A study of the anisotropy of stress in a fluid confined in a nanochannel

A study of the anisotropy of stress in a fluid confined in a nanochannel
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  A study of the anisotropy of stress in a fluid confined in a nanochannel Remco Hartkamp,  A. Ghosh, T. Weinhart, and S. Luding   Citation: J. Chem. Phys. 137 , 044711 (2012); doi: 10.1063/1.4737927   View online:   View Table of Contents:   Published by the  American Institute of Physics.   Additional information on J Chem Phys Journal Homepage:   Journal Information:   Top downloads:   Information for Authors:    THE JOURNAL OF CHEMICAL PHYSICS  137 , 044711 (2012) A study of the anisotropy of stress in a fluid confined in a nanochannel Remco Hartkamp, 1,a) A. Ghosh, 2,b) T. Weinhart, 1,c) and S. Luding 1,d) 1  Multi Scale Mechanics, Faculty of Engineering Technology, MESA +  Institute for Nanotechnology,University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands 2 Van Der Waals-Zeeman Institute, University of Amsterdam, Valckenierstraat 65,1018 XE Amsterdam, The Netherlands (Received 8 February 2012; accepted 29 June 2012; published online 31 July 2012)WepresentmoleculardynamicssimulationsofplanarPoiseuilleflowofaLennard-Jonesfluidatvari-oustemperaturesandbodyforces.Localthermostattingisusedclosetothewallstoreachsteady-stateup to a limit body force. Macroscopic fields are obtained from microscopic data by time- and space-averaging and smoothing the data with a self-consistent coarse-graining method based on kernelinterpolation. Two phenomena make the system interesting: (i) strongly confined fluids show layer-ing, i.e., strong oscillations in density near the walls, and (ii) the stress deviates from the Newtonianfluid assumption, not only in the layered regime, but also much further away from the walls. Variousscalar, vectorial, and tensorial fields are analyzed and related to each other in order to understandbetter the effects of both the inhomogeneous density and the anisotropy on the flow behavior andrheology. The eigenvalues and eigendirections of the stress tensor are used to quantify the anisotropyin stress and form the basis of a newly proposed objective, inherently anisotropic constitutive modelthat allows for non-collinear stress and strain gradient by construction.  © 2012 American Institute of Physics . [] I. INTRODUCTION Computer simulation studies 1–14 and experiments 15–19 of fluids confined in narrow channels or pores show oscillatorydensity profiles close to the wall. Particularly, when the chan-nel width or pore diameter is of the order of a few molecu-lar diameters,  σ  0 , such variations can occur over the wholesystem, leading to a highly inhomogeneous and anisotropicsituation. In such systems, not only density but also stressandtransportpropertieslikediffusion,viscosity,andheatcon-ductivity become functions of the position and direction. 20–31 Furthermore, slip between the fluid and the wall can becomeof significant importance in narrow pores. The effect of thechannel width and wall roughness has been studied widely inrecent years 1,3,32–41 Consequently, the flow behavior or,e.g., the heat transfer characteristics of such systems devi-ate from the predictions for classical Navier-Stokes fluids, forwhich the global transport properties are implied to be homo-geneous (i.e., independent of position) and isotropic. 22 Various simulations and experiments have been per-formed on confined fluids with the aim to understand anddescribe the flow behavior of the system by looking atrelevant global and local physical quantities. While someexperiments 16,19 could predict the effective global propertieslike relaxation time, frictional force or shear response of ultra-thin films, the extraction of local values of state variables (likedensity, pressure, and temperature) is still beyond the reachof experimental measurements. On the other hand, such lo-cal quantities can be extracted rather easily from simulations. a) Electronic mail: b) Electronic mail: c) Electronic mail: d) Electronic mail: Several numerical studies in the past years have been devotedto gain understanding of the properties of dense fluids in ananochannel. For example, Sofos  et al. 6 performed a thor-ough study of the density, velocity, and temperature profilesof a simple liquid in channels of several widths, temperatures,body forces, and average fluid densities. One of their find-ings is that, while a dense fluid becomes homogeneous in thecenter of a wide channel, a fluid with low average densityremains inhomogeneous, due to wall-effects. Recently, Long etal. 12 studiedinfluenceoftheconfinementonthenormalandtangential stresses for argon in a carbon nanochannel. Theyfound that the normal stresses can be positive or negative,depending on the channel