Institute for Condensed Matter Theory and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Transcription

1 Shear-thinning of molecular fluids in Couette flow Bharath V. Raghavan 1 and Martin Ostoja-Starzewski 2 1) Department of Mechanical Science and Engineering,University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 2) Department of Mechanical Science and Engineering and Institute for Condensed Matter Theory and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (Dated: 27 January 2017) We use Non-equilibrium Molecular Dynamics (NEMD) simulations, the Boltzmann equation, and continuum thermomechanics to investigate and characterize the shearthinning behavior of molecular fluids undergoing Couette flow, interacting via a Lennard-Jones (LJ) potential. In particular, we study the shear-stress under steadystate conditions and its dependency on fluid density and applied shear-strain rate. Motivated by kinetic theory, we propose a rheological equation of state that fits observed system responses exceptionally well and captures the extreme shear-thinning effect. We notice that beyond a particular strain-rate threshold, the fluid exhibits shear-thinning, the degree of which is dependent on the density and temperature of the system. In addition, we obtain a shear-rate dependent model for the viscosity which matches the well established Cross viscosity model. We demonstrate how this model arises naturally from the Boltzmann equation, and possesses an inherent scaling parameter that unifies the rheological properties of the LJ fluid. We compare our model with those in the literature. Finally, we formulate a dissipation function modeling the LJ fluid as a quasilinear fluid. 1

2 I. INTRODUCTION Current developments in the field of micro and nanoscale systems have led to observations of interesting properties in fluids. At the molecular level, classical (continuum) fluid dynamic models (i.e. the Navier-Stokes equations) often breakdown due to their inability to account for the discrete nature of the fluid 1, and thus its observable properties. In such cases, one often has to resort to Molecular Dynamics (MD) simulations in order to account for the discrete nature of the fluid. Non-equilibrium Molecular Dynamics (NEMD) and in particular the SLLOD algorithm are useful techniques for studying the rheological properties of molecular fluids. Shearthinning has been widely observed in molecular fluids under SLLOD dynamics. Heyes 2 used macro-rheological equations to model the viscosity of a LJ fluid undergoing SLLOD dynamics. Additionally, Heyes 3 also studied the structural features of the flow that result in shear-thinning such as the shear-rigidity modulus and the structural anisotropy of the flow. He investigated the transition from an amorphous liquid to a string phase that induces shear-thinning. Furthermore, it was found that the viscosity was proportional to γ 0.5 in the nonlinear regime of the flow 4,5, and via mode-coupling analysis 6. Recently, experimental verification has been presented for shear-thinning in NEMD simulations where the authors fitted experimental and simulation results for the viscosity of squalane to a single Carreau equation 7. NEMD has seen widespread applicability in determining the rheological properties of alkanes 8 11, polymer melts 12, and colloidal suspensions 13. These studies have extensively investigated the nonlinear shear-thinning regime of the flow, and the structural features at the onset of shear-thinning. A framework of using Monte-Carlo simulations and NEMD simulations using the SLLOD equations to study the non-newtonian shear viscosities of simple liquids under shear has been proposed 14. The results from Monte-Carlo simulations and MD simulations compare excellently. This theory was extended by removing the limitation of requiring isothermal conditions 15, and was subsequently used to extensively study the rheological properties of the LJ fluid. Applying the Generalized Boltzmann equation, a nonlinear expression for the viscosity of the LJ fluid similar to the Ree-Eyring formula, but absent empirical parameters, was derived. It was discovered that the scaled shear-viscosities at various state-points collapsed onto a single curve given by this expression, alluding to the 2

3 existence of a universal scaling function for all temperatures and densities 15. The classical Navier-Stokes (NS) equations for Couette flow consider a linear dependence between shear stress and deformation rate. As a consequence, in studies covering the transition to turbulence, there are instances where transition occurs at a lower Reynolds number compared to that predicted by linear stability analyses 16. This is believed to be due to a drop in internal friction between fluid layers that results in shear-thinning. Conversely, it has been found that the onset of turbulence can be delayed in Couette flow experiments by dissolving a small amount of polymer molecules in water without altering its Newtonian nature 17. Motivated by this need for a more general constitutive stress-deformation model, we resort to kinetic theory and MD to study the steady-state shear stress and its dependency on the density and temperature of the system, and the applied shear strain rate. We propose a rheological equation of state to capture the shear-thinning feature observed in the stress-deformation plot. Finally, we compare our model to other equations of state established in the literature. From this equation, we obtain a model for the viscosity of the LJ fluid that depends entirely on state parameters (i.e. the density and temperature) and the applied strain-rate. II. SIMULATION METHODOLOGY A. Theoretical background The problem of simulating homogeneous flows driven by boundaries in real physical systems (e.g:-couette or elongational flows) is that a microscopic simulation explicitly including the walls invariably induces density inhomogeneities into the fluid 18. This profound shortcoming gave rise to NEMD, which currently is well established as an accurate method for simulating homogeneous shear flows. Two well-known algorithms in this category are the DOLLS and SLLOD algorithms. These produce a microscopic description consistent with macroscopic observations. The first of the homogeneous NEMD algorithms was based on the DOLLS Hamiltonian 19 defined as 3

