GEORGE A. MILLERAND CHINGS. LEE
4644
Brillouin Spectra of Dilute Solutions and the Landau-Placzek Formula by George A. Miller and Ching S. Lee School of Chemistry, Georeia Imtitute of Technology, Atlanta, Georgia $03~98 (Receivd July 8, 1968)
The intensity ratio of central to side peaks has been measured in the Brillouin spectra of several dilute solutions. The solvents, chosen for their known deviation from the Landau-Placzek formula, were benzene, carbon tetrachloride, water, and ethylene glycol, with toluene included as a %ormal” liquid. The concentration dependence of the intensity ratio was shown to be affected by a deviation from the Landau-Placzek relation. If one wishes to determine activity coefficients of binary solutions by measuring intensity ratios, careful consideration must be given to these deviations. The resolving of the Brillouin spectrum or fine structure of Rayleigh scattering offers an interesting new approach to the study of the intensity of light scattered by solutions. Since concentration fluctuations contribute to the unshifted central component of the scattered light, one can substitute the intensity of the Doppler-shifted side peaks for that of the incident beam as a reference. Then the Rayleigh ratio or the turbidity is replaced by the ratio of the intensities of the central peak to the two side peaks as the experimental quantity to be determined. One of us has developed the relationship of this ratio to concentration for binary solutions’ from the standard thermodynamic theory of light scattering. In the simplest form the result reduces to the well-known Landau-Placzek (LP) formula J = y - 1 for a pure liquid, where J is the intensity ratio and y is the ratio of heat capacities at constant pressure and constant volume. However, for many liquids the L P formula is not an accurate one. Light-scattering intensity measurements provide a tool for studying the nonideality of solutions or, as is more common, for determining the weight-average molecular weight of polymers in solution. I n either case, if the Brillouin spectrum approach is to be used, it is necessary to be able to treat those instances in which the pure solvent does not obey the LP formula. Wit,h this in mind, we have measured intensity ratios for several deviant liquids with small amounts of a second component dissolved in them. The solvents studied were toluene (normal liquid), benzene and carbon tetrachloride (internally relaxing liquids), water (liquid with very small thermal expansion), and ethylene glycol (structurally relaxing liquid). The concentrations of the solute did not exceed 10 mol 70 and were much less for the more nonideal solutions studied; hence we were working always within or close to the Henry’s law region of ideality. I n what follows, we discuss the relevance of these measurements to the L P formula. I n most accurate form, the thermodynamic theory yields, for an ideal, binary solution, having a solute mole fraction of 22 The J O U T Wof ~ Physical Chemistry
J -
where T is the absolute temperature, P is the pressure, n is the refractive index, R is the gas constant, CY is the thermal expansivity, v is the molar volume, and PT and ps are the two compressibilities. The partial differentiation always concerns the triad P, T , and x2. A good assumption for many liquids is that
bP
-
which leads to the simplified form where
Equations 1 and 3 may be thought in an approximate way to arise from the equation
+
(entropy fluctuation term) (concentration fluctuation term) J = (pressure fluctuation term) We will take eq 3 as representing the behavior of a “normal” solution. Deviations from normal behavior may result first of all from the incorrectness of the assumption implied in eq 2, in which case the full eq 1 must be used. Then there will be cases where the solution is a relaxing fluid for which the simple thermodynamic theory is inadequate. For discussing deviant behavior of various kinds, we can write a slightly more general form of equation for the intensity ratio (1) G. A. Miller, J. Phys. Chem., 71,2305 (1967).
