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J. Phys. Chem. B 2010, 114, 16487–16493

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On the Use of Excess Entropy Scaling To Describe Single-Molecule and Collective Dynamic Properties of Hydrocarbon Isomer Fluids Ravi Chopra,† Thomas M. Truskett,‡ and Jeffrey R. Errington*,† Department of Chemical and Biological Engineering, UniVersity at Buffalo, The State UniVersity of New York, Buffalo, New York 14260, United States, and Department of Chemical Engineering and Institute for Theoretical Chemistry, UniVersity of Texas at Austin, Austin, Texas 78712, United States ReceiVed: August 19, 2010; ReVised Manuscript ReceiVed: October 8, 2010

We use molecular simulation to study the ability of excess entropy scaling relationships to describe the kinetic properties of four hydrocarbon isomers: n-octane, 2,2-dimethylhexane, 2,5-dimethylhexane, and 3-methyl3-ethylpentane. Four dynamic properties are considered: translational and rotational diffusivities, a characteristic relaxation time for rotational motion, and a collective relaxation time stemming from analysis of the coherent intermediate scattering function. For each of the dynamic properties considered, reduced data collapse onto a species-specific common curve when expressed as a function of the thermodynamic excess entropy. Because each isomer exhibits a quantitatively distinct excess entropy scaling relationship, straightforward corresponding states principles do not provide an effective means to predict dynamic properties. I. Introduction Hydrocarbons are an important class of chemical compounds. They serve as a feedstock for the production of numerous fine chemicals, form the building blocks of many biological molecules, and play a central role in the energy field. In many design applications, scientists and engineers are interested in the kinetic properties of hydrocarbon liquids, such as diffusivities, viscosities, and thermal conductivities. In this work, we examine the prospects of using excess entropy scaling relationships1-5 to describe the transport properties of these fluids. Within excess entropy scaling strategies, one establishes a link between a kinetic property of interest and the excess entropy. The central idea is that the temperature and density dependence of an appropriately reduced transport property can be expressed as a single-valued function of the excess entropy.1,4 Provided such a relationship exists, one can use an estimate of the excess entropy to predict the dynamic properties of a fluid at a specified state point. Much of the work in this area has been focused on fluids characterized by spherically symmetric interparticle interactions (e.g., atomistic fluids).6-21 These studies have demonstrated that excess entropy scaling techniques provide a robust means to quantitatively describe the dynamic behavior of bulk and confined fluids that behave “simply”, that is, fluids for which measures of mobility monotonically decrease upon isothermal compression. For such fluids (e.g., hard sphere, Lennard-Jones, etc.), bulk transport properties follow a “quasiuniversal” scaling function of the excess entropy introduced by Rosenfeld,4 which is exponential in form for conditions that range from dense gases to the freezing transition. More recently, Krekelberg et al. have introduced a “generalized Rosenfeld” scaling procedure17 wherein transport properties are nondimensionalized using knowledge of both thermodynamic and kinetic fluid behavior in the low particle density limit. This approach, which guarantees excess entropy scaling behavior at low density, has also been shown to accurately characterize the dynamic * Corresponding author. E-mail: [email protected]. † University at Buffalo, The State University of New York. ‡ University of Texas at Austin.

