1306
BRUNO H, ZIMM
OPALESCENCE OF A TWO-COMPOKEiXT LIQUID SYSTEM NEAR THE CRITICAL MIXING POINT’ BRUNO H. ZJMM Department of Chemistry, University of California, Berkeley 4, California Received December 19, 19.49
Near any critical point large fluctuations of some of the properties of the system occur. If these fluctuations are accompanied by changes in the refractive index, the system is no longer optically uniform and the amount of light scattered will be large. A discussion of the resulting opalescence was given by Smoluchowski (12) in 1908, followed by a more complete treatment by Einstein (1) in 1910. The latter’s contribution has stood to this day as the best basic treatment, and has been extended, but not modified, by later authors. In the present case we have a two-component one-phase liquid system near the critical mixing point. Ordinarily small fluctuations in composition will occur as a result of the thermal motion of the molecules, but they will seldom be large because of the free-energy increase accompanying a fluctuation away from homogeneity. Near the critical mixing point, however, the free-energy changes very slightly with composition so that the fluctuations may easily become large. Putting these ideas into quantitative form, Einstein derived essentially the following equation:
where
the “turbidity,” the fraction of the incident light scattered per unit length, XO = the wave length of the light in vacuo, N , = Avogadro’s number, R = the gas constant, T = the absolute temperature, fl = the compressibility, p = the density, n = the refractive index, Vl = the molar volume of component 1, q2 = the volume fraction of component 2 (see below), and al = the activity of component 1. 7 =
The extension of this equation to include angular variations of intensity is discussed later. The first term of this equation, involving the compressibility, is very small in a mixture near the critical mixing point. Essentially, therefore, equation 1 may be looked on as a relation between the slope of the activity isotherm, d In 1 Presented a t the Symposium on Critical Phenomena, which was held under the auspices of the Division of Physical and Inorganic Chemistry a t the 116th meeting of the American Chemical Society, Atlantic City, New Jersey, September 22, 1949.
OPALESCENCE OF LIQUID SYSTEM NEAR CRITICAL MIXING POINT
1307
al/dp2, and the turbidity, 7.Thermodynamic quantities may thus be determined by light-scattering measurements. I n fact, because of the reciprocal relation between the turbidity and the slope of the isotherm, this method is eminently suited to measuring the slope near the critical point, where it is so small that other methods fail. Let us consider what might be expected from a theoretical basis for the isotherms in the neighborhood of a critical point. For simplicity, we shall restrict our remarks to a system of two components that forms two liquid phases below the critical (mixing) point and one phase above. We shall use the volume fractions, 91 = N1v1/V and p2 = N272/V to describe the composition, where N1 and N Z are the numbers of moles of components 1 and 2 , and TZare the partial molar volumes of these components, and V is the total volume. Of course, p1 p2 = 1. To describe the thermodynamic potentials we shall use the activities, al and a2. In discussing the slopes of the isotherms, we shall use principally the quantity d In UI/ d In p1, temperature and pressure being assumed constant, of course. In a system that obeys Raoult's law d In al/d In p1 is always unity. In other cases it may be represented by a power series in 'pz,
+
where the coefficients B,, are functions of the temperature only for any given system, provided the system is of one phase. The appearance of a second phase creates a singularity in d In a l l d In p1, which may be considered mathematically as a point of divergence of equation 2. The graphical representation of equation 2 for a hypothetical system showing a classical critical point is shown in figure la. At high temperatures (curve I) d In al/d In e has a minimum at some volume fraction in the middle of the diagram. As the temperature is lowered the minimum dropsloweruntil it touches zero (curve 11). This is the critical point. At still lower temperatures a phase transition occurs before the curve reaches zero (curve 111),as signified by the discontinuous drop to zero, where the second phase appears. Mathematically speaking, there is a critical temperature a t which the function first vanishes, and at this