J, Phys. Chem. 1992, 96, 11004-11009
11004 S. A., Ed.%; John Wiley and Sons: New York, 1975.
(78) Kamb. 8. Science 1965, 150, 205. (79) Ben-Naim, A. Wurer und Aqueous Solurions. Inrroducrion to u Moleculur Theory; Plenum Press: New York, 1974. (80) Rowlinson, J. S . Truns. Furuduy Soc. 1951, 47, 120. 1969, 3, 144. (81) Barker, J. A,; Watts. R. 0. Chem. Phys. h??. (82) Ben-Naim, A.; Stillinger, F. H. Srrucrure and Trunsporr Processes in Wurer und Aqueous Solutions; Home, R. A., Ed.; Wiley-Interscience: New York, 1972; Chapter 8. (83) Rahman, A,; Stillinger, F. H. J . Chem. Phys. 1971, 55, 3336. (84) Stillinger, F. H.; Rahman, A. J . Chem. Phys. 1974, 60, 1545. (85) Popkie, H.; Kistenmacher, H.; Clementi, E. J. Chem. Phys. 1973,59, 1325. (86) Matsuoka, 0.;Clementi, E.; Yoshimine, M. J . Chem. Phys. 1976,64, 1351. (87) Bcrendsen, H. J. C.; Postma, J. P. M.;Gunsteren, W. F.; Hermans, J. Inrermoleculur Forces; Pullman, B., Ed.; Reidel: Hingham, MA, 1981; p 331. (88) Jorgensen, W. L. Chem. Phys. h i t . 1983, 79, 926. (89) Jorgensen, W. L. J. Chem. Phys. 1982, 77,4156. (90) Perram. J. W. Hydrogen Bond Statistics. In The Hydrogen Bond. Recenr Developments in Theory und Experiments; North-Holland Publishing
Co.: Amsterdam, 1976; Chapter 7, p 368. (91) Stryjek, R.; Chappelear, P. S.;Kobayashi, R. J. Chem. Eng. Duru 1974, 19, 334. (92) Knapp, H.; Doring, R.; Oellrich, L.; Plocker, U.; Prausnitz, J. M. Vapor-Liquid Equilibria for Mixturcs of Low Boiling Substances. In DECHEMA Chemisfry Series; Behrens, D., Eckermann, R., Ed.; DECHEMA: Berlin, 1982; Vol. VI, p 539. (93) Clusius, von K.; Piesbergen, U.; Varde, E. Helu. Chim. Acru 1962, 45. ~. 1211. (94) Scheunumann. U.; Wagner, H. Gg. Ber. Bunsen-Ges. Phys. Chem. 1985,89, 1285. (95) Mueller, C . R.; Kcarns, E. R.J. Phys. Chem. 1958.62, 1441. (96) Chevalier, J. L.; Bares, D. J . Chim. Phys. Phys-Chim. Bio. 1969,66, 1448. (97) Ambrosc, D. J. Chem. Thermodyn. 1975, 7, 185. Ambrosc, D.; Lawrenson, I. J. J . Chem. Thermodyn. 1972, 4, 755. (98) Galivel-Solastiouk, F.; Laugier, S.;Richon, D. Fluid Phuse Equilib. 1986,28,73. (99) McGlashan, M. L.; Williamson, A. G.J . Chem. Eng. Dura 1976. 21, 196. (100) Randzio, S. L.; Tomaszkicwicz, I. Thermochim. Acru 1986, 103, 215. ~~~
~
Local Composition Model for SquareWell Chains Using the Generalized Fiory Dimer Theory Costas P. Bokis, Marc D. Donohue,* Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 21 218
and Carol K. Hall Department of Chemical Engineering, North Carolina State Uniuersity, Raleigh, North Carolina 27695 (Received: March 20, 1992)
The generalized Flory dimer (GFD) theory of Hall and co-workersprovides a highly accurate equation of state for hard-chain molecules. It has been extended here to treat chain molecules that interact with a square-well site-site potential by using closed-form expressions for the equations of state for squarewell monomer and dimer fluids based on local composition theory and using the analytic solution of the Percus-Yevick theory for monomers and RISM/MSA calculations for dimas. Comparison with Monte Carlo calculations for square-well molecules of lengths 4, 8, and 16 shows that there is very good agreement between this theory and simulation data.
