A critical mixture of isobutyric acid and water was quenched from the one-phase region into the two-phase region and, after the spinodal ring was well developed, a reverse quench returned the system to the one-phase region. Light-scattering measurements for this process exhibit a clearly nondiffusive relaxation which, at least for early times after the quench reversal, is in good agreement with Ruiz\u27s scheme for the coupling of velocity and composition fluctuations

Easwar, Nalini

Goldburg, W. I.

Joshua, M.

Maher, James V.

English

Smith College: Smith ScholarWorks

Smith ScholarWorks Physics: Faculty Publications Physics 12-20-1982 Evidence for Coupling of Velocity and Composition Fluctuations in a Binary Liquid Mixture James V. Maher University of Pittsburgh Nalini Easwar University of Pittsburgh, neaswar@smith.edu W. I. Goldburg University of Pittsburgh M. Joshua University of Pittsburgh Follow this and additional works at: https://scholarworks.smith.edu/phy_facpubs  Part of the Physics Commons Recommended Citation Maher, James V.; Easwar, Nalini; Goldburg, W. I.; and Joshua, M., "Evidence for Coupling of Velocity and Composition Fluctuations in a Binary Liquid Mixture" (1982). Physics: Faculty Publications, Smith College, Northampton, MA. https://scholarworks.smith.edu/phy_facpubs/37 This Article has been accepted for inclusion in Physics: Faculty Publications by an authorized administrator of Smith ScholarWorks. For more information, please contact scholarworks@smith.edu VOLUME 49, NUMBER 25 PHYSICAL REVIEW LETTERS 20 DECEMBER 1982G. E. Gdowski and B.J. Madix, Surf. Sci. 115, 524(1982); H. Pfnur et al ., Chem. Phys. Lett. 59, 481(1978); J. L. Taylor and W. H. Weinberg, Surf. Sci. 78,259 (1978); D. I. Hagen et al. , Surf. Sci. 57, 632 {1976);C. Kohrt and H. Gomer, Surf. Sci. 24, 77 (1971).G. S. De et al. , Phys. Rev. B 21, 3256 (1980); B.J.Garrison. et al. , J. Chem. Phys. 67, 4317 (1977); J. E.Adams and J. D. Doll, J. Chem. Phys. 74, 1467 (1981);E. K. Grimmelmann et al. , J. Chem. Phys. 74, 5300(1981); J. D. Doll, J. Chem. Phys. 68, 3158 (1978);J. L. Beeby, Surf. Sci. 114, 118 (1982).S. A. Adelman and J. D. Doll, J. Chem. Phys. 61,4242 (1974); B.J. Garrison and S. A. Adelman, Surf.Sci. 66, 253 (1977).J. C. Tully, J. Chem. Phys. 73, 1975 {1980), andAnn. Bev. Phys. Chem. 31, 319 (1980).H. A. Kramers, Physica (Utrecht) 7, 284 (1940);S. Chandrasekhar, Bev. Mod. Phys. 15, 1 (1943).J. N. Allison and W. A. Goddard, III, Surf. Sci. 115,553 (1982).S. Glasstone et al. , The Theory of Rate Pyooesses(McGraw-Hill, New York, 1941), p. 347.Evitlence for Coupling of Velocity and Composition Fluctuations in a Binary Liquiti MixtureJ. V. Maher, N. Easwar, W. I. Goldburg, and M. JoshuaDePartment of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260(Received 30 August 1982)A critical mixture of isobutyric acid and water was quenched from the one-phaseregion into the two-phase region and, after the spinodal ring was well developed, areverse quench returned the system to the one-phase region. Light-scattering meas-urements for this process exhibit a clearly nondiffusive relaxation which, at least forearly times after the quench reversal, is in good agreement with Ruiz's scheme for thecoupling of velocity and composition fluctuations.PACS numbers: 64.60.Ht, 64.70.JaIn a recent series of papers Ruiz and Nelson"have discussed the properties of "active" turbu-lent mixing of two fluids; i.e., mixing which isobservably affected by coupling between velocityand composition fluctuations. As a simple spe-cial case of velocity-composition coupling, Ruizhas performed an approximate calculation for thebehavior of the relaxation of very large composi-tion fluctuations under conditions where the massdiffusion coefficient, D, becomes sufficientlysmall that active coupling to the velocity field cancompete with diffusive relaxation.In this Letter we report the results of an experi-ment designed to look for active coupling effectsnear the critical point in a binary fluid mixture.In each measurement a critical mixture of isobu-tyric acid and water, initially at equilibrium attemperature T, in the one-phase region, wasquenched to temperature T, in the two-phaseregion (at time t= -t,), allowed to remain at T,until the spinodal