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Modified combinatorial factor in the hole theory
Fluid Phase Equilibria 161 Ž1999. 271–281
Modified combinatorial factor in the hole theory Mooho Hong a , Jaeeon Chang a , Hwayong Kim
a, )
, Jiho Park
b
a
b
School of Chemical Engineering, Seoul National UniÕersity, Seoul 151-742, South Korea R&D Center, Kumho Chemicals, 45-25, SungamDong, NamGu, Ulsan GwangyeokSi, 680-140, South Korea Received 1 September 1998; accepted 16 January 1999
Abstract To describe the behavior of liquid state by hole theory, Flory’s combinatorial factor used in the partition function of the hole theory is modified in a general form by assuming that unit segment of chain molecule can occupy more than one unit cell or lattice. From this combinatorial factor and free volume expression, a new equation of state is derived, and is used to calculate the PVT behaviors of 10 pure low molecular weight liquid systems. In addition, the PVT behavior of carbontetrachloridercyclohexane binary system is calculated. Good agreements between the theory and experimental data are obtained. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Liquid state; Combinatorial factor; Equation of state; PVT; Mixture
1. Introduction Hole theory was developed from cell model by adopting the free volume concept by Eyring and Hirschfelder w1x. Especially for liquid state, free volume model in the hole theory is known to be the best in accuracy. The partition function for the hole theory is composed of three parts, combinatorial term, free volume term and energy term. Until recently, free volume modification has been the main issue in the hole theory. Significant contributions have been made by Simha and Somcynsky w2x and others w3–5x. The combinatorial factor plays an important role in accounting for the geometry and arrangement of molecules. Progress has been made by Staverman w6x and Tompa w7x since Flory w8x and Guggenheim w9x. These combinatorial factors as well as Guggenheim’s are often used in lattice model. On the other hand, Flory’s combinatorial factor, which turned out later to be a limiting case of
)
Corresponding author. Tel.: q82-2-880-7406; fax: q82-2-888-7295; e-mail: [email protected]
0378-3812r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 9 . 0 0 1 9 1 - 0
272
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
Fig. 1. Simple molecular arrangement in the cellrlattice.
Guggenheim’s, has been used in the hole theory because of its simplicity. Even though intramolecular bonding for bulky molecules was considered by Staverman et al., all these combinatorial factors are based on the assumption of a single segment per a single cell r lattice. In the hexagonal close packing structure of cellrlattice, as seen in Fig. 1, one segment of molecule occupies one cellrlattice, which most of the hole theories are based on. In this study, we devise a new molecular arrangement in which a cellrlattice is composed of smaller subcellsrsublattices, as shown in Fig. 2. The new molecular arrangement allows segments of different kinds in chain molecules to occupy different number of subcellsrsublattices. With this idea, we extend Flory’s combinatorial factor more general form in which repeating unit segment of chain molecule occupies p unit subcells with this modified Flory’s combinatorial factor and free volume expression appropriate for this system, we derive a new equation of state and applied it to various pure polymer systems w10x, pure low molecular weight liquid systems. Quite good agreement between the theory and experimental data are obtained for low molecular weight liquid materials especially at low pressure. And we applied it to carbontetrachloridercyclohexane binary liquid mixture in order to predict the PVT behavior.
Fig. 2. New molecular arrangement with subcellsrsublattices.
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
273
2. Combinatorial factor The partition function in the hole theory is expressed as:
f s V ÕfCNexp yE0rkT
Ž1.
where, V is combinatorial factor, which determines the number of arrangement of molecules according to their structure and shape, Õf is free volume expressing the mobility and flexibility of molecules, E0 is interaction energy between segments. C and N are the external degree of freedom and the number of molecules, respectively. In this study, we derived a new combinatorial factor based on the following assumptions: Ø The steric hindrance between molecules is neglected. Ø The whole system before arranging molecules is composed of n 0 numbers of smaller unit subcellsrsublattices. Ø N monodisperse s-mers are arranged in this system and other Nh vacant subcells are regarded as holes. Ø One repeating unit segment of chain molecule can occupy a group of subcells, which consists of p subcells. Ø Each group of p subcells which segments of chain molecule occupy is exclusive for other segments of same molecule or other molecules. With these assumptions, the number of ways of arranging i q 1-th s-mer, Õiq1 , assuming that i s-mers already occupy psi unit subcells in total number of n 0 unit subcells, is calculated. The first segment of i q 1-th s-mer can be arranged in n 0-psi vacant cells. The second segment is placed in z neighboring sites of first segment already inserted. The third and other segments are subsequently placed in equal z y 1 neighboring sites of the previous site: Õiq1 s Ž n 0 y psi . z Ž 1 y f i .Ž z y 1 .Ž 1 y f i . PPP Ž z y 1 .Ž 1 y f i .
Ž2.
Eq. Ž2. can be approximated to Eq. Ž 3. : s Ž n 0 y psi . z Ž z y 1 .
sy2
Ž1 y fi .
sy1
( Ž n 0 y psi .Ž z y 1 .
sy1
Ž1 y fi .
sy1
Ž3.
The cell for any segment to be placed in should be vacant. So, it is necessary to calculate 1 y f i , the probability of finding a vacant cell. Although the f i ’s are different from segment to segment in the same molecules, they can be assumed to be the same without serious error when the equal probability approximation w10x is adopted. Because the agglomerated p unit subcells cannot be distinguished, f i can be calculated as follows: p
Ý f i ( pf i s is1
psi n0
Ž4.
so: fi s
si n0
Ž5.
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
274
Inserting Eq. Ž5. into Eq. Ž3., gives: Õiq1 s p
ž
n0 p
/
y si Ž z y 1 .
sy1
ž
n 0 y si n0
sy1
/
Ž6.