BRILLOUIN SPECTRA OF DILUTESOLUTIONS AND THE LANDAU-PLACZEK FORMULA
J =A
+BKx~x~
(4)
If we consider dilute solutions, then the coefficients A , B , and K , which are actually concentration dependent, will be essentially the values a t infinite dilution. Except for the refractive index increment which occurs in K , these coefficients will depend only on the properties of the pure solvent to a good approximation. First, we will consider the error involved in introducing eq 2. Coumou, Mackor, and Hijmans2 have studied this problem in relation to total isotropic light scattering. They introduced the parameter
which becomes zero if eq 2 holds exactly. Equation 1 may be rewritten in terms of x by means of a factor f = - 22
1
YX2
-2
+
(1
- 2)2
to give a form of eq 4 in which
For many organic solvents, x is typically 0.01-0.05,2 which means that A and B are some 2-10% less than y - 1. For water, x is negative, and A and B are ' the about 50% greater than y - 1 a t 25". At 4 density of water goes through a maximum, and the above formulai,ion in terms of x may not be used. 1 = 0, and eq 1 becomes Instead we have a! = y
-
J
I=
.
ik (3
Formally, this fits eq 4 with
However, in the complete coefficient of the concentration term, bn/bT is cancelled out, i.e.
BK =
(-)
bn/bx2 RT bn/bP
which, we will see, allows us to ignore the probably rapid change of bn/bT with concentration in aqueous solutions at 4". Next, we will consider relaxation effects. I n a relaxing liquid the spontaneously generated sound waves, as represented by the pressure fluctuation term, transfer energy to nonpropagating modes of energy, thereby decreasing the intensity of the side peaks and enhancing that of the central ~ e a k . ~The , ~ summed intensity of these peaks remains the same as that predicted by thermodynamic theory. Let R be the value of J for a
4645
pure liquid, as given by thermodynamic theory. R will be approximately equal to y - 1 but is given more precisely by the formulas discussed above. Let J o be the experimental intensity ratio of the pure liquid. The effect of a relaxation process will be that J o > R; the intensity of the side peaks will be multiplied by the factor (R l)/(Jo 1) and that of the central peak by the factor Jo(R l)/R(Jo 1 ) . For dilute solutions in which the solvent is a relaxing liquid, the intensity ratio is given, thus, by eq 4 with
+
+
A = Jo;
+
+
B = R(Jo
+ 1)/(R + 1)
(7) I n some cases the component of the central peak contributed by the relaxation process may be broad enough to appear to contribute partly to the intensity of the side peaks. Therefore, a reasonably consistent procedure should be used to determine the relative intensities of the three peaks, such that the relaxation component contributes the same amount to the side peaks a t different concentrations. Examples of liquids in which the relaxation is with internal modes of energy are carbon tetrachloride5 and benzeneP-8 I n viscous liquids, such as ethylene glycol and glycer01,~~~ the relaxation occurs with structural or intermolecular modes of energy. We find eq 7 quite inadequate for glycol solutions. Since no theoretical treatment of Brillouin scattering from viscous solutions is available, we must consider the coefficient B to be unpredictable for such solutions. Water is an interesting case experimentally, because J o is very small and difficult to measure in the face of small contributions from dust and instrument scattering. Values of Jo calculated by thermodynamic theory agree as well as can be expected with experiment.lO Since there are no relaxation effects here, we can assume that A = B for aqueous solutions. With the proper choice of solute, large values of J may be obtained from relatively dilute solutions. Therefore, B may be measured more accurately than A . At 4O, where J o is truly miniscule (4 X B is still a readily measurable quantity. Our assumption about the limiting behavior of A and B warrants additional comment. I n the limit of (2) D.J. Coumou, E. L. Maokor, and J. Hijmans, Trans. Faraday SOC.,60, 1539 (1964). (3) C. J. Montrose, V. A. Solovyev, and T. A. Litovitl;, J . Acoust. SOC.Amer., 43, 117 (1968). (4) D. A. Pinnow, S. J. Candau, J. T.LaMaochia, and T.A. Litovita, ibid., 43, 131 (1968). (6) W. 5. Gornall, G. I. A. Stegeman, B. P. Stoioheff, R. H. Stolen, and V. Volterra, Phys. Rev. Lett., 17, 297 (1966). (6) J. L. Hunter, E. F. Carome, H. D. Hardy, and J. A. Bucaro, J. Acoust. SOC.Amer., 40, 313 (1966). (7) C. L. O'Connor and J. P. Sohlupf, ibid., 40, 663 (1966). (8) E, F. Carome and 5. P. Singal, ibid., 41, 1371 (1967). (9) D. H. Rank, E. M. Kiess, and U. Fink, J. Opt. SOC.Amer., 56, 163 (1966). (10) C. L. O'Connor and J. P. Sohlupf, J . Chem. Phys., 47, 3 1 (1967).
