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

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 $\eta$ with increasing shear rate $\dot{\gamma }$ (1–16). Shear thinning is a process relied on heavily for, for example, industrial purposes such as lubrication. Revealing the physical origins behind the nonlinear flow behavior is, however, very challenging due to the nonequilibrium and nonlinear nature of the phenomenon. Nevertheless, it has been recognized for a long time that shear thinning in glass-forming liquids is associated with emergence of anisotropy in the liquid structure as measured, for example, by the radial distribution function (RDF) $g(\text{𝐫})$ (4–7).

There exist at least two standard approaches connecting $g(\text{𝐫})$ to liquid dynamics in the quiescent (equilibrium) state. The first approach is of first principles origin exemplified by mode-coupling and random first-order transition theory. Recently these theories have been extended to account for nonlinear shear-thinning behavior (10, 14, 17, 18) but unfortunately with varying success.

The second approach is of semiempirical origin connecting dimensionless (or reduced) transport coefficients to the excess entropy $S_{ex}$ (over an ideal gas). This scaling approach was first proposed by Rosenfeld (19, 20) and was motivated by hard-sphere perturbation theory providing a quasi-universal relationship for the dimensionless diffusivity as $\stackrel{\sim }{D}=\mathrm{\hspace{0.17em}0.585}\exp [A{S}_{ex}/k_B]$, where $A$ is a constant and $k_B$ is Boltzmann’s constant, for simple atomic liquids. Two decades later, Dzugutov (21) proposed a closely related scaling using an infinite-term expansion (22) of the excess entropy keeping only the two-body term $s_2$ and applying different dimensionless units from Rosenfeld. The two-body approximation provided the relation $\stackrel{\sim }{D}$ = $0.049\exp [s_2/k_B]$. However, for supercooled liquids, the exponential relationships are known to break down as opposed to the scalings themselves (23, 24).

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 $\exp [S_{ex}/k_B]$ (22). Other attempts at explaining these scalings have also been proposed starting from mode-coupling theory or cell-theory models (25, 26).

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 $\stackrel{\sim }{X}$ and excess entropy (or two-body excess entropy) are all invariant along an isomorph, leading to relationships of the type $\stackrel{\sim }{X}$ = $g(S_{ex})$ or $\stackrel{\sim }{X}$ = $h(s_2)$.

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 $g(\text{𝐫})$ rather than calculating the full excess entropy that may require sophisticated free-energy methods (24).

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 $|s_2|$ means low configurational entropy—that is, higher structural order.

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 ${\tau }_{\alpha }$ as a function of the two-body excess entropy in the nonlinear regime for a square-well and single-component Lennard–Jones (LJ) fluid. Breakdown of equilibrium relations in sheared fluids is expected due to their perplexing nonlinear behavior. On the other hand, previous studies by Yamamoto and Onuki (8) observed an approximate common relationship between the bond-breakage time and the spatial correlation length of dynamical heterogeneity (calculated from bond-breakage events) in the quiescent and sheared state for a binary soft-sphere mixture. The observed scaling demonstrates a possible mapping between the equilibrium and nonequilibrium dynamics via the dynamical correlation length. However, later on it was shown that dynamical heterogeneity actually becomes strongly anisotropic in the shear-thinning regime as opposed to isotropic in the quiescent state (13).

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 $g(\text{𝐫})$ becomes anisotropic under shear, this fact must be taken into consideration when extending the two-body entropy scaling to sheared fluids. We show below that the two-body excess entropy is able to rationalize nonlinear sheared dynamics when the extensional direction of the flow is considered. More specifically, we find collapse of ${\tau }_{\alpha }$ (or $\eta$) in the quiescent and sheared states for five different systems when plotted against the two-body excess entropy along the extensional direction. The systems range from harsh repulsive to soft and finite pair potentials, thus representing a wide range of realistic systems. The state points under study range from high temperature fluids to deeply supercooled liquids with a decrease in viscosity of about 3 decades with increasing shear rate, thus venturing far into the nonlinear regime.

