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Surface tension theory of pure liquids and polymer melts
Surface Tension Theory of Pure Liquids and Polymer Melts CLAUDIA I. POSER 1 AND ISAAC C. SANCHEZ z Polymer Science and Engineering Department and Materials Research Laboratory, University of Massachusetts, Amherst, Massachusetts 01003 and Center for Materials Science, National Measurement Laboratory, National Bureau of Standards, Washington, D. C. 20234 Received September 5, 1978; accepted November 27, 1978
Using the lattice fluid model in conjunction with the Cahn-Hilliard theory of inhomogeneous systems, an accurate method has been developed for calculating the surface tension of nonpolar and slightly polar liquids of arbitrary molecular weight. For low molecular weight liquids, the surface tension can be computed from the triple point to 0.7 of the gas-liquid critical temperature with an error of less than 5%. The theory is somewhat less accurate for polymer melts because it tends to overestimate the surface entropy. The molecular weight dependence of the surface tension and entropy is closely related to the dependence of liquid density on molecular weight.
I. INTRODUCTION
The lattice fluid (LF) theory (1-3) is able to describe the thermodynamic properties of both low and high molecular weight fluids semiquantitatively. In the present work, the LF model is employed in conjunction with the Cahn- Hilliard (CH) theory (4, 5) of inhomogeneous systems to develop a method for calculating the surface tension of a liquid in equilibrium with its vapor. Motivation for this work arises from a number of sources. First, there is the desire to introduce a unified approach for low molecular weight liquids and polymer melts. Second, a theory for the surface properties of polymer melts that is computationally simple but accurate enough to allow practical use is needed. Finally, surface properties of polymers are important for many techno1 NBS Guest Worker. 2 To whom correspondence should be addressed.
logical applications. Surface tension plays a role in the processing of thermoplastics, in coating processes, in film production, and in the use of polymers as adhesives. The surface tension theory described in this work can be extended to yield a more general interfacial tension formulation for mixtures. The results reported here will thus serve as the foundation for the more general theory. In sections II and III brief summaries of the CH and LF theories will be presented. Derivation of the present theory is contained in section IV. Numerical results and comparisons with experimental data are discussed in section V. II. C A H N - H I L L I A R D THEORY
The CH theory of inhomogeneous systems provides a means for relating a particular equation of state, based on a specific statistical mechanical model, to surface
539 0021-9797/79/060539-10502.00/0 Journal o f Colloid and Interface Science, Vol. 69, No. 3, May 1979
Copyright © 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.
540
POSER AND SANCHEZ
properties. The Helmholtz free energy density, a, of a system containing a density gradient is expressed in terms of an expansion in density, p, and its derivatives as follows (4): a(p,Vp,V~p ....
) = ao(p) +K1V~
+ K~(Vp)~ + . . . .
[1]
where K, = OalOV~[o; K~ =
1/2O~al(OlVp[)~Jo.[2]
ao(p) is the free energy density of a uniform
fluid of density p. The subscripts 0 in Eq. [2] indicate that the derivatives are to be evaluated in the limit of Vp and V2p going to zero. If the density variation is sufficiently slow, the free energy, A, of a system of volume V is given by A = ~ [a0(p) + K(Vp)Z]dV, )V
[3]
K = - d K 1 M p + tq.
[4]
where Using these relations, the surface tension, o-, for a planar interface can be written o- = I+~ [ha + K(dp/dx)2]dx.
[5]
x is the direction perpendicular to the interface and Aa is the difference between the Helmholtz free energy density of a homogeneous fluid of density p and a two-phase equilibrium mixture of the same density. After minimization the following equations are obtained: Pl
or = 2
I
[KAa]llZdp,
[6]
Og
Aa = r ( d p / d x ) 2.
[7]
where pg and p~ are the homogeneous gas and liquid densities. The derivation summarized here involves truncating the expansion given in Eq. [1] after terms higher than second order so that Eqs. [6] and [7[ should become more exact as the critical point is approached. HowJournal o f Colloid and Interface Science, Vol. 69, No. 3, May 1979
ever, Bongiorno, Scriven, and Davis (6) have shown that this approximation is quite successful over the entire coexistence temperature range for simple, nonpolar fluids. The expressions obtained above can be used to calculate both surface tensions and density profiles if K and Aa are known. The LF theory will be used to evaluate both of these quantities. 1II. LATTICE FLUID THEORY
The model from which the L F theory is derived has been described in detail elsewhere (1). The lattice is occupied by N rmers (each mer occupying one lattice site) and there are No vacant sites. The configurational partition function is evaluated by assuming random mixing of r-mers and vacant sites. To within an additive constant the reduced chemical potential, t2, is given by fz - tz/(NrE *) = -{~ + [ ~
+ T[(b-1)ln(1L
h) + l _ l n p ] r ]
[8]
#,/5, b, and/5 are the reduced temperature, pressure, volume, and density, defined as T* ~ e*/k,
[9]
P* =- e*/v*,
[10]
:F =-- T/T*, [' =-P/P*,
=- 1/{~ =- V / V * , V* =- N ( r v * ) =- N ( M W / p * ) ,
[ll]
where e* is the total interaction energy per mer, v* is the close-packed mer volume, V* is the close-packed volume of the N rmers, and M W is the molecular weight. T, P, and V are the temperature, pressure, and volume, respectively. At equilibrium the chemical potential is at a minimum and satisfies the following equation of state: t5~ + / 5 + 7"[ln (1 - P) +(1-
1/r)h] = 0 .
[12]
541
SURFACE TENSION THEORY
The equation of state defines the value of t5 which minimizes the free energy at a given and P. Three parameters, either e*, v*, and r or T*, P*, and p*, suffice to characterize a fluid. These parameters can readily be calculated using PVT data, and have been tabulated for a large number of low molecular weight fluids (1) as well as a number of common polymers (3). A partial list of these parameters is given in Table I. IV. SURFACE TENSION FORMULA
Equation [6] can be conveniently rewritten in dimensionless form (all tilde quantities are dimensionless): J ct51
6- = 2 ] (~Afi)~2dt5 J bg
[13]
6- = or/or*; o'* = e*/v .213
[14]
k =- ~clv.2/3
[15]
A5 = h0(tS) - 5e =
[/~/£(h,Pe)
[16a] -- eel
-
[h/£e
-
/re]
= [h/2(h,/b~)- /2~]
[16b] [16b]
= P c - t 5 2 + T[(1-tS) l n ( 1 - t S ) + ([~/r) In D] - tS/2e, [16C]
where 5, is the reduced Helmholtz free energy density of a two-phase equilibrium mixture of density tS, /2~ is the reduced equilibrium chemical potential of the homogeneous phases, a n d / 5 is the equilibrium vapor pressure. Equation [16c] is obtained through combination of Eqs. [16b] and [8]. The expansion of the free energy density is carded out in reduced variables 5 -~ a l P * = 50(/5) + ~qVZ/5
+ K2(VtS)2 + . . . .
