3143
SURFACE TENSION OF POLYMER SOLUTIONS
The Surface Tension of Polymer Solutions.
I.
Solutions of Poly(dimethylsi1oxanes) by G. L. Gaines, Jr. General Electric Research and Development Center, Schenectady, N e w Yorlc 12301 (Received J a n u a r y 20, 1969)
The surface tensions of solutions of poly(dimethylsi1oxanes) in toluene and tetrachloroethylenehave been measured at room temperature. The poly(dimethylsi1oxanes) studied range in molecular weight from the dimer, hexamethyldisiloxane,to a 1000-cs silicone fluid having Rn = 6350. Measurements cover the entire concentration range from pure solvent to undiluted polymer. Equations for the surface tension of polymer solutions based on a simple lattice model are found to show good agreement with the experimental results over the entire range of composition and molecular weight. Much effort has been devoted to the study of the behavior of polymer molecules a t interfaces. During the past few years several workers have reported measurements of the surface tension of polymer me1ts.l A large body of available data relates to adsorption of macromolecules from solution on solid surfaces.2 By contrast, relatively little work has been reported on the properties of' polymers a t the solution-air interface ; most of that, moreover, concerns the behavior of water-soluble polymers in aqueous solution.a The present work was undertaken to provide information on surface behaxior of polymers in solution in nonpolar organic solvents. Poly(dimethylsi1oxanes) were selected for initial study because of the ready availability of a wide range of homologs and because they have long been known to exhibit surface activity in organic solvents. (In spite of a general awareness of this property, only a few measurements of the surface tension of solutions of poly(dimethylsi1oxanes) have been published.'?5) Measurements of surface tension a t room temperature have been made on solutions of poly(dimethylsi1oxanes) in toluene and tetrachloroethylene, covering a wide range of composition and molecular weight. A simple semiempirical theory has been developed which correlates these data. Experimental Section Solvents, toluene and tetrachloroethylene, were reagent grade, purified before use by passage through columns of alumina and silica gel or by distillation. The measured surface tensions and densities are recorded in Table I. Hexamethyldisiloxane (MIvI),~dodecamethylpentasiloxane (MD3R9), and octadecamethyloctasiloxane (MDeM) were samples fractionated by distillation in this laboratory.' The M A 1 sample was further purified before use by distillation from CaHz to remove any silanol fragments. The other poly (dimethylsiloxane) samples were SF-96 silicone fluids of various viscosity grades, kindly
provided by Dr. F. M. Lewis, Silicone Products Dept., General Electric Company. Their average chain lengths and molecular-weight distributions were estimated by vapor-pressure osmometry and gel-permeation chromatography; the results are indicated in Table I, along with measured surface tensions and densities. These values are all consistent with comparable data in the literature.* Table I : Properties of Solvents and Siloxane Fluids
Material
Toluene Tetrachloroethylene
MM MDaM MDeM SF-96-5 cs SF-96-10 cs SF-96-20 cs SF-96-100 cs SF-96-1000 cs
Density,a g/ml
Surface tension,a dyn/om
0.862
28.1
1 632 0.753 (0.872d) (0.911d) 0.915 0.936 0.949 0.966 0.972 I
31.1 15.3 17.7 18.8 18.7 19.5 19.8 20.4 20.6
nn' ...
ivw/8nc
...
...
...
...
... ...
... ...
...
687 1040 1590 3710 6350
1.16 1.16 1.20 1.68 2.54
"Measured a t 24 f. 1'. 'From vapor pressure osmometry in benzene. From gel permeation chromatography, benzene, 55'. Literature values, ref 6, 7.
(1) R.-J. Roe, J . P h y s . Chem., 72, 2013 (1968), and references cited therein. (2) J. J. Kipling, "Adsorption from Solutions of Non-electrolytes," Academic Press, New York, N. Y., 1965, Chapter 8. (3) J. E. Glass, J . P h y s . Chem., 72, 4450, 4459 (1968), andreferences cited therein. (4) M. H. Gottlieb, i b i d . , 63, 1687 (1959). (5) T. Sato, T. Tanaka, and T. Yoshida, Polymer Letters, 5 , 947 (1967).
