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Intermolecular Potential-Based Equations of State from Molecular Simulation and Second Virial Coefficient Properties Richard J. Sadus*

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Computational Science Laboratory, Swinburne University of Technology, P.O. Box 218, Hawthorn, Victoria 3122, Australia ABSTRACT: The importance of both the Boyle temperature (TB) and temperature maximum (Tmax) for the ability of intermolecular potential-based equations of state to accurately predict second virial coefficient (B) behavior was examined. The TB, Tmax, and B vs T behavior of several Lennard-Jones equations of state, developed from molecular simulation data, were compared with exact theoretical values. The analysis spanned low (T ≤ TB), mid (TB < T ≤ Tmax), and high (T ≫ Tmax) temperatures. The value of TB was accurately predicted by most equations of state studied, whereas most failed to adequately predict Tmax. The ability to accurately predict both TB and Tmax appears to be correlated with the accurate prediction of B values for the entire range of temperatures. This provides a useful criterion to improve future potential-based equations of state. The Mecke et al. and Thol et al. LennardJones equations of state yielded the most accurate results. In some cases, a many-fold improvement in accuracy was observed compared with alternative equations of state. The exact B data were also used to obtain accurate polynomial relationships covering different ranges of temperatures.

1. INTRODUCTION Molecular simulation1 has an indispensible role in evaluating statistical mechanical2 concepts because it directly links the interaction between particles to a macroscopic phenomenon. It has been applied successfully to a wide range of phenomena, such as vapor−liquid3 and solid−liquid4 phase equilibria, thermodynamic properties,5 and protein dynamics.6 For thermodynamic properties, a simple alternative is to use an equation of state7 (EoS) that provides a relationship between pressure (p), temperature (T), and volume (V). Although the relationships between the EoS and all thermodynamic quantities are theoretically rigorous, the accuracy of this approach is limited by the assumptions and approximations used to formulate the EoS. Consequently, there is often considerable uncertainty as to whether or not the EoS adequately represents the modeled particle interactions. It is nevertheless highly desirable to overcome the accuracy limitations of conventional EoS because they are both computational expedient and, unlike molecular simulation data, the results are not subject to statistical uncertainty. The EoS and molecular simulation approaches are not mutually exclusive. Molecular simulation data can be used to improve the accuracy of EoS and molecular simulation studies can be informed by EoS predictions. In particular, it is possible to determine a potential-based EoS using molecular simulation data for a specified intermolecular potential (u(r), where r is the interparticle separation). Several potential-based EoS have been developed8−17 using molecular simulation data for the Lennard-Jones (LJ) potential.18 The focus on the LJ potential is understandable © XXXX American Chemical Society

in view of its important role in statistical mechanics. Obtaining a LJ EoS, or any other potential-based EoS, is a laborious process involving testing against a variety of simulation data over a large p−V−T range. Most LJ EoS8−13,17 use the modified Benedict−Webb−Rubin (MBWR) equation8 as the underlying formula, although some alternative formulations have also been investigated.14,16 The second virial coefficient (B) is particularly useful in evaluating potential-based EoS. For the LJ potential and other types of u(r), B and its attributes can often be evaluated exactly and as such it is not subject to either error or statistical uncertainty. Although B and other virial terms have been used in the development of LJ EoS, some of their key attributes have arguably not been exploited fully. If u(r) involves attractive interactions, B = 0 at the Boyle temperature (TB). B > 0 for T > TB, and this crossover between negative (at low T) and positive B values is frequently interpreted as the onset of dominant repulsive interactions. At a higher temperature (Tmax), B can be expected to pass through a maximum value when dB/dT = 0 and d2B/dT2 < 0. As Tmax ≫ TB, it could be reasonably expected that an EoS that accurately predicts both these properties could be accurate for a large range of temperatures. This is an important consideration because molecular simulation data at high T values are relatively scarce. We test this hypothesis by examining the B vs T behavior of several LJ EoS. The results Received: June 15, 2018 Revised: July 16, 2018 Published: July 20, 2018 A

