Autocorrelation function of scattered light for a binary fluid near the critical mixing point

Autocorrelation function of scattered light for a binary fluid near the critical mixing point


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V o l u m e 9, number 3


November 1973

AUTOCORRELATION FUNCTION OF SCATTERED LIGHT FOR A BINARY FLUID NEAR THE CRITICAL MIXING POINT ~ Cherif BENDJABALLAH* University o/Maryland, Department of Physics and Astronomy, College Park, Maryland 20742, USA Received 10 August 1973

Using a time-to-amplitude conversion method the second order correlation function of the central component in light scattered by a binary fluid was measured very close to the critical point. At k/~ '-. 7 deviation from an exponential decay of the concentration fluctuations turns out to be smaller than one percent.

Questions concerning the structure of the central component of light scattered by a binary mixture near its critical point are of active current interest. Several works emph)ying different techniques such as an optical mixing spectroscopy technique [1,2] and photon counting [3] have been already devoted to this subject. The main object of this letter is to report a measuren~ent o f the autocorielation f u n c t i o n of light scattered from c,,mcentratmn lluclualions in a critical mixture using a time-to-amplitude conversion me dtod. The scattered light is relatively weak so that the method used here is very appropriate and we were able to determine the correlation flmction form with satisfactory accuracy. The experimental syslem consists essentially of two photonmltipliers with a converter operating as a delayed coincidence circuit and a pulse height analyzer [4.5[. The sample and experimental arrangement for stabilizing lhe temperature were the same used by Chang et ai. [6]. The sample could be at a constant temperature to within a few millidegtees over a period of several hot, rs. l h e critical temperature "I"c of the sample was ¢' This research was supported in part by NASA Grant NGR 21-002-285, Office of Naval Research Contract N00014-67A-0239-0014 and NSF Grant GII 33577. * Permanent addre,~s: l.aboratoire d'l"tude des Phdnom&res Al('atoire% B/itiment 210. Univcrsit6 dc Paris XI, 91405 Orsay, France,

determined with a precision better than 1 m°C and for our sample (3-methylpentane-nitroethane) to be

26.377°C. A cw gas laser (Spectra Physics model 120) was scattered by the sample held at a temperalure siightly above Tc ( T - Tc ~ l re°C) at the angle close to 70 °. The beam was collected and passed through a small aperture and was split into two beams of approximately equal intensities which fell on the two fast photomultipLiers (PM) RCA C31024 as shown m fig. 1.


[Laser SP m~el 120]- . . . . . . . . . . . . .0. .


~3m BS~

¢ ,mm ~ /"



• •.


. ~ tram



Fig. 1. F.xperimental setup for measuring correlation %notion. Sample IS) was mounted in a tcmperature controlled thermostat 10102 ); BS = beam splitter; I'M1, PM2 = photomultipliers: SP = standardizer pulse; Sc = scaler;8 = delay;TAC = time to ;unplitude ctmverter; PlIA :: pulse height analyzer.


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The PM's were arranged so that the path distances ~ere~erv nearly equal {25 cm from the bean~ splitter). -Ilw pulses detected from each of the phototubeswere standardized in amplitude and duration witL discriminators {Ortec model 453) and entered the time to amplitude converter {TAC, Ortec model 437 modified to increase the conversion range). The pulses from one of the photomultiptiers then went directly to tlte TAC while the pulses from the other were first delayed by 400psec. This delay allows us to move the center of the correlation function in the center of the TAC ranin order to study the symmetry of the whole curve and also to calibrate the pulse mean count rate. The output of the TAC was directed to a 256 channels INS 600t pulse height analyzer. In fact the TAC instrument measures the probabilib' distribution w(6 t6 + r) of the time intervals bet~een photoelectric pulses according to [7] wt.8 ,5 + r) = (~l~2/(1>) X (I(6~1(8 + r ) e -~2





where a I and a 2 are constants of proportionality inc,,rporatmg the quantum efficiency' of the deteclors, their aeas. and the square of the time resolution of the TAC: ( I ) is the mean value of the stationary scattered [email protected] intensity l(u). At a low mean count rate +?,+ g.



llu) d u ' ~ 1.



so eq. (I) becomes

~t'18i,5 + r ) -



length (range of field fluctuations in the sample). Thus, the scattered field E(t) is gaussian, anU cq. (3)may be rewritten as

q(r) = (1(8) I(6 + r)> =
q: (r) = (I) exp(-- r/rc) ,


where r e is the coherence time of the electric',d field. in the critical regime (k~. ~ 1) the data have to be corrected in order to take into account the fact that the viscosity depends on wavenumber and frequency. The non-local effects modify the magnitude of the relaxation time [6, 12, 13] but do not affect file exponential character of I"I (r). The second effect (called retardation effect) theoretically predicted by l'erl and Ferrel! [11, 15] and then by Lo and Kawasaki [ 16] (called memory effect by them) shuuld lead to nonexponentia~ distortion of the autocorrelation function. We have made several very' careful autocortelation function measurements (see for instance fig. 2 in semilogarithmic scale) at T - Tc < 1 m°C and k~ approximately 7 and compared our data with a single exponential function titling. We studied the reduced moments (rk) of the Fl(r ) distribution defined by Tm




where FE (r)is the electrical field autocorrelation function. In the hydrodynamical regime (k/~ ,~ 1, k being the wavelength of the light in the medium) the fluctuations in electrical susceptibility can be considered to be due only to the concentration fluctuations which decay by diffusion process [8, 9], so that

