Modeling Self-Diffusion Coefficient and Viscosity of Chain-like Fluids Based on ePC-SAFT

: In this work, we developed a new self-diffusion coefficient model for chain-like fluids, which was coupled with the SE equation to simultaneously describe transport properties (i.e., self-diffusion coefficient and viscosity) using the parameters obtained from thermodynamic properties. In modeling, the self-diffusion coefficient model was developed based on the diffusion coefficient of LJ spherical fluids by incorporating a correction function to describe the characteristics of chain-like molecules. Subsequently, the SE equation was used to calculate the viscosity. Based on the molecular parameters in ePC-SAFT (i.e., segment number N , segment diameter σ , and energy parameter ε / k B ), one set of universal parameters was determined from the self-diffusion coefficients and viscosities of 19 n -alkanes (C 2 H 4 − C 20 H 42 ) at various temperatures and pressures. The model reproduces the experimental self-diffusion coefficient data (804 data points) with an average ARD of 8.4% and the experimental viscosity data (1539 data points) with an average ARD of 7.2% for 19 n -alkanes over wide ranges of temperature and pressure. Furthermore, the viscosity and self-diffusion coefficient of the other 17 compounds, including long n -alkanes, branched alkanes, and cyclic compounds, were predicted, and among them, the relatively poor prediction results of branched alkanes and cyclic compounds were further discussed. Finally, the proposed model was extended to ionic liquids, generally providing reliable results for these complex fluids. This study suggests that it is possible to describe the thermodynamic and transport properties with one set of molecular parameters based on ePC-SAFT.