(6) These abbreviations are described by C. B. Hurd, J . A m e r . Chem. Soc., 68, 364 (1946). (7) Cf. W. Patnode and D. F. Wilcook, ibid., 68, 358 (1946); D.F. Wilcock, ibid., 68, 691 (1946). (8) H. W. Fox, P. W. Taylor, and W. A. Zisman, I n d . Eng. Chem., 39, 1401 (1947).
V o l u m e 73, Number 9
September 1969
3 144
G. L. GAINES,JR.
Solutions were prepared by volume, or by. weight for the more viscous fluids. Solution densities were required t o convert weight concentrations to volume fractions, and also for correction of the surface tension measurements. Measurements of density for a number of solutions (of several solvent-polymer combinations) indicated that in all cases the densities of the solutions could be taken as additive in volume fraction within the precision required for these conversions (A0.2%) and they were so assumed for all solutions. Surface tension measurements were made with a Cenco-DuNuoy tensiometer using a 6-cm ring, and the Harkins-Jordan correctionsgwere applied. It was found that with reasonable care such measurements were consistently reproducible to f0.1 dyn/cm. KO time-dependent effects could be noted, except for the most viscous samples (1000-cs fluid at volume fractions of 0.8 or greater) and for very low concentraof the higher molecular tions (volume fractions weight materials (100-cs and 1000-cs fluids). The only datum presented here where any time dependence was observed is the surface tension of the undiluted 1000-cs fluid. The value reported was obtained by reducing the rate of increase of pull in the tensiometer measurement until no further change in the measured maximum pull could be detected. The limited quantity of MD&I available precluded an extensive series of measurements, and data were obtained only for low concentrations of this material in toluene. For all the other materials, measurements were made over the complete concentration range from pure solvent to 100% siloxane, in most cases in both toluene and tetrachloroethylene. All measurements were made in air at 24 f 1O .
Theoretical Considerations The surface tensions of solutions are generally treated in terms of the Cibbs adsorption equation, which for a plane surface at constant temperature may be written dy =
Ei ri ~ F L (
(1)
where y is the surface tension and rdand p i are, respectively, the surface excess relative t o an arbitrarily chosen dividing plane, and the chemical potential of the ith component.’O For two-component solutions, a common convention is to place the dividing surface in such a position that the surface excess of solvent vanishes, whence dr =
rzl dpz
(14
(the superscript (1) indicating the choice of this convention). Although eq l a has been used by some authors in discussing polymer solutions, it is important to note that it is generally not strictly applicable, since polymer The Journal of Physical Chemistry
samples are rarely monodisperse. If a distribution of molecular weights is present, the correct formulation, with the solvent-surface excess set equal to zero, is n
dy =
i - 2
Ftl dpL(
Ob)
It is known that the chemical potential of polymer molecules in solution is molecular-weight dependent. Furthermore, the surface excess may depend on molecular weight; direct evidence for this in the case of poly(dimethylsi1oxanes) is provided by the observation of molecular-weight fractionation in foaming.” For these reasons, it appears that the application of the Gibbs equation to polymer solution surface tensions is not generally instructive, in the absence of additional ihformation on the distribution of species between surface and bulk. (In passing, it is also worth noting that the correct equation for the surface entropy of a “pure” polymer per unit area is
For this reason, it is possible that the observed small temperature coefficients of surface tension of molten polymers1 may reflect in part changes of molecular weight distribution in the surface.) A realistic picture of the interfacial region between a polymer solution and the vapor phase is complicated, of course. This is so because a long flexible polymer molecule may be expected to lie at any given instant with some of its segments a t the liquid surface while others are far away (in terms of solvent molecule diameters) in the bulk of the solution. A number of attempts have been made to analyze this situation by the methods of statistical mechanics. A recent and thorough discussion of this sort has been given by Silberberg,12 who also provides references to earlier work. It is beyond the scope of this report to discuss these theoretical treatments, except to note that, at the present stage of their development, they possess certain disadvantages for the analysis of our results. For mathematical reasons, it is always necessary to introduce certain approximations, and it is not yet clear how serious these are in modifying the behavior to be predicted from the initially realistic model. All involve unknown parameters related to such properties as polymer molecule flexibility, polymer-solvent inter(9) W.D.Harkins and H. F. Jordan, J. Amer. Chem. SOC.,52, 1751 (1930). A simple computer program for applying these corrections was developed; this is available in General Electric R & D Center Report 68-C-334,E. E. Romagosa and G. L. Gaines, Jr., September 1968. (10) For a full discussion, see, e.g., A. W. Adamson, “Physical Chemistry of Surfaces,” 2nd ed, Interscience Publishers, New York, N. Y.,1967,p 78ff. (11) G. L. Gaines, Jr., and D. G. LeGrand, Polymer Letters, 6, 625 (1968). (12) A. Silberberg, J. Chem. Phyls., 48,2835 (1968).