Introduction In recent years, Hall and co-workersI4 have derived continuous-space analogues of the Florys lattice theory. They developed the generalized Floryl (GF)equation of state for freely-jointed tangent hard chains (n-mers). The GF compressibility factor can be written in the form ZHC(v,n)= 1 + aZcp(q,l)
(1)
where ZnP(v,l) is the compressibility factor for a system of hard spheres (monomers), which can be calculated by the CarnahanStarling6 equation, v is the volume fraction (9 = n ~ p d / 6where u is the segment diameter), and a is the ratio of the excluded volume of the n-mer to the excluded volume of the monomer (a = zc(n)/ue(l)). Their derivation was based on the realization that the nary lattice estimate for the probability of ihpertinga segment in a chain fluid is too large for a continuous-space fluid. Hence, they replaced the Flory insertion probability with the insertion probabilitycalculated from the Carnahan-Starling equation. They made the assumption that the probability of inserting a monomer into a chain fluid is the same as the probability of inserting a monomer into a monomer fluid, at the same volume fraction. The ratio of the excluded volumes, a,appears in eq 1 because the insertion of the second and subsequent segments of the chain Author to whom correspondence should be addressed.
0022-3654/92/2096-11004$03.00/0
requires a smaller hole in the fluid than the insertion of the first segment. As noted by Vimalchand and Donohue,' the GF equation has the same functional form as the perturbed-hardchain theory (PHCT).* Honnell and Hall4 extended and improved the GF theory by accounting for chain connectivity through the use of a dimer equation of state. They developed the generalized Flory dimer (GFD)theory for a hard-chain n-mer, which has the form ZHC(v,n)= Z(a2)
+ Y,(Z(v,2)- Z(v,1))
(2)
where Z(q,l) is the hard monomer compressibility factor, calculated from the Carnahan-Starling equation, Z(tj.2) is the hard dimer compressibility factor, calculated from the Tild~aley-Streett~ equation of state, and Y,is a function of the excluded volumes, given by
(3) where ue(l), ue(2), and u,(n) are the excluded volumes of the monomer, dimer, and n-mer, respectively. The GF'D theory assumes that the probability of inserting the whole chain into the chain fluid is the product of the probability of inserting the first site times the product of the conditional probabilities of inserting each subsequent site along the chain, given that the previous site has already been inserted. Q 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 1100s
Local Composition Model for Square-Well Chains
Honnell and Hall4 tested the performance of the GF and the GFD theories in predicting the compressibility factor of freelyjointed, tangent hard 4-mers, 8-mers, and 16-mers. They found that the GFD theory is in very good agreement with the simulation data of Dickman and Hall3 over the entire density range. The GF theory agrees qualitatively with the simulation data, but it overestimates the compressibility factor everywhere. More recently, Yethiraj and Halllo extended the GFD theory to square-well chains. They wrote the probability of inserting a segment of the square-well chain as the product of the insertion probability of the hard core of the segment times the conditional probability of inserting the square-well, given that the hard core has been already inserted. In their derivation they found that the attractive term in the equation of state has the same dependence on the excluded volumes as the repulsive term does, for both GF and GFD theories. Yethiraj and Hall obtained a square-well monomer equation of state from the Omstein-Zernike equation" with the mean spherical approximation (MSA) closureI2and a square-well dimer equation of state from the RISM t h e ~ r y ' ~ . ' ~ with the MSA closure. Although the resulting equation of state is highly accurate when compared to Monte Carlo simulations of square-well 4-mers, 8-men, and 16-men, it has the disadvantage that it is not suitable for engineering correlations, because the square-well monomer and dimer equations of state must be solved numerically, making the full equation of state too complicated for use in phase equilibrium calculations. The purpose of this paper is to examine various closed-form equations of state for square-well monomers and dimers and assess their accuracy in comparison to computer simulation data. Our ultimate goal is to find simple closed-form expressions for Z(q,l) and Z(w2) that, when inserted into the GFD equation of state for square-well chains (eq 2), give an equation of state that can serve as the basis for an engineering correlation of thermodynamic properties. In this work we use a local composition model for square-well monomers and dimers to calculate the attractive contribution to the compressibility factor of squarewell n-mers, and we compare our calculations with molecular simulation data.
