Introduction to Nonequilibrium Molecular Dynamics

Transcription

1 Introduction to Nonequilibrium Molecular Dynamics Dr Billy D. Todd Centre for Molecular Simulation Swinburne University of Technology PO Box 218, Hawthorn Victoria 3122, Australia B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 1

2 Overview In this final Module, we will introduce some of the basic ideas of nonequilibrium molecular dynamics (NEMD). What you have looked at in Modules 7-10 have been exclusively equilibrium molecular dynamics, i.e., MD based upon systems at thermodynamic equilibrium. In Module 11 some examples of NEMD simulation results were presented. In this Module you will become more familiarised with the basic concepts that enable one to simulate nonequilibrium fluids by molecular dynamics methods. By nonequilibrium we mean a fluid which is acted upon by an external field. This field drives the fluid away from equilibrium and, if sufficient time is available, towards a nonequilibrium steady-state (as long as the field itself is predictably well behaved). As there are many subtleties associated with NEMD this Module only serves as a basic introduction. For a more comprehensive description, see references [1-3]. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 2

3 Topic 1: NEMD and homogeneous flow If time permitted, a suitable starting point for the description of NEMD would be to first discuss the Liouville equation, which is the governing equation of nonequilibrium statistical mechanics: f t =.( G f ) G f = f. G G. G G (12.1) where f is the phase-space distribution function and G = (r, p) = (r 1, r 2, r N, p 1, p 2,.p N ). In general, the RHS of Eqn (12.1) is non-zero and the solution of the differential equation is either extremely difficult or impossible to obtain. In fact, it is for this reason that rigorous and robust nonequilbrium Monte Carlo methods do not exist. While a number of MC schemes have been recently proposed (e.g, see refs [4,5]), they rely upon a number of approximations that limit their validity to the weakfield linear regime. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 3

4 Once the time evolution of the distribution function is discussed (with an appropriate derivation of Eqn (12.1)), one might then go on to discuss continuum mechanics principles and linear irreversible thermodynamics. Finally, one could tie in all three of the above topics into one, namely how microscopic dynamics may be related to irreversible thermodynamics, and from this to discuss algorithms which may be useful for NEMD applications. As we do not have time to develop all these themes (an entire set of half a dozen Modules would need to be devoted to them!), we will concentrate only on the end product, namely some useful algorithms and numerical schemes enabling one to perform NEMD simulations of relatively simple fluid systems. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 4

5 1.1 Transport coefficients NEMD involves the simulation of a classical N-body system of atoms or molecules interacting via interatomic potential forces. However, in addition to interatomic forces, an external field drives the system away from thermodynamic equilibrium. Generally NEMD is used to study fluid systems, though in principle there is no restriction for using it to study either gasses or solids under the influence of external forces. The equations of motion for a system of N single component atoms may be formulated as follows: r i = p i m i pi p i = Fi + FE ζ ur mi (, t) (12.2) B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 5

6 where r i represent the laboratory positions of atom i and p i the laboratory momenta. By laboratory, we mean the unconvected momenta (i.e., the total momentum of atom i minus the steaming component due to the fluid flow). F i is the sum of all interatomic forces acting on atom i due to all other N-1 atoms, F E is an external field that drives the system away from thermodynamic equilibrium (e.g., a gravitational field or a driving pressure head), ζ is a thermostat multiplier that may be used to keep the temperature of the system constant, and u(r, t) is the streaming velocity of the fluid. These equations of motion are numerically integrated at each timestep to generate the particle phase-space trajectories of the system. From these trajectories one may use the principles of nonequilibrium statistical mechanics to calculate all the thermodynamic properties of interest of the fluid (e.g., elements of the pressure tensor or heat flux vector), as well as its transport properties. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 6

7 Of particular usefulness is the determination of transport coefficients, such as diffusion, viscosity and thermal conductivity coefficients. Of relevance to computational fluid dynamicists is how NEMD complements computational fluid dynamics (CFD). It does this primarily in two ways: NEMD allows a direct calculation of the transport coefficients of a fluid, assuming only knowledge of the interatomic forces acting between atoms. This means that the output of a typical NEMD simulation (i.e., the transport coefficients) may be used as input for a typical CFD calculation. The transport coefficient is thus calculated abinitio, without recourse to semi-empirical approximations or experimental data. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 7