INTRODUCTION
Transport properties, such as viscosity and self-diffusion coefficient, are crucial physical properties, required in various industrial processes.These properties have a significant impact on the flow behavior of fluids, as well as the rate of heat and mass transfer, making them indispensable in many industrial applications, such as carbon capture and storage 1,2 and chemical enhanced oil recovery. 3The transport properties of fluids are commonly determined through experimental measurement and theoretical calculation.Considering the expensive and time-consuming nature of conducting experimental measurements for fluids under various conditions, especially under high-temperature and high-pressure conditions, 3,4 it is desirable to develop theoretical models to describe the transport properties of fluids.Various methods have been developed and reviewed for the prediction and correlation of viscosity 3,5,6 and self-diffusion coefficient 7 of different fluids.For gases, the kinetic theory of gas proposed by Chapman and Enskog 8 has been developed to widely describe the viscosity and self-diffusion coefficient at low to moderate pressures.For dense gases and liquids, based on the kinetic theory of gas, a number of semiempirical models have been developed to represent their viscosity and self-diffusion coefficient.One common approach is based on a database of transport properties obtained from molecular dynamics (MD) simulations 9−14 for simple "model" molecular fluids, such as hard-sphere (HS) 9,10 and Lennard-Jones (LJ) 11−14 fluids, and also semiempirical equations were developed to describe viscosity 15−17 and self-diffusion coefficient 18−22 of real substances.Another approach can be categorized as applied statistical mechanics models, such as Eyring's model 23 (i.e., absolute reaction rate theory), residual models (i.e., friction theory 24 and free volume theory 25,26 ), scaling models (i.e., excess entropy scaling 27,28 and density scaling 29 ), and so on.In the above-mentioned models, thermodynamic properties such as density, which are typically determined from experimental measurements or calculated from thermodynamic models, are required as input to obtain the transport properties.However, in general, the model parameters in describing the transport properties are different from those in representing their thermodynamic properties.In principle, for a substance, the molecular parameters should be independent of its properties; also considering that the thermodynamic properties of a substance are often studied much more both experimentally and theoretically, developing models with one set of molecular parameters to simultaneously represent thermodynamic and transport properties will be feasible and important.
In general, there are two main theoretical approaches to simultaneously calculate the thermodynamic and transport properties with one set of molecular parameters. Wei and Rowley, 30 as well as Cao et al., 31 determined the parameters of the thermodynamic model from vapor−liquid equilibrium (VLE) data and then predicted the viscosities of binary liquid mixtures via combining with the Eyring's model.In our previous study, 32 we combined the nonrandom two-liquid model (NRTL) with the Eyring's model to investigate how to simultaneously represent the excess thermodynamic properties and viscosities of ionic liquid (IL) mixtures.The models based on Eyring's theory are mainly restricted to describing the excess viscosity of liquid mixtures, and the studies of the self-diffusion coefficient of liquid mixtures are limited.Furthermore, to describe the properties, studies of constituents in the mixture are needed additionally.
Recently, the entropy-scaling theory gained significant attention as a promising method to establish a link between transport properties and residual entropy, where the residual entropy can be determined from thermodynamic models, such as perturbed-chain statistical associating fluid theory (PC-SAFT).A universal relationship between dimensionless transport properties (i.e., scaled viscosity and self-diffusion coefficient) and residual entropy for simple fluids was proposed by Rosenfeld, 27,28 whereas this relationship cannot extend to molecular liquids. 44To improve the model capability based on the entropy-scaling theory, some researchers 33−41 modified the relationship between the residual entropy and dimensionless properties (i.e., viscosity 33−35,37−39 and self-diffusion coefficient 36,40,41 ) by using different mathematical formulations.These modified models can describe the viscosity or self-diffusion coefficient of molecular liquids using a set of generalized parameters, while their adjustable parameters are usually based on either the self-diffusion coefficient or viscosity data.Studies on the simultaneous representation of both properties using entropyscaling models are still rare.Moreover, the above-mentioned semitheoretical models have not been extended to advanced and complex fluids, such as ILs with charges.
Different from the Eyring's models and the models derived from the entropy-scaling concept, on the basis of force fields, molecular simulation is used as a powerful technique for calculating the thermodynamic and transport properties of real molecular systems, and the all-atom (AA) or coarse-grained (CG) force fields have been developed for a variety of molecular fluids, such as hydrocarbons, 43,45−47 electrolytes, 46 and ILs. 48On the other hand, several studies 43,46,47,49 also used an equation of state (EOS), such as SAFT-based EOS, to estimate the CG force fields, which were then used directly in molecular simulations to obtain thermodynamic and transport properties of fluids.For example, the research group of Galliero 43,46 developed two coarse-grained molecular models (i.e., the LJ Chain model 43 and Mie Chain model 46 ) to simultaneously represent thermodynamic properties from bulk to interface and transport properties of short n-alkanes using one set of molecular parameters.Considering the computational intensity and time requirements in molecular simulations, it is desirable to develop semitheoretical models that can simultaneously describe both the thermodynamic and transport properties with one set of parameters for various fluids, including ILs with charges.
In this work, we developed models to simultaneously describe the self-diffusion coefficient and viscosity of substances, including n-alkanes, branched alkanes, cyclic compounds, and ILs by using one set of molecular parameters based on ePC-SAFT owing to its capability to accurately describe the thermodynamic properties of neutral (nonelectrolytes) and charged (electrolytes) substances, especially those of chain-like molecules.The self-diffusion coefficient equation for the LJ spherical fluids was incorporated with a correction term, F(N, ρ*, T*), to describe the characteristics of chain-like molecules, and the Stokes−Einstein equation was developed to obtain the viscosity.In modeling, based on the molecular parameters in ePC-SAFT, one set of universal parameters was determined from the self-diffusion coefficients and viscosities of n-alkanes (C 2 H 4 −C 20 H 42 ) at various temperatures and pressures.Based on the obtained universal parameters, the viscosity and self-diffusion coefficient of branched alkanes and cyclic compounds were predicted and discussed.Furthermore, the proposed models were extended to ILs.

