evaluating viscosity in molecular...

D. Keffer, MSE 614, Dept. of Materials Science & Engineering, University of Tennessee, Knoxville

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Evaluating Viscosity in Molecular Simulation

David Keffer Department of Materials Science & Engineering

University of Tennessee, Knoxville date begun: March 30, 2016

date last updated: April 7, 2016

Table of Contents

I. Purpose of Document ........................................................................................................... 2

II. Types of Viscosity ............................................................................................................... 2

III. Viscosities from Equilibrium Simulation .......................................................................... 4

IV. Viscosity from Non-Equilibrium Simulation .................................................................... 5

IV.A. NEMD Simulations of Shear Flow ............................................................................ 5

IV.B. NEMD Simulations of Extensional Flow .................................................................. 8

IV.C. Some NEMD Examples ........................................................................................... 10

IV.C.1. Rheological, Energetic, Structural and Theoretical Properties .......................... 10

IV.C.2. Visualization ...................................................................................................... 12

IV.C.3. Chain Dynamics ................................................................................................ 14

IV.C.4. Optical Properties .............................................................................................. 14

V. Built in LAMMPS Functionality ...................................................................................... 15

V.A. Equilibrium Simulation ............................................................................................. 15

V.B. Non-Equilibrium Shear Flow Simulation.................................................................. 15

V.C. Non-Equilibrium Extensional Flow Simulation ........................................................ 16


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I. Purpose of Document

The purpose of this document is to provide a practical introduction to the evaluation of viscosities in LAMMPS. The notes begin with some formal theory and conclude with practical implementation.

II. Types of Viscosity

When one derives the microscopic momentum balance, one typically obtains something of the form

( ) Φ∇−⋅∇−∇−∇⋅−=∂∂ ˆρρρ τp

τvvv (1)

where p is the pressure, τ is the viscous stress tensor (sometimes called the extra stress tensor), and Φ̂ is the specific external field imposed by, for example, gravity. If gravity is the source of the external field then we have Φ−∇= ˆg . Again, the functional form of the extra stress tensor must be determined by the choice of constitutive equation. One common constitutive equation is Newton’s law of viscosity. We understand that the LHS is an accumulation term. The first term on the RHS is the convection term, the second term on the RHS represents the momentum transport due to molecular transport due to a gradient in the pressure, the third term is the momentum transport due to molecular transport due to viscous dissipation, and the fourth term is due to an external potential such as gravity. This equation is a the difference of equation (3.2-9) on page 80 of the seminal text book, “Transport Phenomena” by Bird, Stewart and Lightfoot (BSL)† and the continuity equation , equation. There are numerous assumptions in this equation. It assumes, for example, that there is no coupling of the momentum and reaction. However, what we are concerned with here is the form of the extra stress tensor, τ.

Following BSL, a component of the total stress, αβπ , (the flux of β momentum in the α direction) is related to the total pressure, p, and the extra stress tensor, αβτ .

αβαβαβ τδπ += p (2)

The diagonal components, αααα τπ += p , are called normal stresses and the off-diagonal components, αβαβ τπ = , are called shear stresses.

In an undergraduate course, we may have been introduced to a simple case of shear flow where Newton’s law of viscosity can be expressed as


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−=

α

βαβ µτ

drdv

(3)

where βv is the velocity in the β direction and αr is the position in the α direction and where µ is the shear viscosity. In this case, it is clear that the shear viscosity is a proportionality constant between the gradient of the β velocity in the α direction with the corresponding extra stress tensor. BSL follows the convention that there is a negative sign in Newton’s law of viscosity to emphasize the parallel with Fick’s Law and Fourier’s Law. Many text books do not include the negative sign in their convention. The general form for Newton’s law of viscosity expressed in Cartesian coordinates is

αββ

α

α

βαβ δκµµτ

++

−+

+−=

z

z

y

y

x

x

drdv

drdv

drdv

drdv

drdv

32 (4.a)

or in tensor notation

( )( ) ( )δvvv ⋅∇

−+∇+∇−= κµµ

32Tτ (4.b)

where κ is called the dilational viscosity (or the bulk viscosity or the volume viscosity). The dilational viscosity is identically zero for ideal gases. For incompressible fluids, the divergence of the velocity is zero, 0=⋅∇ v , in which case we again don’t need to know κ . There are examples where including κ is essential to capturing the desired physics of a system. For example, compressibility is important in problem involving the propagation of sound in fluids.