width. Furthermore, they observedthat the shear stress is very sensitive to changes in the bulk pressure.These studies, besides leading to deeper insight into thephysics of flow in thin films and channels, also help to com-pute effective transport properties by averaging over localquantities and their fluctuations. In this framework, the con-cept of a “non-local viscosity” was introduced by Bitsanis et al. 42 First, the local average density at any point is ob-tained by averaging the local density over a spherical vol-ume centered around the point. The functional dependenceof shear viscosity on density at a given temperature wasthen expressed using the Enskog theory of hard-sphere fluids.Building further on the method developed by Bitsanis  et al. ,Hoang and Galliero 31 recently presented a study using a si-nusoidally varying external potential to study the non-localviscosity of a simple fluid in a periodic box. Effective vis-cosities obtained by numerically integrating such local func-tionals over the entire domain of variation are shown to bein agreement with the value calculated from molecular dy-namics simulation in different flow situations. A number of  0021-9606/2012/137(4)/044711/19/$30.00 © 2012 American Institute of Physics 137 , 044711-1  044711-2 Hartkamp  et al.  J. Chem. Phys.  137 , 044711 (2012) papers 20,23,24,43 in the last years showed local viscosity calcu-lations from shear stress–strain rate relations as a function of location. For example, Todd  et al. 27 and Todd and Hansen 28 compared local and non-local constitutive relations in narrowrectangular channels with Weeks-Chandler-Andersen (WCA)atoms. 44 Recently, Sofos  et al. 4 and Sun  et al. 45 have applied theGreen-Kubo relation locally in order to find how the trans-port properties are affected by the confinement of a fluid.Sofos  et al. 7 studied the influence of wall roughness onthe average and local shear viscosity and diffusion coeffi-cient. Due to a coarse bin averaging, the layering of atomsnear the walls is not explicitly visible in their results. Also,theirstresscalculationassumesahomogeneousdensityacrosseach bins, which would only be approximately satisfied farfrom the walls. However, a global impression of the shearstress, strain rate and shear viscosity is given across a planarchannel.Travis and Gubbins 23 studied planar Poiseuille flow inmuch narrower slits of pore width 4.0 σ  0  and 5.1 σ  0 . They alsouse the mesoscopic integration of the Navier-Stokes equa-tion to compute shear stress, whereas strain rates are derivedfrom a polynomial function obtained by fitting the stream-ing velocity profile across the channel. The same system hasbeen studied with different interatomic interactions (Lennard-Jones and WCA potential) to probe the effect of these inter-actions on the flow properties. It was found that the layer-ing of a Lennard-Jones fluid is stronger than that of a WCAfluid with the same temperature and density. Highly nonlin-ear shear stress and strain rate profiles were observed acrossthe channel irrespective of the kind of interaction potentialused.Different ways of computing the stress tensor in a con-fined fluid have been discussed and compared by Todd  et al. 20 In their “method of planes” (MOP), local stress is computedfrom the consideration of intermolecular force transfer perunit area across a plane passing through the point of interest.This is compared with the stress calculations obtained fromIrving-Kirkwood real space expressions and mesoscopic in-tegration of the Navier-Stokes momentum conservation equa-tion which does not require any molecular information. TheMOP proves to be an easy method which conveniently avoidsthe singularities which occur in microscopic fields. However,without further modifications of the method, it is not able tocalculate the full stress tensor. Recently, Heyes  et al. 46 haveshown, for the limiting case of infinitesimally thin bins, theequivalence between the MOP and the “volume averaging”(VA) method, introduced by Cormier  et al. 47 Shen and Atluri 48 derived an atomistic stress tensor byusing an approach based on kernel interpolation. This methodis easy to implement and results in a continuous stress field.Furthermore, they show that this method, in contrast to manyother widely used methods, satisfies the conservation of linearmomentum. Goldhirsch 49 discussed in much detail the advan-tages and limitations of calculating macroscopic fields fromsmoothed microscopic data.In the present study, we apply the stress formulation in-troduced by Schofield and Henderson 50 in conjunction withspatial smoothing, as is discussed by Goldhirsch, 49 to amolecular dynamics simulation of planar Poiseuille flow innarrow slits, about 11 atomic diameters wide. While stronglyconfined fluids have been widely studied, finding a constitu-tive relation that holds near the walls as well as in the bulk isstill an open problem. The strain rate profile shows strongeroscillations than the shear stress in the region near the walls.Hence, the ratio between the shear stress and strain rate de-pends on the distance to the walls and is an unsuitable mea-sure for the shear viscosity. Since a tensorial viscosity wouldincrease complexity enormously, a more commonly used be-lief is that the shear stress relates to the strain rate via a convo-lution integral over a non-local viscosity kernel. 