We present experimental results on the critical exponent ψEKE describing the divergence of the Kerr constant of binary liquid mixtures near the critical consolute point. We show that the measured value of ψEKE agrees with the theoretical prediction only if the measurement is performed with a mixture of two liquids presenting a small mismatch in the dielectric constant, and that the measured ψEKE grows as the dielectric constant mismatch increases. Such findings are consistent with a recent model which assumes that the elongation of critical fluctations along the direction of the electric field can become so strong that fluctuations in the direction perpendicular to the electric field may cross over from Ising to mean-field behavior

A. Onuki

B. Scaife

D. Beysens

D. Wirtz

E. Fredericq

J. Goulon

J. S. Høye

J. Ziolo

Mario Giardini

R. Piazza

S. J. Rzoska

Sylwester J. Rzoska

T. Bellini

Vittorio Degiorgio

W. Pyzuk

English

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Relationship between dielectric properties and critical behavior of the electric birefringence in binary liquid mixtures

Rzoska, Sylwester J.

Degiorgio, Vittorio

Giardini, Mario Ettore

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryRzoska, Sylwester J. and Degiorgio, Vittorio and Giardini, Mario Ettore (1994) Relationship betweendielectric properties and critical behavior of the electric birefringence in binary liquid mixtures.Physical Review E, 49 (6). pp. 5234-5237. ISSN 1539-3755Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/PHYSICAL REVIEW E VOLUME 49, NUMBER 6 JUNE 1994Relationship between dielectric properties and critical behavior of the electric birefringencein binary liquid mixturesSylwester J. Rzoska, Vittorio Degiorgio, and Mario GiardiniDipartimento di Elettronica, Uniuersita di PaUia, Uia Abbiategrasso 209, 27100 Pavia, Italy(Received 21 January 1994)We present experimental results on the critical exponent QE&E describing the divergence of the Kerrconstant of binary liquid mixtures near the critical consolute point. We show that the measured value ofits„E agrees with the theoretical prediction only if the measurement is performed with a mixture of twoliquids presenting a small mismatch in the dielectric constant, and that the measured PsKs grows as thedielectric constant mismatch increases. Such findings are consistent with a recent model which assumesthat the elongation of critical fluctations along the direction of the electric field can become so strongthat fluctuations in the direction perpendicular to the electric field may cross over from Ising to mean-field behavior.PACS number(s): 64.70.Ja, 64.60.Ht, 78.20.FmINTRODUCTIONIn the last few years several groups have studied effectsdue to the application of a static electric Geld E in the vi-cinity of the critical consolute point of binary liquid mix-tures. Two different effects have been experimentally in-vestigated: the electrooptic Kerr efFect (EKE) [1—9], andthe nonlinear dielectric efFect (NDE) [10—12]. In theformer case, one measures the Kerr constant B, definedasE2 E2(2)where c, denotes the dielectric constant componentIIparallel to the applied field, and c is the dielectric con-stant in the limit of zero static electric field. The dielec-tric constants are measured at radio frequencies.Under the assumption that E is suf6ciently small, bothB and D take values which are independent of E and aredetermined only by the equilibrium properties of the sys-tem.Both quantities B and D exhibit a marked increase onapproaching the critical consolute point. Theoreticalmodels [13—15] predict a power-law divergence of Dand 8 as functions of the reduced temperaturet = i T T, i /T„T, being —the critical temperature:B„D,= A~at (3)Present address: Institute of Physics, Silesian University, ul.Uniwersytecka 4, 40-007 Katowice, Poland.1 An 1 nil niB=—E2 g E2where hn=nil—n j is the anisotropy in refractive indexdue to the field E, and A, is the wavelength of the lightbeam which is used to probe the induced birefringence.In the latter case, the experiment gives the quantitywhere the