Laser Light Scattering by Polymer Gels BY K. L. WUN, G. T. FEKE AND (the late) W. PRINS Dept. of Chemistry, Syracuse University, Syracuse, New York 13210 Received 6th December, 1973 The frequency-integrated, absolute Rayleigh-Debye scattering measured as a function of scattering angle down to lo, contains information about the supramolecular structural order exhibited by gels. For covalently crosslinked, swollen polymer networks this provides a measure for the degree of spatial non-randomness in crosslinking. The non-randomness index for a series of poly(2-hydroxy- ethyl methacrylate) (PHEMA) networks is found to vary systeniatically by a factor of over a hundred depending on the history of the network formation. Intensity-fluctuation spectroscopy at any given scattering angle provides a probe for the local viscoelasticity of a gel without applying an external driving force.Swollen PHEMA networks are found to exhibit a spectrum of long s) relaxation times. In dilute aqueous agarose systems both above and below the sol-+gel transition a similar spectrum of relaxation times is found. During the transition, the autocorrelation function exhibits oscillatory behaviour for several hours. This is attributed to the mass flow taking place during the microphase separation. Upon inducing the sol-tgel transition in agarose solutions by quenching to various temperatures below the phase boundary, the frequency-integrated Rayleigh-Debye scattering is found to vary systematically a thousand fold. A Bragg maximum appears, caused by the regularly spaced spontaneous concentration fluctuations which occur in a nucleation-free, spinodal phase decomposition.Polymer gels are defined as disperse systems of at least one polymer in at least one diluent and exhibiting the properties of an elastic solid. Polymer gels can be formed by copolymerizing bifunctional and multifunctional monomers in the presence of a diluent. A polymer network formed in the absence of diluent, can subsequently be swollen so as to yield a gel. Gels of a similar nature are obtained if already existing polymer chains are covalently linked by a crosslinking agent. A different type of polymer gel is formed by the physical aggregation of existing polymer chains into a three-dimensional structure, usually with the chains in somewhat ordered conforma- tions. The nature of the diluent may be such that this second type of gel structure superposes itself on a covalently linked structure.The intensity of the light scattered by the polymer chains in gels measured as a function of the scattering vector K (IKj = (47r/4 sin 812) provides information about the supramolecular structural order exhibited by the gels. If a laser beam of high monochromaticity is used as the light source, one can at each K-value also perform intensity fluctuation spectroscopy and thus obtain the frequency content of the Rayleigh scattering. This provides information about the dynamics of the scatter- ing entities. In the following, we apply these two light scattering techniques 10 covalently crosslinked swollen polymer networks and to aggregating polymer solu- tions which form thermoreversible gels.QUASI-ELASTIC LIGHT SCATTERING THEORY AND DATA ACQUISITION The light scattering of most gels is predominantly caused by fluctuations in the Only a minor amount of scattering arises froin fluctuations in We, therefore, consider the gels to be describable by an isotropic mean polarizability. local anisotropy. 