properties of various dense model fluids governed by either hard or soft interparticle interactions (e.g., Gaussian core fluid). Many of the recent investigations22-32 addressing excess entropy scaling have focused on fluids possessing rotational and intramolecular degrees on freedom (e.g., molecular fluids). These studies have shown that, to within a good approximation, appropriately reduced dynamic properties of molecular fluids can also be expressed as single-valued functions of the total (thermodynamic) excess entropy. However, one finds that Rosenfeld’s “universal” scaling function does not extend beyond atomistic fluids. Instead, a unique scaling relationship exists for each molecular species. Relevant examples stem from the experimental work of Abramson and West-Foyle, who have examined the relationship between viscosity and excess entropy for several small-molecule fluids, including nitrogen, oxygen, carbon dioxide, and water.22,23 They demonstrated that reduced viscosity data for these systems approximately collapse onto species-specific common curves when plotted versus the total excess entropy. The Chakravarty group has used molecular simulation to study models for chain fluids,24 water,25 and network-forming ionic melts.25-27 They have shown that both reduced diffusivities and viscosities of linear Lennard-Jones chain fluids scale with the exponential of the excess entropy.24 Galliero and Boned subsequently demonstrated that reduced thermal conductivities of Lennard-Jones chain fluids also collapse to single-valued functions of the excess entropy.28 For each of the chain fluid transport properties examined,24,28 the scaling relationships for different species (chain lengths) were found to be qualitatively similar, but quantitatively distinct. This observation was also reported by Gerek and Elliott,28 who studied the robustness of Rosenfeld scaling relations for the n-alkane homologous series. We recently examined the efficacy of excess entropy scaling relations with models for water30 and for fluids of dumbbell-shaped particles.31,32 Consistent with previous studies, we found that the scaling relationships for dumbbells of different length were not transferable. In this work, we further investigate issues related to the application of excess entropy scaling strategies to molecular fluids. Molecular simulation is employed to determine kinetic

10.1021/jp107878u  2010 American Chemical Society Published on Web 11/22/2010

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and thermodynamic properties of various eight-carbon saturated hydrocarbons, including n-octane, 2,2-dimethylhexane, 2,5dimethylhexane, and 3-methyl-3-ethylpentane. We first examine the extent to which excess entropy scaling concepts can be used to describe the density and temperature dependencies of four dynamic properties: translational diffusion coefficient, rotational diffusion coefficient, a rotational relaxation time, and a collective relation time. In each case, we find that, to within a good approximation, the appropriately reduced dynamic property for a given species collapses to a single-valued function of the excess entropy. We next consider the issue of transferability with respect to the scaling relations. If straightforward transferability strategies were applicable, then a single relation could be used to describe a dynamic property of fluids with chemically and/or topologically distinct molecules. Previous studies, however, suggest that topologically identical molecules do not share the same quantitative scaling relationships (recall the collection of linear molecules studied by Abramson and West-Foyle23). The group of isomers studied here allows us to examine the extent to which a collection of molecules differentiated by molecular topology only share similar quantitative scaling relationships. Our results reveal that excess entropy scaling relations, while useful for a fluid with a particular type of molecule, are not generally transferable among fluids of its isomers. This Article is organized as follows. In the following section, we describe the model examined in this work and the molecular simulation methods used to compute thermodynamic and transport properties of interest. Next, we present our simulation results for the evolution of the dynamic properties with temperature and density and discuss the ability of excess entropy scaling to capture the trends observed. II. Molecular Simulations Molecular Model. We work with the united-atom version of the transferable potentials for phase equilibria (TraPPE) force field.33,34 Saturated hydrocarbons are constructed from C, CH, CH2, and CH3 pseudo atoms with nonbonded interactions described by the Lennard-Jones potential. The set of nonbonded interactions is defined by all pseudo atom pairs separated by more than three intramolecular bonds. Bending and torsion angles are constrained with harmonic and cosine series potentials, respectively. We replace the fixed bond lengths designated within the original TraPPE model with harmonic springs characterized by the stretching potential u(r) ) 0.5Kr(r - ro)2, with ro ) 1.54 Å and Kr ) 2250 kJ/mol · Å2. We refer to the stretching, bending, and torsion potentials as the set of bonded interactions. Thermodynamic Properties. We employ free-energy-based simulation techniques to evaluate the thermodynamic excess entropy. Here, excess properties are defined as the difference between the property of a real fluid and an ideal gas at the same temperature and density. We utilize a two-step process similar to that described in previous reports30-32 to determine the free energy of the real fluid at state conditions of interest. In the first step, we obtain the density dependence of the Helmholtz free energy at relatively high temperature using GC-TMMC simulation.35,36 In the second step, we perform a canonical temperature expanded ensemble37 transition matrix Monte Carlo (TE-TMMC) simulation30-32,38 to evaluate the change in Helmholtz free energy with temperature at constant density. These simulations also enable us to determine the configurational energy at a state point of interest via a straightforward ensemble average. Because the model fluids examined here contain

