- Email: [email protected]
The sol-gel transition
Physica B 156 & 157 (1989) 381-385 North-Holland, Amsterdam CHAPTER
10
GELS AND POLYMERS
THE SOL-GEL
TRANSITION
J.F. JOANNY E. N.S. 46 Alltfe d’ltalie 69364, Lyon Cedex 07, France
Invited paper
We briefly review some recent aspects of the sol-gel transition of chemical gels, physical gels and non-organic silica gels focussing on neutron scattering experiments. These results are compared with the percolation model and the FloryStockmayer theory.
1. Introduction
The polycondensation of difunctional monomeric units leads to the formation of linear macromolecules: after a finite reaction time or equivalently a finite fraction of reacted monomers, a polymer solution with a well-characterized polydispersity is obtained but one never gets an infinite molecule. If some multifunctional units are copolymerized with the bifunctional monomers, branched polymers are formed and eventually, after a finite time, an infinite macromolecule spanning through the sample, the gel, is formed [l-3]. Experimentally, the apparition of the gel is associated with a drastic change in the viscoelastic properties of the system: below the gelation point, in the sol phase, the system is a branched polymer solution with a finite viscosity which thus flows; above the gelation point the system has a finite shear modulus and does not flow, the shear modulus grows from zero at the transition but in general remains as low as 10m6times that of conventional solids. An alternative way of defining the gelation transition is to study the weight average molecular weight M, of the finite molecules: in the sol phase, it increases with the fraction of reacted monomers and diverges at the transition, in the gel phase more and more small molecules are incorporated to the infinite cluster and M, decreases with the fraction of reacted monomers. Although the sol-gel transition is a critical point where one characteristic length scale
(namely the size of the largest molecule) diverges, it is not a thermodynamic transition which may be characterized by the singularities of a free energy and in particular it is not associated with any singularity in concentration fluctuations, it is rather a connectivity transition between a sol where the monomers are not connected and a gel where they are connected. As many other critical points, it is in general characterized in terms of a set of (hopefully) universal critical exponents. In this short review, we first briefly present the current theories of the sol-gel transition and discuss their relevance to chemical gels such as those obtained by polycondensation of monomeric units or by chemical crosslinking of semidilute polymer solutions; we discuss in particular neutron scattering experiments in the vicinity of the transition. The last section is devoted to the presentation of other types of gels, physical gels and non-organic silica gels. 2. Chemical 2.1.
gels, the connectivity
critical
point
Theoretical results
As early as 1941, Flory and Stockmayer [4,5] proposed a theoretical description of the sol-gel transition. Starting from monomers with functionality z 3 3, they studied the formation of random trees at a given fraction of reacted monomers, p. This theory ignores thus loop formation in the macromolecule and also any excluded volume interaction between monomers. Within
0921-4526/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
382
these approximations, one relations for the number with n monomers [6]. A predicted for a critical monomers p, = l/(z - 1) .
J.F.
Joanny
I The sol-gel
can derive recursion of molecules (trees) gelation transition is fraction of reacted (1)
A more refined percolation model for the sol-gel transition was proposed independently by Stauffer and deGennes [2,7]. It is a static lattice model where the monomers lie on all sites of a lattice with a functionality z equal to the number of nearest neighbors. Bonds are put on the lattice with a probability p which mimics the fraction of reacted monomers. Below the bond percolation threshold p,, only finite clusters are formed, above the percolation threshold, an infinite cluster corresponding to the gel appears. The threshold explicitely depends on the nature of the lattice and cannot in general be calculated exactly; it is approximately given by p, = 3122.
(2)
It turns out that the Flory-Stockmayer theory is a mean field version of the percolation model. In particular, it is expected to be valid for the vulcanization problem that corresponds to the crosslinking of polymer chains in the melt [2] (the functionality z is here very large, equal to the number of monomers per polymer). From a fundamental point of view, these two theories are not very satisfactory, they mistreat the solvent and ignore the time evolution of the system. This has prompted the development of kinetic theories [lo], aggregation models [ll] and numerical simulations [12]. Although these have been largely developed over the last few years, we will not describe them here, the agreement between experiments and the percolation model being quite good. In the vicinity of the transition we are interested in two kinds of properties, the polydispersity in mass of the branched polymers and their configuration. The polydispersity is very broad and the number of polymers with IZ monomers at a distance from the critical point decays as a “=P-PC power law
transition
C,,(E)- n-‘f{nin*(&>> ,
(3)
where the exponent T is equal to 512 in the mean field theory and to 2.2 in the percolation model; f is a cutoff function which decays exponentially so that there are no macromolecules in the solution larger than a characteristic mass n* which diverges at the gelation point n * - F ’ /” ,
(4)
the exponent (T equal to 112 in mean field and 0.4 in the percolation model. The branched polymers are fractal objects with a fractal dimension d, related to the polydispersity exponent r, 7 = 3 id, + 1 (percolation), i.e. their radius r increases with the mass II as I/J, r--n
,
(5)
the mean field value of the fractal dimension d, = 4 is larger than the space dimension and is thus not acceptable in the limit of infinite polymers. The two scaling laws (3) and (4) allow the determination of all properties in the vicinity of the gelation transition. For example, the weight average molecular weight is defined as c .“2_
(6) it diverges in the vicinity of the transition as Cy with an exponent y = (3 - T) /a. The elastic modulus in the gel phase grows from zero as E - F’. deGennes has proposed an analogy between gel elasticity and conduction on a percolating cluster of resistors [8] which leads to a value p = 1.7, the equivalent mean field value has been calculated by Gordon, p = 3. Another model relates the elasticity to the bending of bonds between crosslinks of the gel and gives a higher exponent I_L= 3.5 (percolation) [9]. 2.2. Experimental results Experimentally the fraction of reacted monomers or percolation probability is hardly known and it is usually assumed that it varies smoothly with the reaction time t around the percolation threshold or that the distance to the gelation
J. F. Joanny
i The sol-gel
transition is proportional to t - tge,, tge, being the time where the transition occurs. With this assumption, some of the scaling laws have been experimentally tested and seem to agree reasonably well with the percolation predictions. There has been several viscoelastic studies of the sol-gel transition [13]. The exponent p obtained for the elastic modulus is in general in good agreement with the electrical analogy although for some gels obtained by polycondensation a much higher exponent p = 3.2 has been observed [3]. The polydispersity in the sol phase has been studied in great details by Schosseler and Leibler [14] for gels obtained by crosslinking a polymer solution by -y-ray irradiation. This creates chemical crosslinks between the chains whose number is controlled by the irradiation time. The reaction was stopped before gelation and the polydispersity measured by size exclusion chromatography coupled to a light scattering determination of the mass of each fraction. The polydispersity could be described in a scaling form by eq. (3) and the scaling function f was experimentally determined; its universality was checked by varying the molecular mass of the precursor polymer. Here again the agreement with the percolation description was quite good. The last set of experiments which we now discuss in more details are small angle neutron scattering experiments. As we already mentioned, no singularity in the monomer concentration fluctuations is expected at the sol-gel transition, they have the same value in the sol and the gel phase. This in turn means that the scattered neutron intensity from the reaction bath does not give any relevant information on the transition. The only way to obtain a useful information is to dilute the system: this allows to study the individual molecules which grow as the transition is approached. It is important to notice however that the dilution procedure changes the quality of the solvent: it is known in polymer physics that as the monomer concentration is increased, the excluded volume interactions are screened. On the reverse, in the dilution experiment the system keeps memory of the architecture and the polydispersity of the branched poly-
transition
383
mers but not of their configuration, the effect of the excluded volume interaction is increased and the fractal dimension D of the polymers in the diluted solution is lower than their fractal dimension d, in the reaction bath. Such dilution experiments have been performed by Bouchaud et al. [15] on gels obtained by polycondensation of polyurethane monomers (for which the fraction of reacted monomers is actually known from the stoichiometry of the reaction) and by Leibler and Schosseler [16] on the irradiated polystyrene gels. Two experiments are needed to measure the polydispersity of the sol. In the first experiment, the dilution is followed by a fractionation and the scattering experiment is made on a well-defined monodisperse sample; it probes the internal structure of the polymers and the scattered intensity decays with the wavevector q as
Z(q)- 4-O> the fractal dimension D in the diluted solution is equal to 1.98 in ref. [ 151, very close to what one expects for randomly branched polymers (animals) in a dilute solution; the fractal dimension of animals is one of the very few exactly known exponents in three dimensions and is equal to D = 217. In the second experiment, the neutron scattering is made on the polydisperse diluted solution, the intensity is averaged over the mass distribution and is expected to vary with q as Z( s) -
4
-N-r)
.
(8)
A comparison between the two experiments gives an exponent r in good agreement with the percolation model. On the whole, the percolation model gives a reasonably good description of the static properties of the sol-gel transition. Few experiments have considered dynamic properties such as the viscoelastic relaxation times or the viscoelastic spectrum of a gelating system, this could provide a very sensitive test of the model [18]. 3. Physical gels, non-organic
gels
3.1. Physical gels
When a semidilute solution is chemically cross-
384
J. F. Joanny
I The sol-gel
linked (such as in a y-ray irradiation experiment), the chemical crosslinks are covalent bonds with a binding energy of the order of one electron-volt, much larger than the thermal energy kT, they cannot be broken neither by thermal excitation nor by any mechanical constraint and the gelation is irreversible. Instead of inducing chemical crosslinks between polymer chains one may induce physical crosslinks with a binding energy of the order of the thermal energy kT. Physical gels are of major practical importance in food or cosmetics industries and the examples are numerous: _ many biopolymers [19] in solution in water have a helix-coil transition at low temperature, in a semidilute solution two different chains may intertwin in a common helix creating a physical crosslink; - the complexation of some ions by hydrosoluble polymers (Cr”’ by hydrolyzed polyacrylamide for instance) [21] forms ionic bridges between different chains which act as crosslinks; - ionomers [20] are usual polyelectrolytes dissolved in nonpolar solvents, the charges are not dissociated and the polymers carry permanent electrical dipoles along the chain; the strong attractive interaction between these dipoles leads to the formation of multiplets and thus of a physical gel. Contrary to chemcial gelation, physical gelation is reversible; in general it can be induced in two ways, either by increasing the polymer concentration or by decreasing the temperature (in some aqueous systems, the ionic strength may also monitor the transition). One can thus define a gelation line in a temperature-concentration phase diagram. The sol-gel transition is not however the only possible phase transition in these systems: the physical bonds are due to an attraction between different monomers which is equivalent to an attraction between different chains; this provokes at low temperature a demixing phase separation between polymer and solvent. The interplay between the thermodynamic demixing transition and the connectivity sol-gel transition creates very subtle effects that are only partially understood [20,21]. Many physical gels also show strong hysteresis
transition
effects, the properties of the gel often depend on its history; this is due to the existence of many metastable states for the crosslinks and suggests that the physical gelation transition shares many common features with a glass transition. There exists a high experimental body of work by many techniques including neutron scattering on well formed physical gels but the description of the sol-gel transition is far worse than for chemical gels. This is in part due to the experimental difficulty of locating the transition. In many cases, even a small mechanical constraint is sufficient to break crosslinks and even in the gel phase the gel flows; connectivity in these systems is not associated with the existence of a non-zero shear modulus at zero frequency and could turn out to be a purely theoretical concept. 3.2. Non-organic gels The manufacturing of certain ceramic glasses of oxydes such asd SiO,, TiO, and ZrO, often goes through the so-called sol-gel route [22]. This involves first the fabrication of a gel in a solvent and then its transformation by drying and viscous sintering. The properties of the final product being strongly dependent on the heterogeneities of the original gel, it seems of prime importance to control the formation of this gel. Such silica gels may be grown in a water alcohol mixture starting from silicon tetraoxyde monomers. The sol-gel transition in this system has been studied in details by Cabane and coworkers [23] both by light scattering and small angle neutron scattering experiments. The neutron scattering data are quite similar to those of ref. [15] on polyurethane polycondensates: - the intensity scattered from the reaction bath shows no signature of the sol-gel transition. It has a peak corresponding to distances of the order of 20 nm which shifts smoothly to lower q values as time evolves, both before and after the transition; - dilution experiments are consistent with a polydispersity decaying as a power law of the molecular mass with an exponent T = 2.2. The behavior of these non organic silica gels in the vicinity of the sol-gel transition looks thus
J. F. Joanny
I The sol-gel
very similar to that of chemical gels although the precise microscopic arrangement of the molecules does not seem well established. Theoretically, it would be of interest to understand the time variation of the characteristic distance observed in the scattering from the reaction bath which could control the heterogeneities of the gel. Acknowledgements
I am grateful to L. Leibler and F. Schosseler for communicating some of their results prior to publication. References ill P.J. Flory, Principles of Polymer Chemistry (Cornell University Press. Ithaca, 1953). PI P.G. deGennes, Scaling Concepts in Polymer Physics (Cornell University Press. Ithaca, 1978). I31 M. Adam, A. Coniglio and D. Stauffer, Advances in Polymer Science 44 (1982) 105. [41 P.J. Flory, J. Am. Chem. Sot. 63 (1941) 3097. [51 W. Stockmayer, J. Chem. Phys. 12 (1944) 125. 161 G.R. Dobson and M. Gordon, J. Chem. Phys. 41 (1964) 2389.
transition
385
[7] D. Stauffer, Introduction of Percolation Theory (Taylor & Francis, London, 1985). [8] P.G. deGennes, J. Physique 37 (1976) Ll. [9] I. Webman, in: Physics of Finely Divided Matter, M. Daoud and N. Boccara, eds. (Springer, Berlin, 1985), p. 186. [lo] P.G.H. Van Dongen and M. Ernst, Phys. Rev. Lett. 56 (1985) 1396. [ll] T. Witten, in: Chance and Matter, J. Souletie, J. Vannimenus and R. Stora, eds. (North-Holland, Amsterdam, 1987). (121 H. Herman, Phys. Rep. 136 (1986) 154. [13] B. Gauthier-Manuel, in: Physics of Finely Divided Matter, M. Daoud and N. Boccara, eds. (Springer, Berlin, 1985), p. 140. F. Schosseler and L. Leibler, Phys. Rev. Lett. 5.5 (1985) 1110. E. Bouchaud et al., J. Physique 47 (1986) 1273. M. Adam et al., J. Physique 48 (1987) 213. F. Schosseler and L. Leibler, private communication. G. Parisi and N. Sourlas, Phys. Rev. Lett. 46 (1981) 871. M. Daoud and J.F. Joanny. J. Physique 42 (1981) 1359. WI J.F. Joanny, J. Physique 43 (1982) 467. P91 A. Clark and S. Ross-Murphy, Adv. Polymer Sci. 83 (1987) 57. PO1 J.F. Joanny, Polymer 21 (1980) 71. WI E. Pezron et al., Macromolecules, (1988), L. Salome, thesis, Paris (1988). I221 C.J. Brinker et al., J. Non-Cryst. Solids 48 (1982) 47. ~31 B. Cabane, M. Dubois and R. Duplessix, J. Physique 42 (1981) 1359.
10
GELS AND POLYMERS
THE SOL-GEL
TRANSITION
J.F. JOANNY E. N.S. 46 Alltfe d’ltalie 69364, Lyon Cedex 07, France
Invited paper
We briefly review some recent aspects of the sol-gel transition of chemical gels, physical gels and non-organic silica gels focussing on neutron scattering experiments. These results are compared with the percolation model and the FloryStockmayer theory.
1. Introduction
The polycondensation of difunctional monomeric units leads to the formation of linear macromolecules: after a finite reaction time or equivalently a finite fraction of reacted monomers, a polymer solution with a well-characterized polydispersity is obtained but one never gets an infinite molecule. If some multifunctional units are copolymerized with the bifunctional monomers, branched polymers are formed and eventually, after a finite time, an infinite macromolecule spanning through the sample, the gel, is formed [l-3]. Experimentally, the apparition of the gel is associated with a drastic change in the viscoelastic properties of the system: below the gelation point, in the sol phase, the system is a branched polymer solution with a finite viscosity which thus flows; above the gelation point the system has a finite shear modulus and does not flow, the shear modulus grows from zero at the transition but in general remains as low as 10m6times that of conventional solids. An alternative way of defining the gelation transition is to study the weight average molecular weight M, of the finite molecules: in the sol phase, it increases with the fraction of reacted monomers and diverges at the transition, in the gel phase more and more small molecules are incorporated to the infinite cluster and M, decreases with the fraction of reacted monomers. Although the sol-gel transition is a critical point where one characteristic length scale
(namely the size of the largest molecule) diverges, it is not a thermodynamic transition which may be characterized by the singularities of a free energy and in particular it is not associated with any singularity in concentration fluctuations, it is rather a connectivity transition between a sol where the monomers are not connected and a gel where they are connected. As many other critical points, it is in general characterized in terms of a set of (hopefully) universal critical exponents. In this short review, we first briefly present the current theories of the sol-gel transition and discuss their relevance to chemical gels such as those obtained by polycondensation of monomeric units or by chemical crosslinking of semidilute polymer solutions; we discuss in particular neutron scattering experiments in the vicinity of the transition. The last section is devoted to the presentation of other types of gels, physical gels and non-organic silica gels. 2. Chemical 2.1.
gels, the connectivity
critical
point
Theoretical results
As early as 1941, Flory and Stockmayer [4,5] proposed a theoretical description of the sol-gel transition. Starting from monomers with functionality z 3 3, they studied the formation of random trees at a given fraction of reacted monomers, p. This theory ignores thus loop formation in the macromolecule and also any excluded volume interaction between monomers. Within
0921-4526/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
382
these approximations, one relations for the number with n monomers [6]. A predicted for a critical monomers p, = l/(z - 1) .
J.F.
Joanny
I The sol-gel
can derive recursion of molecules (trees) gelation transition is fraction of reacted (1)
A more refined percolation model for the sol-gel transition was proposed independently by Stauffer and deGennes [2,7]. It is a static lattice model where the monomers lie on all sites of a lattice with a functionality z equal to the number of nearest neighbors. Bonds are put on the lattice with a probability p which mimics the fraction of reacted monomers. Below the bond percolation threshold p,, only finite clusters are formed, above the percolation threshold, an infinite cluster corresponding to the gel appears. The threshold explicitely depends on the nature of the lattice and cannot in general be calculated exactly; it is approximately given by p, = 3122.
(2)
It turns out that the Flory-Stockmayer theory is a mean field version of the percolation model. In particular, it is expected to be valid for the vulcanization problem that corresponds to the crosslinking of polymer chains in the melt [2] (the functionality z is here very large, equal to the number of monomers per polymer). From a fundamental point of view, these two theories are not very satisfactory, they mistreat the solvent and ignore the time evolution of the system. This has prompted the development of kinetic theories [lo], aggregation models [ll] and numerical simulations [12]. Although these have been largely developed over the last few years, we will not describe them here, the agreement between experiments and the percolation model being quite good. In the vicinity of the transition we are interested in two kinds of properties, the polydispersity in mass of the branched polymers and their configuration. The polydispersity is very broad and the number of polymers with IZ monomers at a distance from the critical point decays as a “=P-PC power law
transition
C,,(E)- n-‘f{nin*(&>> ,
(3)
where the exponent T is equal to 512 in the mean field theory and to 2.2 in the percolation model; f is a cutoff function which decays exponentially so that there are no macromolecules in the solution larger than a characteristic mass n* which diverges at the gelation point n * - F ’ /” ,
(4)
the exponent (T equal to 112 in mean field and 0.4 in the percolation model. The branched polymers are fractal objects with a fractal dimension d, related to the polydispersity exponent r, 7 = 3 id, + 1 (percolation), i.e. their radius r increases with the mass II as I/J, r--n
,
(5)
the mean field value of the fractal dimension d, = 4 is larger than the space dimension and is thus not acceptable in the limit of infinite polymers. The two scaling laws (3) and (4) allow the determination of all properties in the vicinity of the gelation transition. For example, the weight average molecular weight is defined as c .“2_
(6) it diverges in the vicinity of the transition as Cy with an exponent y = (3 - T) /a. The elastic modulus in the gel phase grows from zero as E - F’. deGennes has proposed an analogy between gel elasticity and conduction on a percolating cluster of resistors [8] which leads to a value p = 1.7, the equivalent mean field value has been calculated by Gordon, p = 3. Another model relates the elasticity to the bending of bonds between crosslinks of the gel and gives a higher exponent I_L= 3.5 (percolation) [9]. 2.2. Experimental results Experimentally the fraction of reacted monomers or percolation probability is hardly known and it is usually assumed that it varies smoothly with the reaction time t around the percolation threshold or that the distance to the gelation
J. F. Joanny
i The sol-gel
transition is proportional to t - tge,, tge, being the time where the transition occurs. With this assumption, some of the scaling laws have been experimentally tested and seem to agree reasonably well with the percolation predictions. There has been several viscoelastic studies of the sol-gel transition [13]. The exponent p obtained for the elastic modulus is in general in good agreement with the electrical analogy although for some gels obtained by polycondensation a much higher exponent p = 3.2 has been observed [3]. The polydispersity in the sol phase has been studied in great details by Schosseler and Leibler [14] for gels obtained by crosslinking a polymer solution by -y-ray irradiation. This creates chemical crosslinks between the chains whose number is controlled by the irradiation time. The reaction was stopped before gelation and the polydispersity measured by size exclusion chromatography coupled to a light scattering determination of the mass of each fraction. The polydispersity could be described in a scaling form by eq. (3) and the scaling function f was experimentally determined; its universality was checked by varying the molecular mass of the precursor polymer. Here again the agreement with the percolation description was quite good. The last set of experiments which we now discuss in more details are small angle neutron scattering experiments. As we already mentioned, no singularity in the monomer concentration fluctuations is expected at the sol-gel transition, they have the same value in the sol and the gel phase. This in turn means that the scattered neutron intensity from the reaction bath does not give any relevant information on the transition. The only way to obtain a useful information is to dilute the system: this allows to study the individual molecules which grow as the transition is approached. It is important to notice however that the dilution procedure changes the quality of the solvent: it is known in polymer physics that as the monomer concentration is increased, the excluded volume interactions are screened. On the reverse, in the dilution experiment the system keeps memory of the architecture and the polydispersity of the branched poly-
transition
383
mers but not of their configuration, the effect of the excluded volume interaction is increased and the fractal dimension D of the polymers in the diluted solution is lower than their fractal dimension d, in the reaction bath. Such dilution experiments have been performed by Bouchaud et al. [15] on gels obtained by polycondensation of polyurethane monomers (for which the fraction of reacted monomers is actually known from the stoichiometry of the reaction) and by Leibler and Schosseler [16] on the irradiated polystyrene gels. Two experiments are needed to measure the polydispersity of the sol. In the first experiment, the dilution is followed by a fractionation and the scattering experiment is made on a well-defined monodisperse sample; it probes the internal structure of the polymers and the scattered intensity decays with the wavevector q as
Z(q)- 4-O> the fractal dimension D in the diluted solution is equal to 1.98 in ref. [ 151, very close to what one expects for randomly branched polymers (animals) in a dilute solution; the fractal dimension of animals is one of the very few exactly known exponents in three dimensions and is equal to D = 217. In the second experiment, the neutron scattering is made on the polydisperse diluted solution, the intensity is averaged over the mass distribution and is expected to vary with q as Z( s) -
4
-N-r)
.
(8)
A comparison between the two experiments gives an exponent r in good agreement with the percolation model. On the whole, the percolation model gives a reasonably good description of the static properties of the sol-gel transition. Few experiments have considered dynamic properties such as the viscoelastic relaxation times or the viscoelastic spectrum of a gelating system, this could provide a very sensitive test of the model [18]. 3. Physical gels, non-organic
gels
3.1. Physical gels
When a semidilute solution is chemically cross-
384
J. F. Joanny
I The sol-gel
linked (such as in a y-ray irradiation experiment), the chemical crosslinks are covalent bonds with a binding energy of the order of one electron-volt, much larger than the thermal energy kT, they cannot be broken neither by thermal excitation nor by any mechanical constraint and the gelation is irreversible. Instead of inducing chemical crosslinks between polymer chains one may induce physical crosslinks with a binding energy of the order of the thermal energy kT. Physical gels are of major practical importance in food or cosmetics industries and the examples are numerous: _ many biopolymers [19] in solution in water have a helix-coil transition at low temperature, in a semidilute solution two different chains may intertwin in a common helix creating a physical crosslink; - the complexation of some ions by hydrosoluble polymers (Cr”’ by hydrolyzed polyacrylamide for instance) [21] forms ionic bridges between different chains which act as crosslinks; - ionomers [20] are usual polyelectrolytes dissolved in nonpolar solvents, the charges are not dissociated and the polymers carry permanent electrical dipoles along the chain; the strong attractive interaction between these dipoles leads to the formation of multiplets and thus of a physical gel. Contrary to chemcial gelation, physical gelation is reversible; in general it can be induced in two ways, either by increasing the polymer concentration or by decreasing the temperature (in some aqueous systems, the ionic strength may also monitor the transition). One can thus define a gelation line in a temperature-concentration phase diagram. The sol-gel transition is not however the only possible phase transition in these systems: the physical bonds are due to an attraction between different monomers which is equivalent to an attraction between different chains; this provokes at low temperature a demixing phase separation between polymer and solvent. The interplay between the thermodynamic demixing transition and the connectivity sol-gel transition creates very subtle effects that are only partially understood [20,21]. Many physical gels also show strong hysteresis
transition
effects, the properties of the gel often depend on its history; this is due to the existence of many metastable states for the crosslinks and suggests that the physical gelation transition shares many common features with a glass transition. There exists a high experimental body of work by many techniques including neutron scattering on well formed physical gels but the description of the sol-gel transition is far worse than for chemical gels. This is in part due to the experimental difficulty of locating the transition. In many cases, even a small mechanical constraint is sufficient to break crosslinks and even in the gel phase the gel flows; connectivity in these systems is not associated with the existence of a non-zero shear modulus at zero frequency and could turn out to be a purely theoretical concept. 3.2. Non-organic gels The manufacturing of certain ceramic glasses of oxydes such asd SiO,, TiO, and ZrO, often goes through the so-called sol-gel route [22]. This involves first the fabrication of a gel in a solvent and then its transformation by drying and viscous sintering. The properties of the final product being strongly dependent on the heterogeneities of the original gel, it seems of prime importance to control the formation of this gel. Such silica gels may be grown in a water alcohol mixture starting from silicon tetraoxyde monomers. The sol-gel transition in this system has been studied in details by Cabane and coworkers [23] both by light scattering and small angle neutron scattering experiments. The neutron scattering data are quite similar to those of ref. [15] on polyurethane polycondensates: - the intensity scattered from the reaction bath shows no signature of the sol-gel transition. It has a peak corresponding to distances of the order of 20 nm which shifts smoothly to lower q values as time evolves, both before and after the transition; - dilution experiments are consistent with a polydispersity decaying as a power law of the molecular mass with an exponent T = 2.2. The behavior of these non organic silica gels in the vicinity of the sol-gel transition looks thus
J. F. Joanny
I The sol-gel
very similar to that of chemical gels although the precise microscopic arrangement of the molecules does not seem well established. Theoretically, it would be of interest to understand the time variation of the characteristic distance observed in the scattering from the reaction bath which could control the heterogeneities of the gel. Acknowledgements
I am grateful to L. Leibler and F. Schosseler for communicating some of their results prior to publication. References ill P.J. Flory, Principles of Polymer Chemistry (Cornell University Press. Ithaca, 1953). PI P.G. deGennes, Scaling Concepts in Polymer Physics (Cornell University Press. Ithaca, 1978). I31 M. Adam, A. Coniglio and D. Stauffer, Advances in Polymer Science 44 (1982) 105. [41 P.J. Flory, J. Am. Chem. Sot. 63 (1941) 3097. [51 W. Stockmayer, J. Chem. Phys. 12 (1944) 125. 161 G.R. Dobson and M. Gordon, J. Chem. Phys. 41 (1964) 2389.
transition
385
[7] D. Stauffer, Introduction of Percolation Theory (Taylor & Francis, London, 1985). [8] P.G. deGennes, J. Physique 37 (1976) Ll. [9] I. Webman, in: Physics of Finely Divided Matter, M. Daoud and N. Boccara, eds. (Springer, Berlin, 1985), p. 186. [lo] P.G.H. Van Dongen and M. Ernst, Phys. Rev. Lett. 56 (1985) 1396. [ll] T. Witten, in: Chance and Matter, J. Souletie, J. Vannimenus and R. Stora, eds. (North-Holland, Amsterdam, 1987). (121 H. Herman, Phys. Rep. 136 (1986) 154. [13] B. Gauthier-Manuel, in: Physics of Finely Divided Matter, M. Daoud and N. Boccara, eds. (Springer, Berlin, 1985), p. 140. F. Schosseler and L. Leibler, Phys. Rev. Lett. 5.5 (1985) 1110. E. Bouchaud et al., J. Physique 47 (1986) 1273. M. Adam et al., J. Physique 48 (1987) 213. F. Schosseler and L. Leibler, private communication. G. Parisi and N. Sourlas, Phys. Rev. Lett. 46 (1981) 871. M. Daoud and J.F. Joanny. J. Physique 42 (1981) 1359. WI J.F. Joanny, J. Physique 43 (1982) 467. P91 A. Clark and S. Ross-Murphy, Adv. Polymer Sci. 83 (1987) 57. PO1 J.F. Joanny, Polymer 21 (1980) 71. WI E. Pezron et al., Macromolecules, (1988), L. Salome, thesis, Paris (1988). I221 C.J. Brinker et al., J. Non-Cryst. Solids 48 (1982) 47. ~31 B. Cabane, M. Dubois and R. Duplessix, J. Physique 42 (1981) 1359.