Ultrasonic Relaxation of Rotational-Isomeric Equilibriain Polymer SolutionsBY €3.4. BAUER, H. HASSLER AND M. IMMENDORFERI. Physikalisches Institut der Universitat Stuttgart, GermanyReceived 1 8th February, I970The sound absorption per wavelength p in solutions of polystyrene in benzene and carbontetrachloride deviates from proportionality with frequency in the low MHz range. This could beaccounted for with good agreement by an additional, Debye-shaped excess absorption. Variousmodels for the relevant relaxation process are discussed; the most probable one is the thermalrelaxation of a rotational-isomeric equilibrium within the polymer chain, consisting of a rotationof a single monomer unit. Under that assumption, from the temperature dependence of positionand amount of the absorption maximum the following data of the rotational potential were calculated :separation of the equilibrium energies AH = 0.9 kcal/mol, activation energy AHlz = 7.5 kcal/mol,frequency factor v = 1 .4 ~ 1OI2 s-l.Sound absorption in liquids is caused by several different mechanisms : (i) Theshear part of the longitudinal wave gives rise to viscous absorption. (ii) Both thetemperature and the pressure part of the wave may force internal equilibria to altertheir position periodically ; relaxation phemonena cause a phase lag with subsequentrelaxational sound absorption. When the equilibria are sensitive to a temperaturechange only due to AH # 0 (i.e.? the volume effect AV of the reaction is negligible)one speaks of thermal relaxation. The reserve case (the heat of reaction AH beingneglected) is coinpressional relaxation.The absorption per wavelength, p = a& of the viscous part (assuming frequency-independent viscosity) as well as of any relaxational part much below the relaxationrate (271fmax = l/z) is proportional to the frequency f.Hence, any deviation of pfrom that proportionality indicates a relaxation process, the absorption of which isaccounted for in the most simple case by an additive Debye absorption maximum :The relaxation rate contains kinetic information, and the absorption maximumthermodynamic information about the processes considered (see, e.g., Lamb,Bauer 2).Sound absorption measurements of various authors in polymer solutions (e.g.,PST, PMMA in benzene, carbon tetrachloride, toluene) 3-7 show a deviation fromp - f in the low MHz region.While the results of Cerf and co-workers 3-5 couldnot be accounted for by any relaxation process (their p exhibits one or two suddensteps instead of a Debye behaviour), our own measurements could be fitted by anequation of the form (1). do not contradict theseresults.Pexcess = 2 l U I n a x ~ z / ( l + m2z2>. (1)Recent studies of NomuraEXPERIMENTALWe used a parallel-path pulse apparatus with broad band transducers of the Sell type.At low frequencies (1-5 MHz) diffraction effects counterfeit additional absorption. Wecorrected for that by the method of B a s 8 The validity of this procedure was verified by23H . - J . BAUER, H. HASSLER AND M.IMMENDORFER 239measurements in the pure solvents. The non-resonant transducers enabled us to determinethe absorption values for a large number of adjacent frequencies, and hence a better and moreaccurate least-squares approximation could be calculated.RESULTSFig. 1 shows a typical result, separated into a single Debye absorption part and acontribution p -f due to viscothermal and vibrational relaxational absorption. Theuse of CC14 as solvent enhances the Debye part relative to the other (fig. 2). The twosets of points correspond to two samples with broad (PST IIIc, 0, technical grade,Mz2200 000) and narrow (PST 430 000, 0, 2 = 430 000) distribution of molecularweight. Apparently there is only a small influence of the chain length on the processbeing observed.-1050 50.20.1f [MHzIFIG.1 .-Absorption per wavelength and excess absorption.To associate the excess absorption with an internal mechanism the concentrationand temperature dependence has been investigated (fig. 3). The mean Debye curvesresulting from the least-squares separation method, exhibit a proportionality betweenexcess absorption and concentration. An increase in the temperature shifts theabsorption maxima to higher frequencies and decreases the maximum absorptionslightly.DISCUSSIONThe excess absorption could also be a viscous one, if the additional viscosityinduced in the solution by the polymer molecules relaxed in the frequency regio24d) ULTRASONIC RELAXATIONAFIG. 2.-Dependence onmolecular weight.10 2 0 I 2 5f [MHzl1052m40FIG. 3.-Dependence on X Iconcentration and temperature. H0.50.20.I I II 2 5 10 20f [MHzH.