42,51,52 Toddand Hansen 28 and Cadusch  et al. 53 studied possible shapes of such kernels. Kobryn and Kovalenko 29 studied the viscosityinhomogeneity in confined fluids by using a stress tensor au-tocorrelation function. In the present study, instead of tryingto find a tensorial viscosity and in the attempt to avoid theconvolution integrals, we introduce a general and simple con-stitutive model which uses eigenvalue analysis to relate thestress to the flow (velocity-gradient) field with the main in-gredient being the difference in eigendirections of stress andstrain.The paper is organized as follows. Section II gives adescription of the system and the simulation method. InSec. III, the calculations of microscopic and macroscopic fields are presented. In Sec. IV, a decomposition for a con- stitutive model is discussed. In Sec. V, the results of various simulations are shown and analyzed. In Sec. VI, the relationsbetween variables of the constitutive model and the measuredmacroscopic fields are studied. Finally, in Sec. VII, the pre-sented method and results are discussed. II. MODEL SYSTEM The system is a slit bounded in the  x  -direction by twoparallel atomistic walls as shown in Figure 1. Periodic bound- ary conditions are applied in the  y - and  z -direction. The heightand the depth of the system are 13.68 σ  0 , with  σ  0  the lengthscale of the atoms ( i.e., the distance at which the potentialenergy between a pair of interacting atoms is zero). Eitherwall is composed of two 001 fcc layers. Each layer is a squarelattice, containing 128 atoms fixed at their lattice site, witha spacing of 1.21 σ  0  between the atoms. The separation dis-tance between the walls is  W   = 11 . 1 σ  0 . The width is definedas the distance between the center of the inner wall layers FIG. 1. Left: a snapshot of the system, and right: a schematic cross-sectionindicating the definition of the channel width.  044711-3 Hartkamp  et al.  J. Chem. Phys.  137 , 044711 (2012) (see Figure 1). A flow of liquid argon is simulated in the slit,with  N  = 1536 fluid atoms.We generate planar Poiseuille flow by applying a con-stant body force  f   to the fluid atoms, acting in the negative  z -direction. The body force must be chosen such that the signal-to-noise ratio is large, since otherwise a very large simulationtime is required in order to obtain accurate statistics. On theother hand, if the body force is too large, the response of thesystem becomes very nonlinear and the temperature will varyconsiderably across the channel. 54–58 The interactions between neutral spherical atoms, suchas argon, are well described by a 12-6 Lennard-Jones pairpotential, 59 U  ( r ij  ) = 4 ǫ 0  σ  0 r ij   12 −  σ  0 r ij   6  ,  (1)where  ǫ 0  is the potential well-depth and  r ij   =| r ij  |=| r j   − r i | is the absolute distance between the centers of theinteracting atoms  i  and  j . The potential is truncated at  r  ij  = r  c = 2.5 σ  0  in order to reduce calculation time. The potential isshifted down by the value  U  ( r  c ) in order to avoid a disconti-nuity at the cut-off distance. The force between atoms is F ij   = dU dr ij  r ij  r ij  ,  (2)where  F ij   is the force acting on atom  i  due to atom  j . Interac-tions between wall and fluid atoms are calculated in the sameway as interactions between a pair of fluid atoms.The physical quantities presented in this work are re-duced using the particle mass  m ∗ , interaction length scale σ  ∗  and the potential energy well-depth  ǫ ∗ , which sets theirnon-dimensional values to unity  m 0  = σ  0  = ǫ 0  = 1. The as-terisk is used to denote dimensional quantities. The reducedquantities are: length  r ij   = r ∗ ij  /σ  ∗ , density  ρ  =  ρ ∗ ( σ  ∗ ) 3  /  m ∗ ,number density  n = n ∗ ( σ  ∗ ) 3 , temperature  T  = k  B T  ∗  /  ǫ ∗ , stresstensor  σ   = σ  ∗ ( σ  ∗ ) 3 /ǫ ∗ , time  t   = t  ∗   ǫ ∗ / ( m ∗ ( σ  ∗ ) 2 ), force  f  =  f  ∗ σ  ∗  /  ǫ ∗ , strain rate ˙ γ   =  ˙ γ  ∗   m ∗ ( σ  ∗ ) 2 /ǫ ∗ ,  and viscosity η = η ∗ ( σ  ∗ ) 2 / √  m ∗ ǫ ∗ .The body force that acts on the atoms generates ther-mal energy leading to a temperature rise in the system. Tocontrol the temperature, the generated heat needs to be re-moved from the system. This is done via the Nosé-Hooverthermostat, which couples the atoms to a thermal reservoir. 