deeply supercooled liquids, which exhibit only the emergent slowdown arising from cooperative, activated transport.
By properly accounting for the selfgenerated cooperativity, we will rationalize the otherwise surprising fact that the onset of shear thinning in supercooled liquids occurs at shear rates that are at least two orders of magnitude lower than the typical structural relaxation rate (11). In addition, we will compute the functional dependence of the viscosity on the shear rate.
Before proceeding with the detailed argument, it is instructive to outline the mechanism of shear thinning in deeply supercooled liquids in qualitative terms, based on several key aspects of liquid dynamics in the activated regime established by the Random FirstOrder Transition (RFOT) theory (12): The high viscosity in supercooled melts arises because transient structures form, around each atom, that last longer than several hundred vibrations (13–15), thus rendering momentum transfer almost as efficient as in a fully stable solid. Consistent with this view, the viscosity of a supercooled fluid is proportional to the average structural relaxation time (16), and grows dramatically with lowering the temperature. The relaxation itself occurs through activated reconfiguration of compact regions of modest size—only a few molecular diameters across—while the individual displacements are small—about one tenth of the molecular size (14). (These reconfigurations are responsible for mass transport.) To accommodate the shear, the immediate environment of the compact region will relax more frequently than is typical in the unperturbed state. This is equivalent to the region itself relaxing more frequently than is typical in the unperturbed state. As just mentioned, a shorter structural relaxation time implies a lower viscosity, and hence will lead to shear thinning. The magnitude of this effect is determined by the nearly universal spatial characteristics of the cooperative reconfigurations, quantitatively established by the RFOT theory, and the shear rate itself. No new motions are introduced by the shear, so long as the dynamics are mainly activated. Thus, independent of chemical detail, all deeply supercooled nonpolymeric fluids will exhibit simple shear thinning, not the broader variety of nonNewtonian response often observed in complex fluids. Although the magnitude of shear thinning itself is predicted to be largely independent of the liquid's fragility, several anomalous features of the supercooled state that are particularly prominent in fragile substances will be be partially suppressed by shear, including the nonexponentiality of relaxations (17) and violation of the Stokes–Einstein relation (18).
Microscopic Framework
We will build the microscopic picture of the supercooledliquid response to high shear by using the RFOT methodology [(14, 19, 20); see ref. 12 for a review]. This framework has yielded predictions or quantitative explanations of the signature phenomena associated with the structural glass transition. These include the connection between the thermodynamic crisis and the diverging relaxation times (14, 17, 19), the length scale of the cooperative rearrangements (14), deviations from Stokes–Einstein hydrodynamics (21), aging (20), crossover between activated and collisional transport (15, 22), the cryogenic anomalies in glasses (23–25), and decoupling between momentum and charge transfer (16). Several other, distinct formalisms have yielded qualitatively similar conclusions on the liquid dynamics near the glass transition (26–32).
Molecular motions in deeply supercooled melts and glasses are cooperative so that transporting a chemically rigid molecular unit (“bead”) requires concurrent rearrangement of a large number N* = (ξ/a)^{3} of surrounding beads, where ξ is the volumetric size of the cooperative region and a is the volumetric size of the bead (see diagram of a cooperative transition in Fig. 1). The cooperativity size N* has been computed (14) by using the RFOT methodology (14, 19, 20). It is equal to ≈20 at the temperature T _{c} of the crossover between collisional and activated transport (15) and grows to ≈200 near the glass transition temperature T _{g} (14). At the same time, the relaxation time of a cooperative transition grows by 12 orders of magnitude or so, according to a Vogel–Fulcherlike law (14). Here, F is the corresponding barrier, and τ_{0} is a molecular timescale, roughly a picosecond (33). Note that the RFOT theory is quantitative below T _{c}, when the transport is mostly activated, i.e., at viscosities >10 Poise or so (15). [At higher temperatures, transport is dominated by collisional effects (15).] Note also that the cooperativity size characterizes the length scale of the “dynamic heterogeneity.” The value of this length and its growth with lowering temperature, as predicted by the RFOT theory, have been confirmed by a number of distinct experimental techniques (34–36), also combined with detailed analyses of 4point dynamic correlations (37–39).
Because each region of size ξ with fixed boundaries is guaranteed to make a structural transition, typically, within time τ, one may speak of a “mosaic” of cooperative rearrangements (14). Note that a volume V > ξ^{3}, with fixed boundaries, will thus undergo a relaxation event typically on a timescale The relaxation times τ are actually distributed (17), and especially so in fragile liquids, leading to nonexponential relaxation (17), deviations from Stokes–Einstein hydrodynamics (21), and “decoupling” between momentum and charge transfer (16). In the course of a cooperative transition, the beads move sequentially, each by the Lindemann displacement at the mechanical stability edge, equal to onetenth of the molecular spacing: d _{L} ≃ a/10.