Assuming that the parameter p is in the order of unity, Eq. Ž 6. can be approximated by: Õiq1 ( p
zy1
sy1
ž /
s
Ž n 0 y si .
n0
Ž7.
Eq. Ž7. can be rewritten in a factorial form as: Õiq1 s p
zy1
sy1
n0
Ž n 0 y si . ! Ž n 0 y s Ž i q 1. . !
Ž8.
The total number of arrangement V is then given by: 1 N Vs ŁÕ N ! is1 i Inserting Eq. Ž8. into Eq. Ž9., V is found to be:
VspN
1
n0 !
zy1
N!
Ž n 0 y sN . !
n0
Ž9.
N Ž sy1 .
Ž 10.
After the factorials are approximated by Stirling’s formula, final form of combinatorial factor is expressed occupied site fraction y:
VssN
zy1
pyy
N Ž sy1 .
y
e
p
2
yN
1y
y
yNs y
Ž 11.
p
where: ys
Nsp
Ž 12.
n0
When p is one, Eq. Ž11. is transformed to Flory’s combinatorial factor. 3. Free volume and energy It is known by Eyring and Hirschfelder that the volume change of liquid is closely related by free volume change with increasing hole fraction w1x. Since then, many studies on the free volume has been made. Among them, Simha and Somcynsky proposed a free volume expression by linearly interpolating the solid part free length, v 1r3 y 2y1r6 ÕU1r3, and gas part free length, v 1r3, according to the occupied site fraction y. In this study, we adopted Simha and Somcynsky’s free volume expression modified for our this agglomerated subcell system. That is, the cell volume v is modified to pv : Õf s l f3 s y ž Ž pv .
1r3
y 2y1r6 ÕU1r3 / q Ž 1 y y .Ž pv .
1r3 3
Ž 13.
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
275
Where, ÕU is the characteristic segment volume. As an energy term for Eq. Ž1., Lennard–Jones 6–12 potential was used in the first version: E0 s
1 2
Nqz ´ U 1.011
ÕU
ÕU
4
ž / pv
y 2.409
2
ž /
Ž 14.
pv
where, qz is the surface coordination number of a molecule defined as: qz s s Ž z y 2 . q 2
Ž 15.
4. Equation of state By inserting Eqs. Ž11., Ž13. and Ž14. into Eq. Ž 1. , complete form of partition function is obtained. From partition function and the thermodynamic relation: Eln f P s kT Ž 16. EV T , y
ž /
and with the minimizing Helmholtz energy: EA s0 E y T ,V
ž /
Ž 17.
Equation of state is derived as in the following: 1r3
˜˜ PV
2y Ž yV˜ . s q 1r3 T˜ ž Ž yV˜ . y 2y1r6 y / T˜
1
2
ž / yV˜
1
s 1 sy1 3c y
ž
p q
s
y
ž
ln 1 y
y p
//
q
1.011
Ž yV˜ .
Ž yV˜ . 3y Ž yV˜ .
1r3
1r3
2
y 1.2045
y 2y1r6
1 s
y 2y1r6 y
Ž 18-1.
1
ž /
6T˜ yV˜
2
2.409 y
3.033
Ž yV˜ .
2
Ž 18-2. The characteristic parameters are the same in both versions of EOS and are defined as: T P V T˜s U , P˜ s U , V˜ s U T P V U U ´ qz ´ qz TUs , PU s , V U s NsÕU ck sÕU
Ž 19-1. Ž 19-2.
5. Results and discussion 5.1. Application to polymer melts We applied this EOS to the PVT behavior of 10 pure polymer melts including polystyrene. In Table 1, the result of calculation and the scaling parameters for the equation of state are shown. We
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
276
Table 1 The calculation results for PVT data of pure polymer melts by version 1 of EOS T U ŽK.
Polymer
P U Žbar.
V U Žcm3 rg.
p
AADŽ%. a SS
Polystyrene Polypropylene Linear-polyethylene Branched-polyethylene Polycyclohexylmethacrylate Poly-o-methylmethacrylate Polyvinylacetate Polyepichlorohydrin Polymethylmathacrylate High Mw. linear-polyethylene grand. Avg.
11 596.2 8462.5 8779.2 9406.0 10 794.9 11 878.0 8541.8 10 374.9 10 814.5 9234.4
6740.8 6247.6 7617.8 6561.5 7572.4 7063.6 9039.1 8560.6 8819.3 8965.2
0.968 1.128 1.143 1.174 0.905 0.986 0.819 0.740 0.842 1.129
1.24 1.24 1.24 1.24 1.21 1.23 1.24 1.24 1.24 1.24
b
Ref. This
0.035 0.061 0.070 0.068 0.067 0.046 0.022 0.042 0.010 0.079 0.050
c
0.025 0.059 0.051 0.053 0.056 0.037 0.021 0.035 0.008 0.078 0.042
w10x w11x w12x w12x w12x w10x w11x w11x w12x w12x
a
AADŽ%. is 1rŽ Number of Point. Ý <Ž Vexp yVcalc .r Vexp <=100%. SS is Simha and Somcynsky’s EOS with Flory’s combinatorial factor. c This is EOS by Eq. Ž22..
b
note that the parameter p, which was at first thought to vary with polymer species, can be set to 1.24 in most polymer melts, except for 1.21 in the case of polycyclohexylmethacrylate, which leads to an improvement of about 15%. For polymer systems studied in this work, irrespective of polymer kinds, PVT of polymer melts can be well described with constant p value of 1.24. This fact clearly shows rather universal characteristics independent of the degree of motion and free volume of the repeating segment. Although the result of this EOS is similar to that of Simha and Somcynsky’s, but a substantial improvement is seen at low pressures. 5.2. Application to low molecular liquids A similar procedure was performed to calculate the PVT of 10 low molecular weight liquid systems. The list of these systems is shown in Table 2. Unlike polymers, low molecular weight
Table 2 The calculation results for PVT data of pure polymer melts by version 2 of EOS PSU No.