Self-Diffusion Coefficient Model.
Based on the selfdiffusion coefficient equation for the simple and spherical LJ fluid proposed in our previous study, 20 a new self-diffusion coefficient equation was developed for a chain-like molecule containing N hard-spheres of diameter σ and mass m.It is expressed as (1) where ( 1) ( ) (2) ( 1) 1 0.527 The reduced temperature T* and reduced density ρ* for the chain-like molecule are defined as In these equations, D is the self-diffusion coefficient (m/s 2 ), N is the number of segments per chain, σ is the segment diameter (Å), m is the molecular mass (kg) expressed as m = M/(N•N A ) (M is the molar weight, kg/mol, and N A is the Avogadro constant, 1/mol), k B is the Boltzmann constant (J/ K), T is the temperature (K), ε/k B is the energy parameter (K), ρ N is the number density (1/Å 3 ) expressed as ρ N = N A / (M•ρ), and ρ is density (kg/m 3 ).Three molecular parameters (i.e., N, σ, and ε/k B ) were used in the model.Additionally, the universal parameters in eq 2 (i.e., a, b, c, P 1 , P 2 , P 3 , P 4 , and P 5 ) and eq 4 (i.e., S 1 , S 2 , and S 3 ) were determined from simulated and experimental data, respectively.The process of obtaining model parameters is discussed in section 2.4.
From eq 1, we can see that with N → 1, the correction function (eq 4) approaches 1.0, and thus the equation simplifies into the self-diffusion coefficient equation of the simple and spherical LJ fluid, which is expressed as where the reduced self-diffusion coefficient D* is written as Equation 6 can be used to represent the self-diffusion coefficient of spherical LJ fluids, including gas, liquid, and supercritical fluid, with a total AAD of 4.95%. 20With a generalized expression for the LJ parameters of pure real fluids (i.e., σ and ε/k B ), eq 6 can be extended to real pure substances.The detailed expressions on the diffusion equation of the LJ fluid can be found in the Supporting Information.

Stokes−Einstein (SE) Equation.
In this work, the SE equation was used to connect the self-diffusion coefficient and viscosity.The SE equation is expressed as where η is the viscosity (mPa•s), and C is a numerical constant determined by the hydrodynamic boundary condition.Specifically, for "slip" and "stick" boundary conditions, C is equal to 2 and 3, respectively.In this work, we assumed that the boundary condition is "stick", and C was taken as 3. σ H denotes the hydrodynamic diameter of a hard-sphere particle.

ePC-SAFT.
Following a previous work, 50 the dimensionless residual Helmholtz energy in ePC-SAFT is a summation of energy contributions accounting for hard-sphere (a hs ), chain (a chain ), dispersive (a disp ), and ionic (a ion ) terms: For nonelectrolytes, each substance was modeled as a nonspherical species with repulsive and dispersive interactions with three parameters (i.e., segment number, N, segment diameter, σ, and dispersion-energy parameter, ε/k B ).For electrolytes (i.e., ILs), following our previous work, 51,52 each IL was modeled as an equimolar mixture of IL-anion and ILcation, three parameters (i.e., N, σ, and ε/k B ) were used to represent each IL-ion in the model, and the Lorentz− Berthelot mixing rules were used to characterize a pure IL without any additional binary parameters: In this work, ePC-SAFT was also used to calculate the density of the studied substances under different temperatures and pressures as inputs in eqs 1 and 6.Also, their parameters were used as inputs in modeling the kinetic properties, as described in section 2.4.