3 145
SURFACE TENSION OF POLYMER SOLUTIONS action in the surface, etc. For these reasons, there has been relatively little confrontation between theory and experiment, except in a qualitative sense. In the analysis of the present results, a less ambitious approach has been taken. Starting from an admittedly unrealistic model, we have developed equations relating the surface tension of the solution to composition, polymer molecular weight, and the surface tension of the polymer and solvent. These equations are then fitted to experimental data to fix the arbitrary parameter which they contain. In the subsequent discussion, the ability of this approach to correlate our experimental results over a range of composition and molecular weight, and the significance of the derived parameter, will be discussed. Our starting point is the simple Flory-Huggins lattice model for polymer solutions, together with the parallel-layer assumption applied by Prigogine and M a r e ~ h a l . ’ ~It is assumed that only the first layer of polymer segment) molecules has a com(solvent position different from the bulk, and only configurations of the polymer molecules in which they are arranged parallel to the surface are considered. This model has been used by Defay, Prigogine, and their collaborator~’~to derive expressions for the surface tensions of both athermal and nonathermal polymer solutions. Neither case, unfortunately, proves useful in the present study. The equations for athermal solutions do predict qualitative changes with molecular weight similar to those observed, but they do not give good quantitative agreement with experiment. The equations for nonathermal solutions in full form,16 on the other hand, are sufficiently complex that we cannot evaluate the surface tension directly in terms of bulk concentration. However, we may introduce some simplification by the assumption that the surface layer behaves athermally, ie., that i n the surface, the interaction energy between a polymer segment and a solvent molecule is equal to the arithmetic average of that between two solvent molecules and that between two polymer segments. Nonideality in the system, however, is still allowed for by including a nonathermal interaction term in the expressions for the chemical potentials in the bulk solution. We then proceed to evaluate the chemical potentials in the surface and in bulk exactly as has been done by the prior workers.14 The expressions which resultI6 are
+
~2
= pz0
+ RT In
cpz
- RT(r - l ) p l + r
,-, a(cp1)2
for the chemical potentials of solvent (component 1) and r-mer (component 2 ) in the bulk solution and
pls = p?,’
pzS = p
-1 + RT(1n cpls + rr + RT(1n - (r - l)cpl’)
cp28)
~
~
,
~ cp2’
- yNa - ryNa
(4)
for the chemical potentials in the surface layer, where
pt =
chemical potential of component 1 1
pi = volume fraction of component
r
no superscript-in bulk solution superscript s-in surface layer
= number of segments in r-mer molecule
y = surface tension
a( = P N ) = interaction parameter (see Discussion)
a
or solvent molecule in surface lattice (assumed identical) N = Avogadro’s number
= area per segment
When the chemical potentials in bulk and in the surface are equated, and the surface tensions of the pure components, y1° and yz0, are introduced, we obtain”
where P has been written for a / N , and IC is Boltzmann’s constant. If these expressions are equated, the result is
The set of equations 5 and 6 are sufficient to calculate the surface tension of an r-mer solution of (bulk) volume fraction cp2 provided that yl0, yzO, and r are known and if values can be assigned to a and p. It should be noted that these equations can also be obtained directly from Defay’s equation^,'^ by letting his surface lattice coordination parameter 1 be zero. The derivation from the model presented here, while it is admittedly arbitrary, appears more physically (13) I. Prigogine and J. Marechal, J . Colloid Sci., 7, 122 (1952). (14) R. Defay, I. Prigogine, A. Bellemans, and D. H. Everett, “Surface Tension and Adsorption,” John Wiley & Sons, Inc., New York, N. Y., 1966,Chapter 13. (16) Reference 14, eq 13.24; R. Defay, J . Chim. Phys., 51, 299 (1954). (16) Reference 14,eq 13.4 and 13.11. (17) Cf. ref 14,pp 203.