Chopra et al. intramolecular degrees of freedom, we also perform a series of simulations to determine the properties of the fluid in the ideal gas state. Collectively, these simulations provide the excess Helmholtz free energy at a given temperature and density of interest. The excess Helmholtz free energy and excess energy are then combined via standard thermodynamic relationships to yield the excess entropy. Grand canonical simulations are conducted at a specified temperature T, volume V, and activity ξ ) qzr exp(βµ), where β ) 1/kT is the inverse temperature, k is Boltzmann’s constant, µ is the chemical potential, q represents the component of the molecular partition function stemming from integration over momenta, and zr represents the molecular configurational partition function of a molecule within the reservoir in contact with the real fluid. Here, the reservoir39 is taken to be a collection of “ideal chains”, which are defined as molecules governed by bonded interactions only. This state is different from the ideal gas state, wherein molecular configurations are governed by both bonded and nonbonded intramolecular interactions. We use a reservoir-based39 expanded ensemble strategy40-42 to gradually insert and remove molecules from the simulation cell. Hybrid Monte Carlo moves43 are used to sample the intramolecular degrees of freedom for molecules both in the central simulation cell and within the reservoir. Transition matrix Monte Carlo44,45 and multicanonical sampling46 techniques are employed to promote uniform sampling of the system over a prespecified particle number range. The key quantities extracted from each simulation are the particle number (density) probability distribution Π(N) and the configurational energy U(N), which includes all bonded and nonbonded (intermolecular and intramolecular) energies. ˜ (N) is related to The normalized probability distribution Π the canonical partition function Q(N) and grand partition function Ξ(ξ) as38,47 βµN N ˜ ˜ (N) ) e Q(N) ) Π(0)ξ Q(N) Π Ξ(ξ) (qzr)N

(1)

where the grand partition function is recast using the relationship ˜ (0), which stems from consideration of the N ) 0 Ξ(ξ) ) 1/Π case. The expression above provides a link to the Helmholtz free energy F(N):

ξ + N ln( ) ] [ Π(N) Π(0) qz

βF(N) ) -ln Q(N) ) -ln

r

(2)

Note that the normalized probability distribution has been replaced by the more general unnormalized version. Determination of the excess function requires the Helmholtz free energy of an ideal gas at the same temperature and density of the real fluid:

[

βFig(N) ) -ln

(Vqzig)N N!

]

(3)

where zig represents the molecular configurational partition function of a molecule within the ideal gas state. The excess Helmholtz free energy is now given by:

Entropy Scaling To Describe Hydrocarbon Isomer Fluids

[

βFex(N) ) -ln

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()

Π(N) zig + N ln(ξV) + N ln r - ln N! Π(0) z (4)

]

The last piece of information needed to evaluate Fex(N) is the ratio zig/zr, which is related to the free energy difference between isolated molecules in the ideal chain and ideal gas states. There are a number of ways that one could obtain this quantity. Here, we perform a GC-TMMC simulation restricted to the N ) 0 and N ) 1 states. The molecule in the N ) 1 state is effectively isolated by eliminating long-range corrections to the energy while including all nonbonded intramolecular interactions. The particle number probability distribution from this simple simulation Πig(N) provides the ratio of interest: ig

[ ] ig

z 1 Π (1) ) ξV Πig(0) zr

(5)

which stems from eq 1 with Q(1) ) qzigV. This simulation also provides the configurational energy of a molecule in the ideal gas state uig via an ensemble average of the bonded and nonbonded energies for the N ) 1 state. Finally, we arrive at an expression for the excess entropy Sex(N):

Sex(N)/k ) β[U(N) - Nuig] - βFex(N)

(6)