8 The ab-initio nature of NEMD simulations allows one to extract thermodynamic fluxes directly and to relate them to their conjugate thermodynamic forces. This in turn allows one to examine in greater detail the validity of a host of constitutive equations used for closing hydrodynamic equations. Such equations may be as fundamental as the standard Navier-Stokes equations for momentum and heat transport, or may invoke more sophisticated constitutive equations for modeling the viscoelasticity of complex polymer solutions or melts, e.g., see ref [6]. We will examine the former situation in the next topic, by looking at the example of planar Poiseuille flow, and how this throws into question the validity of the standard Navier-Stokes heat transfer equation. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 8

9 As an example of the direct calculation of a transport coefficient, consider a homogeneous fluid of atoms under planar shear flow in the x-direction, where the velocity gradient is constant in the y-direction. If we are interested in the shear viscosity, this may be extracted directly as η = σ xy + σ 2γ yx (12.3) where η is the shear viscosity, σ xy and σ yx are the xy and yx elements of the stress tensor, and γ is the strain rate imposed by the flow field, given as γ = y u x (12.4) B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 9

10 The angle brackets indicate that the stresses are time averaged, and we note that at steady-state <σ xy > = <σ yx >. The steady-state stresses are calculated directly from the appropriate statistical mechanical expressions (e.g., the Irving-Kirkwood pressure tensor, as discussed in Module 10). Eqn (12.3) is an expression of Newton s law of viscosity. Thus, for a variety of thermodynamic state points, one may calculate the shear viscosity as functions of temperature, density and strain rate. This information could conceivably be fed back into a CFD calculation, which needs η(ρ, T, γ ) as input. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 10

11 One may thus generate a three dimensional grid of input viscosities for a CFD calculation. The output of such a CFD calculation may be, for example, the stress and velocity profiles of a shearing fluid where the density, temperature and strain rate may vary as a function of position. 1.2 Equations of motion for homogeneous flow Eqn (12.2) as currently formulated poses some problems. In order to simulate boundary driven shear flow, for example, one would need suitable boundaries, such as a solid wall. On microscopic length scales this leads to very strong inhomogeneities in the fluid density, which makes analysis of simulation data complex. An alternative may be to apply a spatially varying external field with suitable periodic boundary conditions to generate shear flow, but this also induces strong inhomogeneities making data analysis difficult. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 11

12 Ideally we would like to simulate a small volume of bulk fluid, where boundary effects may be ignored and the fluid is genuinely homogeneous. Indeed, Eqn (12.2) may be reformulated such that these conditions are precisely met. The resulting equations are known as the SLLOD equations of motion [1], and for an atomic fluid are given as r i p i = + i mi r. u p = F p. u ζ p i i i i (12.5) Now, p i is already defined by Eqn (12.5a) to be peculiar (i.e., with respect) to the streaming velocity u. If the flow is planar shear flow, then (12.5) reduces to r i p i = + m i iγ y p = F iγ p ζp i i yi i i (12.6) B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 12

13 where i is the unit vector in the x-direction. In Problem 12.1 you are asked to derive Eqn (12.6), based on Eqn (12.5). These equations now have the beneficial property of transforming a boundary driven flow (such as planar Couette flow) into one that is driven by a smooth mechanical force. The SLLOD equations are in fact rather subtle, and a more full account of them, and their relationship to Newtonian equations of motion, may be found in ref [1]. When implemented with suitable periodic boundary conditions, such as those devised by Lees and Edwards [7], they generate homogeneous isothermal flow, making extraction of the shear, bulk or elongational viscosities a relatively simple task. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 13