Volume 79,Number 13 December 1968
GEORGE A. MILLERAND CHINO: S. LEE
4646 He-Ne laser
cell
(7 -
!I
6328 A cylindrical lens interferometer spherical lens
-- pin hole
0
photomultiplier
Figure 1. Schematic of light-scattering apparatus.
infinite dilution, the derivatives bA/bxz and bB/bxz will not be zero except by accident. The value of B obtained from a determination of J in dilute solutions, even if extrapolated to xz = 0, will contain an unwanted increment, (bA/bx2)/K. This error may be kept small by choosing a solute-solvent pair with a large refractive index increment.
Experimental Section Brillouin spectra were recorded at a scattering angle of approximately 90" as shown in Figure 1. A PerkinElmer Model 6320 He-Ne laser provided 10-mW radiation at 6328 8 with a half-width of about 0.025 cm-1 (0.010 8). The sample was contained in a standard square turbidity cell made by the Phoenix Precision Instrument Co. The cell was surrounded by a close fitting, electrically heated aluminum block for temperature control. The scattered light was gathered with a cylindrical lens of 10-cm focal length, which served to collimate the light parallel to the horizontal plane only. Thus the original scattering angle of a ray was preserved as it passed through the lens. The scattered light was analyzed with a piezoelectrically driven, scanning Fabry-Perot interferometer. A simplified sketch of the interferometer is shown in Figure 2. The actual mounting resembled that of the magnetically driven interferometers described by Fork, et aL11 The 95% reflectant mirrors gave a finesse of about 60. Light emerging from the interferometer was focused by a 50-cm focal length lens onto a pinhole. The central spot of the interferometer ring system was then gathered by an EM1 9592-B photomultiplier with an S-10 response. The signal was amplified through a Schoeffel 34-600 photometer and was displayed on an X-Y recorder. The mirror spacing, d, was set at 10.67 mm, giving a free spectral range of 0.48 cm-' and an over-all instrumental half-width of about 0.032-0.042 om-' (arising from the laser line width, the interferometer finesse, and the size of the pinhole). The voltage drive for the interferometer consisted of a stabilized 350-V power supply coupled with a 10-turn, 250,000-ohm Helipot driven by the multispeed output of a synchronous motor. The same ramp voltage provided the signal for the X axis of the recorder. Thus the piezoelectric element of the interferometer remained The Journal of Physical Chemistry
PM,IM/
P
Figure 2. Simplified diagram of scanning Fabry-Perot interferometer showing piezoelectric ceramic tube (P), multilayer dielectric mirrors (MI, M2), micrometer adjustment for parallelism (X, Y). Mirror spacing (d) is adjustable.
the only real source of possible nonlinearity in the frequency axis of the recorded spectrum. The solutions were made from commercially available reagents without further purification. The benzene, toluene, carbon tetrachloride, glycerol, and ethylene glycol were Fisher Certified reagents. The naphthalene was Eastman Highest Purity, recrystallized from alcohol. The water was ordinary distilled. Organic solutions were made dust free by a single filtration through a Millipore filter of 0-22-p pore size under a slight excess pressure of nitrogen. The turbidity cell was first washed free of dust with the condensing vapors of boiling carbon tetrachloride, and the solution to be studied was then filtered immediately into the cell. Aqueous solutions required repeated filtration (about six times) to remove dust. Some of this dust came from the walls of the cell and could not be removed by steaming, hence the main purpose of repeated filtration was to wash the dust from the cell. Using this fairly well standardized procedure on water gave a good indication of how much dust remained on the average in the solutions. The glycol solutions required two to three filtrations. Naphthalene dissolved in glycol only upon heating the mixture to about 80". The most concentrated solution was definitely supersaturated at 25". The index of refraction of the solvents as a function of temperature and of the solutions as a function of composition was determined on an AbbB-type Bausch and Lomb precision refractometer, using the laser as the illuminating source.