Model Parameters.
A great number of parameters are required in the models in describing thermodynamic and kinetic properties.In this work, the following are used: (1) The universal parameters in eq 2 (i.e., a, b, c, P 1 , P 2 , P 3 , P 4 , and P 5 ).These were taken directly from our previous work, 20 where the self-diffusion coefficients at the range of 0.8 ≤ T* ≤ 4.0 and 0.05 ≤ ρ* ≤ 1.0 were predicted using the MD simulation for simple spherical fluids. 14The corresponding values are listed in Table S1.It should be mentioned that the finite size corrections 53 were not employed in our previous work. 20According to the study by Bell et al., 54 the error in simulating diffusion coefficients without incorporating finite size corrections could be from 1 to 15% for the MD results of Rowley et al. 14 We conducted preliminary studies, indicating that these deviations only have a marginal impact on the development of diffusion models for real substances.
(2) The universal parameters (i.e., S 1 , S 2 , and S 3 ) in the correction term F(N, ρ*, T*).In the previous work, these parameters were commonly determined from the MD simulation results.For example, Yu and Gao 19,55 obtained a correlation term F(N, ρ*) with 3 parameters to account for the chain characteristics using the MD simulation data of hard-chain fluids. 56Resi et al. 21,57 determined a correlation term F(N, ρ*, T*) with 3 parameters using the MD simulation data of LJ chain fluids.He et al. 22 proposed two correlation functions G(N, ρ*, T*) and H(N, ρ*, T*) with a total of 6 parameters to describe the chain characteristics, where the parameters were determined on the basis of the MD data by Resi. 57However, in this work, these parameters were set as adjustable parameters, where both viscosity and self-diffusion coefficient data of pure liquid n-alkanes were used in parametrizing the correction function F(N, ρ*, T*).The details for parametrizing and the corresponding results were described in section 4.
(3) The molecular parameters (i.e., N, σ, and ε/k B ) in ePC-SAFT.In this work, these three molecular parameters for the studied substances were taken from previous work. 50 −52,58 It should be mentioned that the potential function of ePC-SAFT is different from that of the LJ self-diffusion coefficient equation, while it is still reasonable to use the ePC-SAFT molecular parameters directly.This is because (1) in this work, the universal parameters of F(N, ρ*, T*) were determined from the experimental data, instead of the MD simulation data, and as suggested by Liu et al., 59 (2) there exist mathematical relations between the self-diffusion coefficient models represented by different potential functions.In this work, the possible deviations caused by parameters of different potential functions were reflected in parametrizing F(N, ρ*, T*).The ePC-SAFT parameters for the substances studied in this work are listed in Table S3.
(4) Additionally, the hydrodynamic diameter σ H in the SE equation was set as the segment diameter σ taken from ePC-SAFT.

