Entropy scaling for diffusion coefficients in fluid mixtures

Infinite-dilution diffusion coefficients

Figure 2a demonstrates the monovariate scaling behavior for infinite-dilution diffusion coefficients ${\widehat{D}}_2^{\infty ,\circ }$ for a binary Lennard-Jones mixture.

Results are shown for the scaled self-diffusion coefficient of component 2 at infinite dilution ${\widehat{D}}_2^{\infty ,\circ }$; in our notation, the hat ^ and ^∘ refer to specific parts of the scaling, see Methods. The data collapse to a monovariate curve, which is impressive considering the fact that a large range of states was studied, cf. Fig. 2a-inset. The quality of the scaling of ${\widehat{D}}_2^{\infty ,\circ }$ (i.e., how well the data can be described by a monovariate function) is essentially the same as that found for the pure component diffusion coefficient D₂ (see Suppl. Note 5 for quantification). This supports the picture introduced in this work that $D_2^{\infty }$ can be considered as a pseudo-pure component property. This picture is physically meaningful considering the fact that in both cases, the mobility of a single particle in a homogeneous environment is described. The Chapman-Enskog diffusion coefficient agrees well with the low-density simulation results—as indicated by the convergence of ${\widehat{D}}_2^{\infty ,\circ }\to 1$ for s_conf/R → 0.

Figure 3 shows the simulation results for the infinite-dilution diffusion coefficients in the systems acetone + isobutane and ethanol + chlorine.

The simulations comprise states in the gas, liquid, and supercritical regions. Details on the simulations are provided in Suppl. Note 2. For both systems, the infinite-dilution diffusion coefficient shows a monovariate behavior with respect to the configurational entropy confirming the results for the Lennard-Jones model system. Significant deviations from the monovariate behavior are observed for the real systems only for ${\tilde{s}}_{\mathrm{conf}}<1$, i.e. for low-density state points. This might be due to larger uncertainties of the simulations in this region.

In Suppl. Note 6, the results for two more binary Lennard-Jones systems and a third real substance system are presented that support the findings discussed here. For the Lennard-Jones model systems, the “global" parameters from ref. ¹⁷ were sufficient for describing ${\widehat{D}}_2^{\infty ,\circ }$. This finding was not expected and demonstrates the broad applicability of the universal parameters g₁ and g₂ in the correlation that were determined in ref. ¹⁷.

From the models for the pure component self-diffusion coefficient ${\widehat{D}}_2^{\mathrm{pure},\circ }\left({\tilde{s}}_{\mathrm{conf}}\right)$¹⁷ and the infinite-dilution diffusion coefficient ${\widehat{D}}_2^{\infty ,\circ }\left({\tilde{s}}_{\mathrm{conf}}\right)$, both $D_2^{\mathrm{pure}}$ and $D_2^{\infty }$ can be predicted in a wide range of states. For the Lennard-Jones systems, the configurational entropy was taken from the Kolafa-Nezbeda EOS³⁶, which is known to give an accurate the description of Lennard-Jones fluids^37,38. We used the mixture implementation for the Kolafa-Nezbeda EOS from refs. ^17,37. Figure 2b shows the results of the entropy scaling models for D₂ and $D_2^{\infty }$ in comparison to the molecular dynamics (MD) simulation results. Both methods are in good agreement, which is a result of the good performance of the EOS as well as the fact that both properties show a monovariate relation with respect to the configurational entropy.

Hence, reliable analytical models for the limiting cases D₁, D₂, $D_1^{\infty }$, and $D_2^{\infty }$ (cf. Fig. 1) are now available that can be applied in all fluid regions. They are the basis for the next step, the prediction of D₁, D₂, ${-D}_{12}$, and D₁₂ at arbitrary compositions in the mixture.

Diffusion coefficients in mixtures

The model proposed in this work (see Methods) is able to predict all diffusion coefficients in binary mixtures. Figure 4 shows the model predictions for all four diffusion coefficients (D₁, D₂, ${-D}_{12}$, D₁₂) of three Lennard-Jones systems for the entire composition range and different temperatures. For comparison, simulation data from ref. ³⁹ are used. The predictions from the entropy scaling framework and the computer experiment results agree well for all investigated systems and all four diffusion coefficients (see Fig. 4).