DOI: 10.1021/acs.jpcb.8b05725 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B

Table 1. Summary of LJ EoS, their Critical Properties, and T* Range Compared in this Work (Listed in Order of Publication Date) EoS/ref

year

description

T*c

ρ*c

T* range

Nicolas et al. (ref 8) Adachi et al. (ref 9) Koutras et al. (ref 10) Johnson et al. (ref 11) Kolafa and Nezbeda (ref 12) Sun and Teja (ref 13) Mecke et al. (ref 14) May and Mausbach (ref 15) Thol et al. (ref 16)

1979 1988 1992 1993 1994 1996 1996 2012 2016

MBWR MBWR CS HS reference MBWR MBWR (the parameters that affect B* are the same as those in Johnson et al.) MBWR HS + attractive term MBWR (the parameters that affect B* are the same as those in Johnson et al.) multiparameter correlation

1.35 1.2731 1.355 1.313 1.3393 1.3 1.328 1.3145 1.32

0.35 0.2842 0.290 0.31 0.3204 0.31 0.3107 0.316 0.31

0.55−6 0.7−3.0 0.68−1.3 0.7−6 0.7−20 0.45−6 0.7−10 0.7−6 0.66−9.24

are likely to have implications for the future direction of potential-based EoS.

B* =

2. THEORY The value of B at any T can be obtained either directly from u(r) or the compressibility factor (Z = p/ρkT, where ρ = N/V is the number density of N particles and k is Boltzmann’s constant) via the following relationships2 B = B(T ) = 2π

∫0

∞

(1 − e−u(r)/ kT )r 2 dr

i dZ y B = B(T ) = jjjj zzzz k dρ { ρ= 0

(1)

(2)

where ε is the minimum depth of the potential and σ is the distance at which u(r) = 0. Lennard-Jones established that B for his eponymous potential could be obtained18,19 exactly from

i ε y y = 2 jjj zzz k kT {

| | oo o 1 yz i o ij ∑ Γjji − zzy /i!}ooooooo 2{ o i=1 k ~o } o o o o o o o o o ~

x2 T*

+

x3 x x y + 42 + 53 zzzz T* T* T* {

(5)

where the x terms represent the first 5 of the 32 parameters in the MBWR equations. This means that reparameterized variants of the LJ MBWR EoS, which have identical values of the first five terms, such as the Johnson et al.,11 Kolafa and Nezbeda,12 and May and Mausbach15 EoS will all yield the same B* properties. The recent Pieprzyk et al.17 reformulation of the LJ MBWR EoS replaces the contribution of these first five parameters with a theoretically exact relationship25 for B*, which means it is guaranteed to yield correct B* values. Therefore, it is not meaningful to compare it to the other LJ EoS for this particular property. An alternative to the MBWR approach is to transpose the properties of the LJ fluid on a simple “hard sphere (HS) + attractive term” framework. Koutras et al.10 used the Carnahan−Starling (CS) HS EoS, with B* obtained from eq 5. The use of the CS framework results in very different values of x than those obtained from the conventional MBWR approach. Kolafa and Nezbeda12 combined a T-dependent HS term based on a Barker−Henderson26 perturbation treatment with a virial expansion. This later aspects means that we cannot use it to independently obtain B* for the EoS and it is not considered in this work. Mecke et al.14 combined hybrid Barker−Henderson26 perturbation and attractive contributions. We find that B* for the Mecke et al. EoS can be obtained from