[la]
which yields the following expression for K: = (-dK~/db
+ K2).
[4a]
To evaluate k for the LF, we adopt the standard assumption (4, 7, 8) that the entropy of the inhomogeneous system de-
TABLEI Parameters, Calculated from PVT Data, Which Characterize a Number of Low Molecular Weight Fluids and Common Polymers T* (°K)
Methane Ethane Propane n-Butane Isobutane n-Pentane Isopentane n-Hexane Cyclohexane n-Heptane n-Octane n-Nonane n-Decane n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Heptadecane Benzene Chlorobenzene Toluene m-Xylene o-Xylene p-Xylene Carbon tetrachloride Diethyl ether Poly(dimethyl siloxane) Poly(vinyl acetate) Poly(isobutylene) Polyethylene (linear) Polyethylene (branched) Polystyrene (atactic)
224 815 371 403 398 441 424 476 497 487 502 517 530 542 552 560 570 596 523 585 543 560 571 561 535 431 476 590 643 649 673 735
P* (INN/ m2)
p* (kg/m3 × 10-~)
O-* (mN/ m)
248 327 313 322 288 3 I0 308 298 383 309 308 307 304 303 301 299 296 287 444 437 402 385 394 381 813 363 302 509 354 426 358 358
0.50 0.64 0.69 0.736 0.72 0,755 0.765 0.775 0.902 0.80 0.815 0.828 0.837 0.846 0.854 0.858 0.864 0.88 0.994 1.206 0.996 0.952 0.965 0.949 1.788 0.870 1.104 1.283 0.974 0.904 0.887 1.105
57.6 77.7 79.6 83.4 77.0 83.7 82.3 83.7 100 86.3 87.0 87.6 87,9 88,2 88,4 88.4 88.4 87.9 112 116 107 105 107 104 102 92.3 84.3 128 104 118 106 109
pends only on the local density and is independent of the density gradient. If this is true, the entropy retains the same mathematical form as in the homogeneous case and the gradient affects only the potential energy. In the original derivation of the L F potential energy only nearest neighbor interactions were taken into account. Modification of this derivation for homogeneous fluids to include long-range interactions has the effect of changing the definition of e* Journal of Colloid and Interface Science, Vol. 69, No. 3, May 1979
542
POSER AND SANCHEZ 0.6
I
I
F o r this potential, the characteristic L F interaction energy e* is given by
I
e* = 27re0/(n - 3)
0.5
[19]
and Eq. [17] becomes: zb m
E - e * [ - [ 92 V v*
O.4
z
v.2/3 n - 3 ] 6 n 5 [gVz[) "
i.-
Comparison with Eqs. [la] and [4a] leads to
0.3
Kx = _ v , 2 / 3 [ 9 [ ~ n - 3 ] n 5
. '
", {1.2 K2=0; 0.1
0.5
1.0
n
5
.
[20]
For the usual value of n = 6, k = ½. The surface tension and the density profile can then be calculated using Eqs. [7], [13], [16c], and [20].
0 0
k=
1.5
REDUCEDTEMPERATURE,
FIG. 1. Calculated values of the reduced surface tension vs reduced temperature for several values ofr. without further altering the theory, Since e* is in practice obtained from a fit o f PVT data, the results o f the L F theory remain the same for homogeneous fluids whether nearest neighbor interactions are considered alone or in conjunction with long-range interactions. This is not the case, however, for inhomogeneous systems. If long-range interactions are to be included, the potential energy, E , for the inhomogeneous case may be written (4, 6) as E V
2zr
V. RESULTS AND DISCUSSION The calculation of the surface tension for pure fluids involves no free parameters, since the three L F parameters are determined from PVT data. Theoretical values o f p g and iS1 required in Eq. [13] are obtained from the equation of state [12]. Pe, the equilibrium vapor pressure, is evaluated using the equilibrium condition (equivalent to the Maxwell construction): ~[Z,Pe,h](T,Pe)] = / x [ T , P e , h g ( T , P e ) ] • [211 Density profiles are calculated from the integrated form of Eq. [7]:
2 - fed =
-v-.2 D foR2 x
f2
(klA~)J/2d[~
2 =- x/v .1/3, ([9+ R26 72[9) E'(R)dg" [171 R is the distance between interacting mers and e(R) is the interaction potential. We assume a Sutherland-type potential (hard core plus attractive tail), i.e.,
e(R) = - eo(V*l13/R)"
[22]
o
R/v .113 < I RIv *v3 > 1.
[181
Journal of Colloid and Interface Science, Vol. 69, No. 3, May 1979
[23]
where 20 and /50 represent an arbitrarily chosen origin and density (jbg
SURFACE TENSION THEORY tension increases and the surface entropy, -dcr/dT, decreases with increasing molecular weight. Both of these predictions are in qualitative accord with experiment. L a t e r in this section we will discuss the quantitative aspects of these predictions. In Fig. 2 some representative density profiles are shown as a function of temperature. As expected, the interface broadens as the gas-liquid critical temperature Tc is approached. The density profiles for r = 5 and 10 appear to exhibit an approximate c o m m o n point of intersection at a density c o r r e s p o n d i n g to the r e d u c e d critical density. Bongiorno and Davis (9) and other workers (10, 11) have observed the occurrence of such intersections for their theories, but only when they located the x-origing o f their density profiles so as to correspond to the Gibbs dividing surface. The x-origin in Fig. 2 is arbitrarily located at a density of (tS~+ tSg)/2. The observation of an approximate c o m m o n point of intersection at the critical density must thus be considered a fortuitous coincidence. Both Figs. 1 and 2 were obtained using k = ½. In Fig. 3 theoretical and experimental surface tensions as a function of temperature are compared for n-hexane and n-decane. This figure illustrates a general property of the theory: The absolute values of surface tension are underestimated by approximately 10% when the ab initio value of k = 1/2 is used. As can be seen an excellent fit to the data is obtained for k -- 0.61. The values of k required to fit surface tension data for the n-alkanes are shown in Fig. 4 over the temperature range 0.4 < T/To < 0.7 where Tc is the critical temperature (T/Tc = 0.4 is near the freezing point for most alkanes.) Notice that k is relatively constant in this wide and useful temperature range with a value of 0.61 +_ 0.03. This value of k corresponds to n - - 5 . 7 5 in Eq. [20]. For hexane, T/Tc = 0.7 occurs at approximately 90°C. As is evident from Fig. 3, the fit of the theoretical line to the experimental data does not break
543
1.0
I
I
I
I r=5
L
I
I
.8
I--
T = .6
-
j /
0 "r- -r-'-'t ..~ 1.0
-
l
i
I
I
I
I
L
I
L
I
l
I
l
}
r=lO T=.6
'
50.