4 (a) (b) FIG. 1. (a) Sliding brick and (b) deforming cube representations of the Lees-Edwards periodic BCs used in NEMD simulations of molecular fluids undergoing Couette flow. H dolls (r N, p N, t) = ϕ(r N )+ i 2m i p 2 i + i r i. v.p i Θ(t) (1) where ϕ(r N ) is the system potential energy due to the interaction between N atoms, r i and p i are the laboratory position and momentum of atom i, and v is the gradient of the streaming velocity v. Furthermore, we enforce that the flow begins at t = 0 as indicated by Θ(t) which is the Heaviside step function. This Hamiltonian generates the DOLLS equations of motion ṙ i = ṗi m + r i. v 4 (2a)

5 ṗ i = F ϕ i + v.p i (2b) where F ϕ i is the interatomic force on atom i due to all the other atoms. However, to account for linear and nonlinear system responses, a modification was proposed as ṙ i = ṗi m + r i. v ṗ i = F ϕ i + p i. v (3a) (3b) which are the so-called SLLOD equations of motion to indicate the transpose v.p i p i. v. In addition, it should be noted that the SLLOD equations cannot be derived from a system Hamiltonian. To obtain an exact representation of planar Couette flow, we apply the Lees-Edwards periodic boundary conditions 6,23 and inducing a streaming velocity field of the form v(r, t) = i γy v(r, t) = (v x, v y, v z ) = ( γy, 0, 0) (4) where i is the unit vector, y is the channel height, and γ = v x / y is the strain rate such that v(r, t) = γy 0 0 (5) Then, the SLLOD equations are expressed as ṙ i = ṗi m + i γy i ṗ i = F ϕ i i γp yi (6a) (6b) A schematic of the MD simulation methodology is shown in Fig. 1. The two representations are equivalent 18. B. Molecular Dynamics model The molecular dynamics simulations are carried out in two-dimensions in LAMMPS 24, with particles interacting via the Lennard-Jones (LJ) potential, the interaction potential for which is given by 4ϵ[( σ ϕ(r N r ) = )12 ( σ r )6 ], r r c (7) 0, r > r c 5

6 where ϵ is the depth of the potential well, and σ is the finite distance at which the interparticle potential is zero. The dynamics of the simulation is governed by the SLLOD equations [Eq. (6)] with Lees-Edwards periodic boundary conditions. To induce isothermal conditions, a Nose-Hoover thermostat in the canonical ensemble (NVT) is implemented, to yield the modified set of equations as 5 ṙ i = ṗi m + i γy i ṗ i = F ϕ i i γp yi αp i α = 1 [ ] T (t) 1 Q T 0 (8a) (8b) (8c) where α is the thermostatting multiplier, T (t) is the actual kinetic temperature at time t, T 0 is the preset kinetic temperature, and Q is a constant. The equilibrium distribution function for the Nose-Hoover thermostat in the canonical ensemble is given by 6 f c (Γ, ζ) = exp[ β(h Qζ2 )] dγdζ exp[ β(h Qζ2 )] (9) where Q and ζ are thermostat parameters. III. RESULTS AND DISCUSSION A. Shear-thinning model based on kinetic theory We now employ the Boltzmann equation for the Couette flow under investigation. The general expression for the Boltzmann equation is f t + v. f + F f m v = J[f, f] (10) where f is the distribution function, v is the velocity, F is the external force, m is the mass, and J[f, f] is the collision operator 25. For simplicity, we consider the BGK model 26 of the form f t + v. f + F f m v = ν(f f L) (11) where f L is the equilibrium distribution function, and ν is an effective collision parameter. For isothermal, uniform shear flow conditions, we have u i (r) = d ij r j (12a) 6