Full name
MW
18 25 110 111 113 174 175 528 532 537
1-Phenyl-3Ž2-phenylethyl. hendecane 9-n-Octylheptadecane 9Ž3-cyclopentylpropyl. heptadecane 1-Cyclopentyl-4Ž3-cyclopentylpropyl. dodecane 1,7-Dicyclopentyl-4Ž3-cyclopentylpropyl. heptane 1-alpha-Naphthylpentadecane 1-alpha-Decalylpentadecane n-Dodecane n-Pentadecane n-Octadecane
336.5 352.7 350.7 348.6 346.6 338.7 348.6 170.3 212.4 254.5
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
277
materials have finite chain lengths. Thus, molecular parameters such as sr3c, Ž s y 1.rs in Eq. Ž18-2. are not unities. The relation between the external degree of freedom and the chain length is given by Eq. Ž20.: 3c s s q 3
Ž 20.
From Eqs. Ž18-1., Ž18-2. , Ž19-1. , Ž19-2. and Ž20., the molecular parameters for these systems are shown in Table 3, where Large p is another version of equation of state in which the first term Žcombinatorial contribution. of the left hand side in Eq. Ž18-2. is reduced to a simple form: lim p™ `
p y
ž
ln 1 y
y p
/
Ž 21.
s y1
When p value is greater than 5, the second part in combinatorial contribution in Eq. Ž18-2. converges to y1, thus the final form of equation becomes Eq. Ž 22.. This limiting behavior is shown in Fig. 3. This effect is also demonstrated in Fig. 4, which is the plot of average absolute deviations for carbontetrachloride with increasing p: 1
s 1 sy1 3c y
ž
s
/
y1 q
Ž yV˜ . 3y Ž yV˜ .
1r3
1r3
y 2y1r6
1 s
y 2y1r6 y
1
2
ž /
6T˜ yV˜
2.409 y
3.033
Ž yV˜ .
Ž 22.
2
Although the value of p, 1.24, can be used for these systems without serious errors, parameter p can be chosen by parameter optimization, which gives larger value than 1.24 as in the case of polymer
Table 3 The results of PVT calculation for low molecular systems by Simha Somcynsky’s, and Large p version Name Cyclohexane CCl 4 Psu18 Psu25 Psu110 Psu111 Psu113 Psu174 Psu175 Psu528 Psu532 Psu537 n-Nonane Grand avg. a
Temp. ŽK. 313–383 273–323 310–408 310–408 310–408 310–408 311–408 333–408 333–408 310–408 310–408 310–408 303–423
Press. Žbar. 1–2140 1–1977 1–10 336 1–10 336 1–10 336 1–10 336 1–10 336 1–5512 1–5857 1–6891 1–6546 1–5512 1–8000
N.P.a 42 25 134 146 145 155 119 48 54 78 69 53 516
P 1.24 1.24 1.00 1.02 1.24 1.00 1.00 1.24 1.24 1.24 1.24 1.24 1.00
AADŽ%. SS
Opt. p
0.087 0.049 0.169 0.124 0.130 0.147 0.081 0.086 0.085 0.299 0.153 0.127 0.131 0.130
0.080 0.035 § 0.123 0.130 § § 0.071 0.066 0.280 0.128 0.106 § 0.120
N.P. is the number of data points. Opt. p is by optimization of p value in this EOS—Eqs. Ž18-1. and Ž18-2.. c Large p is EOS version Eqs. Ž18-1. and Ž22.. b
Ref. b
Large p 0.088 0.054 0.150 0.104 0.101 0.132 0.077 0.095 0.085 0.296 0.158 0.136 0.118 0.123
c
w14x w15x w13x w13x w13x w13x w13x w13x w13x w13x w13x w13x w16x
278
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
Fig. 3. Combinatorial contribution in Helmholts energy minimization with increasing p.
systems. The reason for this is that the parameter p is related to free volume as in Eq. Ž13.. Especially for n-nonane as shown in Table 3, with the large number of PVT data and the wide range of temperature and pressure, the accuracy of the calculation was improved by about 10% compared to that of Simha Somcynsky’s. From these facts, it can be deduced that in the range of high temperature and low pressure where the kinetic motion of molecules is enhanced, the accuracy of the calculation is much more improved as parameter p increases. Thus, as shown in Table 4, the result of calculation agrees with the experimental data at low pressure.
Fig. 4. AAD plot for carbontetrachloride at atmospheric pressure with increasing p.
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
279
Table 4 The results of PVT calculation for low molecular systems at atmospheric pressure by Simha Somcynsky’s, and Large p version Name
T U ŽK.
P U Žbar.
V U Žcm3 rg.
AADŽ%. a SS b
T U ŽK.
P U Žbar.
V U Žcm3 rg.
AADŽ%. a Large p c
Cyclohexane CCl 4 Psu18 Psu25 Psu110 Psu111 Psu113 Psu174 Psu175 Psu528 Psu532 Psu537 n-Nonane Grand avg.
5552 5553 8845 8142 8317 8243 8775 8828 8747 6738 7333 7666 6134
7519 7974 7591 7184 7637 8173 8014 8878 8605 8978 7763 7723 6421
1.118 0.547 1.054 1.186 1.153 1.112 1.088 1.058 1.113 1.207 1.210 1.200 1.243
0.0465 0.0401 0.0158 0.0192 0.0340 0.0621 0.0183 0.0171 0.0151 0.1626 0.0409 0.0293 0.0427 0.0418
4974 5002 6893 6213 6386 6478 6793 6783 6804 5395 5657 6218 5143
6889 7369 7591 7183 7637 8173 8014 8878 8173 8978 7763 7723 5218
1.116 0.546 1.039 1.167 1.136 1.100 1.074 1.042 1.100 1.202 1.187 1.200 1.228
0.0039 0.0008 0.0089 0.0068 0.0184 0.0271 0.0107 0.0162 0.0459 0.0358 0.0443 0.0187 0.0473 0.0219
a
AADŽ%. is 1rŽNumber of Point.Ý <Ž Vexp yVcalc .r Vexp <=100%. SS is Simha and Somcynsky’s EOS with Flory’s combinatorial factor. c Large p is EOS version Eqs. Ž18-1. and Ž22..
b
5.3. Application to binary liquid mixture To predict the PVT data of mixture with only pure component parameters, mixing rule is necessary. Mixing rule adopted for this calculation is as follows: ² s : s x 1 s1 q x 2 s 2 , ² c : s x 1 c1 q x 2 c 2
Ž 23-1,23-2 .