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) $r^{-12}$ fluid, the Dzugutov liquid (48), the Kob–Andersen binary Lennard–Jones (KABLJ) mixture (49), and the Gaussian core model (GCM) fluid (50). All systems studied are 3D systems, and further details on the models are given in 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 $\dot{\gamma }$ $\equiv$ $\partial v_x/\partial y$. Here $x$, $y$, and $z$ are the flow, shear gradient, and vorticity direction, respectively. We note that the SLLOD algorithm with a Gaussian thermostat places an upper limit on the shear rates attainable in our simulations as an artificial string phase tends to form (53). Data are reported using the natural reduced units for each system (see also Materials and Methods).

Fig. 1 shows the self-part of the intermediate scattering function $F_s(q_p,t)$ and the corresponding structural relaxation time ${\tau }_{\alpha }$ in panel $A$ and $B$, respectively, for the WCA system over a wide range of densities $\rho$ at fixed temperature $T=\mathrm{\hspace{0.17em}0.025}$. We determine ${\tau }_{\alpha }$ from $F_s(q_{\text{p}},{\tau }_{\alpha })$ = 0.2, where $q_{\text{p}}$ is the first peak wave number of the static structure factor. We observe strong non-Arrhenius behavior of the structural relaxation time with increasing density corresponding to a fragile liquid.

Fig. 1.

Quiescent and sheared properties for a weakly size-polydisperse WCA fluid at $T$ = 0.025. ($A$) Self-part of the intermediate scattering function for several densities $\rho$ in equilibrium. ($B$) Structural relaxation times ${\tau }_{\alpha }$ corresponding to the state points in $A$ displaying strong non-Arrhenius behavior of ${\tau }_{\alpha }$ with increasing density. ($C$) Viscosity $\eta$ as a function of shear rate $\dot{\gamma }$ for $\rho$ = 0.750, 0.800, 0.815, 0.830. The arrows indicate the onset of shear thinning, ${\dot{\gamma }}_{\text{c}}$. ($D$) Angular-averaged RDFs for $\rho$ = 0.830 and shear rates $\dot{\gamma }$ = ${10}^{-7}$, ${10}^{-3}$, and ${10}^{-2}$.

Fig. 1 C and D show the effects of shear on the polydisperse WCA system. We observe shear thinning behavior (Fig. 1C) for all densities studied with a change in viscosity $\eta$ of about three decades at the highest density $\rho$ = 0.830. Similarly, we find in Fig. 1D that the angular-averaged RDFs at $\rho$ = 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

$$s_2\equiv -\rho /2\int \left[g(\text{𝐫})\ln g(\text{𝐫})-(g(\text{𝐫})-1)\right]d\text{𝐫},$$ [1]

is able to rationalize the dynamical effects of shear on the WCA system. As the concept of entropy in the out-of-equilibrium situation is still an active research topic (54), one may think of $s_2$ simply as a structural quantity as it is determined entirely by $g(\text{𝐫})$ (see Eq. 1).

Here and in the following, we neglect for simplicity structural anisotropy in the vorticity direction $z$—that is, $g(\text{𝐫})=g(r,\theta )$, where $\theta$ is defined with respect to the positive 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 $F_s(q,t)$ under shear, which we also confirm for our system, and thus we present only ${\tau }_{\alpha }$ averaged over wave vectors for the transverse directions. We note also that we do not apply the dimensionless units of the original Dzugutov scaling in our two-body entropy scaling; we elect to do this as these units are independent of shear rate.

Fig. 2A shows the structural relaxation time ${\tau }_{\alpha }$ plotted against $s_2$ in the quiescent and sheared states (here and henceforth, we plot against the negative of $s_2$). We do not find a common curve for ${\tau }_{\alpha }$ using $s_2$ but instead a shear rate-dependent quantity—that is, ${\tau }_{\alpha }$ = $f(s_2,\dot{\gamma })$, which is consistent with the observations of previous studies (45, 46, 55). In fact, a rather drastic deviation from the equilibrium curve is observed at all densities with increasing shear rate.

Fig. 2.