Volume 79, Number 9
September 1969
3 146
G. L. GAINES,JR.
meaningful (a point which is considered further in the Discussion). Certain special cases of these equations deserve men= 0, of course, the equations reduce to tion. If those given by Prigogine and Marechall3 for athermal r-mer solutions. When T = 1, we obtain
where the X’s are mole fractions (since the volume fractions in a lattice are defined by 91 = nl/(nl T ~ Z and ) cpz = mz/(n1 T ~ J n’s, , representing numbers of molecules). These resemble the SchuchowitskyGuggenheim equation for regular solutions, l8 with the omission of their term containing the coordination number in the surface plane. This, of course, reflects our assumption of zero energy of mixing in the surface layer. For T = 1, p = 0, the equations reduce to Butler’s equations for perfect solution^.^^ On the other hand, for r = Q), (qa”/qz)l/‘ = 1, and it is possible to obtain an analytical expression for y in terms of 91
+
+
have no measurements of the composition of the surface layer, so an appraisal of the significance of these calculated volume fractions in the surface layer is not possible. I n view of the divergence of the present model from reality, however, it appears uncertain whether they will prove to be meaningful. We must also emphasize that the equations, as presently developed, do not allow for molecular weight distribution. At this stage, therefore, they suffer from the same disadvantages in principle as the applications of the Gibbs adsorption equation already mentioned. I n what follows, we assume (without justification) that average molecular weight values may be used. It appears, however, that it should be possible to take account of molecular weight distribution explicitly in the derivation of these equations, as is done in applying the lattice theory to the thermodynamics of polymer solutions.20 Finally, it is worth noting that the equations developed here are thermodynamically consistent, in the sense that they do not predict surface-layer compositions at variance with a two-component Gibbs adsorption equation. Thus, they do not suffer from the deficiency found with the Schuchowitsky-Guggenheim equations, but rather reflect the behavior of expressions for the surfaces of perfect solutions.21
Computations If p1 = PZ = 0.5, eq 6 becomes
Examination of this equation shows that only if P = (yzo - y?)a will it yield a correct end value, y = y10 a t 91 = 1. (This result, like that obtained if T = m is introduced in the athermal solution equations of Prigogine and Marechal, may be rationalized on the basis of the impossibility of introducing solvent molecules into a lattice occupied by a polymer molecule of infinite size.) However, for an arbitrary value of P, it predicts that at extreme dilution, a solution of a very high molecular weight polymer will have a surface tension approaching the value
IcT
-U
(84 By expanding the exponential term, it can be shown that this y* is always Iy1O. For cases where yzo< ?lo, this is not unreasonable physically; when yzo > rlO,however, the significance of eq Sa becomes highly doubtful. (In fact,,for [(y2O - ?I0)u - P ] >> 0, y* becomes negative, an obviously meaningless result.) It is of interest that these equations all permit (in fact, require for their use) evaluation of surface composition as well as surface tension. At present, we The Journal of Physical Chemistry
i.e., the composition of the surface layer at 9 1 = 0.5 is independent of 0. Hence, once a value of a has been calculated from eq 6a can be selected, the value of used, with a measured value of y at cpl = 0.5, to evaluate P, using eq 5. A method of application of these equations now becomes apparent : given experimental values of surface tensions of r-mer, solvent, and a mixture of volume fraction 0.5, and assigning a value to a, the surface lattice spacing parameter, we can calculate in turn the parameter P and the surface volume fraction and surface tensions for solutions of any concentration. For arbitrary values of r, it is not possible to solve eq 5 and 6 algebraically, and recourse must be had to numerical methods. Fortunately, these equations can be cast as
(18) A. Schuchowitsky, Acta Physicochim. URSS, 19, 508 (1944);
E.A. Guggenheim, Trans. Faraday SOC.,41, 150 (1945). (19) J. A. V. Butler, Proc. Roy. SOC.,A135, 348 (1932). (20) P. J. Flory, “Principles of Polymer Chemistry,” Cornel1 University Press, Ithaca, N. Y., 1953, Chapter 12. (21) Reference 14, p 177. Also 8. Ono and S. Kondo in “Handbuch der Physik,” Vol. X, Springer-Verlag, Berlin, 1960,p 159.