146K . L. WUN, G. T. FEKE AND W. PRINS 147 pokdrizability, which exhibits fluctuations, q(r, t ) , in space and time. The Rayleigh ratio for polarized incident light in the Rayleigh-Debye approximation is then given by 1 6n4 y(r, z) exp (iK r) exp ( iooz) exp ( - iwz) dr dz (1) -03 where I(K, w) is the scattered intensity at a scattering vector K and frequency w, Ro is the distance from the scattering volume V to the detector, I , the incident intensity and 2, the wavelength of the incident light in uacuo.The space-time correlation function y(r, z) is defined by Integrating over the scattering volume one has with y(r, z) exp (iK.r) dr. Integrating over the frequency one has 16n4 1 +a3 2: 2n -03 -* = -(q2)- J+a T(K, z) exp ( i w o ~ ) exp (- im) dz da, = ( 16n4p:>(?2>r(K, 0). (3) According to the Wiener-Khinchine theorem the spectral power density is equal to the Fourier transform of the time correlation function of the scattered field <E(K, t)E*(M, t +z)). Upon normalization we thus have J T(K, z) exp (iwoz) exp ( - iwz) dz R(K7 4 - -al +03 (E(K, z)E*(K, t + z ) ) -m<@K, f)E*(K, f)> exp (- iwz) dz. (4) Data acquisition is conveniently carried out by counting the anode pulses generated by the photons arriving at the cathode of a square-law detector (photomultiplier).In our instrumentation, a digital autocorrelator and mini computer are employed,2 covering a time domain of 100 ns-1 s or a frequency domain of 10 M Hz to 1 Hz.148 LIGHT SCATTERING BY GELS The pulse counting correlation function is related to the field correlation function as follows : (n(K, t)n(K, t + z)) = (n(K, t))2 + (n(K, t))”r2((n, z) + 6(z)(n(K, t ) ) . Here n(K, t ) denotes the number of pulses accumulated at K and time t. normalization and rejection of the z = 0 channel (shot noise), one has Upon Eqn (5) shows that experimentally the square of the normalized field correlation function (eqn (4)) is obtained.The same instrumentation is used to determine the frequency-integrated Rayleigh ratios given in eqn (3). The number of pulses per lops channel are collected at a scattering vector K and in the K = 0 position for about 10-30 s, stored and averaged in the minicomputer. The Rayleigh ratio then follows from its definition (eqn (1)) : where Q is the solid angle viewed by the detector, S the cross-section of the beam, d the thickness of the sample and V = Sd. Neutral density filters are used to reduce the (n(0, t ) ) count. Corrections for turbidity, reflection and refraction are applied as usual.4 COVALENTLY CROSS-LINKED SWOLLEN POLYMER NETWORKS For a randomly-crosslinked swollen polymer network the frequency-integrated Rayleigh scattering (in excess of that caused by the diluent) derives from the thermally- induced fluctuations in the polymer segment density.These fluctuations can be evaluated if an expression for the free energy of the gel is available. In a good diluent the equilibrium degree of swelling (and thus the average segment density) is deter- mined by the sum of a (negative) free energy of dilution, which drives the swelling and a (positive) network free energy, which limits the swelling. Employing the Flory- Huggins expression for the former and ideal rubber elasticity for the latter, the Rayleigh ratio for polarized incident light is In this result n is the refractive index of the gel, 4 the volume fraction of polymer, zll the molecular volume of the diluent, x the Flory-Huggins interaction parameter, v the number density of network chains,fthe functionality of a crosslink, qo the refer- ence degree of swelling (which is simply related to the degree of dilution at which crosslinking takes place) and ( R b ) is the mean square radius of gyration of all seg- ments belonging to a given crosslink. Values for the network parameters can be obtained by measuring the degree of swelling and the elasticity modulus of the gels.5 In reality, the crosslinking is often non-random, so that the gel will possess spatial variations in the local degree of swelling, which willenhance the light scattering.One can now quite generally describe the scattering in terms of eqn (3) : sin K r 11: Kr 16n4 R(K) = -(q2)r(K, 0) = - ( q 2 ) y(r)-4nr2 dr. 2;: A2K. L . WUN, G . T. FEKE AND W. PRINS 149 In principle, y(r) is extractable from eqn (8) by Fourier inversion but if y(r) is assumed to be a sum of two or more Gaussians : y(r) = C xi exp (-r2/a;), C xi = 1.