A Nonequilibrium Molecular Dynamics Study of the Rheology of Alkanes S.A. Gupta, S. T. Cui, P. T. Cummingsand H. D. Cochran Department of Chemical Engineering University of Tennessee Knoxville, TN 37996-2200 and
Chemical Technology Division Oak Ridge National Laboratory Oak Ridge, TN 37831-6268 Abstract
We examine the rheological properties of four di erent alkanes - n-decane (C10H22), n-hexadecane (C16H34), n-tetracosane (C24H50) and squalane (C30H62). The simulations of Couette ow are performed for a range of shear rates with 100 molecules in each case using a replicated data version of our code. The number of interaction sites ranges from 1000 ? 3000. We have performed extremely long simulations required to obtain acceptable statistics at low shear rates. The alkanes show a transition from non-Newtonian to Newtonian behavior as the shear rate decreases to low values. KEYWORDS: Nonequilibrium Molecular Dynamics, rheology, alkanes
1 Introduction The use of molecular-based approaches to address the problem of lubricant performance has gained signi cant momentum in the past few years. Much of the work has focused on relatively simple systems, the primary limitation being the high computational cost involved in characterizing these lubricants. The use of simple systems cannot give useful quantitative predictions for actual lubricants. The calculations that we have performed to determine the rheological properties of liquid alkanes in the intermediate molecular
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size range (C ? C ), can be eciently done with codes running on massively parallel supercomputers. Such calculations allow systematic studies of these systems with the aim of identifying the most important properties a ecting their performance as lubricants. They also aid the design of synthetic lubricants with desired properties. Planar Couette ow has been the focus of active research using non-equilibrium molecular dynamics, in part due to its applicability in lubrication science. The theoretical foundations of this non-equilibrium system and the non-equilibrium molecular dynamics (NEMD) techniques developed for its study have been addressed [1]. In most studies of planar Couette ow, a reversible deterministic thermostat such as a Nose thermostat is incorporated into the equations of motion to permit a steady state to develop. The thermostatted SLLOD equations of motion are solved under the so-called Lees-Edwards periodic boundary conditions. The interested reader is referred to [1] for technical details. A common approach in simulations of chain molecule systems, like alkanes, has been to use the method of constraints for the internal degrees of freedom, e.g. bond bending and stretching modes. This method is computationally advantageous since it allows using a larger time step than that permissible for a fully exible model. Another approach to decreasing cpu time is to utilize the short-ranged purely repulsive potential model for the intermolecular site-site interaction. This potential was used in the NEMD simulations of eicosane (C H ) [2]. The disadvantage of the repulsive potential model is that it fails to yield quantitative comparison with experimental data. A more realistic simulation using the Ryckaert{Bellemans model [3] was used to determine the rheological properties of n-hexadecane [4] at a relatively high temperature and density. The calculated Newtonian viscosity was lower than the experimental value by a factor of almost three. To obtain improved quantitative predictions of the equilibrium and non-equilibrium properties of a molecular uid, it becomes essential to use a realistic and exible molecular model, at the expense of increased computational requirements. Such a model was used to predict accurately the vapor-liquid phase envelope of linear alkanes in the range C ? C [5]. The model was subsequently used by Mundy et al. [6] and Cui et al. [7] to predict quantitatively the viscosity of n-decane at a liquid state condition similar to those in the phase envelope calculation (high temperature, moderately high density). Mondello and Grest [8] used the same model to determine the equilibrium properties of linear and branched alkanes at near ambient temperature conditions, obtaining reasonable agreement with experimental results. There is a signi cant practical interest in extending these studies to medium size alkanes (C ? C ) as the main constituents of lubricant basestocks. However, given the enormous computational requirements of such systems, it is essential to develop ecient parallel NEMD algorithms. Our approach uses a fully exible (including stretching, bending, and torsion) united atom model to describe the molecular sites. To aid in computational eciency we use a multiple timestep method based on a phase variable Liouvillean derived from the SLLOD equations of motion [1]. The work of Cui et al [9] implements the algorithm on the Intel Paragon in order to enable the NEMD simulation of long-chains. In this paper, we present our computational approach and the results of such an NEMD study to determine 20
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the rheological properties of four di erent alkanes - n-decane (C H ), n-hexadecane (C H ), n-tetracosane (C H ) and squalane (C H ). The rst three alkanes are linear while squalane is a molecule with six symmetrically placed, methyl side-groups. Parallel computers make it possible to run the extremely long simulations required to obtain improved statistics at low shear rates. 10