² qz: s x 1Ž s1Ž z y 2 . q 2 . q x 2 Ž s2 Ž z y 2 . q 2 . s ² s : Ž z y 2 . q 2 ² M0 : s
x 1 s1 M01 q x 2 s2 M02 ² s:
Ž 24 . Ž 25.
U U4 U U4 ² ´ U :² ÕU :4 s X 12´ 11 Õ 11 q 2 X 1 X 2 ´ 12 Õ 12 q X 22´ U22 ÕU224
Ž 26-1.
U U2 U U2 ² ´ U :² ÕU :2 s X 12´ 11 Õ 11 q 2 X 1 X 2 ´ 12 Õ 12 q X 22´ U22 ÕU222
Ž 26-2.
X 1 s x 1qz 1r Ž x 1qz 1 q x 2 qz 2 . , X 2 s x 2 qz 2r Ž x 1qz 1 q x 2 qz 2 .
´
U 12 s
(
U 11
j ´ ´
U 22
,
ÕU12 s
ß
ž
ÕU1r3 q ÕU1r3 11 22 2
Ž 27.
3
/
Ž 28-1,28-2 .
Where, j and z are binary interaction parameters for energy and volume, respectively. The volumetric binary interaction parameter, z , can be fixed to unity for this system for all three equations
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
280
Table 5 AAD’s for CarbontetrachloriderCyclohexane mixture Mole frac. of CCl 4
SS
This EOS
Large p
0.0000 0.2512 0.3720 0.6506 0.7475 1.0000
0.0465 0.0536 0.0553 0.0536 0.0506 0.0410
0.0704 0.0715 0.0741 0.0742 0.0716 0.0618
0.0039 0.0090 0.0116 0.0118 0.0093 0.0009
This EOS is Eqs. Ž18-1. and Ž18-2. when p is 1.24.
of state, the energetic binary interaction parameter, j , is also close to unity in Simha Somcynsky’s and Large p version. The value of this energetic binary parameter in this equation of state is as small as about 0.95. Mixing rule for p is not necessary because p values for two components are equal to each other. In Table 5, for the system of carbontetrachloridercyclohexane at atmospheric pressure, a good agreement is obtained by Large p version as Eqs. Ž 18-1. and Ž22. . We deduce from this fact that in the region where dynamic motion of molecules increases, parameter p can fully express this effects, close to infinity. Another reason for the good agreement is attributed to the fact that the entropy change of mixing is far much closer to that of real system. In other words, ideal mixing entropy change is guessed to be smaller than that of real system, but more accurate value close to that of real system can be described with the parameter p derived from the new combinatorial factor.
6. Conclusion A generalized form of Flory’s combinatorial factor was derived from the assumption that a segment of chain molecule occupies more than one cellrlattice. From this, an equation of state based on hole theory was derived and applied to polymer, low molecular systems and binary liquid mixtures. Good agreement was obtained and PVT of mixture was well predicted. Especially for low molecular weight liquid systems at low pressure, Large p version of this equation of state is recommended from the fact that optimized parameter p goes to large value. Parameter p introduced in this study has a clear physical meaning and the degree of dynamic motion of molecule is accounted for by this parameter.
7. List of symbols A 3c k N Nh P p
Helmholtz energy external degrees of freedom Boltzmann constant the number of molecules the number of holes pressure the number of cells occupied by segments
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
s T V X y z
the number of segments in one molecule temperature volume surface fraction volume fraction occupied by segment coordination number
Greek letters ´ energy parameter in Lennard–Jones potential f partition function V combinatorial factor v volume of unit cell Subscripts f h
free volume hole
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x
H. Eyring, J. Hirschfelder, J. Phys. Chem. 41 Ž1937. 249–257. R. Simha, T. Somcynsky, Macromolecules 2 Ž1969. 342–350. C. Zhong, W. Wang, H. Lu, Fluid Phase Equilibria 27 Ž1993. 660–664. E. Nies, A. Stroeks, Macromolecules 23 Ž1990. 4088–4092. J. Park, H. Kim, Fluid Phase Equilibria 144 Ž1998. 77–86. A.J. Staverman, Recl. Trav. Chim. Pays-Bas 69 Ž1950. 163–171. H. Tompa, Trans. Faraday Soc. 48 Ž1952. 363–370. P.J. Flory, J. Chem. Phys. 10 Ž1942. 51–62. E.A. Guggenheim, Mixture, The Oxford University Press, Amen House, London, 1952, 183–196. A. Quach, R. Simha, J. Appl. Phys. 42 Ž1971. 4592–4601. P.A. Rodgers, J. Appl. Polym. Sci. 48 Ž1993. 1061–1079. O. Olabisi, R. Simha, Macromolecules 8 Ž1975. 206–210. W.G. Cutler et al., J. of Chem. Physics 29 Ž1958. 727–731. J. Jonas et al., J. of Phys. Chem. 84 Ž1980. 109–112. F.I. Mopsik, J. of Chem. Physics 50 Ž6. Ž1969. 2559–2563. R.K. Crawford et al., J. of Chem. Physics 50 Ž1969. 3171–3183.