index c denotes the critical part of the giveneffect, A~ and AD are critical amplitudes, and the criticalexponent is given bytP =d v —2y -=3(0.63)—2(1.24) =0.59,where d is the dimensionality of the space, and v and yare the critical exponents for the correlation length andosmotic compressibility, respectively.However, the available experimental results give valuesof PExE in the range 0.60—0.88 [1—8] and of QNDEaround 0.4 [10—12]. The value of the critical exponent isthe only, but fundamental, point of disagreement betweentheory and experiments. A possible explanation of thedisagreement was proposed in Ref. [16] by assuming thatthe elongation of critical fluctuations along the directionof E can become so strong that fluctuations in the direc-tion perpendicular to E may crossover from Ising tomean-field behaviors. This would yield the following crit-ical exponents: QNDz=0. 37 and /EKE=0. 85. In princi-ple, if the model is correct, one should be able to see thecrossover from QExE=0. 59 to QExE=0. 85 with a specificbinary mixture by simply increasing the value of E. Inpractice, as discussed below, it appears to be easier tochange the magnitude of the induced anisotropy, at fixedreduced temperature and fixed field, by acting on thedielectric constant mismatch between the two com-ponents of the liquid mixture.In this article we present experimental results for theelectro-optic Kerr effect. We show that the measuredvalue of /EKE agrees with the theoretical prediction onlyif the measurement is performed with a mixture of twoliquids presenting a small mismatch in the dielectric con-stants, and that the measured JETE grows as the dielec-tric constant mismatch increases. All available data indi-cate the existence of a clear correlation between the mea-sured /EKE»d the quantity (ei —sz) je, where s, and szare the dielectric constants of the two separate com-ponents of the liquid mixture, and e is the dielectric con-stant of the mixture. Such findings are consistent withthe model discussed in Ref. [16].1063-651X/94/49(6)/5234(4)/$06. 00 49 5234 1994 The American Physical Society49 RELATIONSHIP BETWEEN DIELECTRIC PROPERTIES AND. . . 5235EXPERIMENTAL RESULTSA general description of electric birefringence can befound in Ref [17]. Our measurement of the Kerr con-stant as a function of the reduced temperature was per-formed with an apparatus similar to that used in previousexperiments [5—8]. We have investigated three binarymixtures, all possessing an upper consolute point: (a)(+)-1-phenylethanol and dodecane, (b) 2-phenylethanoland decane, and (c) n-butylbenzene and propylene car-bonate. The three mixtures present very different valuesfor the mismatch in dielectric constant: (a) s, —e2=-5, (b)s, —e2—= 10, and (c) e, —e2-50. The critical temperatureT, and the critical mass fraction of the first componentx, were established by the visual method. The obtainedvalues are (a) T, =21.6'C, x, =0.545; (b) T, =44.9'C,x, =0.51, and (c) T, =18.9'C, x, =0.41.The sensitivity of EKE and NDE to molecular proper-ties of liquids makes the estimation of the noncriticalbackground effect rather difficult. In this work the non-critical background contribution was determined by themethod employed in previous NDE [18] and EKE [4]studies. By exploiting the fact that in the three investi-gated solutions —(i )-1-phenylethanol and dodecane, 2-phenylethanol and decane, and n-butylbenzene and pro-pylene carbonate —the background is due mainly to thefirst component, the background contribution was ob-tained from electric birefringence measurements per-formed with reference solutions containing carbon tetra-chloride as the second component, each prepared withthe same volume fraction of Kerr-active component as inthe critical solution. The choice of carbon tetrachlorideis motivated by the fact that by itself it presents a negligi-ble Kerr efFect, and, at the same time, the reference solu-tions used are not close to phase separation points.In Fig. 1 we show the critical Kerr constant measure-ments, performed as function of the reduced temperature,in mixtures (a) and (c). The straight lines represent a fitwith the power-law behavior predicted by Eq. (3). Thebest-fit exponents are QExE=0. 