14 It is perhaps instructive to think of Eqn (12.5) in the following way. In the top equation, r i is the position vector of atom i. Atom i will change its position in time due to two processes: thermal diffusive motion (represented by the peculiar momentum p i ), and the local streaming convecting velocity, represented by the term coupling u with r i. The peculiar momentum is in turn obtained by integrating the second equation. This depends on the sum of all intermolecular forces acting on particle i, F i. However, as the momentum solved for is only the thermal (peculiar) momentum, we must subtract out the contribution due to the convecting fluid, namely the second term that couples the streaming velocity ( u ) with the peculiar momentum, p i. The last term involving the thermostat multiplier is as before, and acts as a friction coefficient to remove viscous heat generated by the flow. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 14

15 For planar elongational flow, where the fluid expands at a rate of ε the x direction, and contracts at the same rate in the y direction (the fluid remains static in the z direction), Eqn (12.5) reduces to in p r = + iε x jε y i i i i mi p = F iε p + jε p ζp (12.7) i i xi yi i where i and j are the unit vectors in the x and y directions, respectively. In Problem 12.2 you are asked to derive the SLLOD equations (12.7) for planar elongational flow, as well as uniaxial and biaxial elongational flows. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 15

16 The thermostat multiplier, ζ, given in Eqn (12.5) is derived in the usual manner, and for a Gaussian thermostat one gets: ζ = N i= 1 p.f ( p. u) i i i N i= 1 p 2 i For planar shear flow, this equation reduces to (12.8) ζ = N ( F.p i i γ pxipyi ) i= 1 N i= 1 p 2 i (12.9) In Problem 12.3 you are to derive Eqn (12.9), as well as the corresponding equations for the Gaussian thermostat multiplier for planar, biaxial and uniaxial elongational flows. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 16

17 1.3 Lees-Edwards periodic boundary conditions If you program Eqn (12.6) (for planar shear flow) into your existing equilibrium MD program you would find that your program won t run for long and that your results will not make much sense. Recall on p.13 it was stated that the SLLOD equations when implemented with suitable periodic boundary conditions will generate homogeneous isothermal flow. The reason why your code would not run for long is that it would be implementing the SLLOD equations while still using equilibrium periodic boundary conditions! To appreciate why equilibrium periodic boundary conditions are unsuitable for planar shear flow (indeed, for any type of flow!), consider the following diagram depicting atoms in a simulation cell with an imposed shearing field, in this case a strain rate γ. Note that the origin (0, 0, 0) is taken as the centre of the middle cell (cell number 5). B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 17

18 1 2 3 u = iu x L u = -iu x ( ) x u y = γ y y x Figure Schematic representation of a shearing cell under Lees- Edwards periodic boundary conditions B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 18

19 Note that any position in the cells 1, 2 and 3 moves relative to the corresponding position in cells 4, 5, 6 at a velocity of + γ Li, where L is the cell length. In time t this corresponds to a shift of + γ Lt in the x direction. We note here that in Figure 12.1 the central cell (5) is the actual simulation cell, whereas all others are periodic images of it. Thus, any particle that crosses the boundary of cell 5 at y = L/2 re-emerges through the y = -L/2 boundary shifted by an amount γ Lt in the x direction. Similarly, any position in the cells 7, 8 and 9 moves relative to the corresponding position in cells 4, 5, 6 at a velocity of γ Li. Thus any particle that crosses the boundary at y = -L/2 re-emerges at the y = L/2 boundary, shifted in the x direction by + γ Lt. Mathematically, this is expressed as ( t) = ( t δt) γ Lt ( particlecrosses y = L/2boundary ) r r i i i mod L ( t) = ( t δt) + γlt ( particlecrosses y = L/2boundary ) i i mod L r r i (12.10) B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 19

20 The code for applying these periodic boundary conditions might thus read as dxd += gamma_dot*delta; dxd -= int(dxd); // ensure max(dxd) = 1 (etc.) for (i=1; i<=np; i++) { // apply pbcs x0[i]-=dxd*nint(&y0[i], &cube)*cube; //nint is a function that returns //the nearest integer of y i /L pbcs(&cube, &x0[i], &y0[i], &z0[i]); } ( ) Here dxd is the total strain γ Lt. gamma_dot is the strain rate, γ. In a similar manner, one must also apply these modified pbcs to all pairs of distances r ij. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 20