281
Modified combinatorial factor in the hole theory Mooho Hong a , Jaeeon Chang a , Hwayong Kim
a, )
, Jiho Park
b
a
b
School of Chemical Engineering, Seoul National UniÕersity, Seoul 151-742, South Korea R&D Center, Kumho Chemicals, 45-25, SungamDong, NamGu, Ulsan GwangyeokSi, 680-140, South Korea Received 1 September 1998; accepted 16 January 1999
Abstract To describe the behavior of liquid state by hole theory, Flory’s combinatorial factor used in the partition function of the hole theory is modified in a general form by assuming that unit segment of chain molecule can occupy more than one unit cell or lattice. From this combinatorial factor and free volume expression, a new equation of state is derived, and is used to calculate the PVT behaviors of 10 pure low molecular weight liquid systems. In addition, the PVT behavior of carbontetrachloridercyclohexane binary system is calculated. Good agreements between the theory and experimental data are obtained. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Liquid state; Combinatorial factor; Equation of state; PVT; Mixture
1. Introduction Hole theory was developed from cell model by adopting the free volume concept by Eyring and Hirschfelder w1x. Especially for liquid state, free volume model in the hole theory is known to be the best in accuracy. The partition function for the hole theory is composed of three parts, combinatorial term, free volume term and energy term. Until recently, free volume modification has been the main issue in the hole theory. Significant contributions have been made by Simha and Somcynsky w2x and others w3–5x. The combinatorial factor plays an important role in accounting for the geometry and arrangement of molecules. Progress has been made by Staverman w6x and Tompa w7x since Flory w8x and Guggenheim w9x. These combinatorial factors as well as Guggenheim’s are often used in lattice model. On the other hand, Flory’s combinatorial factor, which turned out later to be a limiting case of
)
Corresponding author. Tel.: q82-2-880-7406; fax: q82-2-888-7295; e-mail: [email protected]
0378-3812r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 9 . 0 0 1 9 1 - 0
272
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
Fig. 1. Simple molecular arrangement in the cellrlattice.
Guggenheim’s, has been used in the hole theory because of its simplicity. Even though intramolecular bonding for bulky molecules was considered by Staverman et al., all these combinatorial factors are based on the assumption of a single segment per a single cell r lattice. In the hexagonal close packing structure of cellrlattice, as seen in Fig. 1, one segment of molecule occupies one cellrlattice, which most of the hole theories are based on. In this study, we devise a new molecular arrangement in which a cellrlattice is composed of smaller subcellsrsublattices, as shown in Fig. 2. The new molecular arrangement allows segments of different kinds in chain molecules to occupy different number of subcellsrsublattices. With this idea, we extend Flory’s combinatorial factor more general form in which repeating unit segment of chain molecule occupies p unit subcells with this modified Flory’s combinatorial factor and free volume expression appropriate for this system, we derive a new equation of state and applied it to various pure polymer systems w10x, pure low molecular weight liquid systems. Quite good agreement between the theory and experimental data are obtained for low molecular weight liquid materials especially at low pressure. And we applied it to carbontetrachloridercyclohexane binary liquid mixture in order to predict the PVT behavior.
Fig. 2. New molecular arrangement with subcellsrsublattices.
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
273
2. Combinatorial factor The partition function in the hole theory is expressed as:
f s V ÕfCNexp yE0rkT
Ž1.
where, V is combinatorial factor, which determines the number of arrangement of molecules according to their structure and shape, Õf is free volume expressing the mobility and flexibility of molecules, E0 is interaction energy between segments. C and N are the external degree of freedom and the number of molecules, respectively. In this study, we derived a new combinatorial factor based on the following assumptions: Ø The steric hindrance between molecules is neglected. Ø The whole system before arranging molecules is composed of n 0 numbers of smaller unit subcellsrsublattices. Ø N monodisperse s-mers are arranged in this system and other Nh vacant subcells are regarded as holes. Ø One repeating unit segment of chain molecule can occupy a group of subcells, which consists of p subcells. Ø Each group of p subcells which segments of chain molecule occupy is exclusive for other segments of same molecule or other molecules. With these assumptions, the number of ways of arranging i q 1-th s-mer, Õiq1 , assuming that i s-mers already occupy psi unit subcells in total number of n 0 unit subcells, is calculated. The first segment of i q 1-th s-mer can be arranged in n 0-psi vacant cells. The second segment is placed in z neighboring sites of first segment already inserted. The third and other segments are subsequently placed in equal z y 1 neighboring sites of the previous site: Õiq1 s Ž n 0 y psi . z Ž 1 y f i .Ž z y 1 .Ž 1 y f i . PPP Ž z y 1 .Ž 1 y f i .
Ž2.
Eq. Ž2. can be approximated to Eq. Ž 3. : s Ž n 0 y psi . z Ž z y 1 .
sy2
Ž1 y fi .
sy1
( Ž n 0 y psi .Ž z y 1 .
sy1
Ž1 y fi .
sy1
Ž3.
The cell for any segment to be placed in should be vacant. So, it is necessary to calculate 1 y f i , the probability of finding a vacant cell. Although the f i ’s are different from segment to segment in the same molecules, they can be assumed to be the same without serious error when the equal probability approximation w10x is adopted. Because the agglomerated p unit subcells cannot be distinguished, f i can be calculated as follows: p
Ý f i ( pf i s is1
psi n0
Ž4.
so: fi s
si n0
Ž5.
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
274
Inserting Eq. Ž5. into Eq. Ž3., gives: Õiq1 s p
ž
n0 p
/
y si Ž z y 1 .
sy1
ž
n 0 y si n0
sy1
/
Ž6.
Assuming that the parameter p is in the order of unity, Eq. Ž 6. can be approximated by: Õiq1 ( p
zy1
sy1
ž /
s
Ž n 0 y si .
n0
Ž7.
