fr2/g where r is a characteristic length of the magnetic

64.70M Nous The J. Physique 44 (1983) 985991 AOÛT 1983, 985 Classification Physics Abstracts 46.30J 64.80 Critical behaviour of the elastic constant and the friction coefficient in the gel phase of gelatin J.C. Bacri and D. Gorse (*) Laboratoire d Ultrasons (**), Université Pierre et Marie Curie, Tour 13, 4 place Jussieu, 75230 Paris Cedex 05, France (Rep le 23 février 1983, révisé le 24 mars, accepté le 28 avril 1983) 2014 Résumé. avons étudié le comportement critique de la constante élastique et du coefficient de friction dans la phase gel de la gélatine. Les exposants critiques trouvés, 1,8 pour la constante élastique et 1 pour le coefficient de friction, sont en bon accord avec une théorie de 3dpercolation. Abstract. 2014 critical behaviour of the elastic constant and the friction coefficient have been measured in the gel phase of gelatin. The critical exponents, 1.8 for the elastic constant, and 1 for the friction coefficient are in good agreement with 3dpercolation theory. 1. Introduction The aim of this paper is to describe the critical behaviour of a typical physical gel, gelatin, near the solgel phase transition. At high temperatures, in the sol phase, gelatin appears as a viscous polymeric solution of random coils. As the temperature decreases, gelation occurs because of two competing phenomena : one is an intramolecular phenomena where the triple helicoidal structure is recovered [1] from the initial random coil, the other is an intermolecular phenomena which is a random bonding between adjacent polymeric chains. It is assumed that going from the sol to the gel phase involves a critical point. At the critical temperature, bonding has spread from one edge to the other of the sample; the gelatin is now considered, in the gel phase, to be a 3delastic medium (collagen network) immersed in the solvent (water). Two different theories describe the solgel phase transition : in the classical gelation theory of Flory [2], the gel is viewed as growing up like a spanning tree (Cayley tree). Then excluded volume effects and cyclic bonds are omitted The classical expectation for the exponent t of the network elastic constant is (*) Present address : C.E.N. Saclay, S.P.A.S., F91191 Gif sur Yvette Cedex. (**) Associated with the Centre National de la Recherche Scientifique. t 3. The deviations from this idealistic tree model stipulated above are taken into account in the percolation model [3]. The prediction is 1.7 t 1.8 in 3dpercolation theory. The gel phase is investigated here by the method of magnetic probes (ferrofluids [14]) immersed in the polymeric solution. Under a static magnetic field (H 100 G) the ferrofluid particles rotate and induce a shear deformation in the network (the solvent remaining motionless). The motion equation of the gel network is given by u is the transverse displacement vector of the elastic medium from its equilibrium position, p the density of the network, G the shear elastic constant ot the network and f is defined as the friction coefficient between the network and the solvent. We deduce from equation (1) that the ferrofluid particle relaxation time, when the field is cutoff, will be proportional to fr2/g where r is a characteristic length of the magnetic particle (cf. 6.1). The elastic energy of the ferrofluid particle is equal to 2 I C02 (cf. 3.2) where 0 is the rotation angle of the ferrofluid particle under H, and C is the product of the shear elastic constant G and the perturbed volume. So we are able to obtain, independently, the variations of these two characteristic coefficients of the gel, G and f, around Tg, as functions of the temperature. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004408098500

For 986 2. Experimental method. The experiment uses a method which has been described elsewhere [4]. A small number of magnetic particles (ferrofluids) are added to the gelatin sample. When applying a uniform magnetic field H (by means of two coils surrounding the sample) these particles tend to line up along the field ; a macroscopic birefringence An is induced in the sample. In the sol phase, we obtain the critical variation of the viscosity il by measuring the birefringence relaxation time, at the instant the field H is switched off. In the gel phase, the measurement of the birefringence level leads to the critical variation of the elastic constant; the birefringence relaxation time gives the friction coefficient f. Our ferrofluids are obtained by a new chemical method [5]. Every elementary magnetic grain (100 A diameter) is a macroanion immersed in an aqueous solution. The balance between the repulsive electrostatic energy and the attractive magnetic dipoledipole interaction energy leads to an agglomeration of few grains. The resulting agglomerate has the shape of a prolate ellipsoid of major axis 2 a N 1 400 A and minor axis 2 b 300 A [6]. 3. Theoretical expression of the birefringence induced by the ferrofluid particles. We suppose that each grain has its own internal birefringence. An assembly of grains in a liquid is optically isotropic because of the brownian motion, but in the presence of a magnetic field the suspension becomes birefringent. This optical birefringence is related to the anisotropy of the electrical susceptibility tensor x. This tensor for an uniaxial dielectric particle located by 0, (p is : where and G 1 is the dielectric constant of the liquid, 8jj and 81 the principal dielectric constant of the particle, v its volume and L its depolarizing factor, with The anisotropic part of the electrical susceptibility of a suspension of these particles in a liquid is Xan r(x I I Xi)(( f(0, qj) )ø,qj where r is the volume fraction of the particles (r N v where N is the number of particles per unit volume) and f( 0, qj)) represents the average value of/(0, T) with respect to 0 and T. The birefringence (which we shall call An from now on) is proportional to the anisotropic part of the susceptibility tensor, An oc xan. 3.1 APPLICATIONS. 3.1.1 In the sol phase. temperatures above the gelatin threshold, the gelatin sample is a liquid The ferrofluid particles are submitted to thermal motion. If we apply a magnetic field H along the Zaxis (the Yaxis being along the laser beam), the magnetic particles line up in the field (Fig. 1); they make the sample uniaxial. We can write Fig. 1. (J and 9 angles definition in sol phase. dependent because of the revolution symmetry around Zaxis. Thus with In this case the distribution function P(O, qj) is not w where JlH cos 0 is the magnetic energy and fl 1 /k T.