-?'7"_-
.6
..I///,1
o gll
"~T --= ,9-- -
/'/
I/ .4
/ 1.0 '
Ib)
I
i
I
I
I
i
i
TF....~ = .6 ,8
-
.75"-
.6
.4
.2
6
I -8
-6
I -4
it/ -2
i
I
I
[ (c) I
0
2
4
6
REDUCED DISTANCE, F I G . 2. C a l c u l a t e d d e n s i t y p r o f i l e s a t t h r e e v a l u e s o f the reduced temperature 7" f o r r = 5, r = 10, a n d r = ~o.
down drastically above this temperature, but rather begins to lose accuracy gradually as the temperature increases. Journal of Colloid and Interface Science, Vol. 69, No. 3, May 1979
544
POSER AND SANCHEZ I
a
f
I
I
i
i
;
I
I
I
i
• 20
,
~
i
t
i
I
J
I
n-HEXANE
i° 30
n-DECANE
20
10
i
I
J
I
I
i
i
i
I
I
50
lOO T,
I
i
i
J
150
°C
FIG. 3. Surface tension vs temperature for n-hexane and n-decane. The lines are theoretical with the dashed lines calculated using k = 0.50 and the solid lines calculated using k = 0.61. The points are experimental values taken from Ref. (17).
I f k is treated as an empirical p a r a m e t e r for a variety o f n o n p o l a r liquids (including b r a n c h e d alkanes, b e n z e n e , the xylenes, diethyl ether, and c a r b o n tetrachloride), we find that k is again relatively constant (k = 0.62 ___ 0.05) o v e r the same range o f T/Tc. Therefore, the present theory with ~: = 0.62 can be used to describe surface tension b e h a v i o r for a variety of nonpolar liquids o v e r a wide t e m p e r a t u r e range with an e r r o r of about 5%. Figure 5 illustrates the type of a g r e e m e n t obtained b e t w e e n e x p e r i m e n t a l and theoretical sUrface tension vs t e m p e r a t u r e values for chlorobenzene and diethyl ether. F o r polar liquids, is less than 0.62 and is t e m p e r a t u r e dependent. Since k 1/~ not only scales the surface tension, but also the surface entropy, -d~/dT, the d e m a n d s on the C H theory Journal of Colloid and Interface Science, Vol. 69, N o . 3, M a y 1979
are quite stringent. This is readily a p p a r e n t in Fig. 3 where the calculated curve for = V2 has a w e a k e r t e m p e r a t u r e dependence than for k = 0.61. Thus, it is significant that a constant value o f k can be used o v e r a wide t e m p e r a t u r e range, because it lends support to the basic validity of the C H formalism. The molecular weight d e p e n d e n c e of the surface tension o f n-alkanes has received considerable attention f r o m p o l y m e r scientists (12-15). T w o alternate empirical equations have been p r o p o s e d to describe the molecular w e i g h t - s u r f a c e tension relationship for p o l y m e r s b a s e d on n-alkane data. Wu (12) p r o p o s e d the use of O- 1 / 4 =
ks
Orm TM - - ~
Mn
,
[24]
545
SURFACE TENSION THEORY I
I
1
where or= is the surface tension at infinite molecular weight, M~ is the number average molecular weight, and ks is a positive constant. LeGrand and Gaines (13) have used Ke o- = cr~ [251
I
n-ALKANES
Mn2/3 '
• o
• i°ll
I
~°o.to
•
where Ke is an empirical constant. Theoretical surface tension values for nalkanes from pentane to heptadecane were fitted to both Eqs. [24] and [25] by linear regression. The standard deviation in surface tension obtained using Eq. [24] is 0.12 mN/m while Eq. [25] yields a standard deviation of 0.11 mN/m. Thus, the two equations are indistinguishable in describing the molecular weight dependence of surface tension values obtained from the present theory. Figure 6 compares the experimental data for n-alkanes to the theoretical lines ob-
.6--
.S - -
.4 .3
i
I
I
I
.4
.5
.6
.7
T/To
FIG. 4. Values of ~crequired to fit experimental data for n-alkanes vs temperature reduced by the critical temperature. Experimental data used to calculate were taken from Ref. (18).
4o
I
I
I
I
I
I
I
I
CHLOROBENZENE
~
2o
D z o
~.