temperature and below a singularity (discontinuity) of the function makes its appearance. The initial appearance of these two phenomena a t the same temperature is remarkable, since each imposes stringent, and apparently unrelated, conditions on the coefficients B, of equation 2 . I t was shown by McMillan and Mayer (8),following earlier work by Mayer (7), that the coefficients B, are complicated but definitely determined functions of the intermolecular potentials. It is conceivable that for a particular system the B, might satisfy simultaneously the conditions for the divergence and the vanishing of the function, but it would seem really remarkable that they should do it for all systems of diverse molecular types in which the values of the individual coefficients differ greatly. Mayer therefore suggested that the isotherms might more reasonably be expected to show the behavior illustrated in figure lb. Here the critical point is the point at which the minimum of an isotherm just
1308
BRUNO H. ZIMM
touches zero (curve 11), but it is not the point at which divergences of the series of equation 2 occur. Divergence, with the appearance of a second phase, occurs for the first time at a lower temperature. The two phenomena are separated by a temperature interval. In this interval a new type of isotherm (curve 1V) appears,-one in which d In aJd In goes continuously to zero, and then remains at zero for a finite interval. There is thus predicted to be a region between the critical point and the temperature at which two distinct phases appear within which there is a second-order transition, the first derivative of the activity going continuously to zero while the second derivative has a discontinuity. The prediction of this anomalous region is not rigorous; it is based on the plausible argument that the two apparently un-
FIG.1. Possible isotherms: (a) classical; (b) according to Msyei
related events of the vanishing of equation 2 and its divergence will not first occur at the same temperature. On this basis the classical isotherms displayed in figure l a would be the result of a miraculous coincidence. Most of the previous investigators who have studied critical opalescence (4, 5 , 11) did so before Mayer’s theory was advanced. Their experiments show no sign of a second-order transition, but it is possible that the transition occurs over such a small temperature interval that it was overlooked. Mason and Maass (6), on the other hand, found anomalous behavior in a one-component system. It therefore seemed worthwhile to carry out a new investigation with the primary purpose of searching for the second-order transition. EXPERIMENTAL
For the experimental investigation the system of carbon tetrachloride and perfluoromethyIcycIohexane, C,Fl,, was chosen, since Hildebrand and Cochran
OPALESCENCE
OF LIQUID SYSTEM NEAR CRITICAL MIXING POINT
1309
(3) had demonstrated a critical point at the convenient temperature of 26.9'C. Moreover the system is nonpolar, a fact that might lead to important simplifications in a theoretical treatment. The materials used were purified with some care. The carbon tetrachloride was Mallinckrodt's analytical reagent grade. It was distilled through a jacketed twenty-plate column, the first fifth (b.p. 76.8'C., corrected) being discarded; the succeeding third (b.p. 76.84'C., corrected) was retained and used without further treatment. The perfluoromethylcyclohexane, C7F14,was the same material used by Hildebrand and Cochran (3) ;it had originally been obtained from the Jackson Laboratory of E. I. du Pont de Nemours and Company. Upon distillation through the twenty-plate column, the first fifth of the material came over below 76.45'C. and mas discarded. About one-half distilled from 76.45' to 76.55'C. and was kept. (The boiling point reported in the literature (2) is 76.32'C.) The remaining portiop boiled over a range of 0.4"C. The distilled material was then twice redistilled in a simple one-plate still over one-tenth its weight of 100-mesh silica gel that had been heated to red heat in air for 30 min. to drive off water. I t was hoped that this treatment would remove small amounts of impurities that might have escaped fractionation. A mixture of 58 per cent by weight of carbon tetrachloride was made with a sample of the perfluoromethylcyclohexane after fractionation and after each treatment with silica gel. The temperature of phase separation was determined for each sample. These were all found to be 28.23'C. f 0.02', showing that no change in the