For Definition 987 After integration over 0 and T we get (with fllth a) : Experimentally, we measure : with A Xil XI. 3.1.2 In the gel phase. theoretical and experimental simplifications, the magnetic particles, quenched in the polymeric network by a strong magnetic held, are all aligned in the direction of the laser beam (Zaxis). A magnetic field H is applied perpendicular to the Zaxis (Xaxis). The measured birefringence is the result of an equilibrium between the magnetic energy,uh sin 0 cos qj and the elastic energy 2 C02 due to the gel network : 0 and rp describe the position of the magnetic moment in the field H (Fig. 2) ; C is the product of the shear elastic constant G times the volume perturbed by the particle rotation. In this geometry (Fig. 2) the birefringence is obtained, Fig. 2. of the angles 0 and T. X and Z are respectively the directions of the applied magnetic field H and the light propagation direction which is also the initial position of magnetic dipole p in the gel phase. making a thermal average over all the contributions of the ferrofluids, taking into account their individual orientations. The distribution function is here : Experimentally we measure For the critical determination of the elastic constant, it is useful to make a limited expansion of On around Tg when PC 1 with a low magnetic field (f3 p,h 1). We find : with f(flc) 0.33(pC)2 + 0.15 j8c + 1. We note that, surprisingly, An has a maximum for pc j:. 0. This maximum must appear at a temperature smaller than Tg (where pc 0). 4. Experimental conditions. Gelatin is an example of weak gel [7]; the crosslinks between the neighbouring peptide chains fluctuate with time, at thermal equilibrium. Here, the evolution of the parameters of the gelatin is studied as a function of temperature. We avoid the evolution in time of the gelatin with a good choice of the temperature variation rate. To this end, a temperature cycle is made on a gelatin sample (with concentration 1.5 % by weight) prepared with ferrofluids. We record the birefringence level versus T Tg. Tg is determined by the sudden

Birefringence Experiment Experiment Experimental 988 Fig. 3. function of T T g for two temperature variation rates. variation (or stabilization) of the birefringence level starting from the sol (or the gel) phase. By examination of figure 3, the critical variations of the birefringence presents no detectable difference (exact overlap) when the temperature is lowered or increased at a rate smaller or equal to 3 x 104 K/s. However, with such a rate, the Tg value determination varies by 5 K depending on which is the initial phase. This is due to the fact that the gelation process depends on the whole history of the sample from the origin (sol or gel). 5. Critical variation of the elastic constant, gel elasticity; the relaxing birefringence signal is also recorded The advantage of this experimental procedure is that, close to Tg, H is so small that the magnetic energy JlH and the elastic energy 2 Cø2 are of the same order of magnitude as kt and therefore the gel is not destroyed by the small rotation of the magnetic probes. Another important point is that Tg cannot be deduced from this experiment. In the magnetic field variation observed there is no great variation near Tg. Thus we need another experiment to determine Tg. 5.2 Tg DETERMINATION. Experiment : The following experiment is performed; the sample is prepared in the same way as in 5. l, but we keep the magnetic field constant during the experiment We record the birefringence level On and its relaxation time as function of temperature. Observation : Figure 5a shows the typical experimental variation of An as a function of temperature. The remarkable feature is that we observe near Tg a maximum in the birefringence level. This maximum is predicted by the limited expansion of the theoretical value of An (Eq. 3), but the physical explanation is hot simple. At the field cut off, the birefringence relaxation time T(T) is plotted versus T (Fig. 5a2). We see that the maximum of the relaxation curve coincides with a temperature T1 for which the birefringence level To get the critical variation of the elastic constant, we proceed in the following way. 5.1 EXPERIMENT. The sample (gelatin + ferrofluids) is quenched at T 5 OC under a high magnetic field (Ho 5 000 G) in order to line up all the ferrofluid particles in the direction of the laser beam (Fig. 2) ; then this field is turned off. We impose a rotation of the magnetic particle by applying a magnetic field H perpendicular to the initial orientation. For each temperature, we adjust the value of the magnetic field in order to keep the birefringence level On constant The variation of the field H is given in figure 4 versus temperature. When H is switched off, An relaxes because of the restoring force due to the! I Fig. 4. In the gel phase, we adjust the value of the magnetic field H(G) in order to keep the birefringence level constant (An is here equal to 0.11 x 102) for each temperature. When increasing T, the network comes loose, and the field necessary to get the same birefringence level becomes smaller and smaller. Fig. 5at. «An(T), H constant». Birefringence level variation as a function of temperature, H remaining constant. The maximum is predicted by the theoretical value of An (Eq. 3). Fig. 5a2. decrease of the relaxation time as a function of temperature, when H constant «An constant, H(T»). Decrease Fig. 5b. of the magnetic field H(T) as a function of temperature.