For both self-diffusion coefficients, for which the simulation results have a smaller statistical uncertainty than for the mutual diffusion coefficients, the entropy scaling model describes most simulation data within their uncertainty, despite the strong non-ideality that leads to extrema at about x₂ ≈ 0.4 mol mol⁻¹ (a–d in Fig. 4) or even to the occurrence of a liquid-liquid equilibrium gap (i–l in Fig. 4)—the phase behavior of these model systems was comprehensively studied in ref. ⁴⁰. The simulation data for ${-D}_{12}$ scatter more and have larger error bars than those for D₁ and D₂. The entropy scaling framework predicts the ${-D}_{12}$ simulation data very well in the entire composition range. In particular, the strongly non-ideal behavior is captured accurately by the entropy scaling framework. We have compared the entropy scaling framework presented in this work to results from the established Vignes and Darken model (see Suppl. Note 8). The entropy scaling model outperforms the empirical models. Also the temperature and pressure dependency of the diffusion coefficients are very well described by the model (cf. Fig. 4 and Suppl. Note 7, respectively). For the predictions of the Fickian diffusion coefficient, the trends are correctly predicted by the entropy scaling predictions, but deviations are observed, cf. Fig. 4d,h,l. The deviations between simulation results and entropy scaling are larger for the Fickian diffusion coefficient, which can be attributed to the inclusion of the thermodynamic factor adding additional complexity to the model predictions (and the MD sampling³³). In contrast to the other three diffusion coefficients (D₁, D₂, ${-D}_{12}$), the Fickian diffusion coefficient D₁₂ exhibits the opposite extremum behavior, which is a result of the underlying thermodynamic factor behavior. Due to the coupling of entropy scaling and EOS, the model is inherently consistent and can be applied over a wide range of states. As an example, Fig. 4i, j includes a system with weak cross-interactions resulting in a liquid-liquid equilibrium at the considered conditions. The entropy scaling model proposed in this work is able to predict not only the different diffusion coefficients in the bulk phases, but also describes the diffusion coefficients of the coexisting phases of the liquid-liquid equilibrium. Also, at the upper critical solution point, the thermodynamic factor becomes zero and so does the Fickian diffusion coefficient (x₂ ≈ 0.38 mol mol⁻¹). This is as expected for the Fickian diffusion coefficient at a critical point³⁷ and correctly captured by the model.

Figure 5 shows the results for the self-diffusion coefficients in three real substance systems: n-hexane + n-dodecane, acetone + chloroform, and nitrobenzene + n-hexane. Figure 6 shows entropy scaling predictions for the Fickian diffusion coefficient in three real substance systems: toluene + n-hexane, acetone + chloroform, and toluene + acetonitrile. For all systems, only very limited experimental data are available (this is common for practically all real mixtures), which is used for the validation of the predictions. The thermodynamic properties of the systems were modeled by the PC-SAFT EOS⁴¹, and the component-specific models for the pure substances were taken from the literature (see Suppl. Note 3 for details). The Berthelot combination rule parameter ξ₁₂ for the systems acetone + chloroform and nitrobenzene + n-hexane were adjusted to match the respective vapor–liquid equilibria qualitatively.

The entropy scaling predictions of both self-diffusion coefficients and the experimental data are in good agreement. Only for the system acetone + chloroform, some deviations are observed. For the systems n-hexane + n-dodecane and nitrobenzene + n-hexane, the model predicts the experimental data very well; the mean relative deviations are $\overline{\delta D_1}=1.24\%$ and $\overline{\delta D_2}=2.91\%$ (n-hexane + n-dodecane) and $\overline{\delta D_1}=4.96\%$ (nitrobenzene + n-hexane), respectively. For the system n-hexane + n-dodecane, the two components strongly differ in their molar mass, which yields a non-linear behavior of D₁(x₂) and D₂(x₂). This behavior is predicted well by the framework. Also, for the system nitrobenzene + n-hexane, the non-ideality of the diffusion coefficients is well captured by the model. For the system acetone + chloroform, the mean relative deviations are $\overline{\delta D_1}=8.58\%$ and $\overline{\delta D_2}=9.69\%$.

Figure 6 shows the results for the Fickian diffusion coefficient D₁₂ in three systems, namely toluene + n-hexane, acetone + chloroform, and toluene + acetonitrile. All results are at ambient pressure, but at different temperatures and shown as a function of the mole fraction. The Fickian diffusion coefficient increases with increasing temperature. The entropy scaling predictions are in good agreement with the experimental data ($\overline{\delta D_{12}}=3.02\%$ for toluene + n-hexane, $\overline{\delta D_{12}}=3.67\%$ for acetone + chloroform, and $\overline{\delta D_{12}}=10.27\%$ for toluene + acetonitrile). For the system toluene + n-hexane, the entropy scaling model yields good agreement. For the systems toluene + acetonitrile and acetone + chloroform, some deviations are observed—especially in the vicinity of the extremum values.