light scattering maximumreached a wave number 4 -2000 cm ', and thenabruptly returned to T, (at time t =0). Underthese circumstances, Ruiz's primary prediction'is thatink (t) = 2ln + C„,(t),2Dvwhere k (t) is the time-dependent wave numberof the maximum in the light-scattering angulardistribution, A is a constant of the motion, t = 0is the time of return to the one-phase region, vis the kinematic viscosity, and n is the coeffi-cient of the composition-gradient-squared termin the Landau-Ginzburg Hamiltonian. C „,(t) isthe global strength of the concentration fluctua-tions, Pq(t); i.e. ,A quantitative test of Ruiz's prediction requiresa good estimate of the pa.rameter R —= uC„, (t = 0)/2Dv. One renormalization-group calculation, 4which uses an Ornstein- Zernike form for thesusceptibility of the order parameter, gives R=B(e,/e, )' where B is of order unity, e, =(T,—T,)/T„e., =(T, —T, )/T„and P ~s. This re-sult is readily obtained on use of n = e'/r p,where p is the mass density and g is the order-parameter susceptibility. Taking g=c ~/k, T,D(T,) =kBT/6nvp $(T,) Iwhere the correlationlength g(T) =(3.57 A)e ' ' for isobutyric acid+water], ' C „,(t = 0) = c,', and v = 2P gives B = 3v,An alternative renormalization-group calcula-tion of R yields a result which is similar in tem-perature dependence and in order of magnitude,Qc 1982 The American Physical SocietyQQLUME 49, NUMBER 25 PHYSICAL REVIEW LETTERS 20 DECEMBER 1982viz. , R = 5o(T,) $(T,)g(T,)/k, T, where the surfacetension is o=k~T/$'. Using the latter estimateof R we can recast Eq. (1) ask (0) e2 ' c„,(t)k. (t) ~, c...(o) ' (2)where a=2.8.The measurements were performed by directinga, laser beam (A. =6328 A) through a reentrant cellof thickness 1 mm and reflecting the emerginglight from a mirror into a photomultiplier. Themirror was rotated through an angular range of+10 about the orientation which directed the cen-tral beam into the photomultiplier. In this waythe photomultiplier was used to measure angulardistributions. Each angular distribution meas-urement required 1 sec and successive measure-ments were separated by integral multiples of 4sec. The laser output and the turbidity were con-tinuously measured by directing the small frac-tions of the beam reflected from thin glass slides(fixed at the input and output points of the cell'stemperature bath) onto a laser intensity meterand a photodiode, respectively. The quenchesand their reversals were accomplished not bychanging temperature from T, to T, and back butrather by abruptly varying the pressure appliedto the liquid mixture. Since it is known' thatchanges in pressure in isobutyric acid-water mix-tures change the critical temperature T, by -57mK/atm and T hardly at all, these pressurechanges are equivalent to changing the tempera-ture. In the discussion below we shall consis-tently use the temperature notation defined above.Signals from the photomultiplier and photodiodewere digitized and used to construct angular dis-tributions proportional to the structure functions(k, t) -=&I q-(t)l').The temperature control on the sample's waterbath kept temperature fluctuations within + 0.2mK over periods of hours. Compromise wasnecessary to keep fractional temperature uncer-tainties as small as possible and yet stay closeenough to T, to keep D very small (D = 2.5lx 10 'x @0 6 cm'/sec). ' We have chosen to make meas-urements with T, —T, =2, 4 and 6 mK; thus inthe one-phase region, the equilibrium towardwhich our reverse-quenched systems relax ischaracterized by e, -lo ' and D-5xlo cm /sec.The behavior of the system was investigated forquench depths T, —T, =2, 4, 6, and 8 mK. Thetime scales for spinodal decomposition are knownto be quite different for such different quenchdepths; in each case we measured angular dis-tributions as the spinodal light-scattering ringdeveloped and reversed the quench when k de-creased to =2000 cm '. Up to the reversal ofquench, the system always showed the behaviorpreviously reported for spinodal decompositionin this system. ' In particular the structure fac-tor was of the scaling form S(k, t) =k '(t)E(k/k„(t)), where I'(x) is independent of time andquench depth.To achieve reproducible results in this experi-ment, it was necessary to stir the mixturethoroughly and then wait for