Elastic Now Birefringence 989 measured is equal to that of Tg ; ðn(t1) An(Tg). We found experimentally that Tg T1 0.5 ± 0.1 K with quite good reproductibility. Thus we have obtained an interesting method for the determination of Tg ; in the case (5.2), where, in order to get the critical variation of the elastic constant, we measure simultaneously H and the birefringence relaxation time as a function of temperature, we just need to make a translation of 0.5 K from the maximum of the birefringence relaxation time curve to get Tg. Tg is located 0.5 K above T1, which corresponds to the maximum of the curve T(7). 5. 3 EXPERIMENTAL RESULTS. we look at figure 3 giving H(T) for An cost The theoretical expression of the birefringence level is given by equation 2. For each value of the field (H) we get the corresponding elastic constant C. Figure 6 shows a loglog plot of elastic constant as a function of T Tg. We obtain almost a straight line over one decade in T Tg. We have fitted the curve C(T Tg) for three different values of the magnetic moment of the agglomerate (the three kinds of symbols observed on Fig. 6). If we consider that the mean value of the moment is about 5.0 x 1016 erg/g, we have to notice that a relative dispersion in the value of that moment is irrelevant; no variation in the slope of the curve can be observed The mean value of this slope give the exponent t, corresponding to C Tg Tit, t 1.7 but if we consider the asymptotic slope, the value of t is near 2. So a reasonable value of the elastic critical exponent is t 1.8 ± 0.2. 6. Critical variation of the friction coefficient f. Now, we try to explain the divergence of the birefringence relaxation time r obtained in the experiments described above in the gel phase. 6.1 RELAXATION PROCESSES. Two different relaxation processes are known in a two medium system (network + solvent) of this type. We consider the shear movement produced by the rotation of the probe. We introduce two transverse displacement vectors from this equilibrium position, one for the network, u(r, t), the other for the solvent, v(r, t). The coupled motion of the gel network and the solvent is described by p, G and f are respectively the density of the network, its shear elastic constant and the friction coefficient between the network and the solvent. There are two simple solutions of equation 4. If the network and the solvent are moving in phase, u v, and the return to equilibrium is due to a damped (because of viscous terms not considered in Eq. 4) sinusoidal motion [8] we did not observe experimentally. If the network is moving alone, v 0; at low frequencies, equation 4 becomes V u f/gu. And the relaxation time r is found to be proportional to fr2 / G (or fr2 / C as C oc G) where r refers to the hydrodynamical radius of the ferrofluid agglomerate. 6.2 EXPERIMENTAL RESULTS. We define a critical exponent for r (T (Tg T) "). Thus, the critical, variation of f is obtained : it goes to zero at T g like Cr. From a loglog plot (Fig. 7) we find x 0.8 + 0.1. 7. Discussion. Two kinds of results have been obtained for the elastic constant and the relaxation time, near T g in the gel phase. The experimental exponent of the shear elastic constant C is centred on the value t 1.8 + 0.2. This value is in good agreement with the theoretical value using the analogy between the macroscopic conductivity of 3ddimensional lattice of random conducting bonds [9] and the elastic constant. On the other hand, it is very different from the classical result t 3 [10]. Only a few experiments have been performed in rela Fig. 6. constant variation as a function of T T g in a loglog plot (+) J.l 10is erg/g; (A) It 2.5 x 1016 erg/g; (0) u 1016 erg/g. The straightline represents t 1.7 and the dashline t 2 0. Fig. 7. relaxation time in the gel phase as a function of T Tg on a loglog plot..