7 d = d ij = j u i = γδ ix δ jy T = 0 (12b) (12c) In order to analyze SLLOD dynamics under Lees-Edwards periodic BCs using the Boltzmann equation, we use a Galilean transformation to transition to the local rest frame (Lagrangian frame) of the fluid. Consequently, we define the peculiar velocities of the fluid particles as 14,15,25 V = v u(r) and r = r ut. In the local rest frame of the fluid, f(r, v, t) = f(v, t). Using Eqs. (12) in the BGK approximation in Eq. (11), and assuming the external thermostatting force is of the form F = mαv,where α is the thermostatting mulitplier related to the heating rate, we obtain which upon simplification yields f t [ f f t + v. r r r + f V αv f v + [I dt] V f r ] V r V v = ν(f f L) (13) [dv + αv ] f V = ν(f f L) (14) Under the conditions of statistical spatial homogeneity, we can neglect spatial derivatives in Eq. (14) to obtain f t [d ijv j + αv i ] f V i = ν(f f L ) (15) At this point, we note that the pressure tensor is the second moment of the peculiar velocity and is expressed as P := mv V f(v, t)dv (16) where denotes the vector outer product. Multiplying Eq. (15) by V i V j and integrating yields P ij t + d ik P jk + d jk P ik + 2αP ij = ν(p ij pδ ij ) (17) Expanding Eq.(17) and bearing in mind the following; P 13 = P 23 = P 33 = 0 under steady-state conditions p = 1 3 [P 11 + P 22 + P 33 ] = 1 3 trp 7

8 all time derivatives vanish under steady-state conditions we replace P 11 = p to obtain the set of coupled differential equations governing the timeevolution of the pressure tensor as P 22 Neglecting all the time-derivatives yields, p t γp αp = 0 (18a) + 2αP 22 + νp 22 = νp (18b) t P 12 + γp αP 12 = νp 12 (18c) t 2 3 γp αp = 0 (19a) 2αP 22 + νp 22 = νp (19b) γp αP 12 = νp 12 (19c) Eq. (19c) yields In addition, Eq. (19a) yields the thermostatting multiplier as P 12 = γ P 22 ν 1 + 2α (20) ν α = γp 12 3p (21) Subsituting Eq. (21) in Eq. (20), noting the BGK approximation that ν = p η 0, where η 0 is the Newtonian viscosity of the fluid, and that under steady-state conditions P 22 = p we have η 0 γ P 12 = [ ν γp 12 3p Substituting P 12 = η 0 γ in Eq. (22), and noting that P 12 = τ 12 yields τ 12 = ] (22) η 0 γ ν 2 γ 2 (23) where τ 12 is the shear stress (which is the negative of the shear component of the pressure tensor). To find a suitable expression for the effective collision term ν in terms of reduced LJ parameters (i.e. the temperature T, density ρ, molecular mass m, and viscosity η 0, we know that 27 ν = 1 ( ) 1 8kB T 2 λ πm (24) 8

9 where λ is the molecular mean free path. The mean free path for extended potentials is defined as 28,29 λ = η 0 ρ ( ) 1 πm 2 2k B T Substituting Eq. (25) in Eq. (24) and converting to reduced LJ units 15, we have (25) ν = νσ (m/ϵ) 1 2 = 4ρ T πη 0m (26) Thus, in reduced LJ units, the shear-thinning equation of state is τ 12 = τ 12 ( σ 3 /ϵ ) = η 0 γ ( γ /ν ) 2 (27) B. Comparison of Numerical and Analytical results The data of primary relevance in this study is the mean value of the steady-state shear stress (i.e. the negative of the shear component of the pressure tensor, the latter being the default output in LAMMPS). After the initial equilibration period, we extract mean values for time-steps at t = increments. We divided the total simulation into blocks of M = 1, 000 time-steps which yields a total number of N = 24, 500 blocks. For each block, we calculate the section average which uses the 1, 000 data points associated with that block given by P 12 ( γ) B = M i=0 P (i) 12 (28) and the cumulative average which accounts for all the previous section averages expressed as P 12 ( γ) C = N i=0 P 12 ( γ) (i) B (29) The steady-state is achieved when fluctuations in the cumulative average are no longer statistically significant. We run simulations for various state-points (Table I) and compare the proposed shear-thinning model, and the associated viscosity model, with those available in the literature. The simulations are carried out for a system size of N = 200 particles, and strain rates 0.0 γ = γσ m/ϵ 2.5 with γ = 0.5 (in LJ reduced units), and a cutoff radius of r c = 2 1/6 σ. The system is run for a total of time steps with t = ; it is sufficiently equilibrated for time-steps and then run for time-steps for data extraction. 9

