Nonempirical Free Volume Viscosity Model for Alkane Lubricants under Severe Pressures

Viscosities $\eta$ and diffusion coefficients $D_s$ of linear and branched alkanes at pressure $0<P<0.7\mathrm{GPa}$ and temperature $T=500–600\mathrm{K}$ are calculated from molecular dynamics simulations. Combining Stokes-Einstein, free volume, and random walk concepts results in an accurate viscosity model for the considered $P$ and $T$. All model parameters (hydrodynamic radius, random walk step size, and step frequency) are extracted from equilibrium molecular dynamics via microscopic ensemble averages rendering $\eta (P,T)$ a parameter-free predictor for lubrication simulations.

Figure 1

Hydrodynamic radius of complex molecules. (a) Considered molecular structures: $n$-dodecane (1, red), $n$-hexadecane (2, orange), ${\mathrm{C}}_{10}$-dimer (3, green), ${\mathrm{C}}_{10}$-trimer (4, light blue), and ${\mathrm{C}}_{10}$-tetramer (5, dark blue). This color code applies to all plots in Figs. 1, 2, and 4. (b) Shear viscosity $\eta$ and self diffusion coefficient $D_s$ for pressure $0<P<0.7\mathrm{GPa}$ and temperature $T=500$ or 600 K from EMD simulations: Quantitative agreement with the Stokes-Einstein relation Eq. (1), assuming a slip boundary condition ($n=4$) and a molecule radius $R_h=\sqrt{⟨a⟩/\pi }$, with $⟨a⟩$ the molecule’s mean cross section. (c) Definition of the configuration dependent cross section area $a≔\mathrm{\Delta }V/ϵ$ ($\mathrm{\Delta }V$ newly occupied volume after a small virtual displacement $ϵ$), and resulting probability distributions $q(a)$; lines mark the mean $⟨a⟩$.

Figure 2

Diffusion and free volume. (a) One ${\mathrm{C}}_{10}$-trimer’s trajectory (85 ps; $1\mathrm{frame}/\mathrm{ps}$) at 0.6 MPa and 600 K. Light blue and large violet spheres: current positions of C atoms and their COM, respectively; small violet: COM positions for all $t$. The COM diffusion (arrows) can be described by a random walk between caged positions due to the confining presence of the surrounding molecules (not displayed). (b) Self diffusion coefficient $D_s$ vs inverse of mean free volume per molecule $v_f=v-v_{\text{mol}}$ (colors as in Fig. 1. Lines are best fits of $D_s=D_0\exp (-v_c/v_f)$ on 600 K data. (c) Fit results $\widetilde{D_0}$ and $\widetilde{v_c}$; error bars: 68% confidence interval.

Figure 3

Diffusion of oriented $n$-hexadecane in an unconstrained bath. (a) Superimposed configurations of a bath and an oriented molecule ($\pm F$ along $\theta =0$, $\pi$; COM motion subtracted; $1\mathrm{frame}/\mathrm{ns}$). (b) Mean cross section $⟨a(\theta )⟩$ for unconstrained or constrained (circle or filled circle, respectively) molecules [with $⟨a⟩$ from Fig. 1]. (c) Radial distribution function $g_{\mathrm{COM}}(\theta ,r)$ around oriented molecules (lines, shifted for each $\theta$) with estimated cage radius $⟨r_c(\theta )⟩$ (red dots); same for isotropic $g_{\mathrm{COM}}(r)$ around bath molecules (bold line, open circle). (d) Oriented molecules’ mean squared displacement $⟨r^2(t)⟩$ in the least dense system. (e) Anisotropic self diffusion coefficient $D_s(\theta )$ (for two densities: filled circle and diamond) normalized with static structure properties according to Eq. (5); the direction averaged values $D_s=[2D_s(\pi /2)+D_s(0)]/3$ ($\times$) coincide with the isotropic $D_s$ of the bath ($+$) and of the unperturbed system (square).

Figure 4

Scaling of viscosity and self diffusion coefficient with molecule structure properties [colors as in Fig. 1]. (a) COM radial distribution functions (lines, shifted for better visibility) with estimate of $⟨r_c⟩$ (small filled circle). (b) Time autocorrelation function of the ${\mathrm{C}}_{10}$ dimer end-to-end vector orientation $\mathbf{e}(t)$ at 500 K (symbols = different densities) with fits of Eq. (6) (lines); all other molecules in Supplemental Material [32] Fig. S8. (c) Correlation times of Eq. (6) and predicted attempt and waiting times of the random walk diusion model: ${\tau }_{\beta }$ vs ${\tau }_0=⟨r_c{⟩}^2/(6D_0)$ (full symbols) and ${\tau }_{\alpha }$ vs $\mathrm{\Delta }t=⟨r_c{⟩}^2/(6D_s)$ (empty symbols); data for ${\tau }_{\beta }$ shows the mean value from all densities; (d) Scaled viscosity $\log (\eta /{\eta }_0)$ with ${\eta }_0=1.5k_{B}T{\tau }_{\beta }/(⟨r_c{⟩}^2\sqrt{\pi ⟨a⟩})$ vs ratio of critical to free volume $⟨r_c⟩⟨a⟩/v_f$ (no fitted parameter); (e) Experimental data for $n$-dodecane at $T=473\mathrm{K}$ from Ref. [35] (filled diamond) and prediction from simulations (line).