Figure 7 shows the predictions for the Maxwell-Stefan diffusion coefficient phase diagrams of the system toluene + n-hexane at two pressures: p = 0.1 MPa (a) and p = 4 MPa (b). The diffusion coefficients of the coexisting phases can be obtained in a straightforward manner by the framework. Additional results (including supercritical, metastable, and unstable states) are presented in Suppl. Note 10. At p = 0.1 MPa, both components are subcritical (see inset in Fig. 7a). The Maxwell-Stefan diffusion coefficient is shown for three isotherms (340, 360, 385 K). As shown in the inset (see Fig. 7a), one isotherm is entirely in the liquid phase (340 K), one isotherm passes through the vapor–liquid equilibrium (360 K), and one isotherm is entirely in the gas phase (385 K). The Maxwell-Stefan diffusion coefficient in the gas phase is significantly larger than that in the liquid phase. In the liquid phase, the diffusion coefficient strongly depends on the composition. In the gas phase, the Maxwell-Stefan diffusion coefficient exhibits only a minor dependency on the composition. This is in line with the Chapman-Enskog theory²—which is inherently incorporated into our framework. At p = 4 MPa (cf. Fig. 7b), n-hexane is supercritical, and a critical point exists at about T = 547.18 K (see inset). Four isotherms are plotted: One subcritical isotherm (T =  540 K), the critical isotherm (T = 547.18 K), one isotherm passing through the vapor–liquid equilibrium (T = 570 K), and one isotherm entirely in the gas/supercritical phase (T = 590 K). The isotherm at T = 540 K exhibits a transition between the two pseudo-pure component limiting cases—from a liquid state at x₂ = 0 to a supercritical state at x₂ = 1 mol mol⁻¹. At the critical point, the diffusion coefficient exhibits a large gradient with respect to the composition. Due to the thermodynamic consistency established by the EOS, the framework correctly predicts the Fickian diffusion coefficient to be zero at critical points (see Fig. 4).

In Suppl. Note 10, it is shown that also diffusion coefficients of metastable and unstable states as well as at spinodal states can be described by the framework (which is for example relevant for nucleation). This significantly exceeds the capabilities of all presently available diffusion coefficient models.

Entropy scaling of infinite-dilution diffusion coefficients

The infinite-dilution diffusion coefficient $D_i^{\infty }\left(T,p\right)$ is treated as a pseudo-pure component property such that the corresponding scaled property ${\widehat{D}}_i^{\infty ,\circ }\left({\tilde{s}}_{\mathrm{conf}}\right)$ exhibits a monovariate function with respect to the reduced configurational entropy ${\tilde{s}}_{\mathrm{conf}}$ (cf. Eq. (6)). The modeling methodology is described in the following. There are many scaling approaches available in the literature that aim at establishing a monovariate behavior of transport coefficients. Here, we use the scaling described in ref. ¹⁷. The pure component scaling described in ref. ¹⁷ (which follows refs. ^13,16) is adapted here to model the pseudo-pure component diffusion coefficients at infinite dilution. Thus, we propose that the infinite-dilution diffusion coefficient is transformed using

where ρ_N is the number density of the solvent at given T and p. The reference mass M_CE is adopted from the Chapman-Enskog (CE) theory as

where M_i and M_j are the molar masses of the pure components. The modified Rosenfeld-scaled diffusion coefficient $D_i^{\infty ,\circ }$ exhibits some scattering in the zero-density limit for s_conf → 0. Therefore, the scaling is further modified using the Chapman-Enskog diffusion coefficient $D_{\mathrm{CE},i}^{\infty ,\circ }$, which is computed using a Lennard-Jones kernel (see Suppl. Note 1 for details). Finally, the CE-scaled infinite-dilution diffusion coefficient ${\widehat{D}}_i^{\infty ,\circ }$ is described by a split between the low-density (LD) and the high-density (HD) region as

where W is a smoothed transition function given by $W\left({\tilde{s}}_{\mathrm{conf}}\right)=1/\left(1+\exp \left(20\left({\tilde{s}}_{\mathrm{conf}}-{\tilde{s}}_{\mathrm{conf}}^{\times }\right)\right)\right)$ with ${\tilde{s}}_{\mathrm{conf}}^{\times }=0.5$. The function W establishes a smooth transition from the low-density region to the high-density region. The choice of ${\tilde{s}}_{\mathrm{conf}}^{\times }$ is heuristic. The resulting CE-scaled infinite-dilution diffusion coefficient can be described by a simple continuous, monovariate function ${\widehat{D}}_i^{\infty ,\circ }={\widehat{D}}_i^{\infty ,\circ }\left({\tilde{s}}_{\mathrm{conf}}\right)$. Here, a function with two adjustable, system-dependent parameters ${\alpha }_2^{\left(D_i^{\infty }\right)}$ and ${\alpha }_3^{\left(D_i^{\infty }\right)}$ was used given as

where $g_1^{\left(D\right)}=0.6632$ and $g_2^{\left(D\right)}=9.4714$ are global parameters fitted to the self-diffusion coefficient of the Lennard-Jones fluid¹⁷. The system-dependent parameters ${\alpha }_2^{\left(D_i^{\infty }\right)}$ and ${\alpha }_3^{\left(D_i^{\infty }\right)}$ were fitted to reference data of the pseudo-pure component, i.e. the diffusion coefficients at infinite dilution. This scaling approach was tested using model systems and real substance systems (see Results). Therefore, MD simulations were carried out using the simulation engine ms2⁴⁵.