10 TABLE I. LJ state points and parameter estimates in Eq. (27) obtained via curve fitting. ρ T η0 2/(3ν 2 ) Consistent with well-established observations made in the literature, the LJ fluid exhibits shear-thinning, the degree of which depends on the fluid state point as illustrated in Fig. 2. At low densities, the fluid only experiences moderate shear-thinning (Fig. 2a) for the range of strain-rates considered. With increasing density, the fluid exhibits extreme shear-thinning as shown by its stress-deformation response in Fig. 2b. The level of shear-thinning tends to set in quicker as the fluid density increases. For instance, extreme shear-thinning occurs at γ 1.0 at the triple point (i.e ρ = and T = 0.722), but only at γ 1.5 for ρ = 0.8 and T = 1.0. Furthermore, an increase in temperature tends to delay the onset of extreme shear-thinning. Comparing two state points at ρ = 0.8 and T = 1.0 and ρ = 0.8 and T = 1.1, the latter requires a strain rate of γ 2.0 compared to γ 1.5 for the former. In either case, we see that Eq. (27) is able to capture he stress-deformation response extremely well. In order to evaluate the effectiveness of our proposed model, we compare it with widely used empirical formulae available in the literature and its goodness-of-fit measured by the r 2 value. The models of interest are; The Sisko equation τ 12 = a γ b + c γ (30) The Ree-Eyring equation τ 12 = k BT V τ sinh 1 ( τ γ) (31) where V τ is the shear stress activation volume and τ is the relaxation time. 10

11 3 2.5 (0.4,1.5) (0.6,1.0) (0.6,1.5) 2 τ γ (a) τ (0.8442,0.722) (0.8,1.0) (0.8,1.1) (0.9,1.1) γ (b) FIG. 2. (a) Moderate and (b) extreme shear-thinning in the stress-deformation curve for the LJ fluid at various state points. The curve-fit is performed using Eq. (27) and is represented by the dotted lines. The Bair-Winer equation τ 12 = τ L 12 ( 1 e η 0 γ τ L 12 ) (32) where τ L 12 is the limiting shear stress; a concept frequently used in high-pressure lubri- 11

12 cation. In all the cases, the rheological equations of state from the literature studied in this article provide excellent models for cases of moderate shear-thinning, as illustrated in Fig. 3, as observed at state points of (ρ, T ) = (0.4, 1.5), (0.6, 1.0) and (0.6, 1.5). However, these models fail to accurately capture the dramatic shear-thinning feature of the LJ fluid at higher densities of (ρ, T ) = (0.8, 1.0), (0.8, 1.1), (0.8442, 0.722) and (0.9, 1.1). We observe that for moderate shear-thinning, the Sisko equation of state, Eq. (30), is comparable in effectiveness to Eq. (27) and fits the data with r However, the former fails to fit the data at state points where dramatic shear-thinning occurs. On the other hand, Eq. (27) provides an exceptionally good fit for all the LJ state points considered, and in particular is able to capture the extreme shear-thinning effect. The Ree-Eyring [Eq. (31)] and Bair-Winer [Eq. (32)] equations of state, despite being widely considered more versatile than the former equations, still suffer from the same drawbacks as being unable to capture dramatic shearthinning. In general, these equations are excellent models at low strain-rates, i.e. below the threshold where extreme shear-thinning sets in. The models are able to successfully capture the stress-deformation curve with r for moderate shear-thinning, which falls to around 0.7 r for extreme shear-thinning stress-deformation behavior. The poor performance of these models at higher strain rates can be attributed to their empirical nature,where parameters are obtained by fitting experimental data, or their derivation from macro-viscoelastic models. In order to obtain a strain-dependent viscosity model, we note that τ 12 = η( γ) γ (33) Thus, using Eq. (27) in Eq. (33) we find that η 1 = η0 ( γ ) 2 (34) 3 ν We compare our model with the well-established Cross model given in the literature as η = η 0 η 1 + β γ n + η (35) where β and n are fitting constants, and η is the infinite shear-rate limiting viscosity. Interestingly, Eq. (34) yields the Cross viscosity model, with n = 2 and η = 0, absent 12