solubility behavior of the perfluoromethylcyclohexane was occurring with further treatment.' Solutions were made by weighing small amounts of the purified materials into glass tubes, which were then sealed until ready for use. The compositions are given in table 1, being expressed both as weight fractions and as volume fractions. The volume fraction of component 2, 9 2 , is given by 9 2 = w2/hV, wherq W Z is the weight of component 2 (carbon tetrachloride), p2 is its density, and V is the total volume; likewise for a. The light scattering was measured in an apparatus previously described (13). In essence, this consists of a 3-ml. glass bulb suspended in a beam of light and containing the solution, with the scattering observed by a revolving photoelectric photometer. The small bulb was surrounded by 1-propanol to reduce reflections and to provide temperature control. The 1-propanol was heated by water from a thermostat flowing through a copper coil and was stirred by an electric motor when necessary. For those solutions showing extremely high opalescence, a special small bulb was made to keep the light path short to reduce secondary scattering. In this bulb the total light path did not exceed 3 mm. at any angle at which observations were made. The apparatus was calibrated by means of a standard polystyrene solution, whose absolute scattering power had been measured with
* The presence of an impurity in the original material was indicated by the low temperature of phase transition, 26.9'C., found by Hildebrand and Cochran (3) and also by the fact that the original material discolored very dilute potassium permanganate solution. The latter reaction is reported to result from the presence of incompletely fluorinated compounds (2). The purified material showed no reaction with potassium permanganate over a period of 16 hr. The purified carbon tetrachloride likewise showed no reaction.
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BRUKO H. ZIMM
exceptional care by Dr. C. I. Carr in this 1aboratoryF Some care was taken to keep dust out of the solutions by using carefully cleaned equipment and minimizing the number of transfers. The dust originally present in the materials was removed in the distillations described above. In applying Einstein's formula (equation 1) the following simplification was made: If the relation of Gladstone and Dale for the refractive index is assumed, TABLE 1
Compositions and separation temperatures of miztures MIXTURE NO.
per cent
1 2 21 3 31* 4 41 5 6
* Critical
OC.
27.9 36.85 42.2 47.25 52.7' 58.4 62.5 67.8 79.95
22.9 26.8 28.0 28.3 28.3; 28.3 28.0 26.8 20.6
0.317 0.400
0.451 0.502 0.560'
0.614 0.653 0.707 0.820
mixture. TABLE 2 d In a l / d In PI
. .. . . .. . . . .. .
Y I X T U P E NO.
-4
1
3%
0.325 0.283 0.244 0.192 0.180 0.169 0.156 0,122'
0.0875 0.0916 0.0386 0.0470 0.0222 0.0258 0.00695 0.00428 0.00610 0.0031 0.0016 0.0030
J
6
0.133 0.0936 0.0589 0.0331 0.0290 0.0240 0.0165'
0.309 0.282 0.245 0.205 0.198 0.191 0.182 0.166 0.158'
I
'C.
50
40 35 30 29 28 26.8 22.9 20.6
0.0419 0.0333 0.0249 0.0171'
* Value just previous t o phase separation.
and moreover it is assumed that there is no volume change on mixing the two components, then dn/dqp,= n2 - nl,where n2and nlare the refractive indices of the pure components. Further, it is convenient to express the results in terms of This standardization, part of the general program of standardization of our equipment, waa done b y three independent methods: a transmission measurement, a measurement of
the scattering a t right angles t o the incident beam, and a measurement of the total scattering in an integrating sphere. All methods gave results which were in agreement.
OPALESCENCE OF LIQUID SYSTEM NEAR CRITICAL MIXING POINT
1311
d In al/d In p1, a quantity that remains finite for all values of the composition. From the definition of the volume fraction it can be shown that d In al/d In p1 = -91 d In addaz. The light-scattering measurements on each solution were converted to d In al/d In p1and plotted on a logarithmic scale against the logarithm of the temperature less an appropriate number. In this way smooth, slightly curved lines were obtained from which the value of d In al/d In 91 could be obtained at any desired temperature. The interpolated results are shown in table 2 and figures 3 and 4. Some of the original experimental data are given in table 4 and figure 5. The deviation of the points from the smooth curve mas usually less than 5 per cent, and appeared to be entirely erratic.
