Viscoelastic Relaxation in Supercooled LiquidsBY A. J. BARLOW AND J. LAMBDept. of Electrical Engineering, The University of Glasgow, Glasgow, W.2Received 2nd Jamary, 1967A study has been made of the viscoelastic behaviour of a range of supercooled liquids each ofwhich has a steady-flow viscosity -q fitting the equation,where TO is the reference temperature at which no free-volume is available for molecular translationalmotion. For all the liquids investigated, the limiting shear rigidity modulus G , varies with tempera-ture according to the relationship,lnq = A'+B'/(T-To),- = - + 5 (T-To),G , Go GIwhere c10 is the coefficient of thermal expansion and Go the shear modulus at temperature TO, and G1is a shear modulus associated with the weakening of the intermolecular force field due to an increasein free-volume with increasing temperature.The complete linear viscoelastic behaviour can becalculated from a knowledge of the dependence upon temperature of the density, steady-flow viscosityand limiting shear rigidity modulus. Details are given of the new liquid model on which thesepredictions are based, together with ample experimental confirmation.Viscoelastic relaxation in liquids can be observed by using alternating shear stressin which the frequency is an experimental variable. At sufficiently low frequencies ofthe applied shear, all liquids behave as Newtonian, the response being entirely viscouswith a phase difference of 90" between the sinusoidal oscillations of shear stress andthe resulting strain.At sufficiently high frequencies it is possible to achieve conditionssuch that the molecules of the liquid are sensibly unable to flow during the short timeperiod of the rapid stress alternations. Stress and strain are then in phase and theliquid behaves essentially as an elastic solid with a shear rigidity modulus G,, whichis of the order 1010 dyne/cm2. At intermediate frequencies the phase angle betweenstress and strain changes progressively with increasing frequency from 90" at lowfrequencies to zero at high frequencies and this constitutes the viscoelastic relaxationregion.In practice, by use of the principle of time-temperature superposition, it is possibleto employ temperature as an equivalent and supplementary variable to frequency,thereby enabling measurements to be made over the complete region of viscoelasticrelaxation. This implies that the variation with temperature of the limiting para-meters q, the steady-flow viscosity, and G,, the shear rigidity modulus, must be deter-mined over the temperature range employed.Although the frequency range whichcan be covered by the experimental systems developed in this laboratory extends fromapproximately 104 to 109 c/sec 1-5 better accuracy is obtained for the techniquesoperating above 106c/sec provided that the liquid has a steady-flow viscosity qexceeding 10 cpoise. For this reason it is generally necessary to work with liquidswhich can be appreciably supercooled.The Arrhenius equation fails to describe the relatively steep temperature depen-dence of the viscosity of most liquids having viscosities above about 0.1 poise and is22224 VISCOELASTIC RELAXATION I N SUPERCOOLED LIQUIDScompletely inapplicable to supercooled liquids.The viscosities of such liquids canbe described by the Doolittle free-volume equation 6 :where A and B are constants, Vf is the free-volume in the liquid and VO is the occupiedvolume. For the supercooled liquids considered here, the density is a linear functionof temperature and under this condition, eqn. (1) can be modified to givean equation identical with one proposed empirically by Tammann and Hesse 7 andsubsequently established on theoretical grounds.899The temperature TO, which can be calculated from viscosity data for the liquid overthe non-Arrhenius region, is considered as the fundamental reference temperature forall transport and relaxation processes and, at this temperature, no free-volume wouldbe available for molecular translational motion in the equilibrium liquid.However,in practice, the slowness of molecular motion below the glass transition temperatureT’,. precludes the attainment of an equilibrium liquid at TO( < T,) in an experiment offinite duration. For present purposes, T, is evaluated as that temperature at whichthe steady-flow viscosity reaches 1013 poise, this temperature being close to values ofTg recorded by differential thermal analysis.10Complete information on the details of the work reported here, is given in twoforthcoming papers.119 12 The purpose of the present paper is to review the presentstate of knowledge in this field and to outline the information