13 3.5 3 Eq. (30) Eq. (31) Eq. (32) 2.5 τ γ FIG. 3. Demonstration of the ability of existing rheological equations [Eq. (30)-(32)] to capture moderate shear-thinning in the LJ fluid and their inability to model extreme shear-thinning. The data markers are the same as used to identify the state points in Fig. 2 empirical parameters since the parameters in Eq. (34) are completely determined by the fluid s density (ρ ), temperature (T ), and Newtonian viscosity (η0). To serve as a validation for our simulations, Eq. (34) yields a viscosity of η 0 = 3.76 at the LJ triple point of (ρ, T ) = (0.8442, 0.722), which is comparable to the value of η 0 = 3.73 calculated in the literature 30. The Cross model has been previously used to model the strain-rate dependent viscosities of simple fluids, but in an empirical sense 3,31. It should be noted that our model has been completely derived from hydrodynamic and kinetic theories. A similar result has been obtained for the Ree-Eyring viscosity formula, which arose naturally from the Generalized Boltzmann equation. In our model, we find that the effective collision parameter, ν, behaves as a universal scaling parameter that facilitates collapsing all the data onto a universal curve, suggesting the existence of a universal scaling law for the LJ fluid, as shown in Fig. 4. The slight scatter is attributed to the inaccuracy of NEMD in accessing small strain-rates for viscosity calculations. Ideally, one would resort to the use of Transient Time Correlation functions 32. Additionally, we attribute the discrepancy between our calculations and those avaliable in the literature to the truncation distance. In most studies, a truncation of r c = 2.5σ was used which would account for the attractive component in the LJ potential as well, compared to r c = 2 1/6 σ in this work. 13

14 10 1 (0.6,1.0) (0.8442,0.722) (0.4,1.5) (0.6,1.5) (0.8,1.0) (0.8,1.1) (0.9,1.1) η /η γ /ν FIG. 4. Scaled viscosities for the LJ fluid at various state points for a system size of N = 200 particles. The fit is given by Eq. (34). C. Quasilinearity based on continuum thermomechanics Note that Eq. (27) may allow one to determine a dissipation function which should equal the irreversible rate of entropy production 33. Within the framework of thermomechanics, we know that the dissipation function is defined as A (d) kl (d ij)d ij = Φ(d ij ) (36) Where the dissipative forces A (d) kl define a symmetric second-order tensor and d ij is the deformation rate. For uniform shear flow, the dissipative forces are the shear stress σ (d) ij that σ (d) ij (d ij)d ij = Φ(d ij ) = so η 0 γ 2 ( γ ) 2 (37) 3 ν Adopting thermodynamic orthogonality (or principle of maximal rate of entropy production ), we have σ (d) ij = µ Φ d ij (38) where µ is a constant of proportionality to be determined by the condition of thermodynamic orthogonality. Under uniform shear, the invariants of the strain-rate tensor are d (1) = 0, 14

15 d (2) = γ 2, and d (3) = 0. Therefore, on account of these side conditions, Eq. (38) becomes σ (d) ij = µ [ ] Φ λ1 d (1) λ 2 d (3) (39) d ij where λ 1 and λ 2 are lagrange multipliers. Since d ij is a deviatoric tensor, then so is σ (d) ij. Thus, we have tr[σ (d) ij (39), we obtain which simplifies to On account of the constraints, we have ] = 0. Furthermore, we also have det [σ(d)] = 0. Upon expanding Eq. [ σ (d) Φ d (1) ij =µ + Φ d (2) d (1) d ij d (2) d ij + Φ d (3) d (3) d ij λ 1 d (1) d ij λ 2 d (3) d ij [ σ (d) Φ ij =µ (d ij d (1) δ ij ) d (12) ] λ 1 δ ij λ 2 (d ik d kj d 2 δ ij ) ij ] (40) (41) 3λ 1 λ 2 (d ik d kj 3d 2 ) = 0 3λ 1 + λ 2 d 2 = 0 λ 1 + λ 2 d (2) = 0 (42a) (42b) Solving the system of equations, Eq. (42a) and (42b), we obtain λ 1 = λ 2 = 0, so that σ (d) ij Multiplying both sides of Eq. (43) by d ij yields For uniform shear, we obtain = µ Φ d (2) d ij (43) Φ = µ Φ d (2) d ij d ij (44) Φ = 2µ Φ ( γ 2 ) γ2 (45) Substituting the explicit form for Φ in Eq. (45) and taking the derivative with respect to γ 2, we obtain µ = 1 + (2/3) ( γ2 /ν 2 ) 2 (46) Thus we see that a dissipation function of the form in Eq. (37) along with Eq. (43) suggest that a molecular fluid behaves at least like a quasi-linear fluid 33 with a constitutive relationship of the form σ d ij = 2η(d (2) )d ij σ d 12 = 2η( γ 2 ) γ (47) where the fluid viscosity η needs to atleast depend on d (2) = γ 2. 15