10
DIETHYL ETHER
01"'--
20,
s
0
20
i
I
i
I
I
I
I
40
60
80
100
120
140
160
180
T,°C
FIG. 5. Surface tension vs temperature for chlorobenzene and diethyl ether. The lines are theoretical curves calculated using k = 0.62 and the experimental points were taken from Ref. (17). Journal of Colloid and Interface Science, V o l . 69, N o . 3, M a y 1979
546
POSER AND SANCHEZ 30
zs o
zo==
15-
12 .01
I
I
I~
.02
.03
.04
I .05
.06
MW -z/3
FIG. 6. Surface tension vs molecular weight to the - ~ power for n-alkanes. Lines were calculated from a linear regression analysis of theoretical values obtained with k = 0.61; experimental points for pentane through eicosane were taken from Ref. (19). rained from fits of Eq. [25] to the calculated surface tension values at two different temperatures. G o o d agreement is obtained at both temperatures. Inspection of Fig. 1 reveals that surface tensions are inherently molecular weight (MW) dependent. This effect is directly related to density. As r increases (MW is roughly proportional to r) the liquid density increases, Opl/Or)f~> O, and the gas density decreases, O~g/Or)~ < 0. Thus, as the bulk density difference and density gradient inI
crease with r at a fixed reduced temperature, a concomitant increase in surface tension occurs. The experimentally observed M W dependence is, of course, further complicated by the fact that T*, the reduction temperature, increases with r which means that ir is not constant for a homologous series at a given temperature. As r increases, 6- approaches a limiting value, 6-~, which implies a corresponding states principle for p o l y m e r melts. In Fig. 7, 6-o° is plotted as a function of ]~ and corn-
I
I
I
POLYETHYLE,E 0.1 u.J
=
• LINEAR POLYETHYLENE o POLYISOBUTYLENE • POLYSTYRENE
~-~
zx POLY(VINYLACETATE) • 0 0.5
POLY(DIMETHYLSILOXANE) I 0.6
I 0.7
I 0.8
I 0.9
1.0
REDUCED TEMPERATURE,
FIG. 7. Reduced surface tension vs reduced temperature of polymers. The theoretical line was calculated using r = ~ and k = 0.55 and experimental values were taken from Refs. (12, 14). Journal of Colloid and Interface Science, Vol. 69, No. 3, May 1979
547
SURFACE TENSION THEORY
pared with experimental data for six polymers. A value of k = 0.55 yields a good fit to the data. All of the data were taken between 413 and 453°K; absolute values of tr range from a high of 32.1 mN/m for polystyrene to a low of 12.1 mN/m for poly(dimethyl siloxane) (1 mN/m = 1 dyn/cm). Surface tension measurements of polymer melts are difficult and have uncertainties of 2 to 5%. The maximum error between experiment and theory (for k = 0.55) in Fig. 7 is about 10%. Although the ~r data for the polar polymer poly(vinylacetate) correlate well with those of the nonpolar polymers as shown in Fig. 7, the polar polymers of methyl methacrylate, n-butyl methacrylate, and ethylene oxide do not (not shown). The required value of k for the methacrylates is about 0.42. Experimentally, the surface entropy, -dtr/dT, of a polymer melt is smaller than for a similar low MW liquid. This effect is qualitatively apparent in Fig. 1. The reason behind this phenomenon can again be understood in terms o~" density. A polymeric liquid has a smaller thermal expansion coefficient, ~, than a similar low MW liquid; that is, OalOr)~ < O. Thus, the interracial T A B L E II E x p e r i m e n t a l and Calculated Values of - d t r / d T for n - A l k a n e s Surface entropy n-Alkanes
(mN/m - °Kp
Exp. (mN/m- °K)°
Difference (%)
5 6 7 8 9 10 11 12 13 14 17
0.1140 0.1084 0.1070 0.1045 0.1020 0.0997 0.0978 0.0962 0.0947 0.0930 0.0886
0.110 0.103 0.104 0.096 0.094 0.092 0.089 0.088 0.087 0,086 0.084
3.6 5.2 2.9 8.9 8.5 8.4 9.9 9.3 8.8 8,1 5,5
Calc.
TABLE III Experimental and CalculatedValues of -dcr/dT for Polymers Surface entropy Polymer P E (linear) P E (branched) PIB PS PVA PMMA PnBMA PEO PDMS
Calc. Exp. (mN/m - °K)" (mN/m- °K)~ 0.0820 0.0729 0.0725 0.0722 0.0920 0.0914 0.0826 0.0975 0.0609
0.057 0,067 0.066 0.072 0.066 0.076 0.059 0.076 0.048
Difference (%) 44 9 9 0 39 20 40 28 27
a U s i ng k = 0.55. b D a t a from Ref. (12).
density gradient and o- change more slowly with temperature for a polymer liquid. There is another reason why the surface entropy should be smaller for a polymer liquid than for a low MW liquid. The interface restricts the number of conformations of polymer molecules near the interface which should lower the surface entropy (16). We have neglected this effect and thus should overestimate -dcr/dT for polymers. Tables II and III show the experimental and calculated values of -do-/dT for nalkanes and polymers, respectively. Notice the greater error for polymers which points to the significance of the conformational entropy effect. VI. C O N C L U S I O N S
Usin g k = 0.61. b D a t a from Ref. (18).
In summary, combining the LF model with the CH theory yields a unified theory of surface tension applicable to nonpolar and slightly polar liquids of arbitrary molecular weight. Without any adjustable parameters, calculated surface tensions are usually 10-15% lower than experimental ones over a broad temperature range. By setting k equal to 0.62 for low molecular weight liquids and 0.55 for polymers, much better agreement is obtained between Journal of Colloid and Interface Science, Vol. 69, No. 3, May 1979
548
POSER AND SANCHEZ
theory and experiment. The theory correctly predicts the molecular weight dependence of the surface tension for nalkanes. REFERENCES I. Sanchez, I. C., and Lacombe, R. H., J. Phys. Chem. 80, 2352, 2368 (1976). 2. Sanchez, I. C., and Lacombe, R. H., Macromolecules 11, 1145 (1978). 3. Sanchez, I. C., and Lacombe, R. H., J. Polym. Sci. PL, 15, 71 (1977). 4. Cahn, J. W., and Hilliard, J. E., J. Chem. Phys. 28, 258 (1958). 5. Cahn, J. W.,J. Chem. Phys. 30, 1121 (1959). 6. Bongiorno, V., Scriven, L. E., and Davis, H. T., J. Colloid Interface Sci. 57, 462 (•976). 7. van der Waals, J. D., and Kohnstamm, Ph., "Lehrbuch der Thermodynamik," Vol. I, p. 207. Maas & van Suchtelen, Leipzig, 1908. 8. Debye, P., J. Chem. Phys. 31, 680 (1959). 9. Bongiorno, V., and Davis, H. T., Phys. Rev. A 12, 2213 (1975).
Journal of Colloid and Interface Science, Vol. 69, No. 3, May 1979
10. Co, K. U., Kozak, J. J., and Luks, K. D., J. Chem. Phys. 66, 1002, (1977). 11. Toxvaerd, S.,J. Chem. Phys. 55, 3116 (1971). 12. Wu, S., "Polymer Blends" (D. R. Paul and S. Newman, Eds.), Vol. I, pp. 244-295. Academic Press, New York, 1978. 13. LeGrand, D. G., and Gaines, G. L., Jr., J. Colloid Interface Sci. 31, 162 (1969). 14. Gaines, G. L., Jr., Polym. Eng. Sci. 12, 1 (1972). 15. Siow, K. S., and Patterson, D., Macromolecules 4, 26, (1971). 16. Roe, R. J., J. Phys. Chem. 64, 2809 (1965). 17. Vargaftik, N. B., "Tables on the Thermophysical Properties of Liquids and Gases," 2nd Ed. Wiley, New York, 1975. 18. Timmermans, J., "Physico-Chemical Constants of Pure Organic Compounds," Vol. I and II. American Elsevier, New York, 1950 and 1965. 19. Rossini, F. D., "Selected Values of Properties of Hydrocarbons and Related Compounds." Research Project 44, American Petroleum Research Institute, Carnegie Press, Pittsburgh, 1953.
