# Structural predictor for nonlinear sheared dynamics in simple glass-forming liquids

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Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved November 13, 2017 (received for review June 28, 2017)

## Significance

Fluidity is the key dynamical physical property of the liquid state and is characterized by the transport coefficient, viscosity. For slow flow, the viscosity of a liquid is generally independent of the flow rate. For fast flow, however, the behavior can be much more complicated; for example, supercooled liquids are known to exhibit a drastic decrease in viscosity under high flow rates, thus flowing more easily and dissipating less energy. Despite its fundamental and industrial importance, the physical mechanism behind this phenomenon known as “shear thinning” remains unknown. Here, we show that shear thinning is characterized only by the flow-induced change of the liquid structure along the extensional direction of the flow, providing a hint at understanding this long-standing unresolved problem.

## Abstract

Glass-forming liquids subjected to sufficiently strong shear universally exhibit striking nonlinear behavior; for example, a power-law decrease of the viscosity with increasing shear rate. This phenomenon has attracted considerable attention over the years from both fundamental and applicational viewpoints. However, the out-of-equilibrium and nonlinear nature of sheared fluids have made theoretical understanding of this phenomenon very challenging and thus slower to progress. We find here that the structural relaxation time as a function of the two-body excess entropy, calculated for the extensional axis of the shear flow, collapses onto the corresponding equilibrium curve for a wide range of pair potentials ranging from harsh repulsive to soft and finite. This two-body excess entropy collapse provides a powerful approach to predicting the dynamics of nonequilibrium liquids from their equilibrium counterparts. Furthermore, the two-body excess entropy scaling suggests that sheared dynamics is controlled purely by the liquid structure captured in the form of the two-body excess entropy along the extensional direction, shedding light on the perplexing mechanism behind shear thinning.

Liquids displaying slow dynamics, such as glass-forming liquids, exhibit striking nonlinear behavior known as shear thinning if subjected to sufficiently strong shear and the behavior is characterized by a power-law decrease of the viscosity η with increasing shear rate

There exist at least two standard approaches connecting

The second approach is of semiempirical origin connecting dimensionless (or reduced) transport coefficients to the excess entropy

Dzugutov argued in his original paper that this kind of structure–dynamics relationship should arise from a proportionality between the frequency of local structural relaxation and the number of accessible configurations, the latter being reduced by

The semiempirical scalings have more recently, when interpreted more general than an exponential relationship, been explained using the isomorph theory for Roskilde-simple (RS) liquids (24, 27⇓⇓–30). Briefly, RS liquids have isomorphs in the phase diagram that are invariance curves of structure and dynamics in dimensionless units. Reduced transport coefficients

For simple spherical particle fluids, the two-body excess entropy is found to account for more than 90% of the total excess entropy (31, 32). The two-body excess entropy scaling has found widespread use in predicting the dynamics of, for example, metallic and hydrogen-bonding liquids such as water (23, 33⇓⇓⇓–37). From a practical point of view, it is easier to calculate the two-body excess entropy requiring only knowledge of

Previously, we showed that the two-body excess entropy is also a useful measure of glassy structural order associated with slow dynamics in glass-forming liquids (38⇓⇓⇓⇓–43), although it cannot directly identify local structural order responsible for glassy dynamics without time averaging due to the lack of information on the many-body correlations (44). Intuitively, this appears natural since a large value of

The key question is thus whether or not the two-body entropy scaling can be extended to nonequilibrium sheared fluids. Krekelberg et al. (45, 46) explored this topic in previous studies but did not find a common relationship for the structural relaxation time

In this report, we study which structural features are responsible for acceleration of dynamics under shear based on the two-body excess entropy scaling that connects dynamics to the two-body structural (excess) entropy. Since

The first half of the paper focuses on a weakly size-polydisperse Weeks–Chandler–Andersen (WCA) system (38, 47) with 8% polydispersity to avoid crystallization. The second half of the paper presents results for an inverse power-law (IPL) *Materials and Methods*.

We use molecular dynamics (MD) computer simulations (51) in the constant particle number, volume, and temperature (NVT) ensemble for probing equilibrium dynamics and SLLOD dynamics with a Gaussian thermostat and Lees–Edwards periodic boundary for simulating Couette shear flow (52). The shear rate is defined by *Materials and Methods*).

Fig. 1 shows the self-part of the intermediate scattering function

Fig. 1 *C* and *D* show the effects of shear on the polydisperse WCA system. We observe shear thinning behavior (Fig. 1*C*) for all densities studied with a change in viscosity η of about three decades at the highest density ρ = 0.830. Similarly, we find in Fig. 1*D* that the angular-averaged RDFs at ρ = 0.830 show visible changes with increasing shear rate tending to wipe out the structural order developing with density in the supercooled liquid (38). The effect of shear on the WCA system is thus highly nontrivial and shows significant nonlinear behavior.

In the spirit of two-body excess entropy scaling, we proceed to consider if the total two-body excess entropy**1**).