21.60' 0
.50
.25
.75
1.00
%,,
FIG.2. Phase diagram of the systkm GFI,-CClr. Scales on lower left axes apply to lower curve; those on upper right to upper curve.
The temperature of phase separation was determined for each mixture by slowly warming the two-phase mixture with shaking in a water bath until a homogeneous solution was obtained. When the mixture was cooled, noticeable supercooling occurred only in mixtures 1, 2., 5 , and 6 (table 1) before the two phases separated again. When the temperature of phase separation was plotted against composition the different mixtures showed erratic deviations of the order of 0.02" from a smooth curve. Believing that the deviations might be caused by differences in the amount of air dissolved in the various mixtures, it was decided to determine the sensitive portion of the phase diagram with one mixture only. About 0.1 ml. of mixture No. 33 was sealed in a vial bearing a straight piece of 2-mm. tubing as a side arm. This vial was suspended in a thermostat regulated within 0.002", in such a way that it could be tilted from outside the bath. When the mixture
1312
BRUNO H. ZIMM
had been made t o separate into two phases at any desired temperature, the tube could be tilted and a portion of the upper phase poured into the side arm. On subsequent cooling of the tube in another bath, the mixture in the side arm would separate again into two phases of composition that could be estimated sufficiently .05
.04
.O? .02
.01
u
&TI l
0
3
.2
.I
0
.3
.5
.6
.0
FIG.3 . Activity isotherms of mixtures of GFu (component 1) and CClr (component 2) at indicated temperatures. Filled circles are points of observed phase transition.
well from the rough phase diagram already determined. By measuring the relative amounts of the two new phases with a telescope and scale it was then possible to calculate the composition of the original upper phase, and from the total composition the composition of the original lover phase could also be determined. In this way the upper curve of figure 2 was obtained. The results of these experiments are shown in table 3.
OPALESCENCE OF LIQUID SYSTEM NEAR CRITICAL MIXING POIXT
1313
.OOIl
c
1
5
0.4
FIG.4. The critical region of figure 3 TABLE 3 Consolute mixtures
T. 28.310* 28.303
0.551 0.531
0.571 0.586
28.173 28.135 27.99 27.92 27 76 27.70
0.491 0.480 0.467 0.463 0.452 0.456
1
0.589 0.598 0.604 0.611 0.622 0.635 0.639 0.650 0.646
In all cases a distinct and sharp meniscus was observed between any two phases in equilibrium. In one experiment mixture No. 31 was heated without stirring
1314
BRUNO H. ZIMM
to 0.01' above the temperature at which two phases had appeared on cooling; in 15 min. the meniscus spontaneously diffused, being replaced by a continually spreading band about 1 mm. in \vidth containing a smooth refractive-index gradient. The diffusion was taken as evidence that a single uniform phase was the condition of equilibrium at this temperature. On cooling, the approach of the phase transition was heralded by an increase in the opalescence of the solution. However, the second phase could alvays be seen to form as small discrete drops, appearing quite different on close examination from the general opalescence that preceded it.
FIG.5 FIG.6 FIG.5. Values of d In al/d In 'pl calculated directly from experimental turbidities and plotted against temperature minus critical temperature for the critical mixture. The dotted line is the line of unit slope. FIG.6. Intensity in consistent but arbitrary units against the square of the sine of half the angle of scattering a t the indicated temperatures. Some of the values have beenmultiplied o r divided by constant factors as indicated. All points for vpz = 0.560 escept the dotted ones, which are for p2 = 0.516 and t = 28.29"C. RESULTS
The phase diagram, determined as described above, is shown in figure 2. It is noteworthy that the two-phase region is rounded on top, not flat (8, 10). However, it should also be noted that the curve describing it is not a parabola. Empirically, the curve is given well enough by the equation:
t = tc
- 600(1~p2- PZ. l3
The critical point itself seems to occur at a volume fraction of carbon tetrachloride, q2., of 0.551 and a temperature, t,, of 28.310°C.,the last two decimal places of the temperature having significance relative only to the other temperatures of this report, since the absolute calibration of the thermometers was reliable only to 0.1'.