21 We also need to modify the neighbour list to accommodate for the shearing flow. First, we need to account for the displacement of particles in the x direction. This can be accomplished by the following line of code (see Module 8): xs[i] += delta*(px0[i] + gamma_dot*ys[i]); Next, we update the maximum displacement vector length: smax = sqrt(smax) *gamma_dot*re*tup; Note that tup is the total simulation time. Periodic boundary conditions can then be applied in the same manner as on the previous page. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 21

22 The next issue to address is the implementation of the SLLOD equations into the numeric integrator. As we have developed our equilibrium MD code using a 4 th order Gear predictor-corrector scheme, we shall adapt that to incorporate the new field-dependent equations. In fact, the only difference between the SLLOD numerical equations and the equilibrium ones is the inclusion of the field dependent terms! These are given on the following page. Note that the predictor equations remain the same as before; the equations of motion only appear in the corrector step. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 22

23 for (i=1; i<=np; i++) { // corrector loop for positions xcor = x1[i] (px0[i] + gamma_dot*y0[i])*delta; ycor = y1[i] - py0[i]*delta; zcor = z1[i] - pz0[i]*delta; } x0[i] -= xcor*fo1; y0[i] -= ycor*fo1; z0[i] -= zcor*fo1; x1[i] -= xcor; y1[i] -= ycor; z1[i] -= zcor; x2[i] -= xcor*f21; y2[i] -= ycor*f21; z2[i] -= zcor*f21; x3[i] -= xcor*f31; y3[i] -= ycor*f31; z3[i] -= zcor*f31; x4[i] -= xcor*f41; y4[i] -= ycor*f41; z4[i] -= zcor*f41; B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 23

24 for (i=1; i<=np; i++) { // corrector loop for momenta pxcor = px1[i] (fx[i] - gamma_dot*py0[i] - zeta*px0[i])*delta; pycor = py1[i] (fy[i] zeta*py0[i])*delta; pzcor = pz1[i] (fz[i] zeta*pz0[i])*delta; px0[i] -= pxcor*fo1; py0[i] -= pycor*fo1; pz0[i] -= pzcor*fo1; px1[i] -= pxcor; py1[i] -= pycor; pz1[i] -= pzcor; px2[i] -= pxcor*f21; py2[i] -= pycor*f21; pz2[i] -= pzcor*f21; px3[i] -= pxcor*f31; py3[i] -= pycor*f31; pz3[i] -= pzcor*f31; px4[i] -= pxcor*f41; py4[i] -= pycor*f41; pz4[i] -= pzcor*f41; } B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 24

25 At this point we make several further observations about thermostats. Firstly, we note that the thermostat operates on the peculiar momenta, which in turn are defined to be local with respect to the streaming velocity u. The SLLOD equations guarantee that the streaming velocity profile is linear (as indeed it is in low Reynolds number (Re) planar shear flow). However, as the shear rate increases beyond low Re flows, the actual streaming velocity profile (i.e., the one occurring in nature) no longer remains linear, but in fact develops an S-shaped kink. The SLLOD equations, however, will not generate this kink. They will keep the streaming velocity linear. The kink is removed because the thermostat interprets this deviation in linearity as excess heat and removes it from the system! This is often associated with the onset of so-called string-phases, a phenomenon in which the fluid particles lower their entropy by organising themselves in solid-like rows aligned with the fluid velocity streamlines. They are, however, an artifact of the system, and are a practical demonstration of how care must be taken in the application of thermostats for nonequilibrium systems (see ref [1]). B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 25

26 The second point about thermostats has to do with numerical stability. In actual fact the temperature of the system will tend to drift after long times under the application of a Gaussian thermostat. In fact, Gaussian constraint methods in general, be they applied to temperature, bond lengths, or anything else for that matter, suffer from numerical drift associated with truncation errors on finite precision computers. It is entirely a numerical feature and has nothing to do with the formal rigour of the methodology. As it is a numerical shortcoming, one can compensate for it by a numerical feedback mechanism. This will add a small perturbation to the equations of motion, proportional to the difference between the desired target temperature and the actual kinetic temperature obtained by summing the momenta. The small magnitude of this perturbation ensures that the dynamics of the system remain faithful to the systems we are trying to simulate. Thus, the thermostat multiplier may be written as B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 26