3 147
SURFACE TENSION OF POLYMER SOLUTIONS
28 26 24t
5
30 28 26 24 22 20
t
32 30 28 26 24 22 20
2.
I
18 ? i z 0 UJ z
e u w
l 3 n
I 8 L 3 160
0.2
0.4
0.6
0.8
I.o
18
x)
18 16
$2
0
Figure 1. Surface tensions of solutions of methylsiloxanes in toluene a t 24’. Points: experimental measurements. Solid lines: values calculated from eq 5, with a = 31.6 A2, /3 = 43.5 x 10-16 erg/molecule, and values of r and yo’s from Tables I and 11. 1p1 = volume fraction of siloxane in solution. Note shifts of ordinate axis.
0.2
0.4
0.6
0.8
I.o
42
Figure 2. Surface tensions of solutions of methylsiloxanes in tetrachloroethylene a t 24’. Points: experimental measurements. Solid lines: values calculated from eq 5 with a = 30.5 p = 43.5 erg/molecule and values of r and y o ’ s X from Tables I and 11. (oZ = volume fraction of siloxane in solution. Note shifts of ordinate axis.
A2,
in which form numerical approximation is straightforward. Computer programs have been written22 to accomplish these computations as follows. (1) Compute pls a t p1 = 0.5, from values of ?lo, yzo and a, using eq 6a and then calculate P from the experimental value of y at p1 = 0.5, using eq 5 in the form
(9) (2) Calculate values of plS and y, for any values of from YZO, ylol a , and P. (3) A program has also been written to evaluate eq 8 algebraically for any value of p1. I n order to permit comparison between experimental data and the predictions of eq 5 and 8, certain assumptions must be made about the appropriate parameters. The assumptions which seem most realistic physically, 23 and which have been made, are the following. (1) The lattice spacing parameter, a, is taken to correspond to the area required for a solvent molecule, PI,
and is set equal to
32 30 28 26 24 22
($)*’’,where V , is the molar
volume of the solvent and N is Avogadro’s number.
(2) The polymer molecular weight parameter, r , is assumed to represent the number of statistical segments of size equal to a solvent molecule, and is equated to V , (polymer)/V, (solvent). V , (polymer), the polymer molar volume, is taken to be equal to g n / p ( p = polymer density).
Results and Discussion ( 1 ) Measured Surface Tensions and Comparison with Calculations. The results of the surface tension determinations are summarized in Figures 1-4. Points represent the measurements, while the lines are calculated from eq 5 and 8. Data for the low molecular weight pure compounds are shown on a linear volume fraction scale in Figures 1 and 2, while Figures 3 and 4 present the results for the polymer fluids plotted against log p2. (If the data for the higher molecular weight materials are plotted on a linear volume fraction scale, differences are much less obvious because of the sharp drop in surface tension a t low concentrations followed by a nearly linear change of y with volume fraction in the range p2 = 0.1-1.0.) I n Figures 1 and 2, the calculated curves for solutions of 100-cs silicone fluid, corresponding to those in Figures 3 and 4 but plotted on the linear scale, have been included for comparison. (22) Programs, in BASIC, available from the author on request. (23) Reference 20, pp 498 and 503.