58+0.04 for system (a),and QE~E=O. 83%0.04 for system (c). For the case of sys-tem (b}, we show in Fig. 2 the measured values of the to-tal Kerr constant. An interesting feature of this solutionis the negative sign of the noncritical contribution, due tothe fact that the permanent dipole moment is not parallelto the optical polarizability vector in the 2-phenylethanolmolecule. The full line through the experimental pointsis the best-fit curve describing the superposition of criti-cal and background effects. The obtained value of thecritical exponent is /EKE=0. 64+0.04.10~;PQI1O-4~ I10 3~ I10 2I10-1FIG. 1. The critical part of the Kerr constant of the liquidmixtures (+)-1-phenylethanol and dodecane (open squares), andpropylene carbonate and n-butylbenzene (full squares), mea-sured as a function of the reduced temperature t =(T—T, )/T, .40-30-III20—10-studies of the anisotropy of the structure factor in a criti-cal solution in presence of a static electric field usingsmall-angle light scattering [9].The situation described in this paper bears some simi-larity to studies of critical solutions under shear flow,where the elongation of critical fluctuations due to theaction of the velocity gradient influences the criticalproperties [20]. It has indeed been shown that, in the re-gime of low shear rates S~& 1 (S and ~ are shear rate andrelaxation time, respectively), the results are described bythe Ising values of the critical exponents (y=1.24 andP=0.33, respectively); whereas, for S~) 1, mean-fleld ex-ponents are found (y = 1 and P=0.5, respectively}.The discussion of Ref. [16] assumes that the elongationof critical fluctuations along the direction of E can be-come so strong that fluctuations in the direction perpen-dicular to E may crossover from Ising to mean-Seldbehaviors. The crossover condition can be expressed as:g~~—gi=g, where g is the correlation range in absence ofE, and gl and gi are, respectively, the correlation rangesparallel and perpendicular to E. Taking into accountDISCUSSIONOn approaching the critical consolute point of a binaryliquid mixture, large-scale fluctuations appear. Criticalfluctuations can be described as isotropic droplets whichcan be anisotropically deformed by an external field. Ifthe two components present a mismatch in the dielectricconstant, then the deformation (elongation) can be in-duced by the electric field [17,19]. Convincing experi-mental evidence of such a deformation is given by recent-10-I65I45I50I I I55 60 70T ('C)FIG. 2. The Kerr constant of the liquid mixture 2-phenylethanol and decane, measured as a function of the tem-perature. Continuous lines show the noncritical backgroundcontribution and the best-fitting curve describing the superposi-tion of critical and background contribution.5236 RZOSKA, DEGIORGIO, AND GIARDINI0.90-j 6,70.80—0.70—0.60-n10.500I40I80(s, -s, )I120 160FIG. 3. The measured critical exponent |(EKs plotted as afunction of the quantity (c&—e,2) /c for seven critical solu-tions: (1) (+)-1-phenylethanol-dodecane; (2) 2-phenylethanol-dodecane; (3) phenylacetonitrile-cyclopentane; (4) nitro-benzene-hexane; (5) propylene carbonate-n-butylbenzene; (6)lutidine-water; and (7) butoxyethanol-water.(E,—e~) (n, n2)—~ EKEEn(5)that, at a given temperature, the size of fluctuations fol-lows a probability distribution, and that the inducedshape anisotropy grows with the size of fluctuations, themodel of Ref. [16] assumes that the eflective critical ex-ponent QEitE is determined by the sum of two contribu-tions; the first, due to small fluctuations, which is de-scribed by the nonclassical exponent QE~E=O. 59; and thesecond, due to fluctuations large enough to present thecrossover to mean-field behavior, which is described bythe exponent QE~E=0.85. The actual value of QE~i,would therefore be determined by the relative weight ofthe two families of fluctuations, and should be an increas-ing function of the mean elongation of critical fluctua-tions.In principle, if the model is correct, one