Using the lattice fluid model in conjunction with the Cahn-Hilliard theory of inhomogeneous systems, an accurate method has been developed for calculating the surface tension of nonpolar and slightly polar liquids of arbitrary molecular weight. For low molecular weight liquids, the surface tension can be computed from the triple point to 0.7 of the gas-liquid critical temperature with an error of less than 5%. The theory is somewhat less accurate for polymer melts because it tends to overestimate the surface entropy. The molecular weight dependence of the surface tension and entropy is closely related to the dependence of liquid density on molecular weight.
I. INTRODUCTION
The lattice fluid (LF) theory (1-3) is able to describe the thermodynamic properties of both low and high molecular weight fluids semiquantitatively. In the present work, the LF model is employed in conjunction with the Cahn- Hilliard (CH) theory (4, 5) of inhomogeneous systems to develop a method for calculating the surface tension of a liquid in equilibrium with its vapor. Motivation for this work arises from a number of sources. First, there is the desire to introduce a unified approach for low molecular weight liquids and polymer melts. Second, a theory for the surface properties of polymer melts that is computationally simple but accurate enough to allow practical use is needed. Finally, surface properties of polymers are important for many techno1 NBS Guest Worker. 2 To whom correspondence should be addressed.
logical applications. Surface tension plays a role in the processing of thermoplastics, in coating processes, in film production, and in the use of polymers as adhesives. The surface tension theory described in this work can be extended to yield a more general interfacial tension formulation for mixtures. The results reported here will thus serve as the foundation for the more general theory. In sections II and III brief summaries of the CH and LF theories will be presented. Derivation of the present theory is contained in section IV. Numerical results and comparisons with experimental data are discussed in section V. II. C A H N - H I L L I A R D THEORY
The CH theory of inhomogeneous systems provides a means for relating a particular equation of state, based on a specific statistical mechanical model, to surface
539 0021-9797/79/060539-10502.00/0 Journal o f Colloid and Interface Science, Vol. 69, No. 3, May 1979
Copyright © 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.
540
POSER AND SANCHEZ
properties. The Helmholtz free energy density, a, of a system containing a density gradient is expressed in terms of an expansion in density, p, and its derivatives as follows (4): a(p,Vp,V~p ....
) = ao(p) +K1V~
+ K~(Vp)~ + . . . .
[1]
where K, = OalOV~[o; K~ =
1/2O~al(OlVp[)~Jo.[2]
ao(p) is the free energy density of a uniform
fluid of density p. The subscripts 0 in Eq. [2] indicate that the derivatives are to be evaluated in the limit of Vp and V2p going to zero. If the density variation is sufficiently slow, the free energy, A, of a system of volume V is given by A = ~ [a0(p) + K(Vp)Z]dV, )V
[3]
K = - d K 1 M p + tq.
[4]
where Using these relations, the surface tension, o-, for a planar interface can be written o- = I+~ [ha + K(dp/dx)2]dx.
[5]
x is the direction perpendicular to the interface and Aa is the difference between the Helmholtz free energy density of a homogeneous fluid of density p and a two-phase equilibrium mixture of the same density. After minimization the following equations are obtained: Pl
or = 2
I
[KAa]llZdp,
[6]
Og
Aa = r ( d p / d x ) 2.
[7]
where pg and p~ are the homogeneous gas and liquid densities. The derivation summarized here involves truncating the expansion given in Eq. [1] after terms higher than second order so that Eqs. [6] and [7[ should become more exact as the critical point is approached. HowJournal o f Colloid and Interface Science, Vol. 69, No. 3, May 1979
ever, Bongiorno, Scriven, and Davis (6) have shown that this approximation is quite successful over the entire coexistence temperature range for simple, nonpolar fluids. The expressions obtained above can be used to calculate both surface tensions and density profiles if K and Aa are known. The LF theory will be used to evaluate both of these quantities. 1II. LATTICE FLUID THEORY
The model from which the L F theory is derived has been described in detail elsewhere (1). The lattice is occupied by N rmers (each mer occupying one lattice site) and there are No vacant sites. The configurational partition function is evaluated by assuming random mixing of r-mers and vacant sites. To within an additive constant the reduced chemical potential, t2, is given by fz - tz/(NrE *) = -{~ + [ ~
+ T[(b-1)ln(1L
h) + l _ l n p ] r ]
[8]
#,/5, b, and/5 are the reduced temperature, pressure, volume, and density, defined as T* ~ e*/k,
[9]
P* =- e*/v*,
[10]
:F =-- T/T*, [' =-P/P*,
=- 1/{~ =- V / V * , V* =- N ( r v * ) =- N ( M W / p * ) ,
[ll]
where e* is the total interaction energy per mer, v* is the close-packed mer volume, V* is the close-packed volume of the N rmers, and M W is the molecular weight. T, P, and V are the temperature, pressure, and volume, respectively. At equilibrium the chemical potential is at a minimum and satisfies the following equation of state: t5~ + / 5 + 7"[ln (1 - P) +(1-
1/r)h] = 0 .
[12]
541
SURFACE TENSION THEORY
The equation of state defines the value of t5 which minimizes the free energy at a given and P. Three parameters, either e*, v*, and r or T*, P*, and p*, suffice to characterize a fluid. These parameters can readily be calculated using PVT data, and have been tabulated for a large number of low molecular weight fluids (1) as well as a number of common polymers (3). A partial list of these parameters is given in Table I. IV. SURFACE TENSION FORMULA
Equation [6] can be conveniently rewritten in dimensionless form (all tilde quantities are dimensionless): J ct51
6- = 2 ] (~Afi)~2dt5 J bg
[13]
6- = or/or*; o'* = e*/v .213
[14]
k =- ~clv.2/3
[15]
A5 = h0(tS) - 5e =
[/~/£(h,Pe)
[16a] -- eel
-
[h/£e
-
/re]
= [h/2(h,/b~)- /2~]
[16b] [16b]
= P c - t 5 2 + T[(1-tS) l n ( 1 - t S ) + ([~/r) In D] - tS/2e, [16C]
where 5, is the reduced Helmholtz free energy density of a two-phase equilibrium mixture of density tS, /2~ is the reduced equilibrium chemical potential of the homogeneous phases, a n d / 5 is the equilibrium vapor pressure. Equation [16c] is obtained through combination of Eqs. [16b] and [8]. The expansion of the free energy density is carded out in reduced variables 5 -~ a l P * = 50(/5) + ~qVZ/5
+ K2(VtS)2 + . . . .