The Newtonian viscosity in the deeply supercooled regime is directly related to the average relaxation time and the Lindemann ratio. A detailed argument shows that (16): since the Lindemann ratio, d _{L}/a ≃ 0.1, changes at most by 10% within the relevant temperature range (33). The averaging is with respect to the barrier distribution. The result from Eq. 3 is consistent with the heuristic Maxwelltype expression: η ∼ K _{∞}〈τ〉_{F}, where K _{∞} is the highfrequency elastic constant of the aperiodic lattice, because of equipartition: k _{B}T ≃ K _{∞}(d _{L}/a)^{2} a ^{3}.
Results
A review of the detailed argument (16) that led to Eq. 3 shows that it is also valid in the presence of external shear, which will act solely to modify 〈τ〉_{F}. The shear imposes additional relaxation in the immediate vicinity of each domain of size N*, which results in renewing the domain's environment and thereby provides an additional source of relaxation of the domain itself. Thus, the survival probability of the domain relaxation is the product of the survival probabilities of (i) the domain relaxation in the absence of shear and (ii) the boundary relaxation due to the shear. The shearinduced boundary relaxation is generally nonexponential and spatially heterogeneous, reflecting of the said distribution of barriers. Let us denote the survival probability of this excess relaxation as where we include the possibility that the probability distribution ψ(λ, r) is space dependent. It should be recognized immediately, however, that this distribution is significantly more narrow and spatially homogeneous than the intrinsic barrier distribution in the fluid. This is because each cooperative domain borders with a roughly spherical shell encompassing ≈30 regions that could relax, see Eq. 2 and Fig. 1, and hence samples a significantly averaged out relaxation rate.
In contrast with the externally imposed relaxation, the intrinsic liquid relaxation time at the boundary is approximately 〈τ〉_{F}/27, according to Eq. 2, because a thin spherical layer encompassing a volume ξ^{3} (the shaded region in Fig. 1) will relax every time a concentric sphere of volume ≃ (3ξ)^{3} = 27ξ^{3} relaxes (see Fig. 1). The average survival probability of the intrinsic relaxation of any domain of size N*, in the absence of the shearinduced relaxation, is p(t) = 〈e ^{−t/τ}〉_{F} (the averaging is over the barrier distribution). The survival probability of the shearinduced boundary relaxation is, again, p _{ex}(t) from Eq. 4. The full survival probability of the current state of a domain, in the presence of externally imposed shear, is the product of the two survival probabilities, as already mentioned: Here, F and λ in the subscript signify averaging with respect to the combined distribution of the two quantities, which, note, are generally correlated. The average waiting time, modified for the shear, is thus leading to a lowered viscosity:
We may use this result to compute the viscosity as a function of the actual shear, ε̇, after we connect the latter with the rate field λ. To infer the connection, we equate the rate of energy influx through the domain boundary with the energy dissipation within the domain (ref. 40, chapter 16) whereby we apply the Gauss theorem to the velocity field v : ∫ ∇v ^{2} d S = 2 ∫(∂v _{k}/∂x _{i})^{2} dV. Here, the two integrals are over the surface and volume, respectively. Clearly, (∂v _{k}/∂x _{i})^{2} = ε̇^{2}, thence 2 ∫(∂v _{k}/∂x _{i})^{2} dV = 2ε̇^{2}ξ^{3}. To estimate the surface integral, we note that energy is transferred into the domain only after both the interface region and the domain have relaxed, implying the relaxation time of energy transfer is the sum of the lifetimes of the two processes: τ_{b} = λ^{−1} + τ_{λ}. The displacements at the boundary are smaller than in the domain bulk, and are approximately equal to (a/ξ)d _{L} (23, 25), leading to v ^{2} = [(a/ξ)d _{L}/τ_{b}]^{2}. The gradient obviously does not exceed 1/a, whereas the domain surface area is ξ^{2}, leading to the following approximate estimate of the surface integral: d _{L} ^{2} a/τ_{b} ^{2} = d _{L} ^{2} a/(λ^{−1} + τ_{λ})^{2}, where τ_{λ} is from Eq. 7. Note that the energy influx (40), η ∫ ∇v ^{2} d S, thus turns out not to depend explicitly on the domain size ξ, consistent with the coupling of the cooperative transitions to phonons being independent of ξ (23, 25). This is expected because a single domain is the smallest relaxing unit in the liquid (essentially a twostate system) and its entropy production should depend explicitly only on the relaxation rate and the degree of deviation from equilibrium. Equating the two integrals and averaging over the barrier distribution yields: within a factor of 2 or so. Since the denominator in the lefthand side only varies between 1 and 2, we observe that the relaxation rate at the boundary exceeds the shear rate essentially by a product of two purely geometric factors reflecting the universal Lindemann ratio a/d _{L} ≃ 10 and the size of the cooperative domain: (ξ/a)^{3/2}, the latter varying between about 4 and 14 in the deeply supercooled regime. (See the T dependencies of ξ for several specific substances covering the full strongtofragile range in ref. 15.)