Two-body excess entropy scaling. ($A$) ${\tau }_{\alpha }$ vs. total two-body excess entropy $s_2$. ($B$) Angular-dependent $s_2^{\theta }$ as a function of $\theta$ for three shear rates at $\rho$ = 0.830. ($C$) $s_2^{\text{ext}}$, $s_2^{\text{norm}}$, and $s_2^{\text{comp}}$ as functions of shear rate $\dot{\gamma }$ at $\rho$ = 0.815. ($D$) ${\tau }_{\alpha }$ vs. $s_2^{\text{ext}}$. An excellent collapse is observed using $s_2^{\text{ext}}$. ($E$) ${\tau }_{\alpha }$ vs. $s_2^{\text{norm}}$; a poor collapse is observed. ($F$) ${\tau }_{\alpha }$ vs. $s_2^{\text{comp}}$; a poor collapse is observed.

As mentioned previously, a system under shear is expected to show anisotropy in $g(\text{𝐫})$, and we therefore introduce the angular-dependent $s_2^{\theta }$, which is defined from the equation $s_2\equiv {\int }_0^{2\pi }{s}_2^{\theta }d\theta$ (see Materials and Methods for more details). $s_2^{\theta }$ is a measure for how much the interparticle correlations at $\theta$ impact the translational structural order. Fig. 2B shows $s_2^{\theta }$ as a function of $\theta$ for several shear rates at fixed density $\rho$ = 0.830 (when referring to $s_2^{\theta }$, we always multiply with the trivial 2$\pi$ factor). The angular-dependent two-body entropy $s_2^{\theta }$ displays a distinctive two maxima/two minima pattern. The maxima are located around $\theta$ = 0 and $\pi /2$, where flow effects on the structure are minimal, and the minima around $\theta =\pi /4$ and $3\pi /4$. As the shear rate increases, the amplitude in the fluctuations of $s_2^{\theta }$ also increases, whereas its magnitude decreases. We remind that a smaller value of $|s_2^{\theta }|$ means more structural disorder along $\theta$. The minimum around, say, $\pi /4$ deviates slightly from $\theta =\pi /4$ and moves toward smaller $\theta$ values with increasing shear rate. This effect is caused by the antisymmetric part of the strain-rate tensor (the vorticity component), which causes rotation of the primary distorted structure away from $\theta =\pi /4$ (and similarly, $\pi /2$, $3\pi /4$, and $\pi$). Below we show that the minimum value of $s_2^{\theta }$ around $\theta =\pi /4$ is the key quantity in determining the viscosity under shear. We refer hereafter to the extremum of $s_2^{\theta }$ along the extensional ($\sim \pi /4$), normal ($\sim \pi /2$), and compressional ($\sim 3\pi /4$) axes as $s_2^{\text{ext}}$, $s_2^{\text{norm}}$, and $s_2^{\text{comp}}$, respectively.

Fig. 2C shows $s_2^{\text{ext}}$, $s_2^{\text{norm}}$, and $s_2^{\text{comp}}$ as functions of shear rate $\dot{\gamma }$ at $\rho$ = 0.815 where the two-body excess entropy quickly starts to decrease when the shear-thinning regime is encountered and suggests an intimate link between the two-body excess entropy and nonlinear sheared dynamics. A similar sharp decrease of the total $s_2$ in the shear-thinning regime has also been noted in previous studies (45, 46, 55).

We proceed to explore if the minima/maxima of $s_2^{\theta }$ (see Fig. 2B) are able to rationalize the sheared dynamics better than the total two-body excess entropy $s_2$. Fig. 2 D–F show two-body entropy scaling using $s_2^{\theta }$ along the three characteristic directions: $s_2^{\text{ext}}$, $s_2^{\text{norm}}$, and $s_2^{\text{comp}}$. We find an excellent collapse using the extensional direction but not using the other two directions. This observation implies that ${\tau }_{\alpha }$ = $f(s_2^{\text{ext}})$, where the function $f$ is the same in equilibrium and under shear (an identical conclusion is reached for T = 0.200; see SI Materials and Methods). Furthermore, a similar scaling holds also for the viscosity $\eta$ (see SI Materials and Methods) and implies that shear flow drastically decreases the structural relaxation time ${\tau }_{\alpha }$ but not the shear modulus $G$. The poor collapses of $s_2^{\text{norm}}$ and $s_2^{\text{comp}}$ are consistent with the poor collapse of the total two-body entropy $s_2$ in Fig. 2A as the integral is dominated by the maxima of $s_2^{\theta }$.