EXPERIMENTAL DATA
In this work, we assumed that the SE equation only can describe the relationship between the viscosity and selfdiffusion coefficient of dense fluids, and thus, the experimental data of all studied substances in the compressed liquid state were surveyed.Tables 1 and 2   ) were determined from accurate volumetric and phase equilibrium data, they are reasonable and can be used to represent both thermodynamic and kinetic properties.In this work, based on the ePC-SAFT parameters, the self-diffusion coefficients (804 data points) and viscosities (1539 data points) of these 19 n-alkanes were used to determine the universal parameters (i.e., S 1 , S 2 , and S 3 ) in eq 4 through the objective function OF, which is expressed as follows: where D i,exp and D i,cal are the experimental and calculated selfdiffusion coefficients; η i,exp and η i,cal are the experimental and calculated viscosities; and N is the total number of data points.
To evaluate the model performance, the absolute relative deviations (ARD) of self-diffusion coefficient and viscosity were used and given by Table 3 shows the adjustable parameters in eq 4 and the ARDs of self-diffusion coefficient and viscosity for the studied n-alkanes (8.4% and 7.2%, respectively).It should be mentioned that the data of methane (CH 4 ) was not included in the parametrization process due to its segment number being equal to 1.0.Using the ePC-SAFT parameters of CH 4 (i.e., N = 1.0, σ = 3.7039, and ε/k B = 150.03), the selfdiffusion coefficient was predicted via eq 1 and subsequently compared to the experimental self-diffusion coefficient data (173.4K ≤ T ≤ 454 K and 0.8 MPa ≤ P ≤ 200 MPa) from Greiner-Schmind et al. 60 and Dawson et al., 130 and the corresponding ARD is 9%.The comparison between predicted and experimental results is illustrated in Figure 1 2(a), and their corresponding ARDs are displayed in Figure 2(b).The deviations are mainly observed at low self-diffusion coefficients, ranging from 10 −10 to 10 −9 m 2 /s, especially for the n-alkanes with short chain lengths.To further analyze, we made a specific comparison for the shortchain n-alkanes, i.e., C 2 H 6 , C 4 H 10 , and C 6 H 14 with ARDs higher than 10% (Figure 3).It shows that (1) the selfdiffusion coefficients decrease with decreasing temperature as well as increasing pressure and chain length; (2) the significant deviations for these three n-alkanes are observed at temperatures below 250 K and above 200 MPa, suggesting that the proposed equations may be unsuitable under these relatively harsh conditions.At such conditions, the corresponding reduced temperature T* is 0.8−1.0, and the reduced density ρ* is 0.85−1.0.According to the simulation by Rowley, 14 these ranges of T* and ρ* are in the liquid−solid phase transition region and close to the solid-state region, respectively.In both these regions, it is difficult to obtain D* due to very low molecular diffusion, and thus the capacity of eq 1 in predicting D* is limited.Based on the above analysis, we conclude that the limitations of eq 1 in predicting the low self-diffusion coefficients result in significant deviations at low temperatures and high pressures.Additionally, the accuracy of the ePC-SAFT model in predicting density may become worse with increasing pressures or decreasing temperatures, leading to discrepancies in the self-diffusion coefficients.
To investigate whether the incorporation of the correction function improves the model performance, we calculated the ARDs of the studied n-alkanes (C 2 H 6 −C 20 H 42 ) by using the LJ self-diffusion coefficient equation of real substance proposed in the previous work. 20The calculated details of the LJ self-diffusion coefficient equation for real substances can be found in the Supporting Information.The results of ARDs are illustrated in Figure 4. Compared to the results shown in Figure 2(b) and Figure 4, it can be seen that the model performance is significantly improved when the concept of the chain segment is incorporated.Moreover, Figure 4 reveals that with increasing the chain length of nalkanes, the values of ARDs show an overall increasing trend, indicating the effects of chain segment should be considered in the LJ self-diffusion coefficient equation when extending to the long-chain n-alkanes.This further proves that the correction function used in this work (eq 4) is reasonable.
In addition to the self-diffusion coefficient, the viscosities of the studied n-alkanes at various temperatures and pressures were also obtained.The comparison between calculated and experimental viscosities is illustrated in Figure 5(a), and their corresponding ARDs are shown in Figure 5(b).Overall, the calculated viscosities exhibit excellent agreement with experimental data, and the ARDs of viscosity are below 10% for almost all substances.However, significant deviations can still be observed in the low-viscosity region (close to 10 −2 mPa•s).This is because the viscosities in the gas−liquid transition region for C 2 H 6 and C 3 H 8 were included in parametrizing, while the SE equation is incapable of describing the relationship between the self-diffusion coefficient and viscosity for the gas-like fluids, leading to obvious deviations in the low-viscosity region.
4.2.Prediction of n-Alkanes (C 24 H 50 −C 30 H 62 ), branched alkanes, and other substances.Based on the models with three universal parameters, together with the ePC-SAFT parameters, the self-diffusion coefficients and viscosities of 17 substances, including long n-alkanes, branched alkanes, and other substances, were predicted and discussed.
For long n-alkanes (C 24 H 50 −C 30 H 62 ) and branched alkanes, the model performance in predicting the self-diffusion coefficient is illustrated in Figure 6(a), and their correspond-  ing ARDs and those predicted with the LJ diffusion equation for real substances are shown in Figure 6(b).It can be seen that, for the majority of alkanes, the models exhibit acceptable predictions, and the prediction capacity was obviously improved compared to the spherical LJ diffusion equation for real substances.However, the model performance is relatively poor for 2,2-dimethylbutane and 2,3-dimethylbutane.In a further analysis, we investigated the molecular structures, and these two isomers of hexane with two methyl groups are more like spherical molecules.In contrast, in ePC-SAFT, they were still modeled as chain-like molecules (N ≈ 2.6), resulting in relatively poor predictions.Figure 7 shows the comparison between the predicted and experiment viscosities as well as their corresponding ARDs.Similar to the observations of the self-diffusion coefficient, the model performance in predicting the viscosities is, in general, desirable but relatively poor for 2,2-dimethylbutane and 2,3dimethylbutane.