27 ζ = ζ + ζ new old 0 N 2 pi 3NT i= 1 3NT In this expression, ζ old is just ζ calculated by Eqns (12.8-9), ζ 0 is an arbitrary constant, and T is the desired target temperature. (12.11) You now have the fundamentals to be able to code up the SLLOD equations of motion into your existing equilibrium MD code, implement the Lees-Edwards pbcs, and correct for numerical drift in the temperature. In Problem 12.4 you are now asked to implement these ideas into a complete NEMD program for simulating planar shear flow. Note that the Lees-Edwards pbcs can be modifed to accommodate planar elongational flow, but this is a more complex task not covered here. Interested readers may looks at references [8-11]. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 27

28 Topic 2: Inhomogeneous flow, a case example In this topic, we now look at a case example of inhomogeneous flow, namely planar Poiseuille flow. Consider a 3 dimensional fluid of atoms confined by atomistically detailed walls, such as depicted in figure In this geometry, an external field (e.g., gravity) acts on fluid atoms in the x direction, generating a streaming velocity. The walls are separated in the y direction, and all thermodynamic quantities we are concerned with are functions of this coordinate. The z axis is normal to the page. The interatomic potential used is the shifted and truncated Lennard-Jones potential [12]. This system is described in detail in reference [13]. As the fluid is now inhomogeneous, the SLLOD equations of motion cannot be used. Instead, we re-cast Eqn (12.2) for both wall and fluid atoms. The wall equations are: pi r i = m i ( ) j p = F κ r q ζp jς i i i i i L (12.12) B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 28

29 Topic 2: Inhomogeneous flow, a case example (cont.) wall atoms y if E x fluid atoms Periodic image of wall atoms Figure dimensional atomic fluid undergoing planar Poiseuille flow. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 29

30 Topic 2: Inhomogeneous flow, a case example (cont.) Here κ is a spring constant, and the second term in the momentum equation represents an harmonic tethering potential used to confine wall atoms to vibrate about their equilibrium lattice sites q i. ζ is the thermostat multiplier, and ζ Lj is a constraint multiplier used to fix the center of mass position of each wall layer. This is necessary so that the walls do not expand as the fluid heats up due to viscous heat generation. As viscous heat is removed only through the walls via the wall thermostat, the equations of motion for the fluid atoms are purely Newtonian, i.e. r i = p i m p = F + if i i i E (12.13) where F E is the external driving field. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 30

31 Topic 2: Inhomogeneous flow, a case example (cont.) When solved, the classical Navier-Stokes heat equation predicts a quartic temperature profile. However, this assumes that the thermal conductivity and shear viscosity are constant. Strictly speaking, this is not true, as both these transport coefficients are state-point dependent. If we ignore the first few monolayers of fluid atoms near the wall surfaces, then the characteristic wavelengths of the transport properties of the fluid are large compared to the range of the interatomic potential. Thus we may assume localization of momentum and energy transport, and write the governing constitutive equations as: xy Q ( y) = ( y) ( y) σ η γ ( ) = J ( y) = λ( y) J r j j Qy ( ) T y y (12.14) B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 31

32 Topic 2: Inhomogeneous flow, a case example (cont.) The first constitutive equation is just Newton s law of viscosity, while the second is Fourier s law of heat conduction. Both equations now include the explicit position (hence state point) dependence of the transport coefficients. T(y) is the position dependent temperature of the fluid. The steady-state Navier-Stokes heat equation is thus ( ) JQy y T y 2 σxy ( y) γ ( y) = λ( y) η( y) γ ( y) = 0 (12.15) y y y ( ) which, when solved explicitly for T(y), yields ( ) 4 8 T y T Ty Ty = (12.16) B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 32