The fact that $s_2^{\text{ext}}$ rationalizes nonlinear sheared dynamics opens up for a new and promising way to predicting nonequilibrium dynamics from knowledge of the behavior of the system in equilibrium (i.e., ${\tau }_{\alpha }$ vs. $s_2$ in equilibrium). The extensional direction of the shear flow is obviously an important direction for sheared fluids. Intuitively, the extensional deformation induces a structural change, providing more space necessary for particle motion, resulting in the decrease of viscosity. In general, a link between structure and dynamics (or transport) arises from the fact that the frequency of local structural rearrangements is related to the number of accessible configurations, which is closely related to structural entropy. The available configurations under shear (i.e., the degree of structural disorder) are maximized along the extensional direction. Nevertheless, the microscopic mechanism behind this collapse remains to be understood in more detail. As pointed out in refs. 13, 14, 56, the onset of shear thinning ${\dot{\gamma }}_c$ is much slower than $1/{\tau }_{\alpha }$ in the supercooled regime (see also Fig. 1C). We show here that shear thinning is a direct consequence of the emergence of shear-induced structural anisotropy (measured using $s_2^{\text{ext}}$).

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 $s_2^{\theta }$ as a function of $\theta$ for an IPL $r^{-12}$ fluid, the Dzugutov liquid, the KABLJ mixture, and the GCM fluid for three different shear rates and at fixed $\rho$ and $T$. These four systems are chosen to represent a wide variation in the functional shape of their pair potentials (see Materials and Methods). The first three systems show the characteristic two maxima/two minima pattern in the $\theta$ dependence of $s_2^{\theta }$ as was also observed for the WCA system. The GCM, on the other hand, shows a very deep minimum around $\theta =3\pi /4$, instead of $\theta =\pi /4$, which is expected from the finite value of the pair potential at $r$ = 0 and in fact leads to crystallization at high shear rates. Nevertheless, the variation in $s_2^{\theta }$ with $\theta$ indicates that we may be able to observe a similar scaling as for the WCA system for these four distinctive systems.

Fig. 3.

Two-body excess entropy $s_2^{\theta }$ as a function of $\theta$ at several shear rates for four different systems ranging from harsh repulsive to soft and finite. ($A$) IPL $r^{-12}$ fluid, $\rho$ = 0.850 and $T=0.195$. ($B$) Dzugutov liquid, $\rho$ = 0.800 and $T=0.54$. ($C$) KABLJ mixture, $\rho$ = 1.204 and $T=0.45$. ($D$) GCM fluid, $\rho$ = 2.00 and $T=3.20\cdot {10}^{-6}$.

Fig. 4 shows the two-body entropy scalings of ${\tau }_{\alpha }$ for the four systems of Fig. 3. We find in all cases an almost perfect collapse onto the equilibrium curve using $s_2^{\text{ext}}$ but not using $s_2^{\text{norm}}$ or $s_2^{\text{comp}}$. The results are independent of whether the high-temperature liquid is probed or the deeply supercooled liquid with dramatic changes in structural relaxation time upon shearing. Interestingly, for the GCM, the deep minimum along the compressional direction gives a poor collapse, whereas the weaker minimum along the extensional direction provides an almost perfect scaling.

Fig. 4.

Two-body excess entropy scaling $s_2^{\theta }$ of the structural relaxation time ${\tau }_{\alpha }$ in the quiescent and sheared states for the systems of Fig. 3. (A) IPL $r^{-12}$ fluid, $\rho$ = 0.850. ($B$) Dzugutov liquid, $\rho$ = 0.800, 0.950. ($C$) KABLJ mixture, $\rho$ = 1.204. ($D$) GCM fluid, $\rho$ = 2.00. The sheared curves for the GCM at $T=3.2\cdot {10}^{-6}$ have been shifted with $-0.06$ in the x direction. The shift is performed to account for integration error observed at low shear rates, hindering the trivial collapse onto the equilibrium curve for $\dot{\gamma }\to 0$.