16 IV. CONCLUSION In this article, we simulate LJ fluids at different state points and employ the Boltzmann equation to study the behavior of the steady-state shear stress with applied strain rate. We observe that, in all cases, the fluid exhibits shear-thinning. At low densities, the shearthinning effect is moderate for the range of strain-rates considered. At higher densities, dramatic shear-thinning sets in at much lower strain-rates. In light of these observations, we are motivated to use kinetic theory to find an equation of state that is able to capture this wide range of behaviors. We compare our proposed equation with other models available in the literature to validate its effectiveness. In general, we find that all the models available in the literature performed well when the degree of shear- thinning is moderate. However, for cases of extreme shear-thinning, the Ostwalde-de Waele and Sisko equations of state completely breakdown, while the Ree- Eyring and Bair-Winer equations experience a significant drop in modeling capability. In contrast, the proposed model performs consistently well across all state-points. Additionally, the formalism involving the Boltzmann equation naturally gives rise to a form of the Cross viscosity model as a steady-state solution. Moreover, the effective collision frequency used in the BGK approximation acts as a scaling parameter that seems to unify the behavior of the LJ fluid across all the state-points considered. Finally, a thermomechanical framework involving the dissipation function suggests that the LJ fluid behaves at least as a quasilinear fluid. This offers an insight into the minimum required approximations to be used in the process of constructing a suitable constitutive model to represent a molecular fluid. The derivation shows that the proposed equation provides a suitable shear-thinning model while preserving thermodynamic orthogonality in molecular fluids, which is an important feature of continuum thermomechanics. The framework of thermodynamic orthogonality provides a means for transitioning from molecular to continuum scales and for a link to the fluctuation theorem which grasps spontaneous violations of the second law of thermodynamics on very small length and/or time scales 34. Thus, for the purposes of modeling such fluids as continua, we need to assume that the viscosity is dependent on the strain-rate in a manner consistent with the postulates of continuum thermomechanics. 16

17 ACKNOWLEDGMENTS This work has been supported by the NSF under grant CMMI The authors would like to thank the University of Illinois for computational resources necessary for undertaking this project. Additionally, the authors would like to thank Dr. R. E. Khayat of the University of Western Ontario and three anonymous reviewers for constructive comments on this manuscript. REFERENCES 1 J. Delhommelle and D. J. Evans, Poiseuille flow of a micropolar fluid, Molecular Physics 100, (2002). 2 D. M. Heyes, Shear thinning of the Lennard-Jones fluid by molecular dynamics, Physica A: Statistical Mechanics and its Applications 133, (1985). 3 D. M. Heyes, Non-Newtonian behaviour of simple liquids, Journal of non-newtonian fluid mechanics 21, (1986). 4 D. J. Evans, Molecular dynamics simulations of the rheological properties of simple fluids, Physica A: Statistical Mechanics and its Applications 118, (1983). 5 A. Baranyai and P. T. Cummings, Fluctuations close to equilibrium, Physical Review E 52, 2198 (1995). 6 G. P. Morriss and D. J. Evans, Statistical Mechanics of Nonequilbrium Liquids (ANU Press, 2007). 7 S. Bair, C. McCabe, and P. T. Cummings, Comparison of nonequilibrium molecular dynamics with experimental measurements in the nonlinear shear-thinning regime, Physical review letters 88, (2002). 8 S. Cui, S. Gupta, P. Cummings, and H. Cochran, Molecular dynamics simulations of the rheology of normal decane, hexadecane, and tetracosane, The Journal of chemical physics 105, (1996). 9 S. Cui, P. Cummings, H. Cochran, J. Moore, and S. Gupta, Nonequilibrium molecular dynamics simulation of the rheology of linear and branched alkanes, International journal of thermophysics 19, (1998). 10 A. Berker, S. Chynoweth, U. C. Klomp, and Y. Michopoulos, Non-equilibrium molecular 17

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22 3 2.5 (0.4,1.5) (0.6,1.0) (0.6,1.5) 2 τ γ

23 (0.8442,0.722) (0.8,1.0) (0.8,1.1) (0.9,1.1) τ γ

24 3.5 3 Eq. (30) Eq. (31) Eq. (32) 2.5 τ γ

25 10 1 (0.6,1.0) (0.8442,0.722) (0.4,1.5) (0.6,1.5) (0.8,1.0) (0.8,1.1) (0.9,1.1) η /η γ /ν