OPALESCEXCE O F LIQUID SYSTEM S E A R CRITICAL MIXING POINT
1315
The results of the light-scattering determination of d In al/d In p1 are shown in figures 3 and 4 and in tables 2 and 4. I t can be seen that the isotherms are of the classical type. No sign of a second-order transition below the critical point i s apparent. A discontinuity in the slope is found even at 0.02" below the critical point, while the slope is observed to be finite and different from zero 0.01" above the critical point. In figure 5 it is seen that the opalescence varies almost linearly with temperature above the critical point, as also in the results of Keesom (4) and Mason and Maass (6) on the liquid-vapor critical point in ethylene. The general appearance of these results is also similar to those of Krishnan ( 5 ) and Rousset (11) on other systems. TABLE 4 Turbidity as a function of temperature in critical mixture ( N o . 34) TEYPEPAWPE
TUPBDITY
'C.
47.2 37.3 34.7 32.6 30.9 29.8 29.34 29.00 28.90
0.0026 0.0061 0,0089 0.0144 0.026 0.052 0.071' 0.13'
0.074 0.031
1
1
TEILPEIULTITPE
'C.
28.69 28.50 28.41 28.34 28.33 28.31 28.29 28.14
0.0374 0.0037 0.0027 0.0015
0.222' 0.56' 1.13* 4.0* ' 7
0.m86
0.00034 0.00017 O.ooOo5
O.ooOo3
>%I* 0.00000 !?
0.60t' 0.074$*
0.00032t 0.0026$
0.150*
* Ideal values, calculated
from scattering extrapolated t o zero angle. composition, p1, of 0.516. $ Refers to lower layer with an estimated composition, pl, of 0.480. 0 Droplets visible in mixture.
t Refers to loxer layer with an estimated
ANGULAR DEPENDENCE O F T H E SCATTERING
The Einstein formula as written in equation 1 holds only if the fluctuations in refractive index areuncorrelated over distances of the order of magnitude of the wave length. I n general, as shown by Ornstein and Zernike (9), the scattering, I, is proportional to the integral:
I
- $e
g(r) 4rr2 dr
(3)
where g(r) is the mean correlation of two fluctuations of refractive index occurring at a distance T apart, s = 4?r sin (0/2)/X with X the wave length of the light and 9 the angle. This integral reduces to the Einstein equation 1 if sin w / s r is unity, i.e., if sr is small, over all values of r for which the correlation g(r) is different from zero. Since s is less than 0.003 reciprocal A. for visible light, the Einstein formula is valid without restrictions if the fluctuations are not correlated over distances of more than 100 A. Otherwise, the scattering depends upon s and hence on the angle. That such a thing sometimes occurs is
1316
BRUNO H. ZIMM
demonstrated by the data plotted in figure 6. However, we may always extrapolate to a point at which the Einstein formula is valid simply by extrapolating to the value of the scattered intensity at an angle of zero, since here s is zero and sin W / ~ Tis unity for all values of T . This was the procedure actually followed wherever an angular dependence of the scattering was found. In this way we avoid the necessity of considerations such as those of Ornstein and Zernike (9). One might similarly extrapolate to infinite wave length, but the extrapolation to zero angle is easier experimentally. Since the variation of scattering with angle occurs only when the fluctuations extend over regions of large, almost visible size, it is interesting to note that such variation was only observed within one degree of the critical point at the critical composition, and over a range of less than 10 volume per cent at the critical temperature. This conclusion is at variance with that of Krishnan (j),based on depolarization studies. ,4n attempt was made to determine the depolarization of the scattered light in order to substantiate the conclusions from the angular variation. The depolarization was found to be very small, and the secondary scattering unavoidable in mixtures of the high turbidity encountered in this work made these measurements uncertain. The secondary scattering also made the quantitative interpretation of the angular variation curves difficult. In particular, it would be interesting to compare them to the theoretical calculations of Ornstein and Zernike, but beyond the statement that there is qualitative agreement it is impossible to go without more elaborate experimental precautions. I t is easy to see from the form of the curve in figure 6 that the extrapolation to zero angle fails at the critical point itself, as indeed it should, since here the fluctuations are macroscopic in size, although the scattering is finite at any angle greater than zero. CONCLUSION