Here and in the following, we neglect for simplicity structural anisotropy in the vorticity direction z—that is, *x*-direction (flow direction). As will be illustrated later, this is not a serious simplification but enhances the statistics. Furthermore, previous studies (8, 13) found little anisotropy in

Fig. 2*A* shows the structural relaxation time

As mentioned previously, a system under shear is expected to show anisotropy in *Materials and Methods* for more details). *B* shows

Fig. 2*C* shows

We proceed to explore if the minima/maxima of *B*) are able to rationalize the sheared dynamics better than the total two-body excess entropy *D*–*F* show two-body entropy scaling using *SI Materials and Methods*). Furthermore, a similar scaling holds also for the viscosity η (see *SI Materials and Methods*) and implies that shear flow drastically decreases the structural relaxation time *A* as the integral is dominated by the maxima of

The fact that *C*). We show here that shear thinning is a direct consequence of the emergence of shear-induced structural anisotropy (measured using

Next, we explore whether this two-body excess entropy scaling is a peculiarity of the harsh repulsion of the WCA system (or, say, hard sphere-like systems) or a more general scaling property of sheared liquids. If a general relation can be established, we will gain not only tremendous fundamental insight into nonlinear rheology but also a practical prescription for predicting the dynamics of sheared systems using the equilibrium two-body excess entropy relationship.

Fig. 3 displays *Materials and Methods*). The first three systems show the characteristic two maxima/two minima pattern in the θ dependence of

Fig. 4 shows the two-body entropy scalings of

The latter observation indicates that it is not the property of being the absolute minimum, or the strongest structural disorder, that gives rise to the observed scaling and at the same time also suggests that the extensional direction is a fundamental direction with respect to two-body excess entropy scaling. However, we note that for the GCM the extensional direction is apparently not unique in the sense that it is not the absolute minimum of

These observations thus support the intuitive argument that the extensional direction is the key direction providing mobility, or the reduction in the viscosity. In other words, only the increase in structural disorder along this direction, linked to weaker constraints on particle motion, is responsible for enhanced dynamics under shear.

Additionally, the collapse of equilibrium–nonequilbrium data suggests that an effective temperature representation of the structural relaxation time (or viscosity, see *SI Materials and Methods*) for sheared fluids is possible (see, e.g., refs. 57⇓–59) even though nonaffine four-point correlation functions exhibit strong anisotropy (13). The latter fact implies that the nature of the relationship between structure and dynamics is fundamentally different for the quiescent and sheared states, but the value of the viscosity (or relaxation time) can still be determined solely by

We emphasize that identical values of *A*). Identical values of **1** and **2** beyond the third shell (see Fig. 5*B*), indicating that mobility may be controlled by the structural features captured in the form of

Next, we speculate on the relationship between the isotropic nature of the structural relaxation and the anisotropic two-body structural entropy. Structural relaxation is characterized by the average escape time of a particle from its cage. The decay of density fluctuations around the interparticle distance is isotropic and independent of θ under shear (8, 13). Here, it is worth noting that the shape of the cage itself becomes anisotropic in the shear-thinning regime, but the cage structure is not enough to determine the structural relaxation rate. This is evident from the fact that the integration of *B*). In other words, the structural relaxation in a deeply supercooled liquid is not determined locally and reflects the anisotropy in a mesoscopic length scale beyond the particle size. This suggests that both larger spacing between neighboring particles along the extensional direction and its spatial organization may be a key in determining the structural relaxation, and this relaxation channel is effective for the relaxation of density fluctuations in any direction. This point needs further study in connection with the microscopic mechanism behind shear thinning.

Thus far, we have shown validity of the equilibrium–nonequilibrium mapping for various potentials. WCA and IPL are hard and soft isotropic repulsive potentials, respectively. GCM is an isotropic repulsive potential with a finite value at r = 0, KABLJ is a binary mixture interacting with isotropic attractive potentials, and the Dzugutov potential is an isotropic potential favoring icosahedral structure. The wide variation in the pair potentials studied here indicates a general validity of our findings.

We find, however, also that *SI Materials and Methods*). The observed breakdown may be a natural consequence of the fact that the liquid structure can be perturbed by shear in a nonmonotonic manner in these types of liquids. We therefore argue that *SI Materials and Methods*), where only one of the two length scales effectively dominates, shows the same simple

To conclude, we find that the two-body excess entropy to a very good approximation rationalizes nonlinear sheared dynamics when the extensional direction is used. Future studies should focus on unraveling the microscopic mechanism behind this observation. The results presented here indicate the importance of the liquid structure along the extensional direction in determining the viscosity under shear. We note that although the structure along the extensional direction captured in the form of *SI Materials and Methods*). Another important point is that

An interesting direction for future work is whether or not realistic molecules can be included in the observed scaling approach. We remain very positive on this aspect as, for instance, the GCM fluid is a simplified model for polymers below their entanglement point. Nevertheless, the internal degrees of freedom of the molecules may couple in a nontrivial manner to the two-body excess entropy, and it thus remains to be shown. One advantage of the two-body entropy approach detailed here is that it may be directly verified in experiments as only the angular-resolved RDF is needed under shear.