Predicting diffusion coefficients in mixtures

For predicting the diffusion coefficients D_i(x_j), D_j(x_j), ${-D}_{ij}$(x_j), and D_ij(x_j) in a mixture, the entropy scaling framework is extended using combining and mixing rules. Diffusion coefficients in mixtures are predicted based on the limiting case models, i.e. the models for the self-diffusion coefficient of the pure components and the (infinite dilution) pseudo-pure component. The self-diffusion coefficient D_i in a mixture is calculated using the limiting case entropy scaling model of the pure component self-diffusion coefficient $D_i^{\mathrm{pure}}$ and the limiting case entropy scaling model of the infinite-dilution diffusion coefficient $D_i^{\infty }$ in the solvent j. The Maxwell-Stefan diffusion coefficient is calculated using both infinite-dilution diffusion coefficient limiting case entropy scaling models $D_i^{\infty }$ and $D_j^{\infty }$. Therefore, the scaled mixture diffusion coefficient ${\widehat{\mathrm{\Lambda }}}^{\circ }\in \left\{{\widehat{D}}_i^{\circ },{\widehat{D}}_j^{\circ },{\widehat{-D}}_{ij}^{\circ }\right\}$ is computed as

where ${\tilde{s}}_{\mathrm{conf}}$ is the scaled configurational entropy of the mixture, i.e. ${\tilde{s}}_{\mathrm{conf}}=s_{\mathrm{conf}}/\left(\mathrm{R}{m}_{\mathrm{mix}}\right)$ with m_mix = x_im_i + x_jm_j. The parameters ${\alpha }_k^{\left(\mathrm{\Lambda }\right)}$ (with k ∈ {2, 3}) are calculated using linear mixing rules

for the self-diffusion coefficients of components i, the self-diffusion coefficient of component j, and the Maxwell-Stefan diffusion coefficient, respectively.

The final (unscaled) diffusion coefficient Λ ∈ {D_i, D_j, ${-D}_{ij}$} is calculated as

where ${\mathrm{\Lambda }}_{\mathrm{CE}}^{\circ }$ is the Chapman-Enskog diffusion coefficient of the mixture, ρ_N is the number density of the mixture, and $M_{\mathrm{ref}}^{\mathrm{\Lambda }}$ is the reference mass of the mixture, which is calculated as

for the self-diffusion coefficients D_i and D_j and the Maxwell-Stefan diffusion coefficient ${-D}_{ij}$, respectively. Details on the calculation of the Chapman-Enskog property of the mixture ${\mathrm{\Lambda }}_{\mathrm{CE}}^{\circ }$, the reference mass M_ref, and ${\widehat{\mathrm{\Lambda }}}^{\circ }={\widehat{\mathrm{\Lambda }}}^{\circ }\left(s_{\mathrm{conf}}^{\mathrm{mix}},{\alpha }_{k,\mathrm{mix}}\right)$ are given in Suppl. Note 1.

The Fickian diffusion coefficient is computed from the Maxwell-Stefan diffusion coefficient by the thermodynamic factor as predicted by the EOS, see Eq. (1). Thus, all required quantities are obtained using predictive combination rules and mixing rules and the EOS mixture model. No adjustable mixture parameters are introduced. However, for modeling the diffusion coefficients of a given binary system based on a given EOS model, adjustable parameters have to be determined for the four limiting cases, i.e. the two pure components and the two infinite dilution pseudo-pure components, cf. α parameters in Eq. (10). For predicting the diffusion coefficients in the mixture, no adjustable parameters are required. The non-ideality of the mixture is primarily taken into account by the underlying EOS via the predicted configurational mixture entropy. The combination of the EOS and entropy scaling enables the consistent calculation of the diffusion coefficients (D_i, D_j, ${-D}_{ij}$, D_ij), homogeneous bulk properties (pvT, s_conf, Γ_ij, etc.), and phase equilibria (e.g., vapor–liquid and liquid-liquid equilibria).

Peer review information

Nature Communications thanks Ian Bell and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.