In a purely shear flow, we can introduce the terminology and notation of a shear rate,

α

βγdrdv

= , such that γµταβ −= . In a purely uniaxial extensional (or elongational) flow, we can

introduce the terminology and notation of a tension rate, α

αεdrdv

= , such that

εηεκµταα ε=

−=

32 , in which we have introduced an extensional viscosity,

−= κµη

32

ε .

†Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transporτ Phεnomεna. Second ed. 2002, New

York: John Wiley & Sons, Inc.


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III. Viscosities from Equilibrium Simulation

The derivation for the expression of a viscosity comes from a Green-Kubo integral. See for example Chapters 7 and 8 (especially Table 8.1) of Hansen & McDonald. The general Green-Kubo integral has the form

( ) ( )∫∞

+=0

τJτJdK ττ (5)

where K is a transport property and J is a flux.

For the case of the shear viscosity, the argument of the Green-Kubo integral is the auto correlation function of the off-diagonal component of the stress tensor, where µTVkK B= and

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )αβαβ

βααβαββα

αβ

βααβαβπ

,,

111

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PEKE

N

iii

N

iii

N

iiiii

N

i

ii

ii

N

iii

ppV

τfτrτvτvmτvτvτaτrm

dττdr

τvdτ

τdvτrmτrτv

dτdmττJ

+=

+=

+=

+===

∑∑∑

∑∑

===

==

(6)

The expression after the final equality expresses the equation in terms of the thermodynamic property presented in the section on thermodynamic properties in of the course notes. Substituting into the General Green-Kubo integral we have for the shear viscosity,

( ) ( )∫∞

+=0

ττdTVkB αβαβ πτπτµ (7)

In this expression, the angled brackets indicate an ensemble average, which is an average over all time origins, τ. This latter fact means that every time step in the MD simulation can be used as a time origin in the calculation of the ACF.

In an analogous way, we have for the diagonal components of the total stress tensor, εBTVkK η= ,

( ) ( )∫∞

+=0

ττdTVk εB αααα πτπτη (8)


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A practical example of evaluating this integral can be found in the following reference.

Title: Molecular Dynamics Simulation of Polyethylene Terephthalate Oligomers Authors: Wang, Q., Keffer, D.J., Petrovan, S., Thomas, J.B. Journal: J. Phys. Chεm. B Vol. 114 Issue 2. pp. 786–795. Published 2010 doi: http://doi.org/10.1021/jp909762j

Figure 1. Right: Momentum flux autocorrelation functions as a function of observation time. Left: Shear viscosity obtained from equation (7) as a function of the upper limit of integration.

One sees in Figure 1, that there is some uncertainty for practical system as to where the upper limit of integration should be fixed.

These Green-Kubo integrals are equilibrium properties. Therefore, they give the viscosity in the limit of the strain (either shear or tension) rate going to zero. For Newtonian fluids, where the viscosity is constant and independent of strain rate, that’s all there is to the rheological behavior of the material. For any non-Newtonian fluid, where the viscosity is a function of strain rate, we must turn to something else.

IV. Viscosity from Non-Equilibrium Simulation

Much of non-equilibrium molecular dynamics (NEMD) owes its origin to an interest in non-Newtonian fluids. NEMD algorithms require a careful combination of modified equations of motion and boundary conditions. Two examples are given below, one for Planar Couette Flow, a kind of shear flow, and one for Planar Elongational Flow, a kind of extensional flow.

IV.A. NEMD Simulations of Shear Flow The SLLOD algorithm was developed to describe Planar Couette Flow, a purely shear flow.