Eq. Ž7. can be rewritten in a factorial form as: Õiq1 s p
zy1
sy1
n0
Ž n 0 y si . ! Ž n 0 y s Ž i q 1. . !
Ž8.
The total number of arrangement V is then given by: 1 N Vs ŁÕ N ! is1 i Inserting Eq. Ž8. into Eq. Ž9., V is found to be:
VspN
1
n0 !
zy1
N!
Ž n 0 y sN . !
n0
Ž9.
N Ž sy1 .
Ž 10.
After the factorials are approximated by Stirling’s formula, final form of combinatorial factor is expressed occupied site fraction y:
VssN
zy1
pyy
N Ž sy1 .
y
e
p
2
yN
1y
y
yNs y
Ž 11.
p
where: ys
Nsp
Ž 12.
n0
When p is one, Eq. Ž11. is transformed to Flory’s combinatorial factor. 3. Free volume and energy It is known by Eyring and Hirschfelder that the volume change of liquid is closely related by free volume change with increasing hole fraction w1x. Since then, many studies on the free volume has been made. Among them, Simha and Somcynsky proposed a free volume expression by linearly interpolating the solid part free length, v 1r3 y 2y1r6 ÕU1r3, and gas part free length, v 1r3, according to the occupied site fraction y. In this study, we adopted Simha and Somcynsky’s free volume expression modified for our this agglomerated subcell system. That is, the cell volume v is modified to pv : Õf s l f3 s y ž Ž pv .
1r3
y 2y1r6 ÕU1r3 / q Ž 1 y y .Ž pv .
1r3 3
Ž 13.
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
275
Where, ÕU is the characteristic segment volume. As an energy term for Eq. Ž1., Lennard–Jones 6–12 potential was used in the first version: E0 s
1 2
Nqz ´ U 1.011
ÕU
ÕU
4
ž / pv
y 2.409
2
ž /
Ž 14.
pv
where, qz is the surface coordination number of a molecule defined as: qz s s Ž z y 2 . q 2
Ž 15.
4. Equation of state By inserting Eqs. Ž11., Ž13. and Ž14. into Eq. Ž 1. , complete form of partition function is obtained. From partition function and the thermodynamic relation: Eln f P s kT Ž 16. EV T , y
ž /
and with the minimizing Helmholtz energy: EA s0 E y T ,V
ž /
Ž 17.
Equation of state is derived as in the following: 1r3
˜˜ PV
2y Ž yV˜ . s q 1r3 T˜ ž Ž yV˜ . y 2y1r6 y / T˜
1
2
ž / yV˜
1
s 1 sy1 3c y
ž
p q
s
y
ž
ln 1 y
y p
//
q
1.011
Ž yV˜ .
Ž yV˜ . 3y Ž yV˜ .
1r3
1r3
2
y 1.2045
y 2y1r6
1 s
y 2y1r6 y
Ž 18-1.
1
ž /
6T˜ yV˜
2
2.409 y
3.033
Ž yV˜ .
2
Ž 18-2. The characteristic parameters are the same in both versions of EOS and are defined as: T P V T˜s U , P˜ s U , V˜ s U T P V U U ´ qz ´ qz TUs , PU s , V U s NsÕU ck sÕU
Ž 19-1. Ž 19-2.
5. Results and discussion 5.1. Application to polymer melts We applied this EOS to the PVT behavior of 10 pure polymer melts including polystyrene. In Table 1, the result of calculation and the scaling parameters for the equation of state are shown. We
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
276
Table 1 The calculation results for PVT data of pure polymer melts by version 1 of EOS T U ŽK.
Polymer
P U Žbar.
V U Žcm3 rg.
p
AADŽ%. a SS
Polystyrene Polypropylene Linear-polyethylene Branched-polyethylene Polycyclohexylmethacrylate Poly-o-methylmethacrylate Polyvinylacetate Polyepichlorohydrin Polymethylmathacrylate High Mw. linear-polyethylene grand. Avg.
11 596.2 8462.5 8779.2 9406.0 10 794.9 11 878.0 8541.8 10 374.9 10 814.5 9234.4
6740.8 6247.6 7617.8 6561.5 7572.4 7063.6 9039.1 8560.6 8819.3 8965.2
0.968 1.128 1.143 1.174 0.905 0.986 0.819 0.740 0.842 1.129
1.24 1.24 1.24 1.24 1.21 1.23 1.24 1.24 1.24 1.24
b
Ref. This
0.035 0.061 0.070 0.068 0.067 0.046 0.022 0.042 0.010 0.079 0.050
c
0.025 0.059 0.051 0.053 0.056 0.037 0.021 0.035 0.008 0.078 0.042
w10x w11x w12x w12x w12x w10x w11x w11x w12x w12x
a
AADŽ%. is 1rŽ Number of Point. Ý <Ž Vexp yVcalc .r Vexp <=100%. SS is Simha and Somcynsky’s EOS with Flory’s combinatorial factor. c This is EOS by Eq. Ž22..
b
note that the parameter p, which was at first thought to vary with polymer species, can be set to 1.24 in most polymer melts, except for 1.21 in the case of polycyclohexylmethacrylate, which leads to an improvement of about 15%. For polymer systems studied in this work, irrespective of polymer kinds, PVT of polymer melts can be well described with constant p value of 1.24. This fact clearly shows rather universal characteristics independent of the degree of motion and free volume of the repeating segment. Although the result of this EOS is similar to that of Simha and Somcynsky’s, but a substantial improvement is seen at low pressures. 5.2. Application to low molecular liquids A similar procedure was performed to calculate the PVT of 10 low molecular weight liquid systems. The list of these systems is shown in Table 2. Unlike polymers, low molecular weight
Table 2 The calculation results for PVT data of pure polymer melts by version 2 of EOS PSU No.