derived from standard thermodynamic relations. There are several ways to approximate N,.The simplest is to assume that N,is independent of temperature and depends linearly on density; this results in the classical van der Waals attractive term. It is useful, however, to consider a more sophisticated approach that is grounded in the local compition lattice theory of Lee et al.I6 This serves as a foundation for building a hierarchy of approximations for N , based on statistical mechanics. Though lattice theories do not describe accurately many of the configurational (i.e., density dependent) properties of fluids,lJ' nonetheless, they are very useful in understanding certain aspects of liquid-state behavior, particubrly issues related to the composition and size dependence of the equation of state. In the Lee et a l . I 6 approach, the fluid is modeled as a lattice containing two components: squarewell monomers (labeled 1) and holes (labeled 0). Because of the attractive forces, these two components are not distributed randomly about a central species molecule but rather follow the local composition model
where N,, is the number of molecules of species i surrounding a central molecule of species j , N, is the total number of molecules of species i, Vis the volume, and P = r/u. By defining a maximum coordination number ZM, we can write NII
+ NO1 = Z M
(9)
By solving eqs 8 and 9 simultaneouslyand recognizing that N II is the coordination number Nc, one obtains (10)
where the quantity 9 is defined as
h
l Composition Model Following Sandler and co-workers1s,16 we start from the generalized van der Waals partition function:
where Vis the total volume, A is the De Broglie wavelength, Vf is the free volume, and 4 is the mean potential, defined as (5)
where Ec is the configurational energy. For the square-well potential this is given by
where N,is the coordination number (Le., the number of molecules that have their centers inside the shell (u,Ru) of a central molecule), g(r) is the radial distribution function, u is the hard-sphere diameter, e is the well depth, and R is the well width (taken here to be 1.5, since most computer simulation work has been done with this value). The coordination number is related to the attractive partition function according to
At this point, we need to make assumptions concerning the temperature and density dependence of the radial distribution function. Four models will be discussed here. In the fmt model, Lee et al. assumed that gv.= g$?-f/2kT,where g", is the hard-sphere radial distribution function. Furthermore, they assumed that the density dependence of the ratio of the integrals of gP. in eq 11 canceled out and since uol = ull, 9 = 4 z T .Also, et al. related the ratio No/Nl to the fluid volume by -No= - 1-7, (12) NI P
fee
where p = p/po with po equal to the closed-packed density. Thus they obtain
Substituting eq 13 into eq 10, we obtain the LLS (Lee,Lomhrdo, and SandlcrI6) equation, which has the form
(7) In the above equation, N , is a function of both density and temperature because the radial distribution function g(r) is a function of both. One needs to assume a function for the coordination number and use eq 7 to evaluate QL", and, then, other thermodynamic quantities (pressure, chemical potential, etc.) can be
In the above equation, Y = eftzkT- 1, where e is the well depth. In the second model, Vimalchand et a1.I' (VTED) improved the LLS equation by taking into account the changes in the fluid structure due to the repulsive part of the square-well potential function. They replaced p by tl,the average number of molecules
11006 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
Bokis et al.
TABLE I: P a m " for Em 18 sod 25 i
(monomers) i = 2 (dimers) i =1
a, 1.0422 0.8624
4 1.4302 1.6636
ci -6.5851 4.5768
kT/r=Z.O
in a spherical well with inner radius u and outer radius Ru surrounding a central molecule in a hard-sphere fluid. Therefore, eq 13 becomes
where
Substituting eq 16 into eq 10 results in the VTED equation, which has the form
0 1
7 I
0.00
I 0.15
I 0.30
t
I
I
t
0.45
I 0.60
I
t
0.75
rl
In the LLS equation, the integral on the right-hand side of eq 16 was assumed constant, independent of density and temperature. In the VTED quation, Vimalchand et al. fitted f, to a simple polynomial in reduced density, using as data the analytic solution of the Percus-Yevick theory (Wertheim,'* 1964). The result is
f~ = alv + 617' + c~v'
Figure 1. Coordination number for square-well monomers plotted against with volume fraction (i.e., density). Calculations are made at t = 2.0. Results of the LLS,VTED,mean-field, and local composition theories and computer simulations (Alder et aL20) are compared.
(18)
where the three constants are given in Table I. This equation is more accurate than LLS, yet retains much of its simplicity. In the third model, which is based on mean-field (MF) concepts, we make the assumption that g,, = $,. Equation 15 can be therefore written as
Substitutionof this equation into eq 10 leads to a simple mean-field expression for the coordination number: JVc
= ZMfI
(20)
In the fourth model, which is based on local composition (LC) arguments, we relax the assumption that gr/ = $, by adding a temperature- and densitydependent term similar to one proposed by Barker and Hender~0n.I~ The result is
N c = ZMfl+ fT 1
- eTZMtl) where f is the Mayer f-function, etlkT- 1.