[la]
which yields the following expression for K: = (-dK~/db
+ K2).
[4a]
To evaluate k for the LF, we adopt the standard assumption (4, 7, 8) that the entropy of the inhomogeneous system de-
TABLEI Parameters, Calculated from PVT Data, Which Characterize a Number of Low Molecular Weight Fluids and Common Polymers T* (°K)
Methane Ethane Propane n-Butane Isobutane n-Pentane Isopentane n-Hexane Cyclohexane n-Heptane n-Octane n-Nonane n-Decane n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Heptadecane Benzene Chlorobenzene Toluene m-Xylene o-Xylene p-Xylene Carbon tetrachloride Diethyl ether Poly(dimethyl siloxane) Poly(vinyl acetate) Poly(isobutylene) Polyethylene (linear) Polyethylene (branched) Polystyrene (atactic)
224 815 371 403 398 441 424 476 497 487 502 517 530 542 552 560 570 596 523 585 543 560 571 561 535 431 476 590 643 649 673 735
P* (INN/ m2)
p* (kg/m3 × 10-~)
O-* (mN/ m)
248 327 313 322 288 3 I0 308 298 383 309 308 307 304 303 301 299 296 287 444 437 402 385 394 381 813 363 302 509 354 426 358 358
0.50 0.64 0.69 0.736 0.72 0,755 0.765 0.775 0.902 0.80 0.815 0.828 0.837 0.846 0.854 0.858 0.864 0.88 0.994 1.206 0.996 0.952 0.965 0.949 1.788 0.870 1.104 1.283 0.974 0.904 0.887 1.105
57.6 77.7 79.6 83.4 77.0 83.7 82.3 83.7 100 86.3 87.0 87.6 87,9 88,2 88,4 88.4 88.4 87.9 112 116 107 105 107 104 102 92.3 84.3 128 104 118 106 109
pends only on the local density and is independent of the density gradient. If this is true, the entropy retains the same mathematical form as in the homogeneous case and the gradient affects only the potential energy. In the original derivation of the L F potential energy only nearest neighbor interactions were taken into account. Modification of this derivation for homogeneous fluids to include long-range interactions has the effect of changing the definition of e* Journal of Colloid and Interface Science, Vol. 69, No. 3, May 1979
542
POSER AND SANCHEZ 0.6
I
I
F o r this potential, the characteristic L F interaction energy e* is given by
I
e* = 27re0/(n - 3)
0.5
[19]
and Eq. [17] becomes: zb m
E - e * [ - [ 92 V v*
O.4
z
v.2/3 n - 3 ] 6 n 5 [gVz[) "
i.-
Comparison with Eqs. [la] and [4a] leads to
0.3
Kx = _ v , 2 / 3 [ 9 [ ~ n - 3 ] n 5
. '
", {1.2 K2=0; 0.1
0.5
1.0
n
5
.
[20]
For the usual value of n = 6, k = ½. The surface tension and the density profile can then be calculated using Eqs. [7], [13], [16c], and [20].
0 0
k=
1.5
REDUCEDTEMPERATURE,
FIG. 1. Calculated values of the reduced surface tension vs reduced temperature for several values ofr. without further altering the theory, Since e* is in practice obtained from a fit o f PVT data, the results o f the L F theory remain the same for homogeneous fluids whether nearest neighbor interactions are considered alone or in conjunction with long-range interactions. This is not the case, however, for inhomogeneous systems. If long-range interactions are to be included, the potential energy, E , for the inhomogeneous case may be written (4, 6) as E V
2zr
V. RESULTS AND DISCUSSION The calculation of the surface tension for pure fluids involves no free parameters, since the three L F parameters are determined from PVT data. Theoretical values o f p g and iS1 required in Eq. [13] are obtained from the equation of state [12]. Pe, the equilibrium vapor pressure, is evaluated using the equilibrium condition (equivalent to the Maxwell construction): ~[Z,Pe,h](T,Pe)] = / x [ T , P e , h g ( T , P e ) ] • [211 Density profiles are calculated from the integrated form of Eq. [7]:
2 - fed =
-v-.2 D foR2 x
f2
(klA~)J/2d[~
2 =- x/v .1/3, ([9+ R26 72[9) E'(R)dg" [171 R is the distance between interacting mers and e(R) is the interaction potential. We assume a Sutherland-type potential (hard core plus attractive tail), i.e.,
e(R) = - eo(V*l13/R)"
[22]
o
R/v .113 < I RIv *v3 > 1.
[181
Journal of Colloid and Interface Science, Vol. 69, No. 3, May 1979
[23]
where 20 and /50 represent an arbitrarily chosen origin and density (jbg
SURFACE TENSION THEORY tension increases and the surface entropy, -dcr/dT, decreases with increasing molecular weight. Both of these predictions are in qualitative accord with experiment. L a t e r in this section we will discuss the quantitative aspects of these predictions. In Fig. 2 some representative density profiles are shown as a function of temperature. As expected, the interface broadens as the gas-liquid critical temperature Tc is approached. The density profiles for r = 5 and 10 appear to exhibit an approximate c o m m o n point of intersection at a density c o r r e s p o n d i n g to the r e d u c e d critical density. Bongiorno and Davis (9) and other workers (10, 11) have observed the occurrence of such intersections for their theories, but only when they located the x-origing o f their density profiles so as to correspond to the Gibbs dividing surface. The x-origin in Fig. 2 is arbitrarily located at a density of (tS~+ tSg)/2. The observation of an approximate c o m m o n point of intersection at the critical density must thus be considered a fortuitous coincidence. Both Figs. 1 and 2 were obtained using k = ½. In Fig. 3 theoretical and experimental surface tensions as a function of temperature are compared for n-hexane and n-decane. This figure illustrates a general property of the theory: The absolute values of surface tension are underestimated by approximately 10% when the ab initio value of k = 1/2 is used. As can be seen an excellent fit to the data is obtained for k -- 0.61. The values of k required to fit surface tension data for the n-alkanes are shown in Fig. 4 over the temperature range 0.4 < T/To < 0.7 where Tc is the critical temperature (T/Tc = 0.4 is near the freezing point for most alkanes.) Notice that k is relatively constant in this wide and useful temperature range with a value of 0.61 +_ 0.03. This value of k corresponds to n - - 5 . 7 5 in Eq. [20]. For hexane, T/Tc = 0.7 occurs at approximately 90°C. As is evident from Fig. 3, the fit of the theoretical line to the experimental data does not break
543
1.0
I
I
I
I r=5
L
I
I
.8
I--
T = .6
-
j /
0 "r- -r-'-'t ..~ 1.0
-
l
i
I
I
I
I
L
I
L
I
l
I
l
}
r=lO T=.6
'
50.