(9) Eqn (8) can be integrated to yield (10) 16n4 II;: R(K) = -(<r12>[x Xia?n3 exp (- K2a:/4)]. The typical example shown in fig. 1, demonstrates that a plot of log R(K) versus sin2 8/2 down to 8 = 1" can be resolved into two straight lines corresponding to two Gaussians in y(r). A reliable extrapolation to K = 0 can thus be made and the para- meters al, a2, X and <q2) determined. Upon comparing the experimental (non- random, nr) Rayleigh ratio [(R(0)Inr with that theoretically obtained for a randomly (r) crosslinked gel (eqn (7)), a quantitative non-randomness index (NRI) can be assigned 0.01 L ' I I I I I I I I I I 0 1 2 3 4 5 6 7 0 9 10 I 1 12 103 x sin2 el2 FIG.1 .-The logarithm of the Rayleigh ratio of a covalently crosslinked poly(2-hydroxyethylmetha- crylate) gel (sample 805) fully swollen in ethylene glycol, plotted against sin2 8/2. The dashed lines indicate that a resolution into two Gaussians is possible, so that al, a2, Xand <q2> can be obtained. for poly(2-hydroxyethyl methacrylate) and ethylene- glycoldimethacrylate in the presence of varying amounts of ethylene glycol. The gels were measured in equilibrium swelling and were visually clear. Decreasing dilution of the monomer mixture decreases the NRI more than a hundred-fold. Such a decrease is to be expected because less microgel formation takes place at lower dilution.6 Similarly, increased crosslinking leads to a lower NRI because of a more uniform crosslinking.The NRI has an important bearing on the mechanical behaviour of polymer networks because it reflects the existence of regions of widely different crosslinking. Excessive non-random crosslinking will place an excessive Table 1 shows some results150 LIGHT SCATTERING BY GELS 1.0- n 0 0.5 n d W z 0 strain on a small portion of the polymer chains in a mechanically loaded network, so that poor mechanical performance may result. a a a - a - a a . - * a - *.. a * . a a a m * a m o m - * a m I I 1 I I I a * * : a a - ,-arr. - a , TABLE 1 .-EXPERIMENTAL AND THEORETICAL RAYLEIGH RATIOS AND THE NON-RANDOMNESS CRYLATE) NETWORKS, SWOLLEN IN ETHYLENE GLYCOL. THE FIRST TWO DIGITS IN THE SAMPLE THIRD DIGIT INDICATES THE LEVEL OF CROSSLINKING (IN 10-5 MOL ETHYLENE GLYCOL DI- INDEX (NRI) FOR A SERIES OF COVALENTLY CROSSLINKED POLY(2-HYDROXY ETHYL METHA- DESIGNATION INDICATE THE VOLUME PERCENT MONOMER DURING POLYMERIZATION AND THE ETHYLENE GLYCOL DIMETHACRYLATE PER ML MIXTURE).sample 01 x lO4/cm 4 x lO4/crn X <+> [R(0)lnr/cm-l [R(0)lr/cm-l NRI 205 1.97 5.91 0.767 6 . 4 2 ~ lo-" 1.88 1 . 1 9 ~ 10-4 1 . 5 8 ~ 104 405 1.49 2.86 0.504 4 . 6 4 ~ 10-l' 3.30 24.2 x 10-4 1.37 x 103 703 1.59 4.12 0.517 1 . 8 1 ~ 10-l' 3.50 1 4 . 9 ~ 10-4 2.05 x 103 805 1.01 2.78 0.617 6 . 0 1 ~ 10-l' 2.89 228 x 1.26 x lo2 705 1.15 4.54 0.694 3.71 x 10-lo 3.17 21.1 x 1 . 5 0 ~ lo3 710 1.01 3.11 0.717 1 . 1 8 ~ 5.94 480x 1 . 2 3 ~ lo2 The dynamics of a gel structure, i.e., its viscoelasticity as driven by thermal motion, is reflected in the time correlation function of the scattered field (eqn (4)), the square of which can be determined by intensity fluctuation spectroscopy (eqn (5)).One of us has reported on this before and Tanaka, Hocker and Benedek * have recently developed an approximate model for the dynamical behaviour of a randomly- crosslinked swollen polymer network. Their model predicts a single exponential decay : In this result, E is Young's inodulus and y the frictional constant of the gel. The latter is defined by AP/L = 72, if AP/L is a pressure gradient across the gel of length L as a result of which diluent is set in motion with a velocity i. These authors find that the autocorrelation functions of crosslinked poly(acry1amid) hydrogels are indeed single exponential decays within experimental accuracy. The Ely values which they obtain from the light scattering experiment agree well with the macroscopically determined E and y.ray r)/r(K, 0) = exp (-m) = exp [ - (E/y)K2r]. (12)K . L. WUN, G . T . FEKE AND