The second general tool that we use (TE-TMMC) enables us to determine the change in excess properties with temperature at constant density. A canonical temperature expanded ensemble37 consists of a collection of subensembles that share the same particle number and volume while possessing different temperatures. In this work, we take the inverse temperature β as the order parameter and establish a set of subensembles that range from βmin to βmax in increments of ∆β. The relative probabilities Π(βm) and Π(βq) of the system visiting subensembles m and q provide the ratio of the configurational partition function at βm and βq,37 or equivalently the change in the reduced Helmholtz free energy between βm and βq:

[ ] Π(βm) Π(βq)

βmF(βm) - βqF(βq) ) -ln

(7)

During the TE-TMMC simulation, we also obtain ensemble averages of the configurational energy U(β). To determine the temperature dependence of the excess properties, we must also consider how the ideal gas properties vary with temperature. We obtain the necessary information by performing a TETMMC simulation with a single isolated molecule governed by bonded and nonbonded intramolecular interactions. The probability distribution Πig(β) provides the variation in the reduced Helmholtz free energy of the ideal gas with temperature:

[ ]

βmf ig(βm) - βqf ig(βq) ) -ln

Πig(βm) Πig(βq)

(8)

Configurational energies uig(β) are also collected during the ideal gas simulation. Finally, the change in the excess entropy with temperature for a system with N molecules is given by:

[Sex(βm) - Sex(βq)]/k ) [βmU(βm) - βqU(βq)] -[βmF(βm) - βqF(βq)] -N[βmuig(βm) - βquig(βq)]

(9)

+N[βmf ig(βm) - βqf ig(βq)] Transport Properties. We calculate the translational selfdiffusivity Dt by fitting the long-time behavior of the meansquare displacement of the alkane center of mass using the Einstein relation: 2 lim〈| b(t) r - b(0)| r 〉 ) 6Dtt tf∞

(10)

where b(t) r is the center-of-mass position vector at time t. We calculate a rotational diffusivity using an approach followed by Lee and Lee.48 For each of the pseudo atoms within a molecule, we first define a normalized polarization vector pˆij(t) using the line connecting the center of mass of molecule i and the location of pseudo atom j. For each of these vectors, we define a rotational displacement φ bij(t) as

f φij(t) )

∫0t ∆φfij(t') dt'

(11)

where ∆φ bij(t) is a vector with direction given by pˆij(t) × bij(t)| ) pˆij(t + dt) and with magnitude given by |∆φ cos-1[pˆij(t) · pˆij(t + dt)]. We calculate the rotational diffusivity Dr by fitting the long-time behavior of the rotational mean square displacement:

f (t)| 2〉 ) 4D t lim〈|φ ij r tf∞

(12)

where the average extends over all rotational displacement vectors. To complement the translational and rotational diffusion coefficients, we also calculate a collective relaxation time and dipole relaxation time. The collective relaxation time τ is defined as the time required for the normalized coherent intermediatescattering function F(q0,t) to decay to a value of e-1. The wavenumber q0 corresponds to the approximate location of the first peak in the structure factor. The dipole relaxation time τ2 associated with the rotational motion of an alkane is defined by the time required for the correlation function 〈P2[cos θ(t)]〉 to decay to a value of e-1. P2 is the second Legendre polynomial, and θ(t) is obtained by averaging the angles formed by the polarization vectors at times t and zero, θij(t) ) cos-1[pˆij(t) · pˆij(0)]. Simulation Details. We performed Monte Carlo (MC) and molecular-dynamics (MD) simulations to acquire thermodynamic and kinetic data over the range of state conditions defined by 280 < T < 1000 K and 0.4 < F < 0.8 g/cm3. For all MC simulations, we used a cell with volume V ) 100 000 Å3 and a cutoff distance of 14 Å for nonbonded interactions. For each of the four isomers studied, a single GC-TMMC simulation was completed at T ) 2000 K with ln ξ ) 0. TE-TMMC simulations were completed over the inverse temperature range T-1 ) 1.0 × 10-4 to 3.6 × 10-3 K-1, with a subensemble spacing of ∆T-1 ) 7.0 × 10-7 K-1. MD simulations were completed with the GROMACS package.49-52 The number of molecules was fixed at N ) 500, and the volume was adjusted to obtain the desired