(21)
Figure 1 shows a plot of the coordination number for quarawell monomers at a reduced temperature kT/t of 2.0 versus the volume fraction, 7 (i.e., density). The points represent the results from the 27-constant polynomial of Alder et a1.20 for the Helmholtz free energy and the relation JV, = -2
a(AStt/NkT)
(22) a(1m One sees that the LLS model (eq 14) gives a reasonable estimate for the coordination number. However, since the evaluation of the compressibility factor involves taking the density derivative of JV, the LLS equation does not give the correct compressibility factor because it does not have the correct shape. The VTED model (eq 17) changes the shape of the function but underestimates its values at low densities and overestimates them at high densities. The error at high densities is not very important, compared to the absolute value of the coordination number itself, whereas at low densities, the deviation becomes significant and leads to errors in calculations of other thermodynamic properties. The mean-field model (eq 20) gives the correct high-temperature limit and predicts the shape of the curve quite accurately, but it underpredicts absolute values of the coordination number ever-
0'7 0.6
t // .
.
I
0.0
0.1
~
I
I 0.3
0.2
I
t 0.4
I
i I
0.5
?
Ngwe 2. Ratio of the coordination numbers for square-well monomers obtained from the LLS, VTED, mean-field, and local composition theories divided by simulation data of Alder et a1.,20plotted against the volume fraction. Calculations are made at t = 1.5, 2.0, 3.0, and 4.0. ywhere. The local composition model (eq 21) departs from the ideal-gas limit at the correct rate and is the most accurate at low densities. The low-density region is very important for phase equilibrium calculations, especially calculations of dew points. This model is also the most accurate over the whole density range. Figure 2 shows a plot of the ratio of the coordination number calculated with the above theories and the simulation results for at reduced temperatures the coordination number of Alder et that range from 1.5 to 4.0. It is clear that the LC model (eq 21) is the most accurate over the entire density range. Both eq 17 (VTED) and 21 (LC) reduce to eq 20 (mean field) at high temperatures. The attractive contribution to the square-well monomer compressibility factor may be obtained for the four models considered by
where @ = l/kT. The following expressions were obtained for
q''
I
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 11007
Local Composition Model for Square-Well Chains
Zf"(VTED)
zMriy 1 + CIY
W
n
where Y = - 1 in eqs 24a and 24b, f = et/kT- 1 in eq 24d, ri= +I(df,/dq), and T = kT/c. For the square-well monomer fluid with R = 1.5, ZMhas a value of 18. This has been noted by Vimalchand et al.;" they mention that by using the analytic solution to the Perm-Yevick theory (Wertheim,Is 1964), it can be shown that 2,s- 18 as 7, 1 ( q 0.7405). This can be seen in Figure 1, where the extrapolation of the data for the coordination number goes to a value of about 18 at the closedpacked limit. This may be surprising, since at close-packing 12 molecules surround each central molecule. This is shown schematically in Figure 3a, where six molecules surround a central molecule in two dimensions and 12 will surround the molecule in three dimensions. However, if one considers how many molecules (of diameter a) can surround a central molecule at a distance of r = 1.5a, a different picture results as shown in Figure 3b. Here, it can be seen that eight molecules can pack easily into a two-dimensional shell of r = 1 . 5 and ~ 18 molecules can pack into the three-dimensional shell. Figure 4 shows calculations of the attractive contribution to the squarewellmonomer compressibility factor for the four models studied, at various reduced temperatures. Comparison is made with simulations by Alder et al.20 The LLS equation fails to predict the correct behavior, as expected, especially at high densities, since it does not take into account the changes in the structure of the fluid due to the repulsive part of the potential. For the other three models there is very good agreement at moderate and high densities. At low densities, however, only eq 24d (LC) is accurate. The VTED and the mean-field (eq 24c) models deviate slightly from the data. This deviation appears small in absolute terms but considerable in percentages. This leads to significant errors in the evaluation of second virial coefficients and, hence,phase equilibrium calculations, especially calculations of dew points. The mean-field theory (or equivalently the firstorder perturbation theory) is very accurate at moderate and high densities, where the temperature dependence of the comwibility factor is almost exactly l/T. It is in the low-density range that MF is inadequate. The local composition model that results from eq 21, however, is very accurate over the entire density range for square-well monomers. In order to implement the GFD themy for square-well chains, it is neceaPary to have an equation of state for squarewell dimers in addition to that for squarewell monomers. Since the mean-field (eq 20) and local composition (eq 21) models were the most s u d approaches for squarewell monomers, we have extended them to the treatment of square-well dimers. The function for dimers was evaluated by fitting the same polynomial used in eq 18 to RISM calculations for hard dimers (with the PY closurezi), with the result
- -
tz= azrl + bzt12+ cZv5
(25)
Values of the constants are also given in Table I. The attractive contribution to the square-well dimer compressibility factor from the mean-field approach (eq 20) is
W Figure 3. (a) Molecules at closed packing around a monomer molecule. (b) Area of influence of a monomer molecule with a square-well of width
R = 1.5. 0
-I
-2
d _I
Alder e t a1.