W. PRINS 151 It thus appears that the viscoelasticity of a gel can be measured by means of light scattering without an externally imposed deformation. Since, moreover, the scattered light is collected from a very small volume element (typically cm3), it is the ZocaZ viscoelasticity which is probed. Spatial variations in network dynamics can therefore be investigated simply by moving different portions of the gel into focus. We have investigated several PHEMA gels in ethylene glycol over an angular range of 35" < 6 < 80".We find a to be proportional to K2 in accordance with eqn (12) if we force the data to fit a single exponential decay. However, visual inspection of any of our autocorrelation results (fig. 2) makes it clear that a single exponential fit does not adequately describe the data. A further data analysis was therefore undertaken as follows. Although the precision in each time channel can be set to 1 % simply by continued sampling until this precision is reached, the reproducibility of the data is much poorer. It was observed that thermal and swelling equiIibrium of a newly handled gel in the light scattering cell is only established after long times (typically 24 h). In the early stages an oscillating behaviour of the autocorrelation is sometimes observed, presum- ably caused by convection currents.If the cell is left undisturbed final equilibrium data were fourid to be reproducible to about 8 %. A polynomial fit programme to the natural logarithm of the data points was therefore used with as many terms as required to reach the 8 % reproducibility of the experimental data. Most of our data required at least a third-order fit. The higher-order terms reflect the existence of more than a single exponential. The coefficients of the polynomial fit are simply related to the moments of the function F@, a) which is defined by -)=I r(Ky F(K, a) exp (-az) da. Eqn (1 3) replaces eqn (1 2). If the fitting polynomial at any given K-value is represented by (ao(K) + a,(K)z + a2(K)z2 + .. .) then, according to the method of c~mulants,~ the moments are = [u2F@, a) da = 24K)+&K). The reliability of higher moments than a2@) is not high.g The magnitude of the variance V(K) = I ~ - = ) ~ I + / = ) = l 2 a 2 ~ ) l + / a 1 ~ ) is therefore taken as a measure of the deviation of the correlation function from a single exponential decay. For a single decay V = 0. In our case, V ranges from 0.64 to 2.8, depending on K and on the type of gel. Jt is thus clear that all our gels exhibit more than one relaxation process in their local viscoelasticity. In view of our earlier finding that the PHEMA gels are highly non-random (see table l), it is possible that within the small scattering volume there are in our case indeed several gel relaxation processes.However, even for a randomly cross-linked polymer network one would expect a spectrum of relaxation times because the approximate model underlying eqn (12) is inadequate. It ought to be replaced by the more realistic Rouse-type model of Gaussian network chains. This model generates a viscoelastic spectrum (eqn (13)) with many long relaxation times, caused by the coupling of the polymer chains into a network.1° However, no theo- retical derivation of F a , a) has been undertaken as yet.152 LIGHT SCATTERING BY GELS REVERSIBLE AGGREGATION OF POLYMER CHAINS AND THE SOL-GEL TRANSITION In dilute solutions of polymers the time correlation function of the scattered field is determined mainly by the translational diffusion constant D of the polymer chains, although at sufficiently high molecular weights the amplitude r2(K, 0) of the larger internal modes of motion (a, = 7;') may become sufficiently large to give an addi- tional measurable decay constant at large K-values l1 : We have investigated the aggregation processes in dilute aqueous solutions of agarose (an alternating copolymer of (1 +4) linked 3,6 anhydro-a-L-galactose and (1 4 3 ) linked P-D-galactose) in the concentration range 0.1 to 1.0 wt.