3.1.1 Definition of a gel
Gels are three-dimensional networks made up of molecules, polymers, particles, colloids, etc., that are connected with each other by the specific parts on themsuch as functional groups and associative groups. The connected parts are called cross-links. Gels usually contain many solvent molecules inside their networks, and hence they are close to liquid in composition, but show solid-like mechanical properties due to the existence of the cross-links [1–4].
Although this statement can be adopted as a formal definition of gels, there are many exceptional ones that do not fall neatly into this categorization. For instance, colloidal suspensions exhibit gel-like rheological behavior at high densities. Entanglements of long rigid fibrillar molecules or an assembly of molecules mutually hinder their motion, and lead to gel-like rheology due to jamming of the rigid segments. In such materials, there are no direct cross-links, but geometrical or topological constraints play simi-lar roles to the cross-links, although they are delocalized. Therefore, to define gels by connectivity only is not sufficient to include these materials.
3.1.2 Classification of gels
Gels can be classified by their constituents. Gels made up of aggregated particles or colloids are particulate gels. Networks made up with covalent bonds, H-bonds, etc., of low molecular weight molecules are low-mass gels. Networks formed by cross-linking of the primary polymers are polymeric gels. If the primary molecules are biopolymers, gels are specifically called biopolymer gels [2].
Gels are also classified into chemical and physical ones by the persistence time (life-time) of their cross-links. Chemical gels (or strong gels) have covalent bonds as the cross-links, so that the connection cannot be broken by thermal motion of the constituent
molecules. The topological structure of a chemical gel is therefore preserved as it is pre-pared [5]. Randomvariables with fixed statistical property but different assigned values depending on the sample are called frozen variables. A systemwith randomvariables as structural parameters is called a random system. Chemical gels are examples of ran-dom systems. A general theoretical scheme to treat ranran-dom systems was developed by Edwards and his collaborators [6,7].
On the other hand, physical gels (or weak gels) are networks cross-linked by physical bonds. The binding energy is of the order of thermal energy, and hence cross-links can be reversibly formed and destroyed by a change in temperature. If the cross-links are sufficiently weak to be created and destroyed by the thermal motion of the constituents, the gels are often called transient gels.
In physical gels, the equilibriumbetween connected and disconnected states is reached if the average lifetimeτ of the cross-links is shorter compared with the time of obser-vation. In the opposite case, the topological structure of the network is observed to be frozen [5]. Because gels are reversibly formed and melted by changing temperature and concentration, physical gels are also called thermoreversible gels [3,4]. However, not all gels are clearly classified into chemical and physical ones, but are distributed in-between the two extremities according to their lifetime.
According to this classification, gels with mobile cross-links, such as sliding ring gels [8], are chemical gels because the number of junctions is preserved. Jamming gels with delocalized cross-links should be regarded as viscoelastic fluids with long relaxation times. There are many gels in which both chemical and physical cross-links coexist.
3.1.3 Structure of gels and their characterization Global structure of a network
A gel has the general network structure shown in Figure 3.1. The part of the chain connecting the neighboring cross-links is a subchain. The branching number of a cross-link is the number of subchains connected in it. It is shown by the number beside each cross-link. A chain with one end connected to a cross-link and the other end free to move is called a free end chain or dangling chain. A path circulating around part of the network along the subchains is a cycle. The total number of independent cycles is the cycle rank of the network. An entangled subchain whose topological relation is preserved is a trapped entanglement. The skeletal structure that remains after all free end chains have been removed is the skeleton of the network.
To characterize the structure of gels, parameters such as the number of cross-links µ, the number of subchains ν, and their average molecular weight M, the branching index of the cross-linksφ, the number of free ends νend, and the cycle rankξ should be specified [1].
For networks with cross-links of a fixed branch number φ, the following two geometrical relations hold:
ν = (µφ +νend)/2, (3.1a)
ξ = ν −(µ+νend)+1. (3.1b)
k = 4
ζ = 8 cross-link (functionality 4)
cross-link (functionality 8) subchain
free end cycle
4 7 4
5 6
4 3
4
Fig. 3.1 Global structure (network topology) and local structure (cross-links) of a network. There are several fundamental parameters that characterize these structures.
(a) (b)
Fig. 3.2 Model networks: (a) perfect network, (b) network made by random cross-linking of primary polymers.
A network with no free ends is called a perfect network (Figure3.2(a)). A network formed by pairwise cross-linking of the primary polymers is a polymer network whose number of free ends is twice as large as the number of primary chains (Figure3.2(b)).
Local structure of the cross-links
Cross-link junctions are important for the elastic properties of gels. The number of chains k combined in a junction is its multiplicity, and the length ζ (in terms of the number of repeat units) is the cross-link length (Figure3.1). Figure3.3shows a network formed by multiple cross-linking of trifunctional molecules. The number beside each junction shows its multiplicity. The multiplicity of the cross-links formed in polycondensation reaction, by cross-linking agencies, etc., is usuallyk = 2.