a prolonged intervalprior to the quench. The requisite minimumequilibration time was -25 min, which is muchlonger than the viscous damping time, T„(takingthe viscosity of the mixture to be -10 ' poise,we estimate r, to be 2 min) and also much longerthan the equilibration time required for normalspinodal decomposition experiments. In deter-mining that our results do not depend on thistime, we have found that results obtained withour stirring procedure agree well with resultsobtained after stirring the system 20 mK aboveT„and then leaving the system at T, for hoursbefore quenching. It would be very desirable tounderstand why sloppier stirring methods are in-adequate for the present experiment when theywere perfectly adequate for spinodal decomposi-tion studies.The results of the present experiment are ingood agreement with the only previously availablemeasurements, ' in which Wong and Knobler re-ported that, after reversal of quench, the dimen-sionless wave number for maximum scattering,q =k„g(e), and the dimensionless time, T =D/g't,were related by q -T '. We find a slight con-cavity to a plot of lnq vs lnT but the concavecurves are never far from a straight line of slope- -0.15.Our main experimental result is shown in Fig.1 where ln[k (0)/k (t)] is plotted againstz=(~,/~, )"[1—c ...(t)/c ...(o)];we have approximated C „,(t) with the productof k„(t) and the turbidity, T(t). For the earlystages [T(t) & 0.3T(0)], all measurements fallon a straight line of slope -0.2. At late timeseach curve shows a dramatic departure fromlinearity; note that increasing time correspondsto increasing I". We have not found an empiricalrelation to make both early- and late-time datafall on the same curve.The early-time data appear to support Ruiz's1851VOLUME 49, NUMBER 25 PHYSICAL REVIEW LETTERS 20 DECEMBER 19823E 2OEcj COOJI I I&I0.4 0.8 l. 2 2.0 2.4FIG. 1. Each curve represents the result of a set of measurements at the indicated combination (T T ~ TC t—T2). The error bars show typical standard deviations. I' is defined in the text.predictions' strongly. The empirical slope R=0.2 is slightly more than an order of magnitudesmaller than the renormalization-group predic-tion of 2.8, but this is not at all a shocking dis-agreement. The discrepancy may arise fromRuiz's dropping factors of order unity in his anal-ysis. ' The departures from scaling at late timesare more difficult to assess. It is clear that Eq.(2) was never intended to be used at very latetimes; no noise term is included in Ruiz s driv-ing force so that at the very least his results can-not be applied near equilibrium where thermalfluctuations are important. On the other hand,with the empirical value R = 0.2, Eq. (2) placesvery severe limits on the ratio q (0)/q (t); i.e.,this ratio, which clearly must diverge at verylate times since q (~) -0, is not allowed by Eq.(2) to exceed -1.2 if e,/e, -1.0. The departuresfrom Eq. (2) tend to appear when the turbidityhas dropped to -30% of its value at t=0. Thisoccurs at dimensionless times T = 50-200 depend-ing on ~,. Values of the slope R for short-timedata and values of the dimensionless time, 7.,and the turbidity ratio, T(v)/T(0), measured atthe break from Eq. (2) are listed in Table I forvarious values of &y and &,. While there is asignificant time interval within which Eq. (2) issuccessful, the dramatic "late-time" departuresoccur long before the system approaches equilib-rium even though the crucial dimensionless ra-tio is v/DR 10'.An alternative way to view the results of thepresent experiment is shown in Fig. 2 where sev-eral light-scattering angular distributions aredisplayed for the case T, —T, =2 mK, T —T2=6mK. It is instructive to compare the measure-ments in Fig. 2 with calculations based on theassumption that the composition fluctuations builtup before t = 0 will relax diffusively. Such calcu-TABLE I. For each combination of Ti-T, and T,-T 2, column 3 lists the empirical values of the slope,p, extracted by fitting the short-time data by Eq. (2).Columns 4 and 5 list the dimensionless time and theturbidity ratio when the data depart from linearity.T( TcmKTc T2mK Slope, a 7 (&)/T (0)0.41+ 0.050.23+ 0.030.15+ 0.040.20 + 0.030.24 + 0.060.17+ 0.040.18+ 0.050.29 + 0.040.18+ 0.050.21+ 