The comparison of predicted and experimental self-diffusion coefficients for the other 8 substances is illustrated in Figure 8(a), and their corresponding ARDs together with those predicted by the LJ diffusion equation for real substances are shown in Figure 8(b).Overall, the model prediction performance for these substances is poor (Figure 8(a)).Figure 8(b) illustrates the ARDs given by the models proposed in this and a previous work, 20 respectively.Surprisingly, except for toluene and pyridine, the equation proposed in the previous work 20 provides better predictions than that in this work.In the LJ diffusion equation for real substances, the segment number of the studied substance is always equal to 1.0, i.e., spherical molecules.In this work, the segment number of each studied substance was taken from ePC-SAFT.Figure 9 shows the molecular structures of the investigated substances and their corresponding segment numbers in ePC-SAFT.It is evidenced that the segment numbers for most studied cyclic compounds are close to those of the n-alkanes with the same carbon atoms, which is obviously unreasonable.In ePC-SAFT parametrizing, several set of parameters can be obtained, but not all of them are reasonable.Refitting the ePC-SAFT parameters and taking the reasonable set will improve the model results in this work.This will be done in our future work.
The viscosities of these 8 compounds were also predicted.The results are illustrated in Figure 10(a).Similarly, the overall deviation can be observed, and the predicted viscosities are lower than those of the experimental data.Instead, the predicted self-diffusion coefficient is higher than the experimental data.This is because the viscosity is inversely proportional to the diffusion coefficient in the SE equation.
To further analyze the reason for such a relatively high deviation, we investigated the reliability of the SE relation for these substances via a literature survey.Among the studied substances, based on the work conducted by Jonas et al. 77,80,81,83,131 and McCool et al., 82 the SE relation was found to be valid for benzene, 77,131 cyclohexane, 80 tetramethyl silane, 77,131 carbon tetrachloride, 82 and octafluorocyclobu-  tane, 83 but not valid for pyridine 81 due to its exceptionally low and pressure-dependent constant C. For cyclopentane and toluene, the validity of the SE relation has not been tested due to the limited experimental data on D and η under the same conditions.Combining the results and discussion in the previous paragraph, we conclude that the high deviations observed for these cyclic compounds are caused by their unreasonable ePC-SAFT parameters.
4.3.Extension to Ionic Liquids.For ILs, the presence of electrostatic interactions between their ionic components makes them apart from simple molecules, and thus, the adjustable parameters in eq 4 were determined using their experimental self-diffusion coefficient and viscosity data.The calculated ARD (D) and ARD (η) are 39.4% and 30.9%, respectively.The adjustable parameters and ARDs for each IL are listed in Tables S5 and S6.The comparison of calculated and experimental self-diffusion coefficients and viscosities is illustrated in Figure 11, showing that for most ILs, the calculated self-diffusion coefficients were lower than the experimental values.Similarly, the calculated viscosities were also lower than the experimental data, particularly in the highviscosity region.This observation is opposite to the finding presented in Figures 8(a) and 9(a).This may be caused by the limitation of the conventional SE equation in accurately describing the relationship between the self-diffusion coefficient and viscosity of highly viscous liquids, such as ILs.This limitation has been previously reported in several studies, 132,133 and as a solution, the fractional SE equation was proposed to improve the performance of the conventional SE equation.In this work, we mainly investigated whether it is possible to describe the transport properties of ILs using the proposed model, and thus, no further work on other modified  SE equations was conducted.Furthermore, it is worth mentioning that the proposed model has predictive capabilities in describing transport properties by combining the ePC-SAFT parameters of different ions.As far as we know, there are limited models available for predicting the self-diffusion coefficients of ILs.In contrast, extensive research has been done on models for predicting the viscosity of ILs.The majority of viscosity models were developed by combining machine learning with either a quantitative structure−property relationship (QSPR) model or a group contribution (GC) model. 134,135The performance of these models commonly depends on the machine learning models, the number of ILs and data points, and the overall ARD varies from values lower than 10% to more than 50%. 135In our opinion, considering that the proposed model has only three universal parameters, the ARD (η) of approximately 30.9% obtained in this study can be considered acceptable.atures and pressures.The proposed models represent the experimental self-diffusion coefficients with an ARD of 8.4% and viscosity with an ARD of 7.2% for 19 n-alkanes (804 diffusion data points and 1539 viscosity data points).The results suggest that the ePC-SAFT parameters determined from thermodynamic properties can simultaneously provide accurate kinetic properties (self-diffusion coefficient and viscosity) of n-alkanes.
Based on the obtained universal parameters, the viscosity and self-diffusion coefficients of long alkanes, branched alkanes, and cyclic compounds were predicted and discussed.For alkanes, the predicted ARD (D) and ARD (η) are 20% and 11.5%, whereas the predicted ARD (D) and ARD (η) for cyclic compounds are relatively poor, with ARD (D) of 45% and ARD (η) of 29.9%.We think that the proposed model with the chain-segment feature, compared with the predicted results of the diffusion equation of spherical LJ fluids, has difficulty providing accurate predictions for the substances with other features, and the ePC-SAFT parameter for cyclic compounds may be unreasonable, causing relatively poor predictive results.
In addition to nonelectrolyte molecules, the proposed models were used to simultaneously describe the self-diffusion coefficient and viscosity of ILs, with ARD (D) of 39.4% and ARD (η) of 30.1%.Considering that this predictive model only has three universal parameters, the performance in predicting can be acceptable compared to other predictive models of ILs. 134,135ASSOCIATED CONTENT * sı Supporting Information The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jced.3c00276.
A brief introduction of the LJ self-diffusion coefficient equation for real substances; parameters of the LJ selfdiffusion coefficient equation for real substances; critical