which can be obtainedfrom measurements of viscoelastic relaxation.In q = A + B( V,/ V,), (1)In q = A’+ B’/(T-T& (2)VISCOELASTIC PARAMETERSIt is convenient to describe the Maxwell model for viscoelastic behaviour althoughsubsequently a new model is presented which supercedes previous analyses based onthe assumption of a summation of Maxwell elements.Consider a simple shear in theX- Y plane with transverse displacement u, caused by the shear stress T, and a shearstrain S = du/dz. In order to account for viscoelastic behaviour, Maxwell postulatedthe stress-strain relationship,where z is the viscolastic relaxation time.T+zaT/dt = qdS/dt, (3)For alternating shear stress of the form exp ( j u t ) we replace aJat byjw and obtainAt low frequencies where oz g 1 , T = juqS, and the response is entirely viscous. Athigh frequencies such that uz $1, T = (q/z)S, and the response is purely elastic.Theelastic modulus G , is therefore equal to (y/z), or q = z . G,.T(l + j o z ) = juqS. (4)Defining the ratio T/S as the complex shear modulus, G* = G’+jG”, we obtainG’ = G,co2z2/(1 +w2r2); G = G,uz/(l +w2z2), ( 5 )(6)and the dynamic viscosity,q’ = G/CO = q/(l +u2x2).In practice the quantities G’, G“ or q’ are not measured directly but are obtainedfrom the components RL and XL of the shear mechanical impedance ZJ, which areexperimentally determined. This impedance is defined by ZL = RL +~XL = - T’gA .J . BARLOW AND J . LAMB 225and solution of the wave equation, which is given by equating the net force on avolume element to the product of mass and acceleration, gives the general relationship :HenceFor a Newtonian liquid, RI, = XL = (nfqp)* ; G’ = 0 and q‘ = q, the steady flowviscosity.The predicted behaviour obtained from the Maxwell model for a single relaxationtime, z = q/G,, is shown in fig. 1 ; the viscosity ratio q’/q falls from 0.9 to 0.1 inZ t = pG*. (7)(8) pG’ = Rl-XL and pG” = 2R,X,.1.0-0 . 5 -10log ( q / G , )FIG. l.-Calculated behaviour of the viscoelastic relaxation parameters according to the Maxwellmodel with a single relaxation time, T = q/G,.approximately one decade of frequency.In order to represent experimental resultsin the normalized form shown for the Maxwell model in fig. 1 it is necessary to knowthe variation with temperature of the density p, the steady-flow viscosity q, and theshear rigidity modulus G,. Both p and q can be measured by conventional methodsbut the determination of the temperature dependence of G, requires shear-wavemeasurements to be made at temperatures above Ts in the region where (cq/G,) ismuch greater than unity.EXPERIMENTALThe liquids were obtained from various suppliers and were samples of high purity. Allsamples were dried before use in a desiccator containing P2O5. Densities were measuredto an accuracy of f0.1 % by the standard procedure of weighing a calibrated flask ofknown volume.Viscosities were measured with calibrated suspended-level viscometersaccording to B.S.S. no. 188. Values obtained are estimated to be accurate to f0-5 %226below 100 poise and to f l % above 100 poise. Measurements of the shear mechanicalimpedance were carried out by using techniques described previously.l-6$11 The estimatedaccuracy of measurement for RL and XL is &500 dyne sec/cm3, these being the c.g.s.units in which the shear mechanical impedance is expressed.VISCOELASTIC RELAXATION I N SUPERCOOLED LIQUIDSRESULTSDEPENDENCE O N TEMPERATURE OF SHEAR RIGIDITY MODULUS, G,The shear rigidity modulus G, can only be studied by making measurements inthe region where the liquid is exhibiting purely elastic behaviour, i.e., at relativelyhigh frequencies and in the temperature range between the glass transition temperatureand that at which viscoelastic relaxation becomes significant. In this region thereactive component XL of the shear mechanical impedance tends to zero and hencefrom eqn.(8), G, -Rz/p.T"CFIG. 2.-Measwed values of p/R: against temperature in the elastic region, showing the linearvariation of 1/G, with temperature ; f = 30 Mc/sec. A, sec-butyl benzene ; +, tetra-(2-ethylhexyl) silicate ; v , squalane ; [XI , 6,6,11 , 1 1-tetramethyl hexadecane : 0 , squalene ; 0, tri-chloroethylMeasurements made under these conditions on a wide range of different liquidshave all been found to give results which within experimental accuracy obey therelations hip,(9)Values of To are obtained from viscosity data, a0 is the thermal expansion coefficientand Go the modulus of the close-packed state at temperature TO and GI is a constantmodulus for a given liquid.(10)and with the valid approximation that Go 9 GI, eqn.