Note that either at small or large values of λ, the latter is strictly proportional to the bulk shear rate ε̇, regardless of the detailed distribution of λ or its possible correlation with τ. Specifically,
Even at the intermediate values of λ, the knowledge of the detailed λ distribution is not essential, because, again, the denominator in the lefthand side only varies between 1 and 2. This implies that even if the λ field breaks up into a heterogeneous pattern, the width of the corresponding distribution is necessarily less than onehalf of the average value. This average value of λ can thus be estimated accurately by using Eq. 8 while assuming that the λ distribution is narrow (see also below). The resulting potential error is within the accuracy of the argument and the scatter of experimental data. Indeed, averaging with respect to the specific barrier distribution in supercooled melts adopted in ref. 16 yields the result shown with two thick lines in Fig. 2, along with experimental data for Na_{2} Si_{4} O_{9} and Na_{2} Si_{2} O_{5} (11). The bottom and top curves correspond to the low T ((ξ/a)^{3/2}(a/d _{L}) = 140) and high T ((ξ/a)^{3/2}(a/d _{L}) = 40) limits of the geometric prefactor from Eq. 8, respectively.
In view of the aforementioned nearlinear relation between λ and ε̇ from Eq. 8, i.e.,
The sheardriven decrease in relaxation times, Eqs. 5 and 6, implies lower effective barriers, by virtue of Eq. 1: where F _{λ,max} is the highest possible effective barrier: see the Fig. 3 Inset. It is straightforward to show that the distribution ψ_{λ} of the shearmodified barriers from Eq. 10 is related to the unperturbed barrier distribution ψ according to: where F is understood as a function of F _{λ}via Eq. 10. Both the unperturbed and shearmodified barrier distributions (for a fragile substance near the glass transition) are shown in Fig. 3 for a particular value of λ.
The significant breadth of the barrier distribution in supercooled liquids, and especially so in fragile substances (17), is responsible for several anomalies, including nonexponentiality of relaxations (17), violation of the Stokes–Einstein relation (21), and the decoupling of the momentum and ionic transport (16). The sheardriven decrease of the width of the effective barrier distribution, as in Fig. 3, implies that shear will partially suppress those anomalies. A simple dimensionless quantity that characterizes the barrier distribution breadth and its effects is the product 〈τ〉〈τ^{−1}〉, which directly reflects the decoupling and also exhibits onetoone correspondence with the exponent β of the stretchedexponential relaxation e ^{−(t/t0)β} (16). The quantity is plotted in Fig. 4 as a function of ε̇ just above T _{g}, for three values of the barrier distribution width covering the currently known fragility range. Several predictions are contained in Fig. 4. First, we observe that the a degree of decoupling and violation of the Stokes–Einstein relation will decrease at larger shear rates. Second, the relaxation will become more exponential at largerε̇ (or λ). For instance, at rates ε̇ ≃ 0.2〈τ〉^{−1}, a fragile liquid with the exponent β = 0.40, will behave as though it were moderately strong, amounting to an effective β ≃ 0.56.
Discussion
Eqs. 7 and 8 reveal the microscopic origin of shear thinning and its early onset in the deeply supercooled regime. Shear induces additional relaxation of the environment of the compact reconfiguring regions and thus facilitates the motions that correspond to the high barrier side of the unperturbed relaxation spectrum. The rate of the excess, shearinduced relaxation is amplified by about two orders of magnitude, relative to the bulk shear rate ε̇, owing to the smallness of individual displacements during structural rearrangements and the relatively large cooperativity size.
According to Eq. 7 and Fig. 3, the shearthinned viscosity is dominated by the slow end of the barrier distribution, and is therefore practically insensitive to the detailed shape of the distribution. This is consistent with the Newtonian viscosity—which is strictly proportional to 〈τ〉 —being also dominated by slow regions, but is in contrast to the ionic conductivity, for instance, which is realized primarily by the fast regions (16). Because the fragility of a liquid is determined by the width of the barrier distribution (17), the theory predicts that shear thinning should be similar in all supercooled nonpolymeric liquids regardless of their fragility.