Volume 79, h’umber 9
September 1969
3 148
G. L. GAINES,JR.
t----7; 20 22
I
20 24 v)
c-
V P
>
26
z 24
& i
0
v)
z
E
22
I-
26
V w
w
u 2 a
26
2 a a
24 30
22
24
20
20
28
22
26
26
20
24
a
v)
24
22 2o
20 I I 0.001
v)
I I111111
I
I l l l l l l l
0.I
0.01
I
I
Illld
I
92 Figure 3. Surface tension of solutions of poly(dimethylsi1oxanes) in toluene at 24'. Open points: experimental measurements, materials cited in Table I. Solid points: measurements on solutions of a blended fluid having same viscosity and surface tension as 20-cs polymer, but broader molecular weight distribution. Solid lines: values calculated from eq 5 with a = 31.6 A2, p = 43.5 X 10-16 erg/molecule, and values of r and y o ' s from Tables I and 11. Dashed line: values calculated from eq 8 (mol wt = m), with same values of a and p, and y20 = 20.4 dyn/cm (100-cs fluid). Value of ylo, pure solvent surface tension, indicated for each curve. p2 = volume fraction of polymer in solution. Note shifts of ordinate axis.
The general behavior in both solvents is similar. For all the siloxane solutions, there is appreciable deviation of the surface tension from linearity with volume fraction. As the polymer chain length increases, the decrease in surface tension at low concentrations becomes greater, until for the highest molecular weight materials studied the surface tension appears to be nearly constant (3-5 dyn/cm below that of the pure solvent) from the lowest concentrations studied to volume fractions of polymer -0.1. While we have not as yet examined the effect of molecular weight distribution in detail, one series of measurements (cf. Figure 3) serves to demonstrate its importance. A blend of 10-cs and 100-cs fluid (-76% by volume 10-cs) was prepared to approximate the The Journal of Physical Chemistry
t
0.001
0.01
0.I
I
+2
Figure 4. Surface tension of solutions of poiy(dimethylsi1oxanes) in tetrachloroethylene a t 24'. Points: experimental measurements. Solid lines:o values calculated from eq 5 with a = 30.5 A2, p = 43.5 X 10-16 erg/molecule, and values of r and y o ' s from Tables I and 11. Dashed line: values calculated from eq 8 (mol wt = m), with same values of a and p, and yzo = 20.4 dyn/cm (100-cs fluid). Values of ylol pure solvent surface tension, indicated for each curve. p2 = volume fraction of polymer in solution. Note shifts of ordinate axis.
viscosity, surface tension, and density of the 20-cs fluid. (The number-average molecular weight of this blend was actually about 25% less than that of the 20-cs fluid.) Solutions of this blend in toluene were examined. As the data in Figure 3 indicate, the surface tensions of these solutions were substantially the same as those of the 20-cs fluid over the concentration range cpz = 0.3-1.0, but at lower concentrations there is a progressive difference. The surface tensions of these dilute solutions clearly show the effect of the higher molecular weight fraction present in the blended polymer. To compare the experimental results with theoretical predictions, it is necessary to fit the data to eq 5. Using values of a and r evaluated from the molar volumes as already described, together with the data in Table I for -yl0 and -@, and the measured (or interpolated) values of solution surface tensions at cp2 = 0.5 for each
3149
SURFACE TENSXON OF POLYMER SOLUTIONS siloxane-solvent combination, the values of P were estimated using eq 6a and 5 . These values, tabulated in Table 11, show considerable scatter. No significant trend, either with molecular weight or between the two solvents, appears. These values of p, furthermore, are quite sensitive to errors in the experimental values of ?lo, yzo and the surface tension of the 50% solution. It seems reasonable, therefore, to assume that the scatter merely represents experimental uncertainty, and to use an average value of p in testing eq 5 . The average of all the p values is 43.5 X 10-la erg/molecule (=O.lOBkT at 24’). This value has been used, with the appropriate values of a and r and the yo’s, to calculate all of the solid curves in Figures 1-4. Table IT: Parameters for Fitting Eq 5 and 8 Solvents
Vm, ml
a, A.2
Toluene Tetrachloroethylene
106.9 101.6
31.6 30.5
Siloxanes
Vm, ml
MM MDaM MDeM 5 cs
215 44 1 667 751 1112 1676 3840 6530
10 cs 20 cs 100 cs 1000 cs
-8, ergs/moleculex 10’8 Tetra-7 = Vm/Vm (solvent)chloroTetrachloroethylToluene ethylene Toluene ene
2.01 4.13 6.24 7.02 10.4 15.7 38.9 61.1
2.12 4.34 6.57 7.39 10.9 16.5 37.8
...