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Figure 1. Evolution of (a) the translational diffusivity, (b) the rotational diffusivity, (c) the inverse of the collective relaxation time, and (d) the excess entropy with density. Squares, circles, diamonds, and triangles represent n-octane, 2,2-dimethylhexane, 2,5-dimethylhexane, and 3-methyl-3-ethylpentane, respectively. Within each plot, the lower (blue) and upper (red) data sets correspond to temperatures of 300 and 600 K, respectively.

density. The temperature was controlled using a Nose´-Hoover thermostat,53,54 and the system was propagated in time using the velocity-Verlet method55,56 with a time step of 1 fs. Statistical uncertainties were determined by performing four independent sets of simulations. The standard deviation of the results from the four simulation sets was taken as an estimate of the statistical uncertainty.

Chopra et al.

Figure 2. Relationship between reduced translational diffusivity and excess entropy. The shape of the symbol denotes the density as indicated in the legend. From top to bottom, the curves correspond to n-octane (red), 2,2-dimethylhexane (blue), 2,5-dimethylhexane (maroon), and 3-methyl-3-ethylpentane (green).

dynamic properties, such as temperature and density. More specifically, we work with the following quantities:

(F/M)1/3 (kT/M)1/2

(13)

1 (kT/M) (F/M)1/3

(14)

1 1 1 ) τ* τ (kT/M)1/2(F/M)1/3

(15)

1 1 1 ) τ*2 τ2 (kT/M)1/2(F/M)1/3

(16)

D*t ) Dt

III. Results and Discussion Kinetic and thermodynamic properties were collected in the general vicinity of the compressed liquid regions of the four hydrocarbons studied here, conditions for which exponential excess entropy scalings for dynamic properties might be expected to apply. The critical temperatures and densities of the isomers span the ranges 550-580 K and 230-240 kg/m3, respectively. Detailed information regarding the liquid-vapor saturation properties can be found in publications from the Siepmann group.33,34 Figure 1 provides the density dependence of the translational diffusivity Dt, the rotational diffusivity Dr, the collective relaxation time τ, and the excess entropy sex at temperatures of 300 and 600 K. The properties often evolve with isomer identity in a direction dictated by the sphericity of the molecule. On the basis of molecular topologies, we rank the degree of sphericity as 3-methyl-3-ethylpentane > 2,2dimethylhexane ≈ 2,5-dimethylhexane > n-octane. The translational diffusivity increases slightly as the isomer becomes less spherical. In contrast, the rotational diffusivity increases substantially as the isomer adopts a more spherical shape. The collective relaxation time, which is often associated with the shear viscosity, also increases with molecular sphericity (τ-1 is plotted in Figure 1). Finally, as the isomer becomes less spherical, the strength of the structural correlations of the fluid, measured by -sex, increases considerably. The trends observed here are similar to those reported by Lee and Lee for five-carbon saturated hydrocarbons (n-pentane, isopentane, and neopentane).48 We now examine the possibility of using excess entropy scaling relations to describe the dynamic properties of the four eight-carbon isomers. We adopt a Rosenfeld-style approach4 in which kinetic properties are reduced by macroscopic thermo-

D*r ) Dr

1/2

In what follows, we examine the extent to which sex can be used to forecast a given reduced dynamic property. ex relationships for the four isomers Figure 2 shows the D*-s t ex points approximately studied here. We find that all D*-s t collapse onto species-specific common curves. The general form ex curves is consistent with what has been reported of the D*-s t for fluids comprising linear Lennard-Jones chains24 and dumband sex bell-shaped particles.31 At moderate densities, ln(D*) t show a near-linear relationship, whereas at relatively high ex densities, one finds substantial curvature in the ln(D*)-s t relationship. For the fluids studies here, this trend is particularly ex curve noticeable with n-octane, for which the ln(D*)-s t becomes concave at large -sex. The results provided here add to a growing list of fluids for which excess entropy scaling concepts provide a good description of dynamic behavior. The data provided in Figure 2 clearly show that excess entropy scaling relationships, while robust for fluids of a given type of molecule, are not generally transferable among fluids of its various isomers. In fact, assuming that two pure fluids of different isomers at the same density and temperature have the