These results might be criticized on the basis that the materials used were impure. In view of the process used, it does not seem that large amounts of impurities could be present unless they were isomers of the perfluormethylcyclohexane. In this case they would also all have the same solubility properties and hence should not, change the behavior of the system. Perhaps the small amounts of water and air that were dissolved in the samples from their contacts with the atmosphere might have a more serious effect. It must remain for future work to assess the importance of this criticism. The general appearance of our data is similar to that of Krishnan ( 5 ) and Rousset (ll),though at variance with the results of Mason and Maass (0) on a one-component system. The failure of a second-order transition to be observed is a major surprise, in view of the theoretical likelihood of this transition. I t is apparent that this is not a contradiction of the theory, but if substantiated it is an uncomfortable demonstration that the critical phenomenon is less well understood than had been hoped. I t mould be necessary that there exist relations among the higher coefficients, B,, of equation 2 such that the divergence of the series first occurs
THEORY OF CRITICAL POISTS
1317
a t a temperature very close to, if not identical with, the critical temperature. Since similar behavior seems to be found experimentally in other systems (5,l l ) , it would mean that these relations among the coefficients must hold regardless of the details of the intermolecular forces. Our experiments were designed to answer experimentally the question of whether the theoretically expected second-order transition can be found. The answer, at least as concerns the system studied, seems to be a clear negative. Xot the slightest evidence was found for any anomalous behavior near the critical point. The results then pose a difficult problem for the theorist,-the problem of explaining why the predicted second-order transition fails to appear. REFERENCES (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)
EINBTEIS,A , : Ann. Physik 33, 1275 (1910). FOWLER, R. D., et a l . : Ind. Eng. Chem. 39, 375 (1947). HILDEBRAND, J. H., AND COCHRAN, D. R . F.: J. 4 m . Chem. SOC.71, 22 (1949). KEESOM,W. H.: Ann. Physik 56, 591 (1911). KRISHNAN, R. S.: Proc. Indian Acad. Sci. 2, 21 (1934); 6, 577 (1937). MASON,Y. G., AND MAASS,0.: Can. J. Research 26B, 592 (1918). MAYER,J. E.: J. Chem. Phys. 6, 67 (1937). MCMILLAK, W. G., AND MAYER,J. E.: J. Chem. Phys. 13, 276 (1945). ORNSTEIN,L. S., AND ZERNIKE,F.: Physik. 2. 27, 261 (1926); 19, 134 (1918) RICE,0. K : J. Phys. & Colloid Chem. 64, 1293 (1950). ROWSET,-4.: Snn. phys. 6, 5 (1936). SMOLUCHOWSKI, M.: Ann. Physik 26, 205 (1908). ZIMM,B. H . : J. Chem. Phys. 16, 1099 (1948).
THE THEORY O F CRITICAL POINTS', L. TISZA Deportment of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts Received December 19, 1949 I. IhTFiODUCTION
The existence of critical points was discovered by Andrews (1) in connection with his classical investigation of the carbon dioxide isothermals. Andrews was mainly concerned with the nature of the liquid, vapor, and gaseous phases of matter. His main conclusion and at the same time the definitive solution of the question was that there is only a single fluid condition of matter which can split into two coexisting forms of different density, provided the temperature is below 1 Presented a t the Symposium on Critical Phenomena, which was held under the auspices of the Division of Physical and Inorganic Chemistry a t the 116th Meeting of the American Chemical Society, Atlantic City, Kew Jersey, September 22, 1949. * This work waa supported in part by the Signal Corps, the Air Materiel Command, and the Office of Naval Research.