Full name
MW
18 25 110 111 113 174 175 528 532 537
1-Phenyl-3Ž2-phenylethyl. hendecane 9-n-Octylheptadecane 9Ž3-cyclopentylpropyl. heptadecane 1-Cyclopentyl-4Ž3-cyclopentylpropyl. dodecane 1,7-Dicyclopentyl-4Ž3-cyclopentylpropyl. heptane 1-alpha-Naphthylpentadecane 1-alpha-Decalylpentadecane n-Dodecane n-Pentadecane n-Octadecane
336.5 352.7 350.7 348.6 346.6 338.7 348.6 170.3 212.4 254.5
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
277
materials have finite chain lengths. Thus, molecular parameters such as sr3c, Ž s y 1.rs in Eq. Ž18-2. are not unities. The relation between the external degree of freedom and the chain length is given by Eq. Ž20.: 3c s s q 3
Ž 20.
From Eqs. Ž18-1., Ž18-2. , Ž19-1. , Ž19-2. and Ž20., the molecular parameters for these systems are shown in Table 3, where Large p is another version of equation of state in which the first term Žcombinatorial contribution. of the left hand side in Eq. Ž18-2. is reduced to a simple form: lim p™ `
p y
ž
ln 1 y
y p
/
Ž 21.
s y1
When p value is greater than 5, the second part in combinatorial contribution in Eq. Ž18-2. converges to y1, thus the final form of equation becomes Eq. Ž 22.. This limiting behavior is shown in Fig. 3. This effect is also demonstrated in Fig. 4, which is the plot of average absolute deviations for carbontetrachloride with increasing p: 1
s 1 sy1 3c y
ž
s
/
y1 q
Ž yV˜ . 3y Ž yV˜ .
1r3
1r3
y 2y1r6
1 s
y 2y1r6 y
1
2
ž /
6T˜ yV˜
2.409 y
3.033
Ž yV˜ .
Ž 22.
2
Although the value of p, 1.24, can be used for these systems without serious errors, parameter p can be chosen by parameter optimization, which gives larger value than 1.24 as in the case of polymer
Table 3 The results of PVT calculation for low molecular systems by Simha Somcynsky’s, and Large p version Name Cyclohexane CCl 4 Psu18 Psu25 Psu110 Psu111 Psu113 Psu174 Psu175 Psu528 Psu532 Psu537 n-Nonane Grand avg. a
Temp. ŽK. 313–383 273–323 310–408 310–408 310–408 310–408 311–408 333–408 333–408 310–408 310–408 310–408 303–423
Press. Žbar. 1–2140 1–1977 1–10 336 1–10 336 1–10 336 1–10 336 1–10 336 1–5512 1–5857 1–6891 1–6546 1–5512 1–8000
N.P.a 42 25 134 146 145 155 119 48 54 78 69 53 516
P 1.24 1.24 1.00 1.02 1.24 1.00 1.00 1.24 1.24 1.24 1.24 1.24 1.00
AADŽ%. SS
Opt. p
0.087 0.049 0.169 0.124 0.130 0.147 0.081 0.086 0.085 0.299 0.153 0.127 0.131 0.130
0.080 0.035 § 0.123 0.130 § § 0.071 0.066 0.280 0.128 0.106 § 0.120
N.P. is the number of data points. Opt. p is by optimization of p value in this EOS—Eqs. Ž18-1. and Ž18-2.. c Large p is EOS version Eqs. Ž18-1. and Ž22.. b
Ref. b
Large p 0.088 0.054 0.150 0.104 0.101 0.132 0.077 0.095 0.085 0.296 0.158 0.136 0.118 0.123
c
w14x w15x w13x w13x w13x w13x w13x w13x w13x w13x w13x w13x w16x
278
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
Fig. 3. Combinatorial contribution in Helmholts energy minimization with increasing p.
systems. The reason for this is that the parameter p is related to free volume as in Eq. Ž13.. Especially for n-nonane as shown in Table 3, with the large number of PVT data and the wide range of temperature and pressure, the accuracy of the calculation was improved by about 10% compared to that of Simha Somcynsky’s. From these facts, it can be deduced that in the range of high temperature and low pressure where the kinetic motion of molecules is enhanced, the accuracy of the calculation is much more improved as parameter p increases. Thus, as shown in Table 4, the result of calculation agrees with the experimental data at low pressure.
Fig. 4. AAD plot for carbontetrachloride at atmospheric pressure with increasing p.
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
279
Table 4 The results of PVT calculation for low molecular systems at atmospheric pressure by Simha Somcynsky’s, and Large p version Name
T U ŽK.
P U Žbar.
V U Žcm3 rg.
AADŽ%. a SS b
T U ŽK.
P U Žbar.
V U Žcm3 rg.
AADŽ%. a Large p c
Cyclohexane CCl 4 Psu18 Psu25 Psu110 Psu111 Psu113 Psu174 Psu175 Psu528 Psu532 Psu537 n-Nonane Grand avg.
5552 5553 8845 8142 8317 8243 8775 8828 8747 6738 7333 7666 6134
7519 7974 7591 7184 7637 8173 8014 8878 8605 8978 7763 7723 6421
1.118 0.547 1.054 1.186 1.153 1.112 1.088 1.058 1.113 1.207 1.210 1.200 1.243
0.0465 0.0401 0.0158 0.0192 0.0340 0.0621 0.0183 0.0171 0.0151 0.1626 0.0409 0.0293 0.0427 0.0418
4974 5002 6893 6213 6386 6478 6793 6783 6804 5395 5657 6218 5143
6889 7369 7591 7183 7637 8173 8014 8878 8173 8978 7763 7723 5218
1.116 0.546 1.039 1.167 1.136 1.100 1.074 1.042 1.100 1.202 1.187 1.200 1.228
0.0039 0.0008 0.0089 0.0068 0.0184 0.0271 0.0107 0.0162 0.0459 0.0358 0.0443 0.0187 0.0473 0.0219
a
AADŽ%. is 1rŽNumber of Point.Ý <Ž Vexp yVcalc .r Vexp <=100%. SS is Simha and Somcynsky’s EOS with Flory’s combinatorial factor. c Large p is EOS version Eqs. Ž18-1. and Ž22..
b
5.3. Application to binary liquid mixture To predict the PVT data of mixture with only pure component parameters, mixing rule is necessary. Mixing rule adopted for this calculation is as follows: ² s : s x 1 s1 q x 2 s 2 , ² c : s x 1 c1 q x 2 c 2
Ž 23-1,23-2 .