The original citation is given here:

http://doi.org/10.1021/jp909762j


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Title: Nonlinear-response theory for steady planar Couette flow Authors: Denis J. Evans and G. P. Morriss Journal: Phys. Rev. A Vol. 30 Issue 3. pp. 1528-1530. Published 1 September 1984 doi: http://doi.org/10.1103/PhysRevA.30.1528

Couette Flow describes a flow due to velocity in the x-direction with a linear profile in the y direction, as shown in Figure X.

The SLLOD equations of motion are

uqpq

∇⋅+= ii

ii

mdτd

(9.a)

upFp

∇⋅−= iii

dτd

(9.b)

where iq and ip are the position and momentum of particle i and u∇ is the velocity gradient. In Planar Couette Flow, the velocity is given by

[ ] [ ] [ ]TTx

Tzyx cyuuuu 0000 +=== γu (10.a)

where γ and c are constants of the flow field. The gradient of the flow field is thus,

=

∂∂

∂∂

∂∂

∂

∂

∂

∂

∂

∂∂∂

∂∂

∂∂

=∇00000000 γ

zu

yu

xu

zu

yu

xu

zu

yu

xu

zzz

yyy

xxx

u (10.b)

Figure X. Couette Flow. Source: https://en.wikipedia.org/wiki/Couette_flow

http://doi.org/10.1103/PhysRevA.30.1528

https://en.wikipedia.org/wiki/Couette_flow


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Thus the equations of motion become

yii

xixi qmp

dτdq

,,, γ+= (11.a)

i

yiyi

mp

dτdq ,, = (11.b)

i

zizi

mp

dτdq ,, = (11.c)

yixixi pF

dτdp

,,, γ−= (12.a)

yiyi F

dτdp

,, = (12.b)

zizi F

dτdp

,, = (12.c)

Thus we observe the consistency between the equations of motion and Figure 1. The time evolution of the x coordinate depends not only on the peculiar velocity but the flow field as well. Since the flow field is a function of y position, the equation of motion in the x direction becomes a function of variables in the y direction.

These equations of motion allow for simulation of a steady state flow. If we want to simulate a bulk fluid, we need to modify the traditional periodic boundary conditions. The appropriate BCs for Planar Couette Flow are called Lee-Edwards Boundary Conditions.

Title: The computer study of transport processes under extreme conditions Authors: A.W. Lees and S.F. Edwards Journal: J. Phys. C: Solid State Phys., Vol. 5, pp. 1921-1929 Published 1972 url: http://iopscience.iop.org/0022-3719/5/15/006

These boundary conditions allow for the row of periodic images in the positive and negative y

dimension to slide with velocity 2

yγ± respectively, where y is the dimension of the

simulation volume in the y dimension.

http://iopscience.iop.org/0022-3719/5/15/006


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IV.B. NEMD Simulations of Extensional Flow

The rigorous algorithm for simulation of Planar Elongational flow is described as proper-SLLOD.

Title: A proper approach for nonequilibrium molecular dynamics simulations of planar elongation flow

Authors: Baig, C., Edwards, B.J., Keffer, D.J. Journal: J. Chem. Phys. Vol. 122 article #114103 Published 2005 doi: http://doi.org/10.1063/1.1819869

In fact the proper SLLOD algorithm has been shown to be rigorous for any arbitrary, steady-state flow and reduces to SLLOD for Planar Couette Flow.

Title: An examination of the validity of nonequilibrium molecular-dynamics simulation algorithms for arbitrary steady-state flows

Authors: Edwards, B.J., Baig, C., Keffer, D.J. Journal: J. Chem. Phys. Vol. 123 article #114106 Published 2005 doi: http://doi.org/10.1063/1.2035079

The proper SLLOD equations of motion are

uqpq

∇⋅+= ii

ii

mdτd

(13.a)

uuqupFp

∇⋅∇⋅−∇⋅−= iiiii m

dτd

(13.b)

It turns out that this additional term in equation (13.b) is zero for Planar Couette Flow so this expression reduces to equation (9.b) in that case.