40 59
45 44
..
..
46 24 48 24 45
38 67 37 49
..
The agreement between experiment and the theoretical curves calculated in this way is qualitatively quite satisfactory. I n fact, considering the obvious deficiencies of the model and the wide range of composition and molecular weight, it is surprisingly good (we shall return to this point later). The quantitative discrepancies increase with increasing molecular weight, especially a t low concentrations. The equations per se do not seem to provide any rationale for this behavior, since a very large value of p would be required to force an improved fit in this region, with a concommitant worsening of agreement a t higher concentrations. One possible cause might be the effect of molecular weight distribution, which has already been mentioned. Some support for this idea may be provided by the dashed curves in Figures 3 and 4. These have been calculated from eq 8, the expression for r = , using the same values of a and p, together with y&’= 20.4 dyn/cm, corresponding to the measured value for the 100-cs silicone fluid. It is apparent that this change has essentially no effect on the calculated values for 0.2 < cpz < 1.0, where there is good Q)
agreement between theory and experiment in any event. In the low concentration range, however, the curves are closer to the experimental calculated taking r = values than those based on the measured number-average molecular weight. (Since y&’ differs only slightly for 100- and 1000-cs fluids, essentially the same result would be found for the higher molecular-weight material.) (2) Signi$cance of the Model. In view of the good agreement between experiment and the predictions of the equations based on an admittedly unrealistic model, it seems appropriate t o consider whether these equations may actually be of more general validity than that model would imply. It is well known that the application of simple lattice theories to polymer solution thermodynamic properties often does not give good agreement with experiment over extended ranges of concentration. It is also generally acknowledged that such theories in fact should not be expected to be valid at low concentrations where the distribution of polymer segments in solution is nonuniform.20 Why, then, do the present equations, which should suffer from all the deficiencies of the simple lattice theory, besides the highly unrealistic “athermal monolayer’’ assumption, work so well? It is, of course, possible that the semiquantitative agreement found in the present cases is purely fortuitous or reflects some peculiarity of the particular polymersolvent systems examined. Further work will be needed to appraise the general applicability of the equations, but preliminary measurements in one quite different system, poly(isobuty1ene)-tetralin, also indicate reasonable areement between theory and experiment. It should be remarked that the assumption of ideal surface behavior (athermal mixing) has been applied with good results by FowkesZ4to several types of surface effects. In the cases examined by him, however, a surface monolayer model was much more realistic, since only small molecules or polymer molecules actually constrained to lie in the surface (spread monolayers) were considered. Hence, while our derivation formally resembles Fowkes’ treatment, it seems that the question of the generality of the resulting equations may not be related. An alternative possibility may be that it is the excess nature of the surface properties-Le., that the surface effects reflect diflerences in molecular interaction between bulk and surface-that leads to enhanced applicability. Thus, if deficiencies occur in the evaluation of the chemical potentials in bulk solution from the lattice model, similar and compensating errors will arise in the calculated chemical potentials in the surface layer, and the result of comparing them will be more nearly correct than either alone. This speculation leads to the idea that the parameter P is to be associated with Q)
(24) F. M. Fowkes,J. Phys. Chem., 66, 385 (1962); 68, 3515 (1964). v o h m e 76,Number 9 September 1969
3150 an excess intermolecular interaction, the difference between interaction energy in bulk and in the surface. We have so far avoided emphasizing that, in the derivation of eq 5, we have used eq 3, which are identical to those for the chemical potentials in polymer solutions derived by Flory and Hugginsn26 From the derivation, therefore, our parameter p should be expected to be The considerations related to the familiar Flory XI. just advanced, however, suggest that this relationship ought not to be invoked. If p indeed represents a differbetween and surface interaction, there is no reason to expect it to resemble, even qualitatively, X I
NOTES for the same system. These points are a t the moment highly speculative. Further work is obviously required to assess their significance.