N
-3 kT/r=3.0
*
t -5
0.0
1 1 0.1
1
1 0.2
,
1 0.3
1
1 0.4
1
1
0.5
7
Figure 4. Attractive compressibility factors, Ptt, for square-well monomers obtained with the LLS,VTED, MF,and LC models are compared with computer simulations (Alder et al.*O).
Equation 21 results in a slightly more complicated expressions for qtt:
r2
where = -q(dc2/dq). For dimers, we found that a value of ZM= 17 gives better agreement with RISM calculations. This is not unexpected because, for dimers, one of the neighboring segments must be a segment on the same molecule. Figure 5 shows a plot of the attractive compressibility factors for squarewell dimers versus the volumc fraction, at various reduced temperatures for the mean-field (eq 26) and the local composition (eq 27) models. The results are compared with RISM calculations with the MSA closureZ2(which are very accurate in comparison to simulations). At high densities, both models give identical results, whereas at low densities (less than 0.2) the local composition model is more accurate than the mean-field. I.€-GFD Equation of State for Squarewell Chain Molecules The compressibility factor for a square-well chain fluid of n segments may be written as a sum of the i.deal-gas, repulsive, and attractive contributions: Z(7.n) = 1
+ Eep(q,n) + Ftt(q,n)
(28) In the GFD theory,I0 both the attractive and repulsive contri-
Bokis et al.
11008 The Journul of Physical Chemistry, Vol. 96, No. 26, 1992 0
0.0
-3 -6
-1.5
j -9 N
-I?
-3.0
kT/c=2.0 kT/c=3.0
A
-
LC-CFD 2
-18
-4.5
N
I
-5 -6.0
-7.5
-
-10
-
kT/c=2.O kT/c=3.0 kT/c=4.O
-
MF LC
*
z: &
*
____ -
-15
HC for SI 8-merr kT/c=ZO kT/c=J.O A
-20
-30
-9.0
0.0
0.2
0.1
0.3
0.4
0.5
1)
butions to the equations of state for chain molecules have the same dependence on the excluded volumes and, therefore, they can be written as
-
-q \
0.6
Figure 5. Attractive compressibility factors for square-well dimers obtained with the local composition model are compared with RISM/MSA calculations.
1K
LC-CFD
-25
-I8
;::: -40
1
1
0.2
0.1
/
\,
L;7;3; for S I 16-men MC
0.0
&
0.4
0.3
0.5
1)
Figure 6. Attractive compressibility factors for square-well 4-mers, 8mers, and 16-mers, obtained from the LC-GFD equation of state. Also shown are Monte Carlo simulation data from Yethiraj and Hall.2'
-
L
IO
(29b) where ZcP(q,l) and ZreP(q,2) are the monomer and the dimer repulsive compressibility factors, calculated with the CarnahanStarling6and the Tildesley-Streett9 equations of state, respectively. Here, the attractive monomer and dimer compressibility factors, Zatt(q,l) and P"(q,2), are taken from eqs 24d and 27, respectively, which result from the local composition model described above.