Results A representative selection of recorder traces of the Brillouin spectra of the various solutions is given in Figures 3-7. Some variation in finesse of the interferometer due to slight misalignment is to be observed. The occasional spikes were caused by a dust particle passing through the incident light beam. Undoubtedly, small amounts of finer dust together with a trace of instrumental scattering added perhaps a few per cent to the intensity of the central peak in all our experi(11) R. L. Fork, D. R. Herriott, and H. Kogelnik, Appl. Opt., 3, 1471 (1964).
BRILLOUIN SPECTRA OF DILUTE SOLUTIONS AND THE LANDAU-PLACZEK FORMULA
4647
iil
Figure 3. Spectrum of light scattered at 25" from toluene (lower curve) and a solution of 100 g of naphthalene/kg of toluene. Figure 6. Spectrum of light scattered at 25" from water (lower curve) and a solution of 300 g of glycerol/kg of water.
Figure 4. Spectrum of light scattered a t 25' from benzene (lower curve) and a solution of 200 g of naphthalene/kg of benzene. Figure 7. Spectrum of light scattered a t 25" from ethylene glycol (lower curve) and a solution of 15 g of naphthalene/kg of ethylene glycol.
Figure 5. Spectrum of light scattered at 25' from carbon tetrachloride (lower curve) and a solution of 100 g of naphthalene/kg of carbon tetrachloride.
ments. However, our method of following the concentration dependence of J minimizes these sources of error, since, to the degree to which they were constant from one solution to the next, they were cancelled out. By the same token we did not try to identify the true base line of our Brillouin spectra and correct for overlap of adjacent peaks. Rather, we drew our base line through the minimum points on the trace between successive orders and divided the central peak from the Brillouin peaks by vertical lines a t the minimum where they overlap. The areas marked off were then measured with a planimeter. The test of this method was that it produces intensity ratios in reasonable agreement with previously published values. This was indeed the case even for carbon tetrachloride where the internalrelaxation component is broad enough to produce severe Volume 76, Number 18 December 1968
GEORGE A. MILLERAND CHINGS. LEE
4648 Table I : Experimental Intensity Ratios, J
wt of solute (g)/ wt of sol-
CiHe-CiaHa
CaHa-CioHe
CCId-CloHa
Solvent-solute --H&CsH80-
-
-C2HaOz-CioH-
(25O)
(25")
(25O)
40
250
250
50'
0 5 10 15 20 25 40 50 60 75 80
0.432
0.890
0.752
0.140
0.140
0.297 0.705 1.136 1.630
0.206
100
0.836
vent (kg)
0.884
0.982 1.091
0.515 1.070
1.383
0.644
0.700
1.143 0.724
150 200 250 300
1.594 1,238 1.325
1.809
1.560
1.940 2.210 2.440
1.016 1.567 1.855 1.988 2.330
~~
Table II : Solvent Parameters
Liquid
Toluene Benzene Carbon tetrachloride Water Water Ethylene glycol Ethylene glycol
R
25 25 25 4 25 25 50
38 25" 32. 52b 31.490 18.101d 17.993d 35.6* 37.1* I
5.51' 6.27' 5.74f 0. 1717d 0. 97Sd 2.572' 2.659'
0.36O 0.43' 0.45' 0 0.0106d 0. 140h 0. 183h
U. 015'
0.046' 0.028' (-
-1
-0. 158d
0.35 0.38 0.42 3.71 X IO-' 0.0145
a J. S. Burlew, J. Amer. Chem. SOC., 62,696 (1940). G. D. Oliver, M. Eaton, and H. M. Huffman, ibid., 70, 1502 (1948). c J. F. G. Hicks, J. G. Hooley, and C. C. Stephenson, ibid., 66, 1064 (1944). Reference 10. G. C. Parks and K. K. Kelly, J. Amer. C h a . SOC.,47,2089 (1925); M. B. Neumann and I. A. Kurlyankin, J. Ga. C h a . USSR,2,317 (1932); I. B. Rabinovich and P. N. Nikolaev, Dokl. Akad. Nauk SSSR,142,1335 (1962). Measured in this laboratory a t 6328 d. ' Values a t 20" from H. Z. Cummins and R. W. Gammon, J. C h a . Phys., 44,2785 (1966); the temperature variation is not more than O.2%/deg. Calculated from 1 = TabB2/C, using data from various sources: temperature dependence of density, "International Critical Tables," Vol. 111, McGraw-Hill Book Co., Inc., New York, N. Y., 1933, p 112; sound velocity, ref 9; and heat capacity, footnote e. Reference 2. Calculated from data in ref 10 using eq 6.