Entropy Scaling To Describe Hydrocarbon Isomer Fluids

Figure 3. Relationship between (a) reduced rotational diffusivity, (b) reduced rotational relaxation time, and (c) reduced collective relaxation time and excess entropy. Symbol shape and color are consistent with the definitions adopted in Figure 2.

same dynamic properties appears a more accurate strategy as compared to assuming transferability of excess entropy scalings, for estimating transport coefficients of one fluid based on data from the other. As an illustrative example, we consider the prediction of Dt for 3-methyl-3-ethylpentane at T ) 400 K and F ) 750 kg/m3 (sex/k ) -6.00) using n-octane as a reference. If we were to take as our estimate the value of Dt for n-octane at the same T and F, we would obtain Dt ) 3.2 × 10-9 m2/s. If instead we were to use the value of D*t for n-octane at the same excess entropy (sex/k ) -6.00, D*t ) 4.8 × 10-2), we would obtain Dt ) 5.2 × 10-9 m2/s. Our MD result for 3-methyl-3ethylpentane at these conditions is Dt ) 2.7 × 10-9 m2/s, indicating that, at least in this case, simply taking the diffusivity to be the same as that of another isomer at the temperature and density provides a better estimate than what one obtains from an excess entropy-based approach. Note, however, that the former approach generally requires one to collect kinetic data as a function of two variables (T,F), whereas within the excess entropy approach dynamic properties are mapped to a single thermodynamic variable (sex). ex Figure 3 provides the D*-s , τ*2 -sex, and τ*-sex relationr ships. In each case, we find relatively strong correlations between the reduced dynamic quantity and the excess entropy. Again, we find that the scaling relations are isomer-specific. Note that the noise apparent in the τ*-sex relations is somewhat ex due to relatively large uncertainties for τ*. The D*-s r ex relationships are relatively linear at low to moderate -s and display noticeable curvature at large -sex. We also find substantial curvature in the relationships involving an inverse relaxation time and the excess entropy. The general trends

J. Phys. Chem. B, Vol. 114, No. 49, 2010 16491 observed here are similar to what we found with fluids consisting of dumbbell-shaped particles.31 The relationships presented in Figures 2 and 3 point to interesting trends related to the shape of the isomers. At a similar level of structural order (measured by -sex), the translational mobility of n-octane is significantly higher than that of 3-methyl3-ethylpentane. In contrast, at a similar level of structural order, the rotational mobility of n-octane is substantially lower than that of 3-methyl-3-ethylpentane. These results show that as the spherical nature of the molecule increases at constant -sex, the rotational mobility increases at the expense of the translational mobility. We now make a connection with the scaling strategies pursued recently by Dyre and co-workers.21,57-62 They argue that one can generally expect excess entropy scaling relations to successfully predict dynamic properties of so-called “strongly correlating” fluids. In this context, a strongly correlating fluid is one that exhibits strong correlations between the equilibrium fluctuations of the potential energy and the virial. In such systems, appropriately nondimensionalized dynamic quantities such as the diffusion coefficient D*t and the total excess entropy approximately depend on a single scaling variable Fγ/T, a trend followed by soft-sphere particles interacting via inverse-powerlaw pair potentials. Dense fluid states of the monatomic LennardJones fluid, the Kob-Andersen binary Lennard-Jones mixture, a united-atom model for toluene, and an asymmetric LennardJones dumbbell model have been shown to be strongly correlating over a wide range of thermodynamic conditions, whereas models for associating fluids such as methanol and water do not typically display this type of strongly correlating behavior.58 It is interesting to note that reduced dynamic properties of water show robust excess entropy scalings over a wide range of conditions,25,30,63,64 indicating that while such scalings should be expected in strongly correlating fluids, they can emerge in other types of fluids as well. For strongly correlating systems, one can approximately map the thermodynamic, kinetic, and structural properties of the fluid onto those of a fluid governed by an inverse-power potential, with the pair interaction energy described by u(r) ∝ 1/r3γ. Within this approach, the mapping would be considered transferrable among (say) a group of isomers, if all species are described by γ /T the same scaling exponent γ. Figure 4 shows the D*-F t scaling relationships for the four hydrocarbon isomers with species-specific optimal values for γ. We obtained the optimal scaling exponents by maximizing the correlation coefficient γ between the ln(D*) t - ln(F /T) data set for a given isomer and a fifth-order polynomial fit to the same data set. This procedure yielded γ ) 6.5, 6.9, 7.1, and 7.8 for n-octane, 2,5-dimethylhexane, 2,2-dimethylhexane, and 3-methyl-3-ethylpentane, respectively. Similar to the excess entropy scaling relationships, γ /T scaling relationships are robust for we find that the D*-F t fluids of a given type of molecule, but are not generally transferable among fluids of its various isomers. We again note γ / a connection to the sphericity of the molecule. For the D*-F t T scaling relationships, we find that γ increases as the sphericity of the molecule increases. This indicates that the effective inverse-power pair interaction becomes “harder” as the molecule adopts a more spherical shape. This correlation seems physically reasonable, as one expects linear chains to share a softer, more penetrable, two-body potential than their highly branched isomers. Before closing, we briefly examine the temperature dependence of the excess entropy. Rosenfeld suggested that sex should scale with T-2/5 along an isochore, a result based upon the