-?'7"_-
.6
..I///,1
o gll
"~T --= ,9-- -
/'/
I/ .4
/ 1.0 '
Ib)
I
i
I
I
I
i
i
TF....~ = .6 ,8
-
.75"-
.6
.4
.2
6
I -8
-6
I -4
it/ -2
i
I
I
[ (c) I
0
2
4
6
REDUCED DISTANCE, F I G . 2. C a l c u l a t e d d e n s i t y p r o f i l e s a t t h r e e v a l u e s o f the reduced temperature 7" f o r r = 5, r = 10, a n d r = ~o.
down drastically above this temperature, but rather begins to lose accuracy gradually as the temperature increases. Journal of Colloid and Interface Science, Vol. 69, No. 3, May 1979
544
POSER AND SANCHEZ I
a
f
I
I
i
i
;
I
I
I
i
• 20
,
~
i
t
i
I
J
I
n-HEXANE
i° 30
n-DECANE
20
10
i
I
J
I
I
i
i
i
I
I
50
lOO T,
I
i
i
J
150
°C
FIG. 3. Surface tension vs temperature for n-hexane and n-decane. The lines are theoretical with the dashed lines calculated using k = 0.50 and the solid lines calculated using k = 0.61. The points are experimental values taken from Ref. (17).
I f k is treated as an empirical p a r a m e t e r for a variety o f n o n p o l a r liquids (including b r a n c h e d alkanes, b e n z e n e , the xylenes, diethyl ether, and c a r b o n tetrachloride), we find that k is again relatively constant (k = 0.62 ___ 0.05) o v e r the same range o f T/Tc. Therefore, the present theory with ~: = 0.62 can be used to describe surface tension b e h a v i o r for a variety of nonpolar liquids o v e r a wide t e m p e r a t u r e range with an e r r o r of about 5%. Figure 5 illustrates the type of a g r e e m e n t obtained b e t w e e n e x p e r i m e n t a l and theoretical sUrface tension vs t e m p e r a t u r e values for chlorobenzene and diethyl ether. F o r polar liquids, is less than 0.62 and is t e m p e r a t u r e dependent. Since k 1/~ not only scales the surface tension, but also the surface entropy, -d~/dT, the d e m a n d s on the C H theory Journal of Colloid and Interface Science, Vol. 69, N o . 3, M a y 1979
are quite stringent. This is readily a p p a r e n t in Fig. 3 where the calculated curve for = V2 has a w e a k e r t e m p e r a t u r e dependence than for k = 0.61. Thus, it is significant that a constant value o f k can be used o v e r a wide t e m p e r a t u r e range, because it lends support to the basic validity of the C H formalism. The molecular weight d e p e n d e n c e of the surface tension o f n-alkanes has received considerable attention f r o m p o l y m e r scientists (12-15). T w o alternate empirical equations have been p r o p o s e d to describe the molecular w e i g h t - s u r f a c e tension relationship for p o l y m e r s b a s e d on n-alkane data. Wu (12) p r o p o s e d the use of O- 1 / 4 =
ks
Orm TM - - ~
Mn
,
[24]
545
SURFACE TENSION THEORY I
I
1
where or= is the surface tension at infinite molecular weight, M~ is the number average molecular weight, and ks is a positive constant. LeGrand and Gaines (13) have used Ke o- = cr~ [251
I
n-ALKANES
Mn2/3 '
• o
• i°ll
I
~°o.to
•
where Ke is an empirical constant. Theoretical surface tension values for nalkanes from pentane to heptadecane were fitted to both Eqs. [24] and [25] by linear regression. The standard deviation in surface tension obtained using Eq. [24] is 0.12 mN/m while Eq. [25] yields a standard deviation of 0.11 mN/m. Thus, the two equations are indistinguishable in describing the molecular weight dependence of surface tension values obtained from the present theory. Figure 6 compares the experimental data for n-alkanes to the theoretical lines ob-
.6--
.S - -
.4 .3
i
I
I
I
.4
.5
.6
.7
T/To
FIG. 4. Values of ~crequired to fit experimental data for n-alkanes vs temperature reduced by the critical temperature. Experimental data used to calculate were taken from Ref. (18).
4o
I
I
I
I
I
I
I
I
CHLOROBENZENE
~
2o
D z o
~.