1
1
1 1
1 3
3
4 2 2
2
2
2 2
2
3
Fig. 3.3 Multiplicity of the cross-link junctions in the polycondensation of trifunctional molecules.
Numbers beside the junction show their multiplicities. Unreacted groups can be regarded as cross-links with multiplicity 1.
3.1.4Examples of gels Chemical gels
Branched polymers are produced by the polycondensation of multifunctional molecules whose functionality is greater than or equal to 3. If the reaction proceeds to the stage where the products grow as large as the space dimensions of the entire system, a three-dimensional network whose parts are connected by covalent bonds is formed. This is the gel point. The reaction continues after this gel point is passed. The polycondensation reaction is irreversible under ordinary conditions, so that the gelation of chemical gels is irreversible.
Let us use the symbol R{Af} for a monomer unit carrying the numberf of A func-tional groups, R{ABf −1} for a monomer carrying one A functional group andf −1 of B functional groups, etc.
As an example, let us consider esterification of tricalvaryl acid and ethyrene glico-hol (Figure3.4). Let us use the symbol A for a –COOH group, and –B for an –OH group. Tricalvaryl acid is the trifunctional monomer R{A3} and ethyrene glicohol is the bifunctional monomer R{B2}. Since the esterification reaction is,
— C# O
–O–H + HO–CH2—−→ — C
# O
–O–CH2— + H2O
three-dimensional branched polymers are produced (Figure3.4).
CH2COOH
Fig. 3.4 Gelation by the polycondensation of trifunctional and bifunctional molecules.
(a) (b) (c)
(d) (e) (f)
Fig. 3.5 Various types of physical cross-links: (a) hydrogen bonds, (b) dipole association, (c) micellar formation of hydrophobic groups, (d) microcrystalline junction, (e) ion association and complex formation, (f) entanglements of long rigid polymers.
Physical gels
Physical cross-linking often makes complex junctions. The cross-links are not strictly localized but extend in the formof junction zones.
(1) Hydrogen bonding
H-bonds between polymer chains formpairwise cross-links and bridge them(Figure 3.5(a)). The binding energy of an H-bond in a solution is of the order of the thermal energy, so that the bonds may easily break and recombine. If there is strong correlation between adjacent H-bonds, as in the complex formation between poly(ethyrene oxide)
and poly(acrylic acid), there is a tendency to formzipper-like contiguous sequences of H-bonds. These are called cooperative H-bonds. Cooperative H-bonds are often observed in the hydration of biopolymers and water-soluble polymers (Section 1.9).
They can be the origin of the sharp conformational transitions of these polymers.
(2) Dipole interaction
If polymers carry dipole moments that are sparsely dispersed along the chains, they are cross-linked by aggregation of the dipole moments (Figure3.5(b)). The aggregates (multiplets) are surrounded by the chain segments, so that there is an upper limit to the multiplicity. Ionomers, such as metal-sulfonated polystyrene, in a nonpolar solvent, are typical examples.
(3) Hydrophobic association
Water-soluble polymers carrying hydrophobic groups, such as short alkyl chains, flu-orocarbon chains, etc., formgels by micellization of the hydrophobic groups in water (Figure3.5(c)). Micelles serve as the cross-links that can dissociate and associate by temperature, external force, added agents, etc. Water-soluble polymers partially modi-fied by hydrophobic associative groups are called associating polymers. Poly(ethylene oxide) and poly(N-isopropylacryl amide), which carry short alkyl chains (–CnH2n+1, n = 12–25) at their chain ends, are typical examples. They are called telechelic poly-mers as both chain ends are active in forming micelles. Because the molecular weight of the main chain and the length of the associative groups can be tuned, telechelic asso-ciating polymers serve as model systems for transient networks in which the junctions can break and recombine. One of the main purposes of this book is to present the recent developments in the research on associating polymers. These will be detailed at the end of Chapters7and9.
(4) Gels with microcrystalline junctions
When crystallizable polymers are quenched below their melting point, they often form gels with microcrystals involving many chains at their junctions (Figure3.5(d)). Junc-tions may have a fringed micellar structure, folded-chain microcrystals, etc. In order for a microcrystal to stay stable, its size should be above the critical nucleus size, and hence there is a lower limit in the multiplicity and sequence length of the junctions. The crystal structure inside the junction may be different from that of the bulk crystals. Gels with small junctions melt at low temperatures. The way that gels form by avoiding crystal-ization depends on the cooling speed, concentration, and other experimental conditions.
Fast cooling, while keeping the systemaway fromthe thermal equilibriumstate, may often lead to gels – as in the formation of glass. For example, isotactic polystyrene, