Non-linear behaviour of gels under strong deformations

Transcription

1 Nonlinear behaviour of gels under strong deformations S. Daoudi To cite this version: S. Daoudi. Nonlinear behaviour of gels under strong deformations. Journal de Physique, 1977, 38 (10), pp < /jphys: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1977 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Our Nous We which LE JOURNAL DE PHYSIQUE TOME 38, OCTOBRE 1977, 1301 Classification Physics Abstracts NONLINEAR BEHAVIOUR OF GELS UNDER STRONG DEFORMATIONS S. DAOUDI Collège de France, Paris Cedex 05, France (Reçu le 18 mai 1977, accepté le 28 juin 1977) 2014 Résumé. contrainte constante 03C3 considérons un gel de chaînes de polymères flexibles en bon solvant : une forte est appliquée au temps 0. Nous étudions la réponse élastique prompte à des temps t tels que la fraction chaînes/solvent n a pas changé. La théorie est basée sur les lois d échelle établies par divers auteurs pour une seule chaîne en forte déformation. On prédit une relation contraintedéformation à grande déformation 03B1 de la forme 03C3 ~ 03B15/2 en traction et 03C3 ~ 03B13/2 en cisaillement simple. Nous présentons aussi quelques remarques sur les diagrammes de diffusion de neutrons, la birefringence et les coefficients de sédimentation que l on attend pour un échantillon déformé. Ces considérations ne s appliquent pas aux caoutchoucs purs Abstract. consider a gel of flexible polymer chains in a good solvent : a strong constant stress 03C3 is applied at time 0. We study the prompt elastic response at times t such that the chain/solvent fraction has not changed. The theory is based on the scaling laws established by various authors for a single chain under strong deformation. The stressstrain relation at large strain 03B1 is predicted to be 03C3 ~ 03B15/2 in longitudinal deformation and 03C3 ~ 03B13/2 in simple shear. We also present some tentative remarks on the neutron scattering patterns, the birefringence and the sedimentation coefficients, expected for the distorted sample. These considerations do not apply to pure rubbers. 1. Principles. present understanding of the organisation in an entangled polymer solution is based on scaling concepts [1, 2]. The basic parameter for a solution of concentration c is the correlation length 03BE(c). For good solvents this length behaves like where RF is the single chain radius in the same solvent, and c* is the critical overlap concentration (N being the polymerization index). At c c* the different coils are just in contact. A polymer gel can be described in very similar terms : at equilibrium, in a good solvent, the chains tend to separate from each other, but the cross links force them to remain in contact : thus the equilibrium concentration c is essentially equal to c* (apart from a numerical constant). The whole Flory theory of gel swelling [3] is summarized by this sentence : all the dependences on solvent quantity are included in the single chain radius RF, for which the Flory analysis of the single chain problem gives us very good numerical values. For small deformations (characterized by a strain a) the stress in the gel is a linear function of the strain where the elastic modulus scales like [3] : (Tis the temperature : we use units where Boltzmann s constant is unity). Eq. (1. 4) holds for both longitudinal and transverse shear (the only difference being in the numerical prefactors we ignore systematically in the present discussion). Our aim here is to extend eq. (1.3) to large deformations (a > 1). 1.1 MACROSCOPIC VIEWPOINT. For a single chain, we know from the work of Fisher et al. [4] and Pincus [5] that the free energy at strong deformations is of the form where r is the end to end distance, and RF anv [For our purposes v 3/5 and (1 v)1 5/2.] Eq. (1. 5) ignores some minor (logarithmic) corrections which are negligible at strong deformations. Consider first the case of a longitudinal deformation, where the sample changes its length from Lo to alo. Each chain, between its junction points, has an Article published online by EDP Sciences and available at