The significance of eq 2 is that if an EoS is developed for a particular u(r), its accuracy can be compared to eq 1, which alone directly involves u(r). This means the accuracy of the EoS can be assessed both rigorously and unambiguously. The LJ potential18 is ÄÅ ÉÑ ÅÅÅij σ yz12 ij σ yz6ÑÑÑ u(r ) = εÅÅÅjj zz − jj zz ÑÑÑ ÅÅk r { k r { ÑÑÑÖ (3) ÅÇ

l o i3y 1 2 o j z B(T ) = πσ 3 y m oΓjj 4 zz − 4 o 3 n k {

3 ijj jx1 + 2π jk

∞

B* =

ij 3 yz zz jjj + jj 2πρ* zzz 0.3674 y i T * j z c { zz k πρc*jjj0.689 + 0.311 T * z c k { m ij ij T * yz i yzz jj cijjj zzz zzz jj ∑ j T * z zz jj k i = 2,7,10,11,12,15,21 k c { { 0.9702

( )

(4)

where Γ is the gamma function. Eq 4 can be generalized for any combination of exponents,20 and subsequently several alternative and equally accurate expressions have been reported.21−25 It is convenient to use reduced properties, i.e., B* = 3B/2πσ3, T* = kT/ε, and ρ* = ρσ3. Nicolas et al.8 are credited with developing the first LJ EoS that accurately reproduced a range of thermodynamic properties from molecular simulation. The underlying basis of their EoS was the 32-parameter MBWR equation. The MBWR framework is common to many subsequent LJ EoS being reparameterized by Adachi et al.,9 Johnson et al.,11 Kolafa and Nezbeda,12 Sun and Teja,13 May and Mausbach,15 and Pieprzyk et al.17 In most instances, B* for the different parameterized MBWR variants can be obtained from

(6)

The first term in eq 6 is a T*-dependent HS contribution, whereas the selective summation in the second term reflects the fact that not all of the ci parameters contribute to B*. Mecke et al.27 subsequently revised the values of some of the EoS parameters; however, these new values are not involved in eq 6. Thol et al.16 have recently proposed an accurate correlation for the LJ fluid using an extensive set of molecular simulation and virial coefficient data. Applying eq 2 to the Thol et al. EoS yields B


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The Journal of Physical Chemistry B Table 2. Comparison of LJ EoS Calculations and Exact Theory for T*B, T*max, and B* (T*max)a

a

ÄÅ ÅÅ t ij T * yz i 3 ÅÅÅÅ B* = ÅÅ ∑ nijjjj c zzzz + 2πρc* ÅÅÅ i = 2,3,11,12 k T * { ÅÅÇ

Values in brackets are the percentage relative deviations from the exact values.

ij T * yz i ∑ nijjj c zzz j z i = 13,14,19,22,23 k T * { ÉÑ 2 y ti Ñ Ñ ij i y z * i y jj jj jj Tc zz zz zz ÑÑÑÑ 2 j z j z expjj−ηε − jjβi jj zz − γi zz zz ÑÑ j j T* z z zz ÑÑÑ jj i i k k { { { ÑÑ k (7) Ö