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Figure 4. Relationship between reduced translational diffusivity and the inverse-power-potential-based scaling variable Fγ/T. Symbol shape and color are consistent with the definitions adopted in Figure 2.

Chopra et al. eight-carbon saturated hydrocarbons. Molecular dynamics simulation was used to obtain translational and rotational diffusion coefficients, a relaxation time associated with rotational motion, and a relaxation time intended to probe the collective dynamics of the system. Free-energy-based Monte Carlo simulation methods were used to compute the thermodynamic excess entropy. Our calculations were focused on state conditions within the compressed liquid regime. Our analysis suggests that the excess entropy is capable of describing the dynamic properties of relatively small linear and branched hydrocarbons. Rosenfeld-reduced dynamic properties collapse to species-specific single-valued functions of the excess entropy. We find that ln X*-sex curves, where X* is a general kinetic property, often exhibit significant curvature, which is consistent with what we have observed with other molecular fluids. Our analysis also indicates that isomers exhibit quantitatively distinct excess entropy scaling relationships. A possible extension to excess entropy scaling techniques is the development of corresponding states methods to map the scaling relationship of one species to that of another. The generalized Rosenfeld scaling approach17 introduced by Krekelberg et al. may provide a means to accomplish this goal. Acknowledgment. J.R.E. acknowledges financial support of the National Science Foundation, Grant No. CBET-0828979. T.M.T. acknowledges support from the Welch Foundation Grant No. F-1696 and from the David and Lucile Packard Foundation. Computational resources were provided in part by the University at Buffalo Center for Computational Research and the Rensselaer Polytechnic Institute Computational Center for Nanotechnology Innovations. References and Notes

Figure 5. Relationship between excess entropy and T-2/5 for 3-methyl3-ethylpentane along select isochores.

application of measure-free density functional theory to repulsive potentials.65 Such a scaling has proven to be robust for simple liquids65 as well as for ionic melts and water.25 Figure 5 provides the relationship between sex and T-2/5 for 3-methyl-3-ethylpentane. Consistent with previous work, we find that, to within an excellent approximation, sex scales linearly with T-2/5. The other octane isomers (data not shown) exhibit a similar scaling. More generally, it appears that “strongly correlating” fluids that exhibit scaling relations between reduced dynamic properties and the excess entropy also show a linear relationship between sex and T-2/5. IV. Conclusions We have used molecular simulation to investigate connections between dynamic properties and the excess entropy for four

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