Figure 2 .
Figure 2. (a) Comparison of calculated and experimental self-diffusion coefficients for all investigated n-alkanes (C 2 H 6 −C 20 H 42 ); (b) ARDs between calculated and experimental self-diffusion coefficients for all investigated n-alkanes (C 2 H 6 −C 20 H 42 ).The dashed line represents an ARD of 10%.

Figure 3 .
Figure 3. Comparisons of calculated and experimental self-diffusion coefficients for (a) C 2 H 6 , C 4 H 10 , and (b) C 6 H 14 in a wide range of temperatures and pressures.Symbols denote experimental data, and lines denote calculated values.

Figure 4 .
Figure 4. Comparison of predicted and experimental self-diffusion coefficients for all investigated n-alkanes (C 2 H 6 −C 20 H 42 ) using the generalized LJ equation proposed in the previous work.20The dashed line represents an ARD of 10%.

Figure 5 .
Figure 5. (a) Comparison of calculated and experimental viscosities for all investigated n-alkanes (C 2 H 6 −C 20 H 42 ); (b) ARDs between calculated and experimental viscosities for all investigated n-alkanes (C 2 H 6 −C 20 H 42 ).The dashed line represents an ARD of 10%.

Figure 6 .
Figure 6.(a) Comparison of predicted and experimental self-diffusion coefficients for long n-alkanes and branched alkanes; (b) ARDs between experimental self-diffusion coefficients and predicted values obtained from the equations proposed in previous work 20 and in this study.