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2 Model, Theory and Simulation method For a system subject to planar Couette ow, the motion of the particles can be described by the SLLOD algorithm which is known to be valid in both the linear and nonlinear response regimes [1]. The SLLOD equations of motion (incorporating a Nose thermostat) for a system undergoing planar Couette are given by r_ ia = mpiaia + yia x^; p_ ia = Fia ? py;ia x^ ? _pia
(1) _ = pQ p_ = F where ria and pia are the vector coordinates and peculiar momentum of atom a in molecule i, yia and py;ia are its y components, mia is the mass, x^ is a unit vector in the x direction, , p , and Q are the variables related to the Nose thermostat, X F = mpia ? 3NkB T; Q = 3NkB T i;a ia
2
2
is the Nose thermostat time constant, and N is the total number of atoms in the system. The model used for the alkanes is the united atom model previously used by Siepmann et al. [5], [10], Mundy et al. [6], and by Cui et al. [7]. In this model the methyl and methylene groups are treated as spherical interaction sites with the interaction centers located at the center of the carbon atoms. Details of the model can be found in the literature [5], [10]. For completeness, we brie y describe the model. The interaction between atoms on di erent molecules and atoms separated by more than three bonds in the same molecule is described by a Lennard-Jones (LJ) potential, " # ij uij (r) = 4ij r ? rij (2) where ij is the well depth and ij is the zero point of the potential between a site of type i and a site of type j . The Lennard-Jones size parameters are CH3 = 3:93 A, CH2 = 3:93 A o o and CH = 3:81 A. The corresponding well depth parameters =k are 114 K, 47 K and B i h 40 oK respectively. The Lorenz-Berthelot combining rules ij = (ij ) = ; ij = are used for the cross interactions. A cuto distance of 9:825 A for the LJ interaction was 12
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used in this study. The bonded (intramolecular) interactions include a vibration term, a bending term and a dihedral term. The bond stretching and bond angle bending are described by harmonic potentials while the torsional potential is that by Jorgensen et al. [11]. For the simulations of squalane, an ad hoc harmonic potential similar to the bending term, is introduced to prevent umbrella inversion of the sp bond con guration at tertiary carbon branch points. This potential was used by Mondello and Grest [8] and it forces the normal to the plane de ned by the two backbone bonds joining at the CH group to oscillate around its equilibrium position, which forms an angle of 27:25o with the normal to the plane de ned by the three CHn groups connected to the tertiary carbon. The state points studied correspond to the densities of the liquid alkanes at atmospheric pressure and at the respective temperatures [12], [13]. These are listed in Table 1. In an NEMD calculation, the strain-rate dependent viscosity is determined from the constitutive relation (3) = hP xy i ; where hPxy i is the average of the xy component of the pressure tensor P and is the strain rate characterizing the shear eld. We have chosen the x direction to be the ow direction and the y direction to be the ow gradient direction, so that = @ux=@y, where ux is the streaming velocity in the x direction. The pressure tensor P is calculated using the atomic formalism, which is given by X 1 X X(r ? r )f + X r f intra ; P a V = miavia via + (4) 2 i;a i;b ia jb ia;jb i;a ia ia i;a 3
(
( )
)
| {z } i6=j
where the indices i and j refer to molecules i and j , indices a and b refer to interaction sites a and b in molecules i and j , respectively. The shear alignment is calculated using the order tensor de ned by the second rank tensor [14], *
N 1 3 1X ( ei ei ? 1) S= 2 Ni 3
+
(5)
=1
Table 1: State points simulated for the liquid alkanes. Alkane T oK (gm=cm ) n-decane 298 0:7247a n-hexadecane (A) 300 0:770b n-hexadecane (B) 323 0:753b n-tetracosane 333 0:7728b squalane 333 0:7837b a Ref. [12], b Ref. [13] 3
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where ei is the unit vector along the end-to-end direction of the molecule i, 1 is a unit second rank tensor and the summation is over all N molecules in the system. The algorithm used for the NEMD simulations is an extension to sheared systems [7] of the application to alkanes of the multi-timestep Nose dynamics algorithm developed in the Tuckermann-Berne reversible reference system propagator algorithm method [15]. Details can be found in [7]. We use two timestep sizes, with all intramolecular interactions being treated as fast motions and the intermolecular interactions as slow motions. The simulations of Couette ow are performed for a range of shear rates with 100 molecules in each case using a replicated data version of our code. The number of interaction sites ranges from 1000 ? 3000. A timestep of 2:35 fs is used for the slow mode of motion. For decane, hexadecane and tetracosane, a time step of 0:235 fs is used for the fast mode of motion, while the corresponding value for squalane is 0:5875 fs. The strain rate dependent properties of the systems are calculated based on the simulation runs after the systems have reached steady state under the in uence of the shear eld.