Loglog 990 tion to the measurement of the elastic constant critical behaviour in gelatin. As far as we know, there is one, due to PenicheCovas et al. [11] (the initial purpose of which was to find the correct multiplicity of the helical crosslinked zone) which leads to the same exponent value t 1.7 [3]. We note that our experiment was performed with a 1.5 % gelatin system whereas the PenicheCovas experiment was done with a 5.7 % quenched sample. (In their case, the experimental setup was a sphere rheometer). Comparing these results with those in chemical gels, we find that the exponents t obtained are all higher than those of the gelatin system. As an example, for a system made by radical copolymerization of styrenemeta divinyl benzene with a solvent, t was found equal to 2.1 [12]. In another system obtained by polycondensation of hexamethyl diisocyanate, t was found equal to 3.2 [13]. Theoretically, the critical variations of the friction coefficient is not well established But one possibility is that f can be related to the permeability of the gel as 1/,2 (Darcy s law) where, is the correlation length of the system (the mean distance between two reticulation points) and?i is the viscosity of the fluid phase (solvant and finite clusters) [14, 15]. If we assume that q has a critical behaviour r ~ DT Is, we obtain f I AT I2vs. In our experiment the relaxation time in the gel phase is T _ fr2/g. We can suppose that the elastic constant G in a dynamic experiment does not have the same critical exponent as that measured in a static experiment. G I AT It. The entanglements increase the value of the static elastic constant (t t). The critical form of the relaxation time is : s I AT Ix with x s + t 2 v. The experimental value of x is 0.8 ± 0.1. In 3dpercolation theory v 0.88 and we deduce that s + t 2.56 ± 0.1. One hypothesis is that the viscosity has the same critical exponent in the sol and gel phase (s s ). Thus we need s. An experiment quite similar to that previously reported [4] was performed in the sol phase where the birefringence relaxation time i is proportional to the viscosity of the sample. i was recorded as a function of T. A loglog plot is shown in figure 8. The slope is directly the exponent s 0.72 ± 0.1. Then according to the previous hypothesis t 1.84 ± 0.2. This value of the critical exponent of the elastic constant obtained from the relaxation time is very near to the value obtained Fig. 8. plot of the birefringence relaxation time as a function of the temperature difference T T g in the sol phase. The slope is s 0.72 ± 0.1. T is proportional to the viscosity of the polymeric solution T oc n oc 111 T Is. in the static experiment t 1.8 + 0.2. This last feature shows that the entanglements do not affect the elastic constant in the relaxation experiment. 8 Conclusion In gelatin (1.5 % in weight) the critical behaviour of the shear elastic constant, the friction coefficient in the gel phase, and the divergence of the viscosity in the sol phase have been measured. We have used an original experimental method which enables the shear elastic constant to be measured very close to Tg ; it is possible to have near Tg an elastic energy (induced by the magnetic field) of the same order as kt. The time dependence phenomena are very important in gelatin but we have found a reproducibility criterion for a defined temperature rate. With these experimental conditions, the critical exponents found are very close to those of 3dpercolation theory. References [1] VEIS, A., Macromolecular chemistry of gelatin (Academic Press, New York) 1964. [2] FLORY, P. J., J. Am. Chem. Soc. 63 (1941) 3083, 3091, 3096. [3] STAUFFER, D., Physica 106A (1981) 177. [4] DUMAS, J. and BACRI, J.C., J. Physique Lett. 41 (1980) L279. [5] MASSART, R., C. R. Hebd. Séan. Acad. Sci. 291 (1980). [6] JOLIVET, J.C., MASSART, R., FRUCHART, J. M., Nouv. J. Chim. (1983). [7] DE GENNES, P. G., Scaling concepts in polymer physics (Cornell University Press, Ithaca, N. Y.) 1979. [8] TANAKA, T., KOCKER, L. U., BENEDEK, G. B., J. Chem. Phys. 59 (1973) 5151.

991 [9] DE GENNES, P. G., J. Physique Lett. 37 (1976) L1. [10] STAUFFER, D., J. Chem. Soc. Faraday Trans. II 72 (1976) 1354. [11] PENICHECOVACS, C., DEV, S., GORDON, M., JUDD, M., and KAJIWARA, K., Faraday Dics. 57 (1974). [12] GAUTHIERMANUEL, B., and GUYON, E., J. Physique Lett. 41 (1980) L509. [13] ADAM, M., DELSANTI, M., DURAND, D., HILD, G., and MUCH, J. P., Pure Appl. Chem. 53 (1981) 1489. [14] STAUFFER, D., CONOGLIO, A., and ADAM, M., Adv. Poly. Sci. 44 (1982) 103. [15] JOUHIER, B., ALLAIN, C., GAUTHIERMANUEL, B. and GUYON, E., The solgel transition (preprint).