Comparison witb Simulations The GFD local composition expression for square-well chains is compared with the Monte Carlo simulation results of Yethiraj and Hall.22 Simulations were performed in the canonical ensemble by using the Metropolis algorithmaZ3The simulation cell was a rectangular parallelepiped, which was bounded by hard walls on two sides, with periodic boundary conditions employed in the other directions. The pressure of the chain fluid was taken from the contact value of the local density at the hard wall. Simulations were performed for squarewell4-mers, 8-mers, and 1Qmers at volume fractions ranging from 0.1 to 0.4 and at reduced temperatures p = kT/c of 1.5, 2.0, and 3.0. Since the GFD theory provides an accurate equation for the properties of hard-chain molecules, we have calculated the attractive contribution to the compressibility factors from the simulation data with the following equation: p i t
= zMC - zHC GFD
(30) where ZMCstands for the Monte Carlo simulation data for the square-well chainsz2and Z:gD is the hard-chain compressibility factor, Calculated from eq 2. Figure 6 shows plots of the attractive compressibilityfactor calculated with the local composition model (eq 29b) v m the volume fraction for squarewen Cmers, 8-mers, and 16-mers. Very good agreement between the model developed in this work and the simulation data is obtained. We believe that the slight deviation one sees for 16-mers is due to the hard-chain part of the equation (eq 2), which becomes less accurate at low densities for very long chains.
YC for S 1 4-merr
-
kT/c=ZO kT/c=3.0
6 -
-
N
CFD-I LC-CFD
A
----
-
/I I
MC for SW 8-merr kT/c=Z.O e kT/c=3.0 A
-
lol
LC-CFD
kT/r=3.0
A
N
14
-2
1
0.0
0.1
,
1
,
1
0.3
0.2
"
1
I
0.4
I 0.5
rl
Figure 7. Total compressibility factors for square-well 4-mers, &men, and 16-mers, obtained from the GFD-1'O and the LC-GFD equations of state. Also shown are Monte Carlo simulation data from Yethiraj and Hall.*'
Figure 7 shows plots of the total compressibility factors for squarewell Cmers, 8-mers, and 16-mers. Calculations are made by using two different equations of state: GFD-1 is the theory proposed by Yethiraj and Hall,Io in which the monomer attractive equation is from OZ/MSA and the dimer attractive equation is
J. Phys. Chem. 1992,96, 11009-1 1017
11009
properties of squarawell chains can be described by extrapolating the properties of the monomer and the dimer fluids,
from RISM/MSA. LC-GFD is the theory proposed in this paper, where the attractive term is calculated from eq 29b, using the local composition expressians for the monomer and the dimer attractive terms (eqs 24d and 27). The LC-GFD theory gives results that are in very good agreement with GFD-1 and the simulation data. For 4 m e n and &men at low temperatures, the compressibility factor shows negative values for both the GFD-1 and the LC-GFD equations. This is because at low temperature and for a range of density, the compressibilities fall inside the twephase region, where negative values of the pressure can be predicted by any equation of state that has a van der Waals' loop. LC-GFD has the advantage that it is a closed-form expression and therefore is suitable for phase-equilibrium calculations, whereas GFD- 1 requires numerical solution of the OZ and RISM integral equations for the calculation of the compressibility factor at each point of interest.
Acknowledgment. Support of this mearch by the Gas Research Institute under Contract 5089-260-1888 and the US. Department of Energy under Contracts DE-FG-02-87ER13777 and DEFGOS-91ER14181 is gratefully acknowledged. Acknowledgement is also made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. We also are grateful to Arun Thomas who performed the RISM calculations.
References and Notes Dickman, R.; Hall, C. K. J . Chem. Phys. 1986,85,4108. Honnell, K. G.;Hall, C. K.; Dickman, R. J. Chem. Phys. 1987,87,664. Dickman, R.; Hall, C. K. J. Chem. Phys. 1988, 89, 3168. Honnell, K. G.; Hall, C. K. J . Chem. Phys. 1989, 90,1841. (5) Flory, P. J. J . Chem. Phys. 1942, 10, 51. (6) Carnahan, N . F.; Starling, K. E. J . Chem. Phys. 1969, 51, 635. (7) Vimalchand, P.; Donohue, M. D. J . Phys. Chem. 1989, 93, 4355. (8) Donohue, M. D.; Prausnitz, J. M. AIChE J. 1978, 24, 849. (9) Tildesley, D. J.; Strectt, W. B. Mol. Phys. 1980, 41, 85. (10) Yethiraj, A,; Hall, C. K. J . Chem. Phys. 1991, 95, 8494. (1 1) McQuarrie, D. A. Sratistical Mechanics; Harper and Row: New York, 1976. (12) Haye, J. S.;Stell, G. J . Chem. Phys. 1977, 67, 439. (13) Chandler, D.; Andcrsen, H. C. J . Chem. Phys. 1972,57, 1930. (14) Chandler, D. J. Chem. Phys. 1973,59,2742; In Studies in Staristical Mechanics VIII; North-Holland: Amsterdam, 1982. (15) Sandler, S. I. Fluid Phase Equilib. 1985, 19, 233. (16) Lee,K. H.; Lombardo, M.; Sandler, S. I. Fluid Phase Equilib. 1985, (1) (2) (3) (4)