3 Following Schematic It 1302 Pincus [5] we can give a much more physical picture of the distorted system (Fig. 1). Each chain is connected to the network by its two extremities, and feels a stretching force ± 0. We call the stretched chain a string with a length jjj. At sizes r>03be (larger than the mesh of the network) we expect an affine deformation : this implies FIG. 1. representation of a strongly elongated network, showing strings between junction points. Each string is a collection of Pincus blobs. The amount of wiggling of an individual string has been purposely underestimated in this drawing, to show the structure most clearly. extension r ç II arf and a size at rest RF, as shown on figure 1. Then the energy per cm3 is We can relate the force (P to the macroscopic stress a in longitudinal deformation. We have c/n strings per cm3, each carrying a tension 0 over a length ç II. Thus Consider now the inner structure of one string : it can be visualised as a nearlylinear sequence of smaller units (Pincus blobs), each containing gp monomers and having a size çp. At distance r çp the force 0 is a weak perturbation, and we find the correlations of a swollen chain at rest. This implies that the relation between gp and çp still has the excluded volume form where c/n is the chain concentration. We assume that during the time of the experiment, the solvent does not have enough time to permeate in or out of the solution : the concentration c is then unchanged by deformation. In fact, for a sample of width W, the permeation time would be of order Tp W ldc where the cooperative. diffusion coefficient Dc is known both from scaling theories and from experiments using the inelastic scattering of light [6]. With typical values W 1 mm and Dc 10 cm2/s the time rp is of order one day. From eq. (1. 5) it is then easy to find the stress At size r > çp the force 0 becomes a strong perturbation. At the crossover (r çp) we must have Finally we know that the whole string is a nearlylinear sequence of N/gp Pincus blobs, and thus Combining eqs. (1.10)(1.14) we get and where S So a1 is the cross sectional area. Inserting eq. ( 1. 5) we get In this formula one factor a is trivial, and is due to the reduction of crosssection. But the remaining factor rx3/2 is interesting. A similar argument for shear deformation gives This with eq. (1.11) then restores the stress strain relation (1.8). 2. Correlations in the distorted state : longitudinal case. is important to observe first that the fraction of the volume occupied by the Pincus blobs is small at large a. This fraction is indeed the difference between (1. 8) and (1.9) is related to the area on which forces are applied. This area is independent of a for simple shear, and varies like a1 in longitudinal stretch THE LOCAL PICTURE : STRINGS AND PINCUS BLOBS. as can be seen from eqs. (1.10), (1.15). Thus for many properties we may consider the strings as uncorrelated. The average distance d (measured normal to the direction of stretch) between strings is such that d 2 ç II RF, and thus Each string is not quite straight, but has fluctuations (transverse to its own axis) of order çl., where ç BI

4 Conjectured In 1303 corresponds to a random walk of N/gp steps, each step being of order 03BEp. Thus giving ç.l 1".1 RF a1/4. Hence çl./ ç II 1".1 rx 5/4 is small, but 039E./d rxl/4 is larger than unity. These remarks lead us to the following conjecture : the coherent neutron diffraction pattern should be qualitatively similar to the scattering due to independent (strongly aligned) Pincus strings. For a given wave vector q, the coherent scattering intensity I(q) is related to the pair correlation function 3. Stressinduced birefringence. a strongly stretched gel, our description in terms of strings (of length jjj II and width ç 1.) suggests that this inhomogeneous distribution of monomers will give rise to an anisotropic dielectric constant tensor E at a scale larger than çp. In the present section we discuss E using the approach by Landau and Lifshitz [9] for heterogeneous media. We define electric induction fields D and E related by : where 8, is taken to be isotropic and proportional to the local concentration : by a Fourier transform. Our conjecture gives the following features for I(q). a) For low angle scattering I(q) + 1(0) measures the number of monomers in one correlated region Thus the present calculation does not include any effect due to an intrinsic anisotropy of the monomers. Separating the small deviations from the volume average, we may write and b) At intermediate q values I(q) becomes smaller, and anisotropic. For q qll II (along the stretch axis) the intensity drops when q 03BE ot RF 1. For q ql (transverse) the drop is at q ~ 11 rxl/4 Ri 1. c) For q > 03BEp 1 we recover the isotropic scattering from a single chain in good solvents [7] We return with these relations to (3.1) and take the average Taking the divergence of (3.1) we obtain, to first order These features are summarized in figure 2. It must be emphasized that figure 2 should apply only if the crosslinks do not contribute strongly to the coherent scattering. In the opposite limit, a correlation peak might appear in I(qll) for wellcalibrated gels [8] : this would reflect a quasiperiodic repetition of nodes along the stretch direction. If we interpret this as a Poisson equation we find Then, from (3. 5) and the definition of E(D have EE) we Introducing the Fourier transform and using (3.2) leads to For in longitudinal deformation we assume a simple Lorentzian form (at q ç; 1) : FIG. 2. pattern for coherent neutron scattering by a strongly elongated network. q is the wave vector (qll refers to the projection along the direction of stretch). where qll and ql are the projections of q parallel and perpendicular to the stretch direction respectively.