(9) can be written as(1 1-1The modulus GI is therefore associated with the weakening of the intermolecularforce field due to an increase of free-volume. Typical plots of p/RZ against T areshown in fig. 2 : values of G1 are obtained from the slope of the straight line and Gois the intercept at T = To. Behaviour represented by eqn. (9) has been confirmed forover 20 liquids measured.phosphate ; 0, tri (m-tolyl) phosphate. f = 450 Mc/sec ; A , sec-butyl benzene.(1IGaJ = ( W O ) + (ao/WT- To).Since the density can be expressed asP = POL1 - ao(T- Toll,VIG, = (VOIG,) + (VflGl)A . J .BARLOW AND J . LAMB 221RELAXATIONAL BEHAVIOURHaving determined the dependence of G , upon temperature according to eqn. (9)-extrapolated values for G, can be obtained for somewhat higher temperatures cover,ing the relaxation region. It is then possible to normalize the measured values of RLlog10 (q/G,)FIG. 3.-Normalized plots of R&G,)) and of X&G,)) against loglo (oq/G,) for squalene (a), andfor 6,6,11,11 -tetramethyl hexadecane (m). The full curves are calculated from the liquid modelusing eqn. (14) and (15). The corresponding predictions from the Maxwell model are shown by thedashed curves.FIG. 4.-Normalized plots of R~l(pG,)t and XL/(~G,)+ against loglo (o-q/G,). The curves are cal-culated according to the liquid model from eqn.(14) and (15); v, squalane; 0, trichloroethylphosphate ; 0, tri(m-tolyl) phosphate ; x , tris(2-ethyl hexyl) phosphate ; + , tetra (2-ethyl hexyl)silicate ; A, bis (m-phenoxy phenoxy) phenyl ether ; 0, di(isobuty1) phthalate ; 0, di (n-butyl)phthalate ; m, iso-propyl benzene ; 0, n-propyl benzene ; A, sec-butyl benzene228 VISCOELASTIC RELAXATION I N SUPERCOOLED LIQUIDSand XL by dividing by (pG,)* and plotting the quantities R-&G,)+ and X~l(pG,)tagainst (wq/G,). Experimental results for squalene and for 6,6,11,ll-tetramethylhexadecane," represented in this manner, are shown in fig. 3 and compared with thecorresponding behaviour according to the Maxwell model. Clearly the observedrelaxation extends over a wider range of frequency than does that given by theMaxwell model and moreover, a single curve for RL/(~G,)) can be drawn through theexperimental results for the two liquids.The same curve has been found to fit theresults for a number of other liquids 11, 12 (fig. 4), the curves drawn on fig. 3 and 4being calculated from the liquid model, which will now be described.VISCOELASTIC RELAXATION MODEL POR LIQUIDSThe results given in fig. (3) and (4) show that the viscoelastic properties of theseliquids can be represented by two standard curves, one for RL/(~G,)* and one forXL/(~G,)+ against loglo(oq/G,). The curves drawn on these figures have beencalculated from a new model which describes the viscoelastic behaviour of liquidswhich have a viscosity-temperature variation according to eqn.(1).At sufficiently low frequencies where elastic effects are negligible, all liquids behaveas Newtonian fluids with a shear mechanical impedance,l3At sufficiently high frequencies, the liquid behaves as an elastic solid with a shearrigidity modulus G,, and the corresponding shear mechanical impedance for anassumed loss-free system is given byZN = R N + j X N = (1 +j)(n fqp)'. (12)Zs = Rs = (pG,)'+. (13)It is found that the behaviour of the liquids studied is represented by a parallelcombination of ZN and ZS, leading to the following expressions for the shear mechan-ical impedance of the liquid l/ZL = ( 1 / 2 ~ + I/&) from which are derived correspond-ing relationships for the components for the shear modulus G* and of the complexcompliance J*( = l/G*).Z, = RL + j X ,G,J' = 1 +(wq/2G,)-*,G, J" = (G,loq) + (oq/2G,)-*.* This compound was prepared by R.M. Schilsa, Monsanto Chemical Co., St. Louis, U.S.AA . J . BARLOW AND 3 . LAMB 229The curves of fig. 3 and 4 have been plotted by calculation from eqn. (14) and (15) ;there is no disposable parameter.DISCUSSIONThe work reviewed here establishes that the viscoelastic relaxation curves for awide range of pure liquids are identical when represented in terms of normalizedco-ordinates. The standard curves can be predicted from a simple model involvingonly density, viscosity and the limiting shear modulus of the liquid without anydisposable parameter.Many different types of liquid have been studied, includingsimple benzene derivatives, phosphate silicate and phthalate esters, polyphenylethers, and relatively long-chain hydrocarbons. Moreover, this general pattern ofviscoelastic relaxation is not confined to liquids which are normally regarded as super-cooled. The same behaviour has been found for a poly-l-butene liquid 14 of lowmolecular weight, this being monodisperse material with eight repeat units permolecule.It follows that viscoelastic relaxation is governed only by physical variables anddoes not depend directly on the type of molecule involved. All of the liquids whichhave been found to conform to the proposed model have a viscosity-temperaturedependence given by eqn. (1).It is therefore reasonable to suppose that this form offree-volume equation for viscosity must hold for the model to apply.At low frequencies (coq/G, 4 1) the variation with frequency of the componentsof the complex shear modulus according to the Maxwell model are given from eqn.