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 $s_2^{\theta }$. The peculiarity of the GCM is the finiteness of the repulsive potential at $r$ = 0, which may allow particles to be overlapped upon compression. This may be the origin for the emergence of the absolute minimum around $3\pi /4$, which is the compressional direction. We note that a smaller value of $|s_2|$ means larger loss of structural entropy—that is, stronger structural disorder. Inhomogeneous overlap between particles may be the source of disorder, leading to the decrease of $|s_2|$, but such disorder due to inhomogeneous particle overlap does not provide any extra relaxation channels since particles are still compressed along $\sim 3\pi /4$. Even in such cases, thus, $s_2^{\text{ext}}$ correlates well with the relaxation time since only the disorder along the extensional axis can provide an extra relaxation channel. Thus, we may conclude that a general picture arises for all systems only when considering the extensional direction.

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 $s_2^{\text{ext}}$.

We emphasize that identical values of $s_2^{\text{ext}}$ and $s_2$ do not necessarily imply identical RDFs (see Fig. 5A). Identical values of $s_2^{\text{ext}}$ and $s_2$ are obtained only after performing the integrations of Eqs. 1 and 2 beyond the third shell (see Fig. 5B), indicating that mobility may be controlled by the structural features captured in the form of $s_2^{\text{ext}}$.

Fig. 5.

(A) Comparison of $g(r)$ and extensional axis $g^{ext}(r)$ for IPL equilibrium ($\rho =0.850$) and nonequilibrium state points ($\rho =0.850$, $\dot{\gamma }={10}^{-1}$) in which the two state points have approximately the same values of $s_2$ and $s_2^{\text{ext}}$. ($B$) The integration of Eqs. 1 and 2 as a function of the integration limit $r$ for the two state points in $A$. Identical values are obtained only after integration of the third shell.

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 $\theta$ 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 $s_2^{\text{ext}}$ beyond the third shell is necessary for the value of $s_2^{\text{ext}}$ to converge to the corresponding equilibrium value (see Fig. 5B). 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 $s_2^{\text{ext}}$ scaling does not hold for a potential with two characteristic length scales obtained from a ramp-like potential (see 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 $s_2^{\text{ext}}$ scaling should hold at least for simple liquids whose interaction potentials are isotropic and do not have more than two characteristic length scales. This conclusion is actually supported by the fact that the same system (the two length-scale potential) at high density (see SI Materials and Methods), where only one of the two length scales effectively dominates, shows the same simple $s_2^{\text{ext}}$ scaling as observed for the five other systems studied here. Nevertheless, the validity of our equilibrium–nonequilibrium mapping should be checked carefully for even more complex potentials including anisotropic potentials in the future.

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 $s_2^{\text{ext}}$ is important for determining the viscosity, it does not imply the presence of a simple relation between $s_2^{\text{ext}}$ and ${\sigma }_{xy}$. Notably, the integrand of ${\sigma }_{xy}$ includes not only $g(r,\theta )$ but also $dv(r)/dr$ and is thus not a function of $g(r,\theta )$ only (see also SI Materials and Methods). Another important point is that $g(r,\theta )$ starts to deviate from the equilibrium $g_0(r)$ for a shear rate that is much lower than $1/{\tau }_{\alpha }$. This indicates that ${\tau }_{\alpha }$ is not the characteristic relaxation time of $g(r)$.