also obtained for the other EoS values, with deviations of typically ≤2%. The largest discrepancies of approximately 5% were obtained for the Koutras et al. EoS. These good results may partially reflect the fact that T*B = 3.4179 is typically in the mid-range of T* values used to evaluate the EoS parameters (see Table 1). T*max = 25.1526 for the LJ potential is well outside the T* range used to parameterize most LJ EoS. Molecular simulation data for such high T* values are relatively few compared with the greater abundance of low T* data. The comparison given in Table 2 indicates that the LJ EoS struggle to accurately predict T*max. The Nicolas et al. and Sun and Teja EoS substantially underpredict T*max, whereas it is predicted at substantially higher temperatures for the Johnson et al./Kolafa and Nezbeda/May and Mausbach, Adachi et al., and Koutras et al. EoS. In contrast, the Mecke et al. and Thol et al. EoS yield very good (1.2%) and near perfect (−0.08%) agreement, respectively. These very good results are impressive as they are outside of the parameterization T* range (Table 1). It has been previously observed16 that the validity of the Mecke et al. EoS could extend to beyond the T* limits of its parameterization. The accuracy of the Thol et al. EoS probably benefits from including virial coefficient behavior as part of its parameterization. The Mecke et al. EoS yields the best agreement (0.01%) for B* (T*max), while also maintaining very good agreement for T*max. In contrast, there is a greater discrepancy (−3.2%) for the Thol et al. EoS despite yielding the most accurate T*max value. The Johnson et al./Kolafa and Nezbeda/May and Mausbach and Adachi et al. EoS also yield good agreement for B* (T*max), although T*max is predicted at a value that is substantially higher than that obtained from eq 4. Both B* (T*max) and T*max are predicted inaccurately by the remaining EoS. 3.2. Prediction of B* from Low to High T*. The values of TB and T*max provide a convenient way of segmenting the B* vs T* behavior into different T* regions, namely, Regions I (0.5 ≤ T* ≤ T*B), II (T*B < T* ≤ T*max), and III (T*max < T* ≤ 50), which very broadly correspond to low, mid-range, and high values of T*. Graphical comparisons of the predictions of the various LJ EoS (symbols) with the exact results (thick solid line) are given in Figures 1−4. The AAD and RMSD values associated with the different regions are summarized in Table 3. In most instances, the values of T* in Region I (Figure 1) fall well within the parameterization scope (Table 1) of the LJ EoS and as such could be considered the fairest T* region for

t

It is apparent that eq 7 requires several adjustable parameters (t, ε β γ, η) and coefficients (n). In common with the Mecke et al. EoS, the summation is restricted to terms that contribute to B*. Both summation terms in eq 7 refer to residual properties that do not involve either any HS or other reference terms. In eq 7, εi is simply an adjustable parameter, which should not be confused with the minimum well-depth term in the LJ potential appearing in eq 3. A summary of the LJ EoS is given in Table 1. The evaluation of the EoS parameters typically includes using vapor−liquid equilibrium data, and Table 1 also summarizes the predicted critical parameters. It is apparent from Table 1 that there is some variation in both T*c and ρ*c compared with the currently accepted28 molecular simulation values of T*c = 1.316 and ρ*c = 0.316. In all cases, the parameters were evaluated for ρ* ≤ 1.2. In this work, we have used the LJ EoS parameters as originally reported in the literature.8−16 It is useful to quantify the quality of agreement between exact values and EoS calculations. For a given property (X), the agreement between exact values and EoS calculations can be quantified in terms of the average absolute deviation (AAD) for the N data points, i.e., AAD =

1 N

N

∑

Xi ,exact − Xi ,EoS Xi ,exact

i=1

(8)

The root mean square deviation (RMSD) is also useful RMSD =

1 N

N

∑ (Xi ,exact − Xi ,EoS)2 i=1

(9)

3. RESULTS AND DISCUSSION 3.1. Prediction of TB and T*max. Values of T*B, T*max, and B* (T*max) calculated for the various LJ EoS are compared with the exact values obtained from eq 4 in Table 2. The T*B results for the Mecke et al. and Thol et al. EoS are in near perfect agreement with the exact values. Good agreement is C


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Figure 1. Comparison of EoS predictions (symbols) with exact B* values (solid red line) in Region I (0.5 ≤ T* ≤ T*B). Results are illustrated for the Johnson et al./Kolafa and Nezbeda/May and Mausbach (○), Nicolas et al. (△), Adachi et al. (▽), Koutras et al. (×), Sun and Teja (+), Mecke et al. (□), and Thol et al. (◊) EoS. The line through the Koutras et al. EoS results is for guidance only.