² qz: s x 1Ž s1Ž z y 2 . q 2 . q x 2 Ž s2 Ž z y 2 . q 2 . s ² s : Ž z y 2 . q 2 ² M0 : s
x 1 s1 M01 q x 2 s2 M02 ² s:
Ž 24 . Ž 25.
U U4 U U4 ² ´ U :² ÕU :4 s X 12´ 11 Õ 11 q 2 X 1 X 2 ´ 12 Õ 12 q X 22´ U22 ÕU224
Ž 26-1.
U U2 U U2 ² ´ U :² ÕU :2 s X 12´ 11 Õ 11 q 2 X 1 X 2 ´ 12 Õ 12 q X 22´ U22 ÕU222
Ž 26-2.
X 1 s x 1qz 1r Ž x 1qz 1 q x 2 qz 2 . , X 2 s x 2 qz 2r Ž x 1qz 1 q x 2 qz 2 .
´
U 12 s
(
U 11
j ´ ´
U 22
,
ÕU12 s
ß
ž
ÕU1r3 q ÕU1r3 11 22 2
Ž 27.
3
/
Ž 28-1,28-2 .
Where, j and z are binary interaction parameters for energy and volume, respectively. The volumetric binary interaction parameter, z , can be fixed to unity for this system for all three equations
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
280
Table 5 AAD’s for CarbontetrachloriderCyclohexane mixture Mole frac. of CCl 4
SS
This EOS
Large p
0.0000 0.2512 0.3720 0.6506 0.7475 1.0000
0.0465 0.0536 0.0553 0.0536 0.0506 0.0410
0.0704 0.0715 0.0741 0.0742 0.0716 0.0618
0.0039 0.0090 0.0116 0.0118 0.0093 0.0009
This EOS is Eqs. Ž18-1. and Ž18-2. when p is 1.24.
of state, the energetic binary interaction parameter, j , is also close to unity in Simha Somcynsky’s and Large p version. The value of this energetic binary parameter in this equation of state is as small as about 0.95. Mixing rule for p is not necessary because p values for two components are equal to each other. In Table 5, for the system of carbontetrachloridercyclohexane at atmospheric pressure, a good agreement is obtained by Large p version as Eqs. Ž 18-1. and Ž22. . We deduce from this fact that in the region where dynamic motion of molecules increases, parameter p can fully express this effects, close to infinity. Another reason for the good agreement is attributed to the fact that the entropy change of mixing is far much closer to that of real system. In other words, ideal mixing entropy change is guessed to be smaller than that of real system, but more accurate value close to that of real system can be described with the parameter p derived from the new combinatorial factor.
6. Conclusion A generalized form of Flory’s combinatorial factor was derived from the assumption that a segment of chain molecule occupies more than one cellrlattice. From this, an equation of state based on hole theory was derived and applied to polymer, low molecular systems and binary liquid mixtures. Good agreement was obtained and PVT of mixture was well predicted. Especially for low molecular weight liquid systems at low pressure, Large p version of this equation of state is recommended from the fact that optimized parameter p goes to large value. Parameter p introduced in this study has a clear physical meaning and the degree of dynamic motion of molecule is accounted for by this parameter.
7. List of symbols A 3c k N Nh P p
Helmholtz energy external degrees of freedom Boltzmann constant the number of molecules the number of holes pressure the number of cells occupied by segments
M. Hong et al.r Fluid Phase Equilibria 161 (1999) 271–281
s T V X y z
the number of segments in one molecule temperature volume surface fraction volume fraction occupied by segment coordination number
Greek letters ´ energy parameter in Lennard–Jones potential f partition function V combinatorial factor v volume of unit cell Subscripts f h
free volume hole
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x
H. Eyring, J. Hirschfelder, J. Phys. Chem. 41 Ž1937. 249–257. R. Simha, T. Somcynsky, Macromolecules 2 Ž1969. 342–350. C. Zhong, W. Wang, H. Lu, Fluid Phase Equilibria 27 Ž1993. 660–664. E. Nies, A. Stroeks, Macromolecules 23 Ž1990. 4088–4092. J. Park, H. Kim, Fluid Phase Equilibria 144 Ž1998. 77–86. A.J. Staverman, Recl. Trav. Chim. Pays-Bas 69 Ž1950. 163–171. H. Tompa, Trans. Faraday Soc. 48 Ž1952. 363–370. P.J. Flory, J. Chem. Phys. 10 Ž1942. 51–62. E.A. Guggenheim, Mixture, The Oxford University Press, Amen House, London, 1952, 183–196. A. Quach, R. Simha, J. Appl. Phys. 42 Ž1971. 4592–4601. P.A. Rodgers, J. Appl. Polym. Sci. 48 Ž1993. 1061–1079. O. Olabisi, R. Simha, Macromolecules 8 Ž1975. 206–210. W.G. Cutler et al., J. of Chem. Physics 29 Ž1958. 727–731. J. Jonas et al., J. of Phys. Chem. 84 Ž1980. 109–112. F.I. Mopsik, J. of Chem. Physics 50 Ž6. Ž1969. 2559–2563. R.K. Crawford et al., J. of Chem. Physics 50 Ž1969. 3171–3183.
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