60 In nature, heat is transported to the walls and the exchangeof momentum and heat between the wall and the fluid takesplace. We could try to mimic nature by allowing wall atomsto vibrate around their lattice sites and controlling the averagetemperature of the walls. However, since thermal walls wouldlead to a decrease in the near-wall inhomogeneity in which weare interested, we choose to fix the wall atoms and thermostatthe fluid locally next to the walls in order to obtain a con-stant temperature profile 58,61 and avoid the thermal slip 62,63 that would occur when the walls are thermostatted instead of the fluid. Since shear generates most heat in the vicinity of thewalls, the fluid is locally thermostatted in this region, but notin the center (bulk) region. On both sides of the channel, threethermostats are located next to each other, each of width 1.The first thermostat, seen from the wall, begins on a distanceof 0.15 from the center of the inner wall layer. Thus, a re-gion of approximately 4.8 wide, in the center of the channel,is not thermostatted. This approach maintains a rather con-stant temperature profile in the fluid, as long as  f   is not toolarge, while a global thermostat does not always succeed 34,58 due to the strong variation in strain-rate across thechannel. III. OBTAINING MACROSCOPIC QUANTITIES In molecular dynamics simulations, microscopic fields of any system are usually obtained by averaging the propertiesof many individual atoms and interactions. Depending on theproblem, properties can additionally be averaged over spaceorovermultipletimesteps.Thesimplestwaytocomputesuchaverages is to associate physical properties with the center of mass coordinates of each atom. Theoretically, the Dirac deltafunction  δ  is used to assign a physical quantity to the center of an atom. For example, the microscopic mass density at point r  and time  t   is obtained as ρ m ( r ,t  ) = N   i = 1 m i δ ( r − r i ( t  )) ,  (3)where  m i  is the mass of atom  i ,  r i  is its position, and  N   thenumber of fluid atoms. Other quantities can be defined in asimilar fashion. 64 A finite number of point-particles in continuous spaceimplies that the mass is zero everywhere, except at the atoms’center of mass. The discontinuities in this (that lead to sin-gular derivatives) can be avoided by averaging over discretevolumes in space, such as binning. However, information islost in the binning process, i.e., it is impossible to recoverthe raw data from the bin-averaged values. Furthermore, itrequires a large amount of statistics to obtain a smooth micro-scopic field, without averaging out small-scale physical struc-tures, by using bin averaging. These disadvantages of bin-ning can be avoided by using a more convenient smoothingmethod.In this paper we will not use binning, instead wesmoothen the data by replacing the Dirac delta function (seeEq. (3)) by a smoothing kernel that we will denote by  φ .Goldhirsch 49 described the requirements of a kernel in de-tail and states that it is of minor importance which functionis used. The level of smoothing, or smoothing length, on theother hand, can have a large influence on the macroscopicfields. When the obtained macroscopic fields are not stronglydependent on the smoothing length, for a range of values(“plateau”), then the smoothing possibly creates a meaning-ful macroscopic field. The existence of a plateau and theappropriate amount of smoothing strongly depends on thesystem. For a detailed discussion, the reader is directed toGoldhirsch 49 and references therein.In this study, we use a Gaussian kernel to spatiallysmoothen the microscopic data φ ( r ) = 1( √  2 πw 2 ) D e − | r | 22 w 2 ,  (4)  044711-4 Hartkamp  et al.  J. Chem. Phys.  137 , 044711 (2012) wherethedimensionofthesystemisdenotedwith  D ,thevari-ance, w 2 ,determines theamount ofsmoothing,whilepreserv-ing the shape and the area under the curve (    φ ( r ) d  r = 1).The kernel is cut off at a distance of 3 . 0 w  from the center.The smoothing kernel has the dimensions of inverse volume,therefore, integrating the kernel over a volume gives a dimen-sionless quantity. The higher the value of   w , the wider infor-mation is diffused (smeared out). The special case of   w  = 0refers to the “point-particle” case as shown in Eq. (3). For thesystem studied here, the smoothing has to be small enoughsuch that the width of the Gaussian is narrow compared tothe length scales of the spatial inhomogeneities observed instrongly confined fluids, but large enough to eliminate thethermal fluctuations from the macroscopic fields. A value of  w  = 0 . 