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 $r^{-12}$ fluid, Dzugutov liquid (48), KABLJ mixture (49), and the GCM (50). The WCA fluid particles interact with an LJ pair potential $v_{ij}(r)=4ϵ[{\sigma }_{ij}/r^{12}-{\sigma }_{ij}/r^6]$ truncated and shifted at the minimum where ${\sigma }_{ij}$ = $({\sigma }_i+{\sigma }_j)/2$. The size polydispersity for the WCA fluid is 8% to avoid crystallization, and the sizes are sampled from a flat distribution. The Dzugutov pair potential has a characteristic maximum, besides the LJ minimum, around the second-nearest neighbor distance of a face-centered cubic crystal. The KABLJ mixture is a 4:1 binary mixture interacting with the LJ pair potential using the parameters ${\sigma }_{AB}=0.80$, ${\sigma }_{BB}=0.88$, ${ϵ}_{AB}=1.50$, and ${ϵ}_{BB}=0.50$, and the GCM pair potential is $v(r)=ϵ\exp [-{(r/\sigma )}^2]$. In 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., $\sigma$ = 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 $\dot{\gamma }$ = $\partial v_x/\partial y$ is the shear rate. Here $x$, $y$, and $z$ are the flow, shear gradient, and vorticity direction, respectively. For each system, we estimate the highest possible shear rate attainable in our simulations from the strong decrease in the configurational temperature with increasing shear rate (around $\dot{\gamma }$ = 0.01 for the WCA system) (53). We simulate at least $\dot{\gamma }t>\mathrm{\hspace{0.17em}10}$ (usually much more) before sampling data—that is, we sample steady-state shear flow.

Calculation Details.

The viscosity is calculated from $\eta$ = ${\sigma }_{xy}$/$\dot{\gamma }$, where ${\sigma }_{xy}$ is the xy component of the configurational stress tensor. The angular-dependent two-body excess entropy $s_2^{\theta }$ is defined by

$$s_2^{\theta }\equiv -\rho {\int }_0^{\mathrm{\infty }}\left[g(r,\theta )\ln g(r,\theta )-(g(r,\theta )-1)\right]r^{2}dr,$$ [2]

where $s_2={\int }_0^{2\pi }{s}_2^{\theta }d\theta$. The anisotropy of $g(\text{𝐫})$ in the vorticity direction is neglected to enhance statistics with no loss in generality—that is, $g(\text{𝐫})\equiv g(r,\theta )$. The RDF is calculated from

$$g(r,\theta )=\frac{L^3}{2r^2\mathrm{\Delta }r\mathrm{\Delta }\theta N(N-1)}\sum _{i\ne k}\delta (r-|{\text{𝐫}}_{ik}|)\delta (\theta -{\theta }_{ik}),$$ [3]

using a bin size of $\mathrm{\Delta }r=0.01$ and $\mathrm{\Delta }\theta$ = $\pi /50$. The integration of Eq. 2 is performed for each bin ${\theta }_i$ to obtain $s_2^{\theta }$ as a function of $\theta$; we checked that the results are independent of binning size and the integration limit is taken as $r=L/2$, with $L$ being the box length. The structural relaxation time ${\tau }_{\alpha }$ in equilibrium is calculated from $F_s(\text{𝐪},t)=⟨\exp [i\text{𝐪}\cdot \mathrm{\Delta }\text{𝐫}]⟩$ when $F_s({\text{𝐪}}_p,{\tau }_{\alpha })=0.2$. The wave number $|{\text{𝐪}}_p|$ is chosen near the first peak of the static structure factor.

The self-part of the intermediate scattering function under shear is given by $F_s(\text{𝐪},t)=⟨\exp [i\text{𝐪}\cdot \mathrm{\Delta }{\text{𝐫}}_{i,\dot{\gamma }}]⟩$. The affine motion of the flow is subtracted via the approximation $\mathrm{\Delta }{\text{𝐫}}_{i,\dot{\gamma }}\equiv \mathrm{\Delta }{\text{𝐫}}_i$ – $(\dot{\gamma }{y}_{i}t)\text{𝐱}$, where $y_i$ is the absolute coordinate of particle $i$ and 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 $F_s({\text{𝐪}}_p,{\tau }_{\alpha })=0.2$. As the self-part of the intermediate scattering function under shear is almost independent of the direction for q_p, we average only the transverse wave vectors when calculating ${\tau }_{\alpha }$ (and, for simplicity, the equilibrium wave number is probed for all sheared state points).