%. At SUE- ciently high temperatures the pulse counting correlation function may be represented by a single exponential decay with a decay rate determined by the average diffusion constant, D, of the various molecular aggregates of agarose present in the solution. Extreme care has to be taken to eliminate externally induced thermal convection currents.This was accomplished by a temperature controlled jacket with narrow openings for the incident and scattered beams. As the temperature is decreased, B also decreases indicating that more aggregation has occurred. At temperatures slightly above the sol-gel transition temperature, typically 40-50°C, the pulse counting correlation function can no longer be represented by a single exponential decay. Instead, (fig. 3) we obtain what appears to be a sum of 1.0 n 0 0 0 5 10 15 20 25 30 35 40 45 50 5X1O2 71s FIG. 3.-The pulse counting autocorrelation function, P ( K , T/rZ(K, 0) (eqn (9, one dot for each of the 50 channels of 2 ms) for a 0.3 wt. % aqueous agarose solution at 56°C and a scattering angle of 55". Clearly, the data deviate from the best fit single exponential decay (solid curve). decays.The additional decays may reflect internal modes of motion of the large aggregates, or microgel particles. We cannot rule out the possibility, however, that the additional decays result from an increasing polydispersity of the microgel particles. When agarose solutions are cooled to temperatures below the sol-gel transition temperature, we observe the onset of a damped oscillatory pulse counting correlation function (fig. 4). We believe that this phenomenon is most likely associated with the mass flow that accompanies the gelation process. The oscillatory correlation function persists for several hours but eventually disappears as the gelation process is completed.K . L. WUN, G .T . FEKE AND W. PRINS 153 The time evolution of previously published optical rotation and frequency integrated light scattering data l 2 also provides evidence for the long duration of the gelation process. $ 5 h k " O ~ - 0 O- 0 10 20 30 40 50 60 70 80 90 100 3.33 x lo2 71s FIG. 4.-The pulse counting autocorrelation function (eqn (5), one dot for each of the 100 channels of 3 ms) for a 1.0 wt. %aqueous agarose system below the sol+gel transition temperature at a scattering angle of 30". The measurement was obtained 3 h after the initiation of the gelation process. Once the agarose gels have equilibrated, we have been able to measure correlation functions in only the most dilute (-0.12 wt. %) samples. We find a sum of expo- nential decays, similar to that shown in fig.2 for a typical swollen PHEMA network, reflecting a spectrum of viscoelastic relaxation times. In more concentrated samples we found that the signal to noise ratio was too low to obtain measurable results. In the light of our present results it appears that the earlier data quoted as ref. (7), indicating damped oscillatory correlation functions in microphase separated gels of agarose and poly(viny1 alcohol), must be re-interpreted. It was conjectured that the oscillatory correlation functions were due to thermally excited low frequency vibra- tions present in the equilibrium gel. Since we have now found that the equilibrated gels do not exhibit oscillatory correlation functions, it seems that the measurements reported in ref. (7) were obtained from samples in which gelation was not yet completed.In the case of PVA, the incompleteness of the gelation process is connected with the slow crystallization of PVA in the concentrated polymer phase. In the course of a closer examination of the sol + gel transition, we have found the frequency-integrated light scattering to be very strongly dependent on the thermal history. Fig. 5 shows that the frequency-integrated Rayleigh ratios of 1 % agarose gels may differ by as much as a factor of 1000, depending on the