(I> (I(8)1(8 +r)>= -(/-5- rt(r)' (3)

whid~ is prop(n tional to the intensity correlation function l]~r +t,. Therefore. the data stored and displayed in the pulse height analyzer give the form of the correlation fnnction directlv, ttowever, the mean count rate inust be high enough to allow us to accumulate statistically significant data in a reasonable time duration. In our case ~:he scattering volume Vs was still large ,:ompared with the coherence volume Vc defined as I]: = ~3 where ~ is the Orstein -Zernike correlation

November ! 973




where r m is the upper limit in the r range (r m ~. 800 psec). Sinular statistical methods such as cunndants [14], could also be used. Notwidthstanding tile hmg accumulating time of our data, the autocorrelation function of the attenuated laser light has been found constant. At tempera. t u r e f a r f r o m T c ( T Tc = 4 ° C a n d T c T = l ° C ) the auto-correlation function was found to be also constant. The total time resolution (time separation be tween

Volume 9, number 3



' ".'*......,

• - ....) •.'*.... ....;....

•-'.,-,....... ...;..


..',;.... ...





2.. 800

600 ~A sec


Fig. 2. Typical r e d u c e d c o r r e l a t i o n f u n c t i o n m e a s u r e d at 70 ° and p l o t t e d in s e m i l o g a r i t h m i c scale.

November 1973

In the critical regime a non-exponential decay form due to the frequency dependence of the viscosity was expected and estimated to be experimentally detectable but an examination shows that the effect may be smaller than the theoretical prediction [15, 161. As a matter of fact, using, for example, eq. (41) of ref. [ 15] or table 1 of the ref. [16], variation of Ar//r/ (At/ being the frequency dependent part of the viscosity) of about 3% leads to a variation ofAh/h (h = (r2)/ 30 [16] and measurement of l"l(r ) at shorter r in order to have contributions from the r/(r) expansion. Experiment under these conditions will be performed in a future work.

Acknowledgements two adjacent channels) was over 4 tasec. The PM's noise was negligeable and the number of coherence areas was of order unity. The data close to the origin of the time axis and close to zm have been excluced in our computation. tinder these conditions of our experiments we found that the fitting of one single exponential func. tion leads to an agreement between data and exponential decay to about I:: for the first order moment, about 2% for the second order moment, and about 4% for the third order moment, about 2% for the second order moment, and about 4% for the third order moment (see table 1). The differences between the data moments and the calculated moments for long ~'n~, increase for the higher order.

Table 1

= 2.359×10-4sec

(r2~ - - - = 1.66 (r) 2

(r 3 )

~r--~-~ = 3.544

Calculated r m = 800usec

Calculated Tm 2 m ~ c







I wish to thank Professor R.A. Fer~eli and Professor J.V. Sengers for helpful discussions and Dr. R.F. Chang and Mr. H. Burstyn for help during preparation of experiment I would also like to thank Professor C.O. Mley for his hospitality at the University of Maryland.

References I 1] G.B. Benedek, in Polarization, Mati~re et Rayonnement, Livre de Jubil~ en l'hormeur du Professeur A. Kasfler (P.U.F. Paris 1969) p. 49. 121 H.Z. Cummins and H.L. Swinney, in Progress in Optics, Vol. VIII, ed. E. Wolf (North Holland PublishingCo., Amsterdam 1970)p. 133. 13] P.N. Pusey and W.I. Gotdburg. Appl. Phys. Left. 13 (1968) 321• [4] D.B. Scarl, Phys. Rev. 175 (1968) 1661. [5] Ch. Bendjaballah,C.R. Acad.Sc. (Paris) 2;2 (197 l) 1244. 16] R.F. Chang, P.H. Keyes, J.V. Sengers and C.O. Alley. .,rOll. BeE Bun~nges. H~ysik.Chem. 76 (1972) """ 171 F. Davidson and L. Mandel, J. Appl. Phys 39 (1968) 62. f81 R.D. Mountain, Joul. of Research of N.B.S. (USA) 69A {1965) 523. [9] C. Cohen, J, Sutherland and J.M. Deutch, Physics and Chemistry of Liquids 2 ( 1971 ) 213.


] 101 E. Kawasaki and S.M. Lo, Phys. Rev. Lett. 29 flY72) JS. [ Ii 1 K Per1 .Ind R A. I’errell, Ph! s. Rev. Lctt. 29 (1972) 51. { 121 C.C. Lai and S.H. Shen, Phys. Rev. Lett. 29(1972)401. \ 121 B. C’hu, S.F. Lee and \Y. Tscharnuter, Phys. Rev. A7 Il973) 353.

! 141 D.E. Koppel, Jour. Chem. Phys. 57 (I 372) 48 14. [ 151 R. Per1 and R.A. b’errell, Phys. Rev. A6 (I 971) 2358. ] 161 S&f. Lo and K. Kawasaki