Figure 2. Close-up comparison of EoS predictions (symbols) with exact B* values (solid red line) for the lowest T* values of Region I. Results are illustrated for the Johnson et al./Kolafa and Nezbeda/May and Mausbach (○), Nicolas et al. (△), Adachi et al. (▽), Sun and Teja (+), Mecke et al. (□), and Thol et al. (◊) EoS. The lines through the EoS calculations are for guidance only.

comparison. It is apparent from Figure 3 that the Koutras et al. EoS overestimates B* at all temperatures and the discrepancy with the exact data progressively increases at low values of T*. In contrast, the remaining EoS provide reasonably accurate predictions at most temperatures. The distinction in the quality of predictions can be best appreciated by considering the AAD and RMSD data in Table 3. The Mecke et al. EoS yields the most accurate results with AAD and RMSD values of 0.094% and 0.015, respectively. This represents a many-fold improvement in accuracy compared with the other EoS. The Thol et al. EoS also yields very accurate values. The Johnson et al./Kolafa and Nezbeda/May and Mausbach and Adachi et al. EoS also provide accurate predictions (AADs ≈ 1%), whereas both the Sun and Teja and Nicolas et al. EoS are considerably less accurate. Greater inaccuracies in the EoS prediction emerge at very low temperatures. Figure 2 presents a close-up view in the range T* = 0.5−0.65. The Koutras et al. EoS is excluded from

this comparison because its inaccuracy in this T* region was already clearly observed in Figure 1. Figure 2 shows that the predictions of the Nicolas et al. EoS are too low, whereas the Johnson et al./Kolafa and Nezbeda/May and Mausbach, Sun and Teja and Adachi et al. EoS remain accurate in this low T* region. The results for the Mecke et al. EoS are too low, whereas the Thol et al. values are too high. We observed that B* for the Thol et al. EoS was particularly sensitive to the input values of T*c and ρ*c; e.g., a change of ±0.01 could typically triple the AAD value. In the mid-range of temperatures (Region II, Figure 3), a very clear distinction between the qualities of prediction of the EoS emerges. At T* > T*B, the Koutras et al., Nicolas et al., and Sun and Teja EoS significantly underestimate B*. For the latter two EoS, this can be at least attributed to the fact that T*max is too low (see Table 2). In contrast, the Adachi et al. EoS values become slightly too high as the LJ potential T*max is approached, which reflects its somewhat higher T*max value. Both the Mecke et al. (AAD = 0.02%, RMSD = 0.0001) and

Table 3. Comparison of AAD (%) and RMSD B* Values Obtained from eq 4 and LJ EoS Calculations in the T* Range Covered by Regions I (0.5 ≤ T* ≤ T*B), II (T*B < T* ≤ T*max), III (T*max < T* ≤ 50), and All (0.5 ≤ T* ≤ 50) Data

D


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Koutras et al., Nicolas et al., and Sun and Teja EoS to significantly underestimate B* continues at high T* values. It is also apparent for the Nicolas et al. and Sun and Teja EoS that the decline in B* is more rapid than the exact LJ potential data. In contrast, both the Johnson et al./Kolafa and Nezbeda/May and Mausbach and Adachi et al. EoS predict B* values that are higher than the exact values from eq 4 at all T* values in this interval. This is consistent with these equations having Tmax > 30 (see Table 2). The Mecke et al. and Thol et al. EoS are accurate in Region III, with the latter clearly overtaking the former for greatest accuracy (see AAD and RMSD values in Table 3). Eq 4 requires B* → 0 as T* → ∞. Only the Thol et al. EoS (eq 7) satisfies this requirement. At this limit, positive nonzero B* values are predicted for the Sun and Teja (0.153), Koutras et al. (0.193), Mecke et al. (0.338), Johnson et al./Kolafa and Nezbeda/May and Mausbach (0.412), and Adachi et al. (0.404) EoS whereas a small negative value is obtained for the Nicolas et al. (−0.021) EoS. It is therefore conceivable that at extreme values of T*, the Sun and Teja, Koutras et al., and Nicolas et al. EoS may over take the Mecke et al. EoS for second place accuracy behind the Thol et al. EoS. Indeed, an upturn in B* calculated for the Mecke et al. EoS has been reported16 for T* > 1000. Table 3 also summarizes AAD and RMSD data calculated for the complete range of T*, which can be used to rank the overall accuracy of the various EoS. In order of most accurate to least accurate, we find (Thol et al., Mecke et al.) > (Adachi et al., Johnson et al./Kolafa and Nezbeda/May and Mausbach) ≫ Sun and Teja ≫ Nicolas et al. ≫ Koutras et al. It is noteworthy that the best two EoS are not based on the MBWR framework. Although the Thol et al. EoS is best overall, its functional form is complicated and the high degree of correlation of properties means that it requires careful implementation. The Mecke et al. EoS is both simpler than the MBWR and Thol et al. approaches and yields accurate results in most cases. The usefulness of eq 5 indicates that the accurate B* data obtained from eq 4 can be fitted to the following general polynomial relationship