1, as will be used below, has shown to satisfy theseconditions and result in fields which do not strongly dependon the value chosen for  w . A more detailed discussion of coarse-graining can be found in Ref. 65.In addition to spatial smoothing, the steady-state simula-tion data in this paper are averaged over discrete snapshots inorder to increase the statistics. A. Streaming velocity and strain rate The streaming velocity  u  can be calculated from the ratiobetween momentum and mass density u ( r ) = J ( r ) ρ ( r ) ,  (5)where  ρ ( r ) =  N i = 1  m i φ ( r − r i ) is the reduced mass densityand  J ( r ) =  N i = 1  m i v i φ ( r − r i ) the reduced momentum den-sity, with  v i  the velocity of atom  i . The velocity gradient ∇  u can be calculated analytically from the mass and momentumdensity and their gradients by applying the quotient rule toEq. (5). Note that fluctuations, i.e., large gradients in the mass and momentum density blow up in the velocity gradients’fluctuations too. Alternatively, the streaming velocity andstrain rate can be calculated from the displacement field. Av-eraging the strain rate over a time interval   t   offers additionalspatial and temporal smoothing compared to the velocity gra-dient and hence reduces noise. Therefore, we compute the lin-ear displacement field over a time interval   t  , as defined inRef. 66, U lin ( r ,t  ) = 1 ρ ( r ,t  ) N   i = 1 m i U i ( t  ) φ ( r − r i ( t  )) ,  (6)with  U i ( t  ) = r i ( t  ) − r i ( t   − t  ) the displacement of atom  i during time interval   t  . The linear strain can then be com-puted from the displacement gradient, ǫ linαβ ( r ,t  ) = 12  ∂U  lin α  ( r ,t  ) ∂r β + ∂U  lin β  ( r ,t  ) ∂r α  .  (7)In Fig. 3 (Sec. V), we compare the streaming velocity with thedisplacementrate U lin ( r ,t  ) t  − 1 ,andthevelocitygradientwith the strain rate  ǫ linαβ  t  − 1 , where   t   is the time intervalbetween snapshots. As expected, the displacement and strainrates over a time interval   t   are smoother than the velocityfield and its gradient, respectively. B. Temperature The kinetic temperature is computed straightforwardlyfrom the fluctuation velocities  v ′ i  of the atoms following theexpression: T  ( r ) = 2 K ( r ) Dn ( r ) = 1 Dn ( r ) N   i = 1 m i v ′ i  · v ′ i φ ( r − r i ) ,  (8)where  K   is the kinetic energy density,  D  is the dimension of the system,  v ′ i  = v i  − u ( r ) is the fluctuation (or thermal) ve-locity of atom  i , defined as the difference between the labora-tory velocity  v i  and the streaming velocity  u  at the location of the function evaluation  r . The kinetic temperature is kept con-stant in the simulations by means of local thermostatting, 58 see Sec. II. C. Stress calculation Calculatingthelocalstressinstronglyconfineddenseflu-ids has been a much studied subject. 12,20,46,48,50,67–70 Variousexpressions have been derived, differing mostly in their phys-ical interpretation. The first stress tensor for inhomogeneousfluids was introduced by Irving and Kirkwood. 67 In lateryears, a number of methods have been developed to calculatethe local stress tensor in an inhomogeneous fluid. 20,48,50,67–69 The microscopic method, which is introduced by Schofieldand Henderson, 50 is used here in combination with a Gaussiankernel, as also done by, e.g., Shen and Atluri, 48 Goldhirsch, 49 and Weinhart  et al. 65 —see also references therein.The stress can be decomposed into a kinetic energy (dy-namic) and a potential energy (configurational) part:  σ  ( r ) = σ  K ( r ) + σ  U  ( r ). The former part is associated with mo-mentum transport, while the latter accounts for interactionsbetween pairs of atoms. Due to the different nature of bothcontributions, some extreme scenario’s can be identified. In adilute gas, the average distance between atoms is much largerthan in a liquid or solid. Hence, the forces are small and theconfigurational stress is small in comparison to the dynamicstress. In a highly compressed dense solid/liquid, at moder-ate temperatures, the opposite applies: the close packing re-sults in large forces and thus a high potential stress, whereasthe transport of momentum (due to fluctuations) is relativelysmall. In a typical liquid as considered in the following, bothterms are of the same order of magnitude and neither part canbe neglected.A force acting on a fluid in a fixed volume  V   should beequal to the rate of change of linear momentum within  V   andthe force acting on the surface  δV  . The change of momen-tum can be caused by interaction with atoms outside of thevolume, or by atoms which exchange momentum with theboundary of the volume (e.g., by leaving the volume). Thelatter is described by the fluctuating kinetic energy densitypart of the stress tensor, σ  K ( r ) = N   i = 1 m i v ′ i v ′ i φ ( r − r i ) ,  (9)where  v ′ i v ′ i  denotes the tensor (dyadic) product between thethermal velocity vectors. It can be seen that in case of 
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