(5) as G’/G, cc o2 and G”/G, cc o. However, according to the liquid model describedhere the corresponding variations given by eqn. (16) and (17) are G’/G,ccco~ andG”/G, cc m. Hence measurements of the frequency-dependence of G‘/G, in thisregion of the spectrum indicate which model is applicable.It has been suggested that liquids, which have a viscosity given by the Arrheniusequationshould exhibit viscoelastic properties which conform to the simple Maxwell model.Unfortunately most such liquids have values of viscosity which are too low for theirviscoelastic relaxation to be studied by existing experimental techniques.However,certain molten compounds are exceptional in that they have viscosities described byeqn. (20) with values exceeding 100 poise. Two such liquids are molten zinc chlorideand molten boron trioxide, and in each of these a single relaxation process has beenfound, the measured behaviour conforming to the Maxwell m0del.15~ 16Since the liquids reviewed here have viscosities which at higher temperatures aredescribed by the Arrhenius equation, it follows that if this approach is of generalvalidity then a single relaxation time would be expected if the viscoelastic relaxationcould be studied in this region.With decreasing temperature a gradual transitionfrom the Maxwell model to that described by the present liquid model would beexpected to occur as the degree of co-operative molecular motion increases. Evidencefor such a transition may be obtainable from measurements of the relaxation regionmade at high frequency atnd at temperatures close to the lower limit of Arrheniusbe haviou r .Recent measurements on mixtures of two liquids, each component of whichconforms to the model for supercooled liquids have shown that a small departure fromliquid purity, amounting to only a few percent, can give rise to significant deviationsof the values of &/(pG,)+ and XL/(~G,)+ from the standard curves over the visco-elastic relaxation region. However, large amounts of impurity do not give grossIn q = A+B/T, (20230 VISCOELASTIC RELAXATION IN SUPERCOOLED LIQUIDSdeviations, and preliminary evidence indicates that for certain critical mixture ratiosthe relaxational behaviour reverts to the standard curves.This suggests that forsuch compositions the mixture behaves as a homogeneous liquid as far as the pro-pagation of shear waves is concerned.This research has been supported in part by a contract with the National Engineer-ing Laboratory, Ministry of Technology. Assistance has also been given by ShellResearch Ltd. and by Imperial Chemical Industries Ltd. Many helpful discussionshave been held with Dr. G. Harrison and Dr. A. J. Matheson, and thanks are due toMiss A. Erginsav for her diligent experimental work.1 A. J. Barlow and J. Lamb, Proc. Roy. SOC. A, 1959,253,52.2 A. J. Barlow, G. Harrison, J. Richter, H. Seguin and J. Lamb, Lab. Pract., 1961, 10, 786.3 A. J. Barlow, G. Harrison and J. Lamb, Proc. Roy. SOC. A, 1964, 282, 228.4 J. Lamb and J. Richter, Proc. Roy. SOC. A, 1966, 293,479.5 J. Lamb and J. Richter, J. Acoust. SOC. Amer., 1967, to be published.6A. J. Barlow, J. Lamb and A. J. Matheson, Proc. Roy. SOC. A, 1966, 292, 322.7 G. Tammann and W. Hesse, 2. anorg. Chem., 1926,156,245.8 M. H. Cohen and D. Turnbull, J. Chem. Physics, 1959,31,1164.9 S . F. Kumar, Physics and Chem. of Glasses, 1963, 4, 106.10 M. R. Carpenter, D. B. Davies and A. J. Matheson, J. Chem. Physics, 1967, to be published.11 A. J. Barlow, J. Lamb, A. J. Matheson, P. R. K. L. Padmini and J. Richter, Proc. Roy. SOC. A,12 A. J. Barlow, A. Erginsav and J. Lamb, Proc. Roy. SOC. A , 1967,298, 481.13 W. P. Mason, Piezoelectric Crystals and their Applications to Ultrasonics, chap. 14 (Van Nostrand,14 A. J. Barlow, R. A. Dickie and J. Lamb, Viscoelastic Relaxation in Poly-I-Butene Liquids of Low15 G. J. Gruber and T. A. Litovitz, J. Chem. Physics, 1964, 40, 13.16 P. Macedo and T. A. Litovitz, Physics and Chem. of Classes, 1965, 6, 69.1967,298,467.Princeton, 1950).Molecular Weight, 1967, Proc. Roy. SOC. A, 1967, in press