Thus, we have shown that the degree of structural disorder, measured in the form of the two-body excess entropy along the extensional direction, is a fundamental structural predictor for sheared dynamics or more specifically the structural relaxation under shear. An open question is the microscopic mechanism behind this nontrivial relation, implying that the structure along the extensional direction dominates the structural relaxation.

## Materials and Methods

### Systems Studied.

Five different 3D systems are studied in the main text: size polydisperse WCA fluid (38, 47), IPL *SI Materials and Methods*, we also present results for the core soft water (CSW) model, which has a ramp-like pair potential and thus two length scales.

System sizes used in the simulations (see below for simulation details) are, respectively, N = 10,976, 10,976, 10,976, 10,000, and 4,096 for WCA, IPL, Dzugutov, KABLJ, and GCM (N = 1,024 for CSW; see *SI Materials and Methods*). We apply natural reduced units (e.g., σ = 1 and ϵ = 1) for each system when reporting data in this paper. Furthermore, a single-component analysis is applied when calculating the two-body excess entropy and structural relaxation time for the WCA fluid and KABLJ mixture.

### Simulation Methods.

MD computer simulations in the NVT ensemble (Nose–Hoover dynamics) are used to study equilibrium dynamics, and SLLOD equations of motion combined with Lees–Edwards periodic boundary and a Gaussian thermostat are used for sheared dynamics applying the RUMD package (51). The SLLOD equations of motion realize Couette shear flow where

### Calculation Details.

The viscosity is calculated from η = *xy* component of the configurational stress tensor. The angular-dependent two-body excess entropy **2** is performed for each bin

The self-part of the intermediate scattering function under shear is given by **x** is a unit vector in the x direction. This expression has been shown to approximate the full integral well and does not require extensive bookkeeping (13). The structural relaxation time under shear is defined similarly as **q**_{p}, we average only the transverse wave vectors when calculating

## Acknowledgments

We are grateful to Jeppe C. Dyre for stimulating discussions. This work was partially supported by Grants-in-Aid for Specially Promoted Research (25000002) from the Japan Society of the Promotion of Science.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: tanaka{at}iis.u-tokyo.ac.jp or trond{at}iis.u-tokyo.ac.jp.

Author contributions: H.T. designed research; T.S.I. performed research; T.S.I. and H.T. analyzed data; and T.S.I. and H.T. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1711655115/-/DCSupplemental.

Published under the PNAS license.

## References

- ↵
- Li JH,
- Uhlmann DR

- ↵
- Cross MM

- ↵
- ↵
- Hanley HJM,
- Rainwater JC,
- Clark NA,
- Ackerson BJ

- ↵
- Evans DJ,
- Hanley HJM,
- Hess S

- ↵
- ↵
- ↵
- Yamamoto R,
- Onuki A

- ↵
- Bair S,
- McCabe C,
- Cummings PT

- ↵
- ↵
- ↵
- Miyazaki K,
- Reichman DR

- ↵
- ↵
- Lubchenko V

- ↵
- Cheng X,
- McCoy JH,
- Israelachvili JN,
- Cohen I

- ↵
- Zhu W,
- Aitken BG,
- Sen S

- ↵
- Miyazaki K,
- Reichman DR,
- Yamamoto R

- ↵
- ↵
- Rosenfeld Y

- ↵
- Rosenfeld Y

- ↵
- Dzugutov M

- ↵
- Wallace DC

- ↵
- ↵
- Ingebrigtsen TS,
- Errington JR,
- Truskett TM,
- Dyre JC

- ↵
- ↵
- Hoover WG

- ↵
- ↵
- Ingebrigtsen TS,
- Schrøder TB,
- Dyre JC

- ↵
- Bacher AK,
- Dyre JC

- ↵
- Bacher AK,
- Schrøder TB,
- Dyre JC

- ↵
- ↵
- Joy A

- ↵
- ↵
- Chakraborty SN,
- Chakravarty C

- ↵
- Fomin YD,
- Ryzhov VN,
- Gribova NV

- ↵
- Gallo P,
- Rovere M

- ↵
- Mishra RK,
- Lalneihpuii R

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Krekelberg WP,
- Ganesan V,
- Truskett TM

- ↵
- Krekelberg WP,
- Ganesan V,
- Truskett TM

- ↵
- ↵
- ↵
- Kob W,
- Andersen HC

- ↵
- ↵
- Bailey NP, et al.

- ↵
- ↵
- Delhommelle J

- ↵
- Evans DJ,
- Morriss G

- ↵
- Ding Y,
- Mittal J

- ↵
- Furukawa A

- ↵
- Berthier L,
- Barrat JL,
- Kurchan J

- ↵
- ↵
- Schroer CFE,
- Heuer A

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