3 Replicated data code For rapid parallelization, we have used a replicated data strategy. This method is essentially what Plimpton [16] termed an \atomic decomposition" method, and was chosen for its simplicity and e ectiveness in parallelizing an existing serial code. Since the action of the serial code has already been proven, the ability to rapidly parallelize an existing, scienti cally-complex code aids productivity enormously. The general strategy relies upon the fact that the most compute-intensive portion of the code is the calculation of the system of forces at work. There are basically two classes of force interactions considered, i.e., intermolecular and intramolecular. For the intermolecular portion of the force calculation, we require data characterizing the entire system state be present upon every node; hence the term \replicated data" refers to the replication of all of the particle positions. The force is evaluated within a double loop that compares uniquely every particle with every other particle. The use of a neighbor list optimizes the calculation time, by considering only those particles that are within a certain cuto distance from a given particle. The computational work is divided up among nodes by assigning a range of outer loop iterations to each node. These can be worked upon in parallel, and the simplicity of the division permits some straightforward load balancing to take place. At least some form of load balancing is required due to the fact that the quantity of computational work in the double loop structure is not uniformly distributed. In fact, due to the neighbor list being used as an optimization strategy, the earlier iterations will carry much more work than later iterations. To counter this imbalance, we have simply divided the work for a set of N nodes, as if we had 2N available nodes. We then assign iterations from opposite ends of the outer loop, in e ect, giving each node a large and a small portion to work on. This method has the advantage that we can decompose the work according to the number of particles divided by 2, in the case of maximum parallelism. 5
Further parallelism can be realized if we pursue inner loop distribution as well, but this is currently not advantageous. In addition to the e ectiveness of this technique, it is very straightforward to implement. Although currently a static load balancing technique is used, a dynamic action could be easily added for little extra cost, if it were required. For the intramolecular portion of the force calculation, there is a di erent action at work. Due to the multi-timestep approach adopted in this simulation, the intramolecular arrangement changes prior to the calculation of the intermolecular forces. These calculations require only local information regarding the particle positions of a single molecule, but that also xes the maximum decomposition to the molecular level. A ner level of molecular decomposition might be possible for larger molecules, but would probably require repeating calculations to avoid additional communication between nodes. Hence, in the strategy we have implemented, two communications are required to ensure a correct simulation action. One is after the integration step following the intramolecular force evaluation, and the other follows the intermolecular force evaluation. The rst communication collects and distributes all of the particle positions onto all of the processors. The intermolecular forces are then evaluated and these atomic forces are then summed globally, so that all nodes have sucient information to update the system via integration, and commence the next round of intramolecular calculations.
4 Simulation Results The longest simulations hreported in this study are for about 9 million timesteps. The i = in Fig. 1 ifor the four alkanes, dimensionless viscosity = =(m) is plotted h as a function of the dimensionless strain rate = (m =) = . These state points correspond to the equilibrium density at atmospheric pressure and at the respective temperatures. Also plotted in Fig. 1 for comparative purposes, is the viscosity for methane (CH ) at the Lennard-Jones triple point calculated using a short-ranged purely repulsive potential [17]. As can be seen from the gure, the viscosities for the alkanes exhibit a shear thinning behavior over the range of strain rates studied. This is typical of chain molecule uid systems. At high strain rates, the shear thinning follows a power law. The slopes of the log-log plots vary from -0.33 to -0.51 as compared to the experimentally observed slopes of -0.4 to -0.9 for polymeric uids. This likeness suggests that even though the alkane chains studied here are short in comparison to polymer systems, they nevertheless exhibit some of the generality of long-chain systems. Another important observation is that the shear viscosities for decane, hexadecane and tetracosane nearly overlap each other at high strain rates. At high strain rates these fairly short and sti alkane chains are well aligned with each other so they can slide past each other relatively easily. Furthermore, as shown by the calculated alignment angles, the longer chains align themselves better resulting in the above mentioned insensitivity of viscosity with the variation of chain length at high strain rate. For squalane, which has the same backbone chain length as tetracosane, the 6 additional methyl side groups 2
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C1 (90 K) C10 (298 K) C16 (300 K) C16 (323 K) C24 (333 K) C30 (333 K)
η∗
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Figure 1: Dimensionless viscosity () for linear alkanes as a function of the dimensionless shear rate ( ) result in viscosities at high strain rate that are about twice those of tetracosane. The alignment angles for squalane at these strain rates are identical to those for tetracosane within statistical uncertainty.
5 Summary We have developed a replicated data nonequilibrium molecular dynamics code to determine the rheological properties of decane, hexadecane, tetracosane, and squalane. Due to the long simulations required to obtain good statistics for these chain molecules, these simulations can be eciently performed only on a massively parallel supercomputer. Work is currently in progress to study other branched alkanes.
Acknowledgments
The work of HDC has been supported by the Division of Chemical Sciences of the U. S. Department of Energy (DOE) at Oak Ridge National Laboratory (ORNL). This work was sponsored by the Laboratory Directed Research and Development Program of ORNL. The authors acknowledge the use of the Intel Paragon supercomputers in the Center 7
for Computational Sciences(CCS) at ORNL, funded by the DOE's Mathematical, Information, and Computational Sciences Division. ORNL is managed by Lockheed Martin Energy Research Corp. for the DOE under Contract No. DE-AC05-96OR22464. The authors also thank Phil LoCascio of CCS for his participation in this work.
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