33 Topic 2: Inhomogeneous flow, a case example (cont.) The coefficients T 0, T 4, T 8, etc, are functions of the density, viscosity, thermal conductivity and driving field. Details of this derivation, and the assumptions made about the viscosity and thermal conductivity profiles, may be found in [14]. Even with the explicit y dependence of the transport coefficients included, the Navier-Stokes (N-S) heat equation still predicts a temperature profile that is quartic in its leading term. In figure 12.3 the results of NEMD generated temperature profiles are plotted and compared with the theoretical prediction of Eqn (12.16) [14]. Clearly the prediction is poor in the center of the channel and does not correctly capture the profile on either side. Furthermore, the value of the thermal conductivity at the center of the channel (i.e., y = 0) is predicted to be λ(0) = 3.89 (reduced units). Independent NEMD simulations using the Evans heat flux algorithm to compute the thermal conductivity [15] give a known value of at the state point of (ρ(0), T(0)) = (0.836, 0.955). The classical N-S equation thus predicts the wrong temperature profile as well as the wrong value of λ(0)! B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 33

34 Topic 2: Inhomogeneous flow, a case example (cont.) T(y) NS; η(y), λ(y) y Figure Conventional Navier-Stokes prediction of temperature profile (solid line) compared with simulation data (dots). B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 34

35 Topic 2: Inhomogeneous flow, a case example (cont.) One may be tempted to dispute the simulation results, given the long and successful history of classical N-S fluid dynamics. However, it should be noted that microscopic flows are not the common experience of classical macroscopic flows, where the traditional N-S solution appears undisputed. Microscopic flows involve length and timescales that often invalidate the central assumptions of macroscopic fluid dynamics, and so must be treated appropriately at the microscopic level. It is this level which holds the key to new nanotechnologies. In 1992, Baranyai, Evans and Daivis [16] postulated that the heat flux vector couples to an additional term proportional to the strain rate tensor twice contracted into its transpose, i.e. ( ) T J Q = T u: u (12.17) λ ξ B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 35

36 Topic 2: Inhomogeneous flow, a case example (cont.) where ξ is a phenomenological strain rate coupling coefficient. While they were able to explain their results with this assumption, the work used a synthetic thermostatting mechanism to regulate the fluid temperature. In the Poiseuille flow simulations, we suffer from no such artefact. In fact, the fluid is not thermostatted at all. Viscous heat is entirely removed by the walls, and the fluid is described by Newton s equations without any constraint forces, thus more closely representing nature. Substituting Eqn (12.17) into the energy continuity equation one obtains a modified N-S heat equation: ( ) ( ) 2 T y γ y 2 λ( y) + ξ( y) + η( y) γ ( y) = 0 (12.18) y y y y When solved for the temperature, this equation yields B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 36

37 Topic 2: Inhomogeneous flow, a case example (cont.) ( ) T y = T + Ty + Ty + Ty (12.19) The leading term is now quadratic in y. In figure 12.4 the simulation data is now compared with the profile predicted by Eqn (12.19). The agreement is excellent. Furthermore, the thermal conductivity at the center of the channel is predicted to be λ(0) = 6.93, which is within the range of errors of the exact known value of These simulations have thrown into question the validity of Fourier s law of heat conduction. This has now been confirmed and validated by recent independent work [17-19]. These results particularly sound a warning that the use of Fourier s law in the standard N-S heat equation is likely to lead to significant errors in the prediction of thermal properties of fluids. This may be extremely important in the prediction of microporous/microchannel flows, where the effect is far more pronounced than in macroscopic flows [14, 19]. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 37

38 Topic 2: Inhomogeneous flow, a case example (cont.) T(y) MODIFIED NS; η(y), λ(y), ξ(y) y Figure Modifed Navier-Stokes prediction, with strain rate coupling, of temperature profile (solid line) compared with simulation data (dots). B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 38