'
-
'
overlap with Brillouin components. For ethylene glycol the Brillouin components of successive orders overlapped so much that it was necessary to measure the area of two Brillouin peaks combined. The experimental intensity ratios (Table I) were plotted against ~ 1 x 2(Figures 8 and 9) in accordance with eq 4. The coefficient B was calculated from the limiting slope, as estimated visually. The quantities needed to calculate the limiting value of K in dilute solutions (KO)are listed in Tables I1 and 111. Also, values of B = ( J - Jo)/Kozlzz were calculated for each solution and extrapolated to zero concentration for comparison. (See Table IV.) We have included these latter plots for the aqueous solutions (Figures 10 and 11) to show that the dust problem was rather severe in this case. For the other systems studied the values The Journal of Physical Chemistry
of B obtained by the two kinds of plots agreed to better than 5%. Table I1 includes the intensity ratio, R, calculated from thermodynamic theory. For the first three solvents, we used values of z determined by Coumou, Mackor, and Hijmans, which are based on values of &/dT and &/bP at 5460 A. For water we were able to use values of these derivatives at the laser wavelength. We were unable to calculate R for ethylene glycol for lack of a value of bn/bP. Table I11 lists our experimental values of A (ie., Jo) and B and the calculated values of B . For the nonaqueous systems, B was calculated by eq 7. Toluene behaved very nearly like a normal liquid. With carbon tetrachloride there is a 10% discrepancy between the calculated and measured values of B. The overlap
BRILLOUIN SPECTRA OF DILUTESOLUTIONS AND
THE
LANDAU-PLACZEK FORMULA
4649
Table I11 : Solution Parameters Temp, Solvent
Solute
"C
bn/bz14
KO
Toluene Benzene Carbon tetrachloride Water Water Ethylene glycol Ethylene glycol
Naphthalene Naphthalene Naphthalene
25 25 25
0.1502 0.1768 0.2217
16.10 14.68 26.6
G1y cero1 Glycerol Naphthalene Naphthalene
4 25 25 50
0.611 0.580 0.400 0.410
1.502 X 106 3.586 X 10'
a
489
499
x , x I a 103
'
Figure 9. Ex&s intensity ratio of solutions a t 25".
Measured in this laboratory a t 6328 1.
Table IV : Intensity Ratio Coefficients Solventsolute
B
Temp, ---ExptlOC A
25 25 25 4 25
0.432 0.890 0.752
25 C2HsOz-CioH8 CZHEOZ-CIOH~ 50
0.297 0.208
C7Ha-CioHs CsHe-CioHs CClrCioHs HnO-CsHsOs HnO-CsHsOs
B
(calcd)
0.37 0.52 0.47
0.37 0.52 0.52 3.71 x 10-4 0.0145
4.1 x 1 0 - 4 " 0. 0138,b 0. OlSoC 0.37 0.20
1 0
I
*02
x2
.04
I
Figure 10. Intensity ratio coefficient for the concentration term of dilute aqueous glycerol at 25'.