should be ableto see the crossover from $8KE=0.59 to 0.85 simply byincreasing the value of E. In practice, there is a limitedexperimental window in which the measurements arefeasible, because low values of E do not give enough sig-nal, and large values of E may induce heating of the solu-tion. In general, the degree of elongation of critical fluc-tuations in the field E is proportional to the osmoticcompressibility and to the excess of dielectric constant offluctuations over the dielectric constant of the solution.Both factors depend markedly on the distance from thecritical point. According to the theoretical models[13—15], the amplitude A EKE is given bywhere n, and n2 are the indices of refraction of the twocomponents, and n that of the mixture. The amplitudeAERIE is the product of two terms, the first (s, —c,2) /sdescribing the strength of the electric forces which in-duce the deformation of the correlation volume, and thesecond (ni —nz) /n the optical anisotropy due to theinduced deformation. By considering the structure of Eq.(5), it appears to be easy to change the magnitude of theinduced anisotropy, at fixed reduced temperature andfixed field, by acting on the quantity (s& —e2) /s.It is noteworthy that the value of JETE measured with(+)-I-phenyiethanol-dodecane is smaller than any previ-ous measurement of 1(E~E, and to our knowledgerepresents the first measurement of QEitE which is inagreement with the theoretical predictions. The fact thatsuch a result is obtained with the binary mixture possess-ing the smallest value of the quantity (s, —s2) /e is con-sistent with the hypothesis discussed in Ref. [16].In Fig. 3 we have plotted the measured 1(axE as a func-tion of (si —s2) /s for the three systems studied in thiswork. We have also added data taken from Refs. [4], [6],and [7]. Figure 3 shows that a correlation exists betweenthe value of (s, —ez) /e. and the measured critical ex-ponent QE&E. For mixtures presenting large valuesof ( s, —e2) /s, such as water-lutidine [6], water-butoxyethanol [7] and nonionic micellar solutions [5(b)],the measured t/rnKE agrees, within experimental errors,with the asymptotic value 0.85 predicted in Ref. [16].As a conclusion, we have shown in this paper, by usingthe present accurate data and also some literature data,that the efFective critical exponent for the divergence ofthe Kerr constant in a critical binary mixture is a mono-tonically increasing function of the mismatch in thedielectric constant of the two components. When themismatch is small, the data are in agreement with the"Ising" exponent predicted by theory. When themismatch is large, the measured critical exponent takes avalue consistent with the hypothesis that the field-induced anisotropy may become so strong to cause acrossover from Ising to mean-field behaviors.ACKNOWLEDGMENTSWe thank T. Bellini, R. Piazza, and W. Pyzuk for use-ful discussions. We acknowledge financial support fromthe Italian Ministry for University and Research(MURST 40% funds). One of the authors (S.J.R.) wassupported by a grant from the International Center forTheoretica1 Physics, Trieste, Italy, under the TrainingProgramme in Italian Laboratories.[1]W. Pyzuk and H. Majgier-Baranowska, Chem. Phys. Lett.63, 184 (1979).[2] W. Pyzuk, H. Majgier-Baranowska, and J. Ziolo, Chem.Phys. 59, 111 (1980).[3]W. Pyzuk, Chem. Phys. 50, 281 (1980).[4] W. Pyzuk, Europhys. Lett. 17, 339 (1992).[5] p. Degiorgio and R. Piazza, (a) Phys. Rev. Lett. 55, 288(1985); (b) Prog. Colloid. Polymer, Sci. 73, 76 {1987}.[6]R. Piazza, T. Bellini, V. Degiorgio, R. E. Goldstein, S.Leibler, and R. Lipovsky, Phys. Rev. B 38, 7223 (1988).49 RELATIONSHIP BETWEEN DIELECTRIC PROPERTIES AND. . . 5237[7]T. Bellini and V. Degiorgio, Phys. Rev. B 39, 7263 (1989).[8] S. J. Rzoska, V. Degiorgio, T. Bellini, and R. Piazza,Phys. Rev. E 49, 3093 (1994).[9]D. Wirtz, K. Berend, and G. G. 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Relationship between dielectric properties and critical behavior of the electric birefringence in binary liquid mixtures

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