5 the 1304 The dielectric constant anisotropy is then given by where so is the sedimentation coefficient of the undistorted sample From a straightforward integration limited by qçp 1 and after simplifications (a > 1) we finally arrive at : 2) Inelastic light scattering experiments might be easier to perform. They measure a cooperative diffusion coefficient [6, 8] We see that the anisotropy of the dielectric tensor is not proportional to the stress tensor : this is rather exceptional, and is specific of the nonlinear behaviour discussed here. 4. Sedimentation and cooperative diffusion. Experiments measuring the relative friction of polymer/solvent should also give us some information on the pair correlation function y(r), if they can be performed at large a and in the required time scale. 1) Consider first a permeation or sedimentation experiment. This would be extremely hard to perform in practice, but is conceptually simple. In a suitable frame, each monomer is subjected to a force f, the solvent being maintained immobile on the average. We define a sedimentation coefficient s for this case as the ratio of the average monomer velocity v to the force. In terms of the Oseen hydrodynamic tensor (where p, v x, y, z, ns being the solvent viscosity), the drift velocity of one monomer due to the force applied at its surroundings is [10] : Inserting for y(r) the form appropriate for fully aligned, uncorrelated rods (i.e. following the conjecture of section 2) we arrive at Eq. (4.4) may also be derived directly from the friction law for a viscous fluid flowing around a cylinder argument of the In term possibly being slightly different. In practice, using eq. (1.10), (1.15) we may simplify eq. (4.4) to Again this should vary essentially like rx 1 : our conjecture leads to a slowing down of fluctuations under stress. 5. Conclusions. 1) The scaling ideas of Fisher, Pincus, and others on a single chain under strong deformations could possibly be checked on gels, if some gels can stand the strong values of a which are of interest (e.g. a N 10). 2) It is tempting to apply the same ideas to solutions of entangled chains : provided that the experiments are performed in a time scale shorter than the disentanglement time Tr, the entanglements are fixed and the behaviour should be gellike. In fact, in a first draft of the present paper, we were putting the emphasis very much on solutions, since here we. are not limited in the a value by rupture problems. But there is a serious difficulty : in an entangled melt, for instance, the distance between consecutive entanglement points on one chain may be of order k 100 monomer units or more [11]. Thus we may define strings of v units. But, in this case, the strings are strongly overlapping and are not independent of each other. The net result is that the strings are essentially ideal, and the FisherPincus nonlinearities are not expected to occur here. A similar effect may well take place in semidilute solutions : as explained in reference [1, 2] we then have independent blobs of g monomers, where g (ca3)5/4. The number of units between successive entanglements Ve scales like g, but may contain a large factor k : ve(c) kg. Thus the strings of interest may contain a large number of blobs, and should then behave ideally. 3) We have not found many experiments which can be meaningfully compared to our model : mechanical tests on gels are generally restricted to small or just moderate deformations. Data on uncrosslinked systems have been often taken on liquid filaments extruded from a spinneret, but our requirement of rather low (semidilute) concentrations is rarely met. However the data taken by Hudson and Ferguson [12] on polybutadiene in dekalin are relevant. In their experiments, the transit time of one chain through the probed region is probably of order 1/100 of the disentanglement or reptation time [6]. Thus it is indeed the

6 We 1305 prompt response which is measured. Also, the authors have plotted their results (for concentrations 612 %) in terms of stress a versus the integrated deformation a. The plot shows a plateau for intermediate values of a (which are not taken into account here) and then a sharp rise for a > 3, with a a3. This is to be compared with the prediction of eq. (1. 8) a rx5/2. But the number of data points on the high a side is not quite sufficient at present. Acknowledgments. have benefited from many related discussions with G. Jannink and M. Adam. References [1] DAOUD, M. et al., Macromolecules 8 (1975) 804. [2] DE GENNES, P. G., Israel J. Chem. 14 (1975) 154. [3] FLORY, P., Principles of polymer chemistry, Cornell Univ. Press, Ithaca NY, [4] FISHER, M., J. Chem. Phys. 44 (1966) 616. FISHER, M., BURFORD, R. J., Phys. Rev. 156 (1967) 183. Mc KENZIE, D., MOORE, M., J. Phys. 4 (1971) L 82. [5] PINCUS, P., Macromolecules 9 (1976) 386. [6] DE GENNES, P. G., Macromolecules 9 (1976) 587, 594. ADAM, M., DELSANTI, G., JANNINK, G., J. Physique Lett. 37 (1976) L53. CANDAU, S., to be published. [7] EDWARDS, S. P., Proc. Phys. Soc. 85 (1965) 613. [8] For undeformed gels, a broad peak shows up in I(q) : it describes short range correlations between blobs : see Du PLESSIX, R., Th. Strasbourg We expect this peak to be removed (at least for q q) when 03B1 ~ 1, because the strings are less closely packed in this case. [9] LANDAU, L. D., LIFSHITZ, E. M., Electrodynamics of continuous media (Pergamon Press, Oxford) [10] This approach to friction coefficients is described in more detail in BROCHARD, F., DE GENNES, P. G., J. Chem. Phys. (Aug. 1977). [11] FERRY, J., Viscoelastic Properties of Polymers (Wiley, N.Y.) GRAESSLEY, W., Adv. Pol. Sci. 16 (Springer 1974), and to be published. [12] HUDSON, N. E., FERGUSON, J., Trans. Soc. Rheol. 20 (1976) 265.