-J. BAUER, H . HASSLER AND M. IMMENDORFER 241observed. Applying Rouse's theory on the experimental static viscosity of thesolution, one obtains a much smaller relaxation rate l/z = 2nfmax ; in addition onewould not expect a single but a multiple relaxation.Other possible processes within the solvent may be excluded from the discussionas well: non-associated solvents do not show any structural relaxation; and theconcentration dependence locates the process on the chain molecule itself.The most simple equilibria on the chain molecule which could relax, in thefrequency region considered, are of the rotational-isomeric type (as in low-molecularweight substances).Transitions between only two different conformations wouldgive rise to a single relaxation process as observed experimentally. The independenceof the results from the chain length demonstrates that the process responsible occurson a small section of the molecule.MAXIMUM ABSORPTIONThe maximum absorption per wavelength of a single equilibrium (for not toolarge relaxation strengths) is~ ( y - 1 ) AH aexp(-AHIRT) AV Cpp " a x E ? ~ R ( ~ ) (l+aexp(-AH/RT))2(1-T m) '(y = ratio of specific heats, Cp = molar heat capacity, 8 = thermal expansioncoefficient, a = degeneracy).Since it contains the thermal (AH) as well as the compressional (AV) part of therelaxation, no simple statement can be made about either of those quantities. Asis common in the instance of rotational-isomeric relaxation (but questioned recentlyby Crook and Wyn-Jones lo) we neglect the volume effect AV and obtain(3)a exp (-AH/RT) - SC,(l+aexp(-AH/RT))'-R'' T IspFIG.4.Rotational potential.The temperature dependence of p,,, is mainly determined by that of the relaxing heatcapacity SC,, which possesses a maximum, beyond which SC, decreases with increasingT as is found experimentally. The experimental prnax-values and their temperaturegradient are in agreement with eqn (3) if we assume (a) a = 2, i.e., the upper state istwofold degenerate as in the rotational potential of fig. 4 ; (b) the process occursonce in every monomer unit; and take (c) AH = (0.940.1) kcal/mol monomer ;remembering (d) that we neglected the volume effect.An estimation shows that amere compressional relaxation (AH = 0) with a AV/V of only 2 % would yield thesame ,urnax. For the final distinction between the assumed thermal and compressiona242 ULTRASONIC RELAXATIONrelaxation very accurate measurements in solvents with strongly differing heatcapacities are necessary.RELAXATION RATEIf for thermal relaxation and small relaxation strengths, the transition rate fromthe upper to the lower state was calculated from the positions of the maxima by(4)Fig. 5 shows the Arrhenius plot of the results with a frequency factor of 1.4 x 10l2 s-,and an activation energy, AHz1 = 6.6 kcal/mol monomer. The total potentialdifference is AHlz = AH,, +AH = 7.5 kcal/mol monomer.Tz 1 = 2?tfmax/( 1 + cc exp ( - AH/RT)).I oa5 - IIrn u 42.5I o 7530 333 280 O K66 kcal /MOI \ 1.1 x I O ” ~ e R ’0 10 percent PST Ill in C C i 45 percent PST 630 000 in CCI2.5 3 3 51031~FIG.5.-Arrhenius plot of the transition rate rzl.TWO-STATE MODELThere is a difference between rotational isomeric transitions in low-molecular-weight substances and polymers; a rotation about a single bond in the latter forcesone part of the molecule to have a large movement through the solvent that is in-compatible with the high transition rate. Therefore a second rotation about anadjacent bond in the opposite sense is very probable since it then affects only a smalllocal part of the molecule.The regular conformation of isotactic polystyrene is a3-1 helix with the potential sequence gtgtgtgtgt. A change of one gauche position ginto a trans t or minus gauche 3 with an opposite rotation of the next but one bondcould give the twofold upper level :g t g t g t g t g t g tg t g t g t t t g t g tg t g t g t g t t t g tThe effect of the rotations about these two bonds is the rotation of a whole monomerunit, cf. fig. 6 FIG. 6.-Regular (1) and rotated (2) conformation.[To face page 242HA. BAUER, H . HASSLER AND M. IMMENDORFER 243J. Lamb in PhysicaZ Acoustics, W. P. Mason ed., (Academic Press, New York 1965), vol. IIA,chap. 4.H.-J. Bauer in PhysicaZ Acoustics, W. P. Mason ed. (Academic Press, New York, 1965), vol.IIA, chap. 2.R. Cerf, R. Zana and S. Candau, Compt. rend., 1961,252,681 ; 1961,252,2229 ; 1962,254,1061.C. Tondre and R. Cerf, J. Chim. Phys., 1968,65,1105..I. Lang, J. Chim. Phys., 1969, 66,88.H. Nomura, S. Kato and Y. Miyahara, Nippon Kagaku Zasshi, 1967,88,502 ; 1968, 89, 149 ;1969, 90, 250. ' H.-J. Bauer and H. Hassler, Proc. 6th Int. Congr. Acoustics, (Tokyo), paper J-5-11, and KoZloid-Z.,1969,230,194.R. Bass, J. Acoust. SOC. Amer., 1958,30,602.P. E. Rouse, J. Chem. Phys., 1952,21,1272.K. R. Crook and E. Wyn-Jones, J. Chem. Phys., 1969,50.3445