Acknowledgement. I am greatly indebted to Mr. E. E. Romagosa, who performed many of the experimental measurements, and to Dr. D. G. LeGrand for several helpful discussions. The cooperation of Dr. F.
M. Lewis in providing samples is also gratefully acknowledged. (25) Equations 3 are, in fact, eq 26 and 32 of ref 20, with our a = R T , or p = kT.
NOTES
The Surface Tension of
I n connection with a study of the surface tension of poly(dimethylsi1oxane) solutions, we have made some precise measurements of surface tensions of hexamethyldisiloxane-toluene mixtures at 27", using a modification of Warren's differential bubble-pressure method.2 In this technique, two identical bubbling tips connected to the same gas supply are immersed in the two liquids to be compared. The depths of immersion are adjusted until bubbles escape from both tips with the same frequency; the difference in hydrostatic head then exactly balances, and is a measure of, the difference in surface tension between the liquids. I n spite of Adam's favorable appraisal,a the method has been little used, and some comments on its application seem appropriate. The results are also of some interest, in view of recent attention to the surface tension of binary mixturesa4
this grinding was done carefully to leave a small fla area (-0.3 mm wide) around the orifice. A rigid framework was provided to support the capillaries and micrometers used to measure the liquid levels. Both containers holding the liquids were supported on platforms with fine-pitch adjusting screws. Precise determination of the vertical position of the capillary tips was achieved by raising the liquid level to just touch each capillary and the corresponding micrometer (fitted with a pointed tip) simultaneously; this procedure permitted establishment of this position to i 0.001 cm. After adjusting the containers to immerse the tips and to balance the bubble escape, the liquid levels could also be determined with the same precision. It was found that some care was required to eliminate difficulties due to vibration. Temperature control to i 0.1" was provided with a simple air thermostat enclosing the entire apparatus. In the measurements on hexamethyldisiloxane-toluene, no dependence on bubbling rate was found, with rates of 5-60 sec per bubble. Attempts to measure the surface tension of solutions of higher molecular-weight poly(dimethy1siloxanes) (e.g., 100-cs silicone fluid, JZ,,= 3700) were not successful, since rate dependence was observed and we were unable to reduce the rate sufficiently to obtain equilibrium.
Experimental Section The design of our apparatus followed that of Warren closely. Identical capillary tips (inside diameter 0.4 mm) were obtained by breaking a single piece of tubing. It was found advantageous to grind the tips to a conical shape to avoid entrapment of escaping bubbles;
(1) G. L. Gaines, Jr., J . Phys. Chem., 73, 3143 (1969). (2) E.L. Warren, Phil. Mag., 4, 358 (1927). (3) N. K. Adam, "The Physics and Chemistry of Surfaces," 3rd ed, Oxford University Press, London, 1941, p 388. (4) See, e.g., (a) S . K. Suri and V. Ramakrishna, J . Phys. Chem., 72, 3073 (1968). (b) R. L. Schmidt, J. C. Randall, and H. L. Clever, ibid., 70, 3912 (1966), and references cited therein.
Hexamethyldisiloxane-Toluene Mixtures
by E. E. Romagosa and G. L. Gaines, Jr. General Electric Research and Development Center, Schenectady, New York 12301 (Received January 20, 1969)
The Journal of Physical Chemistry