Figure 3. Comparison of EoS predictions (symbols) with exact B* values (solid red line) in Region II (T*B < T* ≤ T*max). Results are illustrated for the Johnson et al./Kolafa and Nezbeda/May and Mausbach (○), Nicolas et al. (△), Adachi et al. (▽), Koutras et al. (×), Sun and Teja (+), Mecke et al. (□), and Thol et al. (◊) EoS. The lines through the EoS calculations are for guidance only.

Thol et al. (AAD = 0.07%, RMSD = 0.0002) EoS are very accurate in this region. These equations also accurately predict both T*B and T*max, which represent the lower and upper T* limits of Region II. The B* vs T* behavior at high T* values (Region III) is illustrated in Figure 4. The trend observed in Figure 3 for the

∑ aiijjj N

B* =

i=0

1 yzi zz k T* {

(10)

We found that it was not possible to obtain a single polynomial relationship that yielded accurate agreement with the exact values (eq 4) for the entire range of temperatures. Instead, accurate agreement with eq 4 could be obtained by analyzing Regions I, II, and III separately. The a coefficients for these regions are summarized in Table 4, and the AAD and RMSD values are given in Table 3. The B* values of Region I proved the most difficult to fit accurately, requiring 10 terms, whereas only 8 and 6 terms were required for Regions II and III, respectively. The accurate representation of Region I was also a challenge for the EoS (see Figure 2). The comparison in Table 3 indicates that eq 10 yields the most accurate overall results. However, both the Mecke et al. and Thol et al. EoS have the advantage of yielding very good results from a single equation.

Figure 4. Comparison of EoS predictions (symbols) with exact B* values (solid red line) in Region III (T*max < T* ≤ 50). Results are illustrated for the Johnson et al./Kolafa and Nezbeda/May and Mausbach (○), Nicolas et al. (△), Adachi et al. (▽), Koutras et al. (×), Sun and Teja (+), Mecke et al. (□), and Thol et al. (◊) EoS. The lines through the EoS calculations are for guidance only.

4. CONCLUSIONS B* relationships for several LJ EoS were reported and used to calculate B*, T*B, and T*max. Comparison with theoretically exact B* values over different regions of T* indicated different E



■

Table 4. Polynomial Coefficients for eq 10 in the T* Range Covered by Regions I (0.5 ≤ T* ≤ T*B), II (T*B < T* ≤ T*max), and III (T*max < T* ≤ 50) coefficient a index

Region I

Region II

Region III

0 1 2 3 4 5 6 7 8 9

0.65266 −1.4488 −3.8897 5.9426 −7.7236 6.5707 −3.7347 1.3462 −0.27855 0.025148

0.45942 4.3199 −90.126 789.2 −4434.3 15094 −28270 22296

0.36609 14.629 −576.5 12977 −1.6477 × 105 8.8844 × 105

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Richard J. Sadus: 0000-0002-6044-0409 Notes

The author declares no competing financial interest.