In Planar Elongational Flow, the velocity is given by

[ ] [ ] [ ]TTyx

Tzyx cycxuuuuu 00 21 +−+=== εε u (14.a)

where ε and 1c and 2c are constants of the flow field. The gradient of the flow field is thus,

http://doi.org/10.1063/1.1819869

http://doi.org/10.1063/1.2035079


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−=

∂∂

∂∂

∂∂

∂

∂

∂

∂

∂

∂∂∂

∂∂

∂∂

=∇0000000

εε

zu

yu

xu

zu

yu

xu

zu

yu

xu

zzz

yyy

xxx

u (14.b)

Thus the equations of motion become

xii

xixi qmp

dτdq

,,, ε+= (15.a)

yii

yiyi qmp

dτdq

,,, ε−= (15.b)

i

zizi

mp

dτdq ,, = (15.c)

2,,,

, εε xiixixixi qmpF

dτdp

−−= (16.a)

2,,,

, εε yiiyiyiyi qmpF

dτdp

−−= (16.b)

zizi F

dτdp

,, = (16.c)

Again, in order to simulate a steady-state flow, we need to couple these equations of motion with appropriate boundary conditions. For Planar Elongation Flow, the appropriate boundary conditions are called Kraynik-Reinelt boundary conditions.

Title: Extensional motions of spatially periodic lattices Authors: A.M. Kraynik, D.A. Reinelt Journal: Int. J. Multiphase Flow Vol. 18 Issue 6 pp. 1045–1059 Published 1992 doi: http://doi.org/10.1016/0301-9322(92)90074-Q

http://doi.org/10.1016/0301-9322(92)90074-Q


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The boundary conditions involve orienting the sample at a special angle then distorting the shape of the periodic unit cell over time until at some fixed period, the particles can be remapped into the original volume. This temporal periodicity in the unit cell occurs many times during the course of a simulation.

IV.C. Some NEMD Examples

IV.C.1. Rheological, Energetic, Structural and Theoretical Properties

Numerous energetic, structural and rheological properties can be obtained from NEMD experiments. We choose a few examples from the following reference.

Title: Rheological and entanglement characteristics of linear-chain polyethylene liquids in

planar Couette and planar elongational flows Authors: Kim, J.M., Keffer, D.J., Kroger, M., Edwards, B.J. Journal: J. Non-Newtonian Fluid Mech. Vol. 152 Issue (1-3) pp. 168-183 Published 2008 doi: http://doi.org/10.1016/j.jnnfm.2007.03.005

In Figure 2, we provide an example of rheological properties from NEMD simulations. One can observe both the Newtonian plateau at low strain rates and the shear-thinning or tension-thinning regime at high shear rates.

Figure 2. Viscosities of linear alkanes from simulations of planar Couette flow (shear) and

planar elongational flow (extensional).

In Figure 3, we provide an example of a structural property from NEMD simulations. The mean distance of a vector connecting the two ends of the polymer chain is described as the end-to-end distance. Its change with respect to strain rate gives quantitative information regarding the amount of chain extension that occurs as a result of the flow field.

http://doi.org/10.1016/j.jnnfm.2007.03.005


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Figure 3. Mean chain end-to-end distance of linear alkanes from simulations of planar

Couette flow (shear) and planar elongational flow (extensional).

In Figure 4, we provide an example of an energetic property from NEMD simulations. In an attempt to understand the molecular origin of rheological behavior, one can examine all of the contributions to the potential energy and observe their changes as a function of strain rate.

Figure 4. Intermolecular energy of linear alkanes from simulations of planar Couette flow (shear) and planar elongational flow (extensional).

In Figure 5, we provide an example of a theoretical property from NEMD simulations. The dynamics of long polymer are described by reptation theory. One of the concepts in reptation theory is the idea that at intermediate times segments of polymer chains move through tube-like


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environments constructed of the material around them. Here the tube diameter is reported as a function of chain length and strain rate.

Figure 5. Tube diameter of linear alkanes from simulations of planar Couette flow (shear) and planar elongational flow (extensional).