0 16d 0. l s d I
a From the zero intercept in Figure 11. * From the limiting slope in Figure 8. from the zero intercept in Figure 10. d Assuming R = y 1.
-
glycerol in H,O
x
0
naphthalene
-3
in
0.I XI X P
Figure 8. Excess intensity ratio of solutions at 25'.
between the central and side peaks was greatest here because of the broadness of the relaxation component. By our method, some of the intensity of this component was included in the side peaks, and, unless this apportionment was the same at all concentrations, a systematic error may have been introduced. Benzene, on the other hand, fit quite well our treatment of relaxatioh, and in our benzene spectra there was very little overlap. For water we assumed no relaxation; i.e., A = B = R. The agreement with experiment is quite good, al-
no*
xp
.04
Figure 11. Intensity ratio coefficient for the concentration term of dilute aqueolts glycerol a t 4".
though we were not able to reduce the dust problem to insignificance. It must be admitted that, at 4",K is very concentration dependent, since small amounts of glycerol destroy the density maximum and greatly increase &/ZIT. In fact we estimate that K decreases by one order of magnitude over the concentration range considered. However, in the product B K , the term &/dT is much less dominant and we could use KO to get a fictitious value of B which varied slowly with concentration and extrapolated readily to the true value at infinite dilution. Our data, nevertheless, are limited, because with our experimental setup we had difficulty operating at 4". To obtain an estimated value of B for ethylene glycol we assumed that R wa.9 equal to y - 1. At 50" we got normal nonviscous behavior with A E B E y 1. At 25" the concentration fluctuation term was enhanced even more than the entropy fluctuation term; i.e., B >A > 7 1. Our treatment of relaxation appears to be inapplicable to viscous solutions.
-
-
Volume 78, Number 19 December 1068
GEORGE A. MILLERAND CHINGS. LEE
4650
Conclusions The intensity ratio of a real binary system follows the general form
J
A
normally in very dilute solutions. If we let c be, say, grams of polymer per kilogram of solvent, then eq 4 becomes
+ BKRT~l/(dpz)/d~z)
Activity coefficients or the excess free energy can be extracted from the concentration dependence of dp2/ ax2. To obtain the latter from experimental values of J, one must know the concentration dependence of the coefficients A , B , and K . For a normal solution these may be obtained from a few subsidiary experiments.' If either component of the solution deviates significantly from the LP formula, some assumptions must be made about A and B . Each system will present its own problems, but some generalizations are possible a t this point. The role of A is minimized by choosing systems with large excess scattering. Conversely, one can study the concentration dependence of A if the excess scattering is so small that any reasonable estimate of B will suffice. In a solution of two miscible liquids, at least one of which undergoes internal relaxation, B will probably not be much different from y - 1. The limiting values of B may be determined experimentally, and the concentration dependence of y - 1 will serve as a good guide to the concentration dependence of B . I n aqueous solutions A is negligibly small and the limiting value of B may be calculated. The important terms in the concentration dependence of B are perhaps CY and bn/bT, and it may be permissible to assume that pT and dn/bP are concentration independent. The determination of the molecular weight (M2) of polymers in solutions by light scattering is carried out
The Journal of Phyeical Chmktru
J
=
Jo
+ BK'M~c
where
and Cp' is the heat capacity of 1 kg of solvent. Since the determination of M2 is directly affected by any uncertainty in the limiting value of B , it would be advisable to measure B using several solutes of known molecular weight. These need not be previously characterized polymers but, rather, simple compounds of low molecular weight like the ones used in the present study. It has been tacitly assumed here that it is not generally practicable to separate the relaxation peak from the rest of the Brillouin spectrum, since this requires a fortuitous relaxation time plus higher resolution and a perfectly aligned interferometer. Indeed such a separation has been achieved with carbon tetrachloride and greatly simplifies the analysis, because addition of the relaxation intensity to that of the side peaks instead of the central peak gives a value of J equal to y - 1.6 Treated thusly, internally relaxing solutions would become normal ones.
Acknowledgment. This research was supported by a grant from the National Science Foundation (GP4027).