In discussing the temperature dependence of the thinning, we note that the experimental data cluster near the low T curve in Fig. 2, which is consistent with the data being taken near the glass transition. The upper curve is a prediction of the present theory near the temperatures of the crossover between activated and collisional transport. We note that in this latter regime, which is dominated by mode coupling effects, Miyazaki at el. (8) report general trends of shear thinning similar to our results. Berthier and Barrat (5) find similar results for shear thinning in simulations of undercooled Lennard–Jones liquids, consistent with the smoothness of the crossover between activated and collisional transport in supercooled liquids (15).
According to Eq. 5, shear shifts uniformly the spectrum of effective relaxation rates of rearranging domain, toward faster rates, by the amount λ from Eq. 8 that signifies the excess relaxation rate of the domain's environment caused by the shear. This implies a new, effective barrier distribution, in which all relaxations whose barrier exceeded the critical value F _{λ,max} from Eq. 10 in the absence of shear, now contribute to the effective barrier distribution below F _{λ,max}. At large enough λ, these contributions form a rather narrow peak, whose lefthand side scales as (F _{λ,max} − F _{λ})^{−1}. This motional narrowing will lead to observable consequences. According to Fig. 4, the magnitude of Stokes–Einstein violation, decoupling between momentum and ionic transport, and the degree of nonexponentiality should all decrease in the presence of shear. Of particular interest are experiments on the diffusion of nanoscopic probes, which would test rather directly the present conclusions that shear does not introduce new motions, but merely modifies the relaxational spectrum.
We may estimate the shear rate above which the present microscopic picture would no longer strictly apply, whereas other processes may take place that lead to material failure. We first recognize that the shearinduced relaxation is, of course, realized by the same motions as the intrinsic relaxations. Therefore, λ may not exceed the intrinsic rate at which the interface relaxes, i.e., the mentioned (〈τ〉/27)^{−1}. At higher rates, transport will no longer consist of concerted bond deformations at the mechanical stability edge, but instead must involve bond breaking. Here, the word “bond” is understood in the precise sense of selfconsistent phonon theories (13, 41), and coincides with its common meaning in molecularly bonded systems. The volume fraction of such “defected” regions is roughly the surfacetovolume ratio of a domain, ∼ a/ξ, i.e., >10%. In other words, at relaxation rates λ such that λτ ≳ (27)^{−1} regions that are relatively weakly bonded will proliferate, which may lead to catastrophic failure in certain experimental geometries, such as during fiber elongation. According to Eq. 7, this critical rate corresponds to shear thinning by a factor of 30 or so, and is indicated on Fig. 2. Most of the silicate fibers analyzed in ref. 11 indeed failed at comparable shear rates, buttressing our microscopic picture. In contrast, we note that many complex fluids, and especially polymers (42, 43) often exhibit shear thinning of several orders of magnitude without breaking, consistent with the presence of additional relaxations in these materials, as mentioned in the beginning of the article.
We emphasize that the present approach is entirely microscopic in that it uses a set of objects derived on a firstprinciples basis, whereas all corresponding quantities can be computed or measured for any specific substance. Note that we did not have to specify the precise chemical composition of the beads because the results depend only on dimensionless ratios ξ/a and a/d _{L}. To compute the absolute value of ξ, which may be useful in designing additives, one needs to know the bead size. This size can be determined by calorimetry, especially if augmented by kinetic measurements. Several dozens of such determinations, for specific systems, can be found in refs. 15, 16, and 44.
Finally, measurements on more liquids and with use of broader temperature ranges are needed to fully test the present microscopic picture. (It appears that such measurements are usually done on liquids containing polymers, to which the RFOT and the present theory would not strictly apply.) Webb and Dingell (11) report measurements of shear thinning on 8 distinct silicates at several temperatures, all of which fall on one universal curve after being rescaled by the typical relaxation time. However, all of these compounds are strong liquids, but the temperature range is relatively narrow, stressing the need for further experimental studies.
Acknowledgments
This work was supported in part by the Donors of the American Chemical Society Petroleum Research Fund, the Grants to Enhance and Advance Research Program at the University of Houston, and the Arnold and Mabel Beckman Foundation Beckman Young Investigator Award.
Footnotes
 ^{1}Email: vas{at}uh.edu

Author contributions: V.L. designed research, performed research, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.
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