0.055060401401201402002202000.390.270.380.270.260.310.260.310.220.33lations are shown along with the measured angu-lar distributions. In this simple diffusion modelthe t = 0 angular distribution is accepted as aninitial condition and the scattering at each wavenumber is assumed to decrease as exp(-2&k't)where D was adjusted to fit the data from 1000to 2000 cm ' in an ensemble of similar measure-ments. The data show clear departures from thesimple diffusion picture. While these departuresare not unexpected, the calculation allows one toclassify the observed phenomena. First, the de-cay at wave numbers in the range 0.6 k„(0)(k& k (0) is roughly exponential but sufficientlyslow that the diffusion constant used in the calcu-lation had to be adjusted to 40% of the accepted'equilibrium value. Second, the decay at smallerwave numbers [=0.4 k (0)] is initially much fast-er than would be predicted by the accepted equilib-rium value while later the scattering at thesewave numbers almost entirely ceases to change.1852VOLUME 49, NUMBER 25 PHYSICAL REVIK%' LETTERS 20 DECEMBER 19821OJDC3IIIIvv I~~tI l» III III(vIpv150seen from Fig. 2.In summary, it is clear that the experimentpresented herein departs significantly from theexpectations of simple diffusion models and there-by shows evidence for an importance of hydro-dynamic effects in the relaxation of large com-position fluctuations. These observed effectsappear to be substantially in agreement with thepredictions of Ruiz' until the diameter of thelight-scattering ring has shrunk to -80U/o of itsinitial (t= 0) value and the turbidity is -30/o ofits initial value.We have benefitted greatly from discussionsand correspondence with D. Jasnow, D. Nelson,T. Ohta, R. Petschek, and R. Ruiz and from thetechnical skills of R. Tobin. This work was sup-ported by the National Science Foundation.iIIIIv IhIIv 'I2602000 0 2000t (cm ')FIG. 2. Angular distributions for light scatteringjust as the quench is reversed and for several latertimes. Solid curves represent measured data whilethe dashed curves result from the diffusion caIculationsdescribed in the text. In the range of k where theforward beam saturates the photomultiplier, the dataare not shown.The rapid early decay and subsequent stability atthese small wave numbers keeps the spinodalring from disappearing at late times as can be'R. Ruiz and D. R. Nelson, Phys. Rev. A 23, 3224(1981}.R. Ruiz and D. R. Nelson, Phys. Rev. A 24, 2727(1981).3R. Ruiz, to be published (two papers}.R. Petschek, private communication.5E. Qulari et al. , J. Chem. Phys. 56, 6169 (1972).6T. Ohta, private communication.VN. C. ong and C. M. Knobler, J. Chem. Phys. 69,725 (1978).B. Chu et al. , J. Am. Chem. Soc. 90, 3402 (1968),and Phys. Rev. A 7, 353 (1973).Y. C. Chou and W. I. Goldhurg, Phys. Bev. A 23,858 (1981};C. M. Knobler and ¹-C.Wong, J. Phys.Chem. 85, 1972 (1981).Interference Phenomena in the Theory of Daumas-Herold Domain WallsG. KirczenowDePa&ment of Physics, Boston University, Boston, Massachusetts 03215(Received 20 September 1982)A theoretical study of dislocations in intercalation compounds reveals that the local de-formation of the host near an intercalant island exhibits damped spatial oscillations, re-sulting in strong interference effects in the elastic energy of Daumas-Herold domainwalls. Explicit calculations for stage 2 predict that a large energy barrier must be over-come before two domains with matching intercalant layers can merge, while staggereddomains should bind to each other. Experimental implications are discussed.PACS numbers: 61.70.-r, 64.60.My, 64.80.GdThere has been for many years a widespreadinterest in polycrystall. ine systems and domainstructures of every sort because of their perva-siveness in nature and their practical importance,and because of their fundamental appeal as a rela-tively well defined exampl. e of metastability. Yetthe list of eases for which both tractable andrealistic models of domain boundaries could be1982 The American Physical Society 1853

Evidence for Coupling of Velocity and Composition Fluctuations in a Binary Liquid Mixture

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Evidence for Coupling of Velocity and Composition Fluctuations in a Binary Liquid Mixture

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