IV.C.2. Visualization

NEMD simulations are also a source of visualizations that communicate in a very clear manner connections between molecular-level mechanisms and macroscopic phenomena. These examples are taken from the following reference.

Title: Visualization of conformational changes of linear short-chain polyethylenes under

shear and elongational flows Authors: Kim, J.M., Edwards, B.J., Keffer, D.J. Journal: J. Mol. Graph. & Mod. Vol. 26 Issue 7 pp. 1046-1056 Published 2008 doi: http://doi.org/10.1016/j.jmgm.2007.09.001

For example, in Figure 6, the evolution of a system from equilibrium under the influence of a shear flow is shown. The white arrow corresponds to the vorticity in the flow.

http://doi.org/10.1016/j.jmgm.2007.09.001


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Figure 6. A linear alkane undergoing shear flow.

In Figure 7, the evolution of a system from equilibrium under the influence of an extensional flow is shown. This flow field has no vorticity.

Figure 7. A linear alkane undergoing extensional flow.


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IV.C.3. Chain Dynamics

The flow field influence chain dynamics. The following example is taken from the reference.

Title: Single-chain dynamics of linear polyethylene liquids under shear flow Authors: Kim, J.M., Edwards, B.J., Keffer, D.J., Khomami, B. Journal: Phys. Lett. A Vol. 373 Issue 7 pp. 769-772 Published 2009 doi: http://doi.org/10.1016/j.physleta.2008.12.062

Figure 8. Single chain dynamics and corresponding relaxation times as a function of shear rate.

In Figure 8, we show the orientation angle and the end-to-end distance for one chain in a melt as a function of time (right). There is some periodicity. A Fourier transform of this behavior yields characteristic relaxation times for the polymer as a function of shear rate as characterized by the dimensionless Weissenberg number.

IV.C.4. Optical Properties

Experimentalists use optical properties to infer information about the state of stress in a flowing fluid. NEMD simulation are capable of generating optical properties such as dichroism and birefringence as well. These examples are taken from the following reference.

Title: A molecular dynamics study of the stress-optical behavior of a linear short-chain

polyethylene melt under shear Authors: Baig, C., Edwards, B.J., Keffer, D.J. Journal: Rheologica Acta Vol. 46 pp. 1171-1186 Published 2007 doi: http://doi.org/10.1007/s00397-007-0199-2

In Figure 9, we show a plot of the xy component of the birefringence tensor vs the stress tensor. The oft-used stress-optical rule assumes a linear dependence between the two. Here we show

http://doi.org/10.1016/j.physleta.2008.12.062

http://doi.org/10.1007/s00397-007-0199-2


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that such a relationship is valid only up to a certain shear rate, after which there is nonlinear behavior.

Figure 9. A plot of the xy component of the birefringence tensor vs the stress tensor. The oft-used stress-optical rule assumes a linear dependence.

V. Built in LAMMPS Functionality

In the directory of LAMMPS examples, there is a directory called VISCOSITY. This contains four examples, each of which compute the shear viscosity of a system. We discuss only two of the four examples.

V.A. Equilibrium Simulation The input in.gk performs an equilibrium simulation from which a Green-Kubo integral is evaluated per equation (7). It is important to note that this implementation suffers from the same egregious approach as does the evaluation of the autocorrelation functions used to calculate the diffusivity as implemented in LAMMPS. Namely, the angled brackets in equation 7, which indicate an ensemble average (an averaging over time origins in this case) is ignored. Only the starting point of the simulation is used as a time origin. Much better statistical reliability is obtained from post-processing based on saved values of the stress tensor.

V.B. Non-Equilibrium Shear Flow Simulation The input in.nemd performs a non-equilibrium simulation using the SLLOD equations of motion and the Lees-Edwards boundary conditions as described above. We have performed no extensive analysis of the implementation in LAMMPS. To our knowledge, there is no obvious drawback with the LAMMPS implementation, as there is


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V.C. Non-Equilibrium Extensional Flow Simulation To my knowledge LAMMPS does not have the capability to simulate extensional flow using the proper SLLOD equations of motion and the Kraynik-Reinelt boundary conditions as described above.

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