Figure 7 .
Figure 7. (a) Comparison of predicted and experimental viscosities for long n-alkanes and branched alkanes; (b) ARDs between predicted and experimental viscosities for long n-alkanes and branched alkanes.
In this work, we developed models to simultaneously describe the self-diffusion coefficient and viscosity of various chain-like substances based on ePC-SAFT.A new self-diffusion coefficient equation was developed based on the diffusion coefficient of LJ spherical fluids by incorporating a correction function, F(N, ρ*, T*), to describe the characteristics of chain-like molecules, and the SE equation was employed to obtain the viscosity.In modeling, based on the molecular parameters in ePC-SAFT, one set of universal parameters were determined from the self-diffusion coefficients and viscosities of n-alkanes (C 2 H 4 −C 20 H 42 ) at various temper-

Figure 8 .
Figure 8.(a) Comparison of predicted and experimental self-diffusion coefficients for the other 8 substances; (b) Comparison of ARDs between experimental self-diffusion coefficient and predicted values obtained from the equations proposed in a previous work 20 and in this study.

Figure 9 .
Figure 9. Molecular structures of n-alkanes (CH 4 −C 7 H 16 ), branched alkanes, and some other 8 substances, as well as their corresponding segment number of ePC-SAFT.

Figure 10 .
Figure 10.(a) Comparison of predicted and experimental viscosities for the other 8 substances; (b) ARDs between predicted and experimental viscosities for the other 8 substances.

Figure 11 .
Figure 11.(a) Comparison of correlated and experimental self-diffusion coefficients for 10 ILs.(b) Comparison of correlated and experimental viscosities for 14 ILs.

Table 1 .
summarize the available experimental self-diffusion coefficient data for 46 real substances, containing 36 simple molecules (1358 data points) and 10 ILs (143 data points), as well as viscosity data for 50 real substances, containing simple molecules (2235 data points) and 14 ILs (1824 data points).The survey Self-Diffusion Coefficient Data for the Studied Substances measured the self-diffusion coefficient data for C 2 H 6 at temperatures from 136 to 454 K, and for C 3 H 8 at temperatures from 112 to 453 K.According to the ePC-SAFT parameters of C 2 H 6 , C 3 H 8 , and C 4 H 10 , their corresponding reduced temperatures are within the ranges 60Number of data points.Dshows that most of the data were determined in a wide range of temperatures and pressures.It should be mentioned that Greiner-Schmid et al.60

Table 2 .
20scosity Data for the Studied Substances T* ≤ 2.37, 0.54 ≤ T* ≤ 2.18, and 0.69 ≤ T* ≤ 2.02, respectively.In the previous work,20the range of 0.8 ≤ T* ≤ 4.0 was set in determining the parameters of eq 2, and the prediction of D* out of this range is questionable, specifically for T* far less than 0.8.Therefore, the experimental self-diffusion coefficient data for C 2 H 6 , and C 4 H 10 at temperatures higher than 150 K and for C 3 H 8 at temperatures higher than 160 K were chosen in this work, and the experimental viscosity data at similar conditions were selected.
a Number of data points.
Based on the corrections of 19 n-alkanes (C 2 H 6 − C 20 H 42 ) together with the results for CH 4 , it can be concluded that, by using the ePC-SAFT model parameters as the input in modeling the diffusion coefficient, a generalized correction function can be obtained to simultaneously describe the selfdiffusion coefficient and viscosity of the studied n-alkanes.The comparison of calculated and experimental selfdiffusion coefficients of n-alkanes (C 2 H 6 −C 20 H 42 ) is further illustrated in Figure , showing good agreement in a wide range of temperatures and pressures.

Table 3 .
Adjustable Parameters of Correction Function Given in eq 4 and Resulting ARDs between Experimental and Calculated Transport Properties of n-AlkanesFigure 1. Comparisons of the calculated and experimental selfdiffusion coefficients for CH 4 in a wide range of temperatures and pressures.Symbols denote experiment data, and lines denote calculated values.