quenching tempera- ture. The sol --+ gel phase transition temperature l2 at this concentration is about 4142°C and rapid supercooling was achieved by switching the cell from one circula- ting thermostat to another in about 5 s. Fig. 5 also shows a maximum in the Rayleigh ratio which shifts to larger scattering angles when the temperature jump is increased.Such a maximum can be either due to the formation of very monodisperse microgel particles which become suspended in the gel matrix in a random array, or to an inter-particle Bragg spacing of regularly placed microgel particles in the gel matrix.12 No maximum is observed when a very dilute, and therefore non-gelling, agarose solution is cooled.12 The scattered in- tensity and its angular dependence indicates that in this case microgel particles of1 54 LIGHT SCATTERING BY GELS larger dimensions than in the hot solution are also formed. Since no maximum is observed, the microgel particles are not very monodisperse. It seems therefore more likely that the scattering maximum observed in the gels is due to a dominant inter- particle Bragg spacing.Such a regular spacing might result from a non-nucleated, spinodal decomposition mechanism along the lines suggested by Cahn's the0ry.l I oc IC r( E 2 n m W & I 0. I - - - 0 32.5 C 29 "C I 2 3 4 s 6 7 8 9 1 0 103xsin2 8/2 FIG. 5.-The logarithm of the frequency integrated Rayleigh ratio (eqn (3)) plotted against sin2 8/2 for 1 % aqueous agarose gels, obtained by cooling the solutions from 80°C to the indicated tempera- tures. Note the appearance of a maximum and its shifting to wider scattering angles with increasing supercooling below the phase boundary of the sol -+ gel transition at 42°C. In order to test whether the gel formation is nucleation controlled ur results from the spontaneous concentration waves of Cahn's theory, the time-evolution of the intensity of scattered light after a temperature jump into the thermodynamically unstable region of the system has to be measured at various K ~a1ues.l~ Such work is in progress but has not yet been completed.Financial support of NSF grant No. GP-33755 is gratefully acknowledged. R. D. Mountain and J. M. Deutch, J. Chem. Phys., 1969,50, 1103. R. W. Wynaendts van Resandt, Master's Thesis (Department of Electrical Engineering, Syracuse University, 1973). E. Jakeman, J. Phys. A. 1970,3,201. R. S. Stein and J. J. Keane, J. PoZymer Sci., 1955, 17, 21. K. L. Wun and W. Prins, J. PoZymer Sci. (Polymer Physics Edition) submitted for publication. K. DuSek, ColZ. Czech. Chem. Commun., 1968, 33, 1100; Brit. Polymer J., 1970, 2, 257. W. Prins, L. Rimai, A. J. Chompff, Macromolecules, 1972, 5, 104. T. Tanaka, L. 0. Hocker and G. B. Benedek, J. Chem. Phys., 1973, 59, 5151. D. E. Koppel, J. Chem. Phys., 1972,57,4814. lo see, e.g., J. D. Ferry, Visco-elastic Properties of PoZymers (J. Wiley and Sons, New York, 1970), l 1 see, e.g., R. Pecora, Ann. Rev. Biophys. Bioengin., 1972, 1, 257. l2 E. Pines and W. Prins, Macromolecules, 1973, 6, 888. p. 437 ff.K . L . WUN, G . T. FEKE AND W . FRINS 155 l 3 J. W. Cahn, J. Chem. Pltys., 1965, 42,93 ; see also J. Goldsbrough, Sci. Prog., 1972, 60, 281. 14J. J. Van Aartsen and C. Smolders, European Polymer J., 130, 6, 1105. Note added in proofi We have shown (G. T. Feke and W. Prins, “ Spinodal Phase Separation in a Macro- molecular Sol+Gel Transition,” Macromolecules, 1974,7, that the Bragg maxima shown in fig. 5 are indeed due to the formation of a supramolecular structure governed by phase separation proceeding via spinodal decomposition. We have explained the shifting and the broadening of the maxima with lower quenching temperatures in terms of the spinodal decomposition mechanism. We have also shown that the absence of a maximum in the upper curve in fig. 5 is the result of phase separation proceeding via nucleation and growth since, in this case, the quenching temperature is higher than the spinodal temperature.