Conelusiolrs In this work we have developed an equation of state for the squarawell fluid. It is b a d on the generalized Flory dimer theory of Hall and co-workers and on a local composition model for the square-well monomer and dimer fluids. We compared the monomer equation against Monte Carlo simulations for spherical molecules and the dimer equation against RISM/MSA calculations. Excellent agreement was observed at various reduced temperatures, over the whole density range. Good results also were obtained by a simple mean-field equation, except at low densities. However, the low-density region is important in phase equilibrium calculations, particularly dew-point calculations. The local composition generalized Flory dimer (LC-GFD) equation of state was tested against Monte Carlo simulation data for square-well 4-mers, 8-mers, and 16-mers at various reduced temperaturea and dcnsitie, and very good agreement was obtained. This suggests that the observation made by Hall and co-workers, that the attractive term of the equation of state has the same size dependence as the repulsive term, is reasonable. Therefore, the
21, 177. (17) Vimalchand, P.; Thomas, A.; Economou, G. E.; Donohue, M. D. Fluid Phase Equilib. 1992, 73, 39. (18) Wertheim, M. S. J . Marh. Phys. 1964, 5, 643. (19) Barker, J. A.; Henderson, D. Rev. Mod.Phys. 1976,48, 587. (20) Alder, B. J.; Young, D. A.; Mark, M. A. J. Chem. Phys. 1972,56, 3013. (21) Thomas, A. Ph.D. Dissertation, The Johns Hopkins University, 1992. (22) Yethirai. A.: Hall. C. K. J . Chem. Phws. 1991. 95. 1999. (23) Metroph, N.; Rknbluth, A. W.; Rknbluth, M. N.; Teller, A. H.; Teller, E. J. Chem. Phys. 1953, 21, 1087.
Ion Migration In Chalcopyrlte Semiconductors Ceula Dagan,**t** T.F. Ciszek,! and David Cahen*.t The Weizmann Institute of Science, Rehovot 76100, Israel (Received: May 26, 1992; In Final Form: September 25, 1992)
Quantitative data are presented that show partial ionic conductivity of Cu or Ag in ternary and quaternary electronic semiconductors, with idealized stoichiometry Cu,Ag,,InSe2. A trend of increasing facility of ionic motion with increasing Ag content was observed. Ionic transference numbers up to 0.13 and 0.55 were measured for CuInSe, and AgInSq,, respectively. This trend can be correlated with the degree of compactness of the structure. It is supported by results from measurements of effective values of chemical diffusion coefficients, obtained by a potentiostatic current decay technique. Those results show a generally higher diffusivity in AgInSe, than in CuInSe2 A clear trend of increasing diffusion coefficient (up to lO-' cm2/s) with decreasing concentration of IB metal was observed. No obvious general correlation is seen between net electronic carrier concentration (or resistivity) and diffusion coefficients, except that overall the highest diffusion coefficients are found for Cu-poor (or Ag-poor) samples which are also the most resistive. The effect of temperature (between 20 and 100 "C) on the diffusivity is small. On the basis of our observations we conclude that diffusion occurs predominantly via a vacancy mechanism and suggest that this possibility of coexistence of significant ionic mobility with true semiconductivity can be useful for electronic doping by native defects.
I. Introduction Chalcogenides, with the general formula IB-III-V12, form a family of h t r u c t u r a l materials with the chalcopyrite structure. They are of practical interest, for use in photovoltaic devices 'The Weiunann Institute of Science. 'Present address: Physics Department, Brooklyn College of CUNY, Brooklyn, NY 11210. I National Renewable Energy Laboratory, Golden, CO 80401,
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(especially Culnsez') and in nonlinear optics (especially AgGa&2). An attractive feature of these materials is the possibility of tailoring their properties by isovalent substitutions and via deviations from the ideal 1:1:2 stoichiometry. This can be achieved because these materials retain their basic structure and semiconductivity over a range of compositions around the ideal one and because the electronic and optical properties of these compounds appear to be governed mainly by native defects. Unfortunately, the chemistry of native defects and the effects of these defects on the Q 1992 American Chemical Society