■

REFERENCES

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levels of accuracy for the various EoS. The Koutras et al. EoS yields poor results at all values of T*. For the remaining EoS that do not accurately predict both T*B and T*max, good agreement is confined only to 0.5 ≤ T* ≤ TB. The Mecke et al. EoS, which accurately predicts both TB and T*max, yielded the most accurate B* predictions in this T* region. At very low temperatures (T* = 0.5−0.65), some deviation from the exact results was observed for both the Mecke et al. and Thol et al. EoS compared with the more accurate results obtained for the Johnson et al./Kolafa and Nezbeda/May and Mausbach, Sun and Teja and Adachi et al. EoS. However, overall, the Mecke et al. EoS provides a many-fold improvement over other EoS. The difference in the predictive quality of the LJ EoS becomes very apparent at both mid (T*B < T* ≤ T*max)- and high (T*max < T* ≤ 50)-temperature regions. In both these regions, highly accurate results are obtained for either the Mecke et al. and Thol et al. EoS, whereas the remaining EoS either significantly under- or overpredict B* values. As T* → ∞, only the Thol et al. EoS correctly predicts that B* → 0, as required by theory. Accurate polynomial relations for B* were also reported, which are specific to the different T* regions. Although many EoS can provide a reasonable estimate of T*B, only the Mecke et al. and Thol et al. EoS can accurately reproduce the theoretically exact values for both T*B and T*max. In particular, very large errors in T*max were observed for the other EoS. The ability to accurately predict both these temperatures appears to be an important consideration for the accurate prediction of B* from low to high temperatures. The ability to predict both T*B and T*max is a useful criterion for the future development of potential-based EoS. Using T*B and T*max to evaluate potential-based EoS has three advantages. First, for the LJ and many other potentials, both T*B and T*max can be calculated exactly from theory. This is important because, irrespective of its accuracy, molecular simulation is always subject to statistical uncertainty. Second, T*B is likely to be well within the range of parameterized temperatures for the EoS, whereas T*max is likely to be well outside of this range. Therefore, the ability to predict T*max provides an indication of whether or not the EoS can be used for out of range temperatures. Third, the requirement that ρ = 0 simplifies the parameterization procedure by not involving many of the EoS parameters. This means, that the reduced range of parameters required for T*B and T*max can be evaluated independently of parameters requited for other properties such as vapor−liquid equilibria. F


Article

The Journal of Physical Chemistry B (21) Glasser, M. L. Second Virial Coefficient for a Lennard-Jones (2n-n) System in d Dimensions and Confined to a Nanotube Surface. Phys. Lett. A 2002, 300, 381−384. (22) Vargas, P.; Muñoz, E.; Rodriguez, L. Second Virial Coefficient for the Lennard-Jones Potential. Phys. A 2001, 290, 92−100. (23) González-Galderón, A.; Rocha-Ichante, A. Second Virial Coefficient of a Generalized Lennard-Jones Potential. J. Chem. Phys. 2015, 142, No. 034305. (24) Eu, B. C. Exact Analytic Second Virial Coefficient for LennardJones. 2009, arXiv:physics/0909.3326. arXiv.org e-Print archive. https://arxiv.org/abs/0909.3326. (25) Heyes, D. M.; Rickayzen, G.; Pieprzyk, S.; Brańka, A. C. The Second Virial Coefficient and Critical Point Behavior of the Mie Potential. J. Chem. Phys. 2016, 145, No. 084505. (26) Barker, J. A.; Henderson, D. Perturbation Theory and Equation of State for Fluids: The Square Well Potential. J. Chem. Phys. 1967, 47, 2856−2861. (27) Mecke, M.; Müller, A.; Winkelmann, J.; Vrabec, J.; Fischer, J.; Span, R.; Wagner, W. Erratum: “An Accurate van der Waals-Type Equation of State for the Lennard-Jones Fluid”. Int. J. Thermophys. 1998, 19, 1493. (28) Potoff, J. J.; Panagiotopoulos, A. Z. Critical Point and Phase Behavior of the Pure Fluid and a Binary Lennard-Jones Mixture. J. Chem. Phys. 1998, 109, 10914−10920.

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