39 Problems Problem 1 Using the general SLLOD equations of motion, Eqn (12.5), derive the specific SLLOD equations (Eqn 12.6) for the positions and momenta of an atomic fluid undergoing planar shear flow. Make use of the fact that the strain rate tensor is given as u u x y u z x x x u u x y uz u = γ 0 0 y y y = u u x y u z z z z where γ u y. x B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 39

40 Problems (cont.) Problem 2 A fluid undergoing pure elongational (extensional) flow may be represented by the following strain rate tensor u u x y u z x x x ε xx 0 0 u u x y uz u = 0 ε yy 0 y y y = 0 0 ε zz u u x y u z z z z Planar elongational flow implies that the fluid expands in one direction (say x) with constant rate ε xx = ε, contracts in another direction (say y) with rate ε, while in the third direction (z) it remains constant. A yy = ε fluid undergoing uniaxial stretching flow implies ε xx = ε, ε yy = ε 2, ε zz = ε 2, while the converse is true for biaxial stretching. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 40

41 Problems (cont.) Problem 2 (cont.) Based on these flow symmetries, derive the SLLOD equations of motion for all three types of extensional flow. Problem 3 (a) Derive Eqn (12.9), based on the derivation of the thermostat multiplier for an equilibrium system, as you did in Problem 9.3(b). (b) From Eqn (12.8) and the previous expressions for the strain rate tensor in Problems 1 and 2, derive the thermostat multiplier expressions for planar shear, planar elongational, uniaxial and biaxial stretching flows. B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 41

42 Problems (cont.) Problem 4 Based on the material learned in this Module, implement the SLLOD equations of motion for planar shear flow into your existing equilibrium MD code. Be sure to implement the Lees-Edwards periodic boundary conditions, as well as the appropriate thermostat terms with proportional feedback. From your program you should now be able to calculate the shear viscosity η, as η = P xy + 2γ P yx What is the value of the viscosity you obtain at the thermodynamic state point of ρ, T, γ = ,0.722,1.0? ( ) ( ) B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 42

43 Reading Material and References 1. D.J. Evans and G.P. Morriss, Statistical Mechanics of Nonequilibrium Liquids (Academic Press, London, 1990). 2. D.J. Evans and G.P. Morriss, Computer Physics Reports 1(6), 297 (1984). 3. S.S. Sarman, D.J. Evans and P.T. Cummings, Physics Reports 305, 1 (1998). 4. V.G. Mavrantzas and D.N. Theodorou, Macromolecules 31, 6310 (1998). 5. B.C. Eu, J. Chem. Phys. 107, 222 (1997). 6. M. Doi and S.F. Edwards, The Theory of Polymer Dynamics (Clarendon, Oxford, 1986). 7. A.W. Lees and S.F. Edwards, J. Phys. C 5, 1921 (1972). 8. B.D. Todd and P.J. Daivis, Phys. Rev. Lett. 81, 1118 (1998). 9. A. Baranyai and P.T. Cummings, J. Chem. Phys. 110, 42 (1999). 10. B.D. Todd and P.J. Daivis, Comput. Phys. Commun. 117, 191 (1999). 11. B.D. Todd and P.J. Daivis, J. Chem. Phys. 112, 40 (2000). B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 43

44 Reading Material and References 12. J.D. Weeks, D. Chandler, and H.C. Andersen, J. Chem. Phys. 54, 5237 (1971). 13. B.D. Todd, D.J. Evans and P.J. Daivis, Phys. Rev. E 52, 1627 (1995). 14. B.D. Todd and D.J. Evans, Phys. Rev. E 55, 2800 (1997). 15. D.J. Evans, Phys. Lett. 91A, 457 (1982). 16. A. Baranyai, D.J. Evans, and P.J. Daivis, Phys. Rev. A 46, 7593 (1992). 17. P. Cordero and D. Risso, Physica A 257, 36 (1998). 18. G. Ayton, O.G. Jepps, and D.J. Evans, Mol. Phys. 96, 915 (1999). 19. P.J. Daivis and J.L.K. Coelho, Phys. Rev. E 61, 6003 (2000). B. D. Todd, Centre for Molecular Simulation, Swinburne University of Technology 44