10
DIETHYL ETHER
01"'--
20,
s
0
20
i
I
i
I
I
I
I
40
60
80
100
120
140
160
180
T,°C
FIG. 5. Surface tension vs temperature for chlorobenzene and diethyl ether. The lines are theoretical curves calculated using k = 0.62 and the experimental points were taken from Ref. (17). Journal of Colloid and Interface Science, V o l . 69, N o . 3, M a y 1979
546
POSER AND SANCHEZ 30
zs o
zo==
15-
12 .01
I
I
I~
.02
.03
.04
I .05
.06
MW -z/3
FIG. 6. Surface tension vs molecular weight to the - ~ power for n-alkanes. Lines were calculated from a linear regression analysis of theoretical values obtained with k = 0.61; experimental points for pentane through eicosane were taken from Ref. (19). rained from fits of Eq. [25] to the calculated surface tension values at two different temperatures. G o o d agreement is obtained at both temperatures. Inspection of Fig. 1 reveals that surface tensions are inherently molecular weight (MW) dependent. This effect is directly related to density. As r increases (MW is roughly proportional to r) the liquid density increases, Opl/Or)f~> O, and the gas density decreases, O~g/Or)~ < 0. Thus, as the bulk density difference and density gradient inI
crease with r at a fixed reduced temperature, a concomitant increase in surface tension occurs. The experimentally observed M W dependence is, of course, further complicated by the fact that T*, the reduction temperature, increases with r which means that ir is not constant for a homologous series at a given temperature. As r increases, 6- approaches a limiting value, 6-~, which implies a corresponding states principle for p o l y m e r melts. In Fig. 7, 6-o° is plotted as a function of ]~ and corn-
I
I
I
POLYETHYLE,E 0.1 u.J
=
• LINEAR POLYETHYLENE o POLYISOBUTYLENE • POLYSTYRENE
~-~
zx POLY(VINYLACETATE) • 0 0.5
POLY(DIMETHYLSILOXANE) I 0.6
I 0.7
I 0.8
I 0.9
1.0
REDUCED TEMPERATURE,
FIG. 7. Reduced surface tension vs reduced temperature of polymers. The theoretical line was calculated using r = ~ and k = 0.55 and experimental values were taken from Refs. (12, 14). Journal of Colloid and Interface Science, Vol. 69, No. 3, May 1979
547
SURFACE TENSION THEORY
pared with experimental data for six polymers. A value of k = 0.55 yields a good fit to the data. All of the data were taken between 413 and 453°K; absolute values of tr range from a high of 32.1 mN/m for polystyrene to a low of 12.1 mN/m for poly(dimethyl siloxane) (1 mN/m = 1 dyn/cm). Surface tension measurements of polymer melts are difficult and have uncertainties of 2 to 5%. The maximum error between experiment and theory (for k = 0.55) in Fig. 7 is about 10%. Although the ~r data for the polar polymer poly(vinylacetate) correlate well with those of the nonpolar polymers as shown in Fig. 7, the polar polymers of methyl methacrylate, n-butyl methacrylate, and ethylene oxide do not (not shown). The required value of k for the methacrylates is about 0.42. Experimentally, the surface entropy, -dtr/dT, of a polymer melt is smaller than for a similar low MW liquid. This effect is qualitatively apparent in Fig. 1. The reason behind this phenomenon can again be understood in terms o~" density. A polymeric liquid has a smaller thermal expansion coefficient, ~, than a similar low MW liquid; that is, OalOr)~ < O. Thus, the interracial T A B L E II E x p e r i m e n t a l and Calculated Values of - d t r / d T for n - A l k a n e s Surface entropy n-Alkanes
(mN/m - °Kp
Exp. (mN/m- °K)°
Difference (%)
5 6 7 8 9 10 11 12 13 14 17
0.1140 0.1084 0.1070 0.1045 0.1020 0.0997 0.0978 0.0962 0.0947 0.0930 0.0886
0.110 0.103 0.104 0.096 0.094 0.092 0.089 0.088 0.087 0,086 0.084
3.6 5.2 2.9 8.9 8.5 8.4 9.9 9.3 8.8 8,1 5,5
Calc.
TABLE III Experimental and CalculatedValues of -dcr/dT for Polymers Surface entropy Polymer P E (linear) P E (branched) PIB PS PVA PMMA PnBMA PEO PDMS
Calc. Exp. (mN/m - °K)" (mN/m- °K)~ 0.0820 0.0729 0.0725 0.0722 0.0920 0.0914 0.0826 0.0975 0.0609
0.057 0,067 0.066 0.072 0.066 0.076 0.059 0.076 0.048
Difference (%) 44 9 9 0 39 20 40 28 27
a U s i ng k = 0.55. b D a t a from Ref. (12).
density gradient and o- change more slowly with temperature for a polymer liquid. There is another reason why the surface entropy should be smaller for a polymer liquid than for a low MW liquid. The interface restricts the number of conformations of polymer molecules near the interface which should lower the surface entropy (16). We have neglected this effect and thus should overestimate -dcr/dT for polymers. Tables II and III show the experimental and calculated values of -do-/dT for nalkanes and polymers, respectively. Notice the greater error for polymers which points to the significance of the conformational entropy effect. VI. C O N C L U S I O N S
Usin g k = 0.61. b D a t a from Ref. (18).
In summary, combining the LF model with the CH theory yields a unified theory of surface tension applicable to nonpolar and slightly polar liquids of arbitrary molecular weight. Without any adjustable parameters, calculated surface tensions are usually 10-15% lower than experimental ones over a broad temperature range. By setting k equal to 0.62 for low molecular weight liquids and 0.55 for polymers, much better agreement is obtained between Journal of Colloid and Interface Science, Vol. 69, No. 3, May 1979
548
POSER AND SANCHEZ
theory and experiment. The theory correctly predicts the molecular weight dependence of the surface tension for nalkanes. REFERENCES I. Sanchez, I. C., and Lacombe, R. H., J. Phys. Chem. 80, 2352, 2368 (1976). 2. Sanchez, I. C., and Lacombe, R. H., Macromolecules 11, 1145 (1978). 3. Sanchez, I. C., and Lacombe, R. H., J. Polym. Sci. PL, 15, 71 (1977). 4. Cahn, J. W., and Hilliard, J. E., J. Chem. Phys. 28, 258 (1958). 5. Cahn, J. W.,J. Chem. Phys. 30, 1121 (1959). 6. Bongiorno, V., Scriven, L. E., and Davis, H. T., J. Colloid Interface Sci. 57, 462 (•976). 7. van der Waals, J. D., and Kohnstamm, Ph., "Lehrbuch der Thermodynamik," Vol. I, p. 207. Maas & van Suchtelen, Leipzig, 1908. 8. Debye, P., J. Chem. Phys. 31, 680 (1959). 9. Bongiorno, V., and Davis, H. T., Phys. Rev. A 12, 2213 (1975).
Journal of Colloid and Interface Science, Vol. 69, No. 3, May 1979
10. Co, K. U., Kozak, J. J., and Luks, K. D., J. Chem. Phys. 66, 1002, (1977). 11. Toxvaerd, S.,J. Chem. Phys. 55, 3116 (1971). 12. Wu, S., "Polymer Blends" (D. R. Paul and S. Newman, Eds.), Vol. I, pp. 244-295. Academic Press, New York, 1978. 13. LeGrand, D. G., and Gaines, G. L., Jr., J. Colloid Interface Sci. 31, 162 (1969). 14. Gaines, G. L., Jr., Polym. Eng. Sci. 12, 1 (1972). 15. Siow, K. S., and Patterson, D., Macromolecules 4, 26, (1971). 16. Roe, R. J., J. Phys. Chem. 64, 2809 (1965). 17. Vargaftik, N. B., "Tables on the Thermophysical Properties of Liquids and Gases," 2nd Ed. Wiley, New York, 1975. 18. Timmermans, J., "Physico-Chemical Constants of Pure Organic Compounds," Vol. I and II. American Elsevier, New York, 1950 and 1965. 19. Rossini, F. D., "Selected Values of Properties of Hydrocarbons and Related Compounds." Research Project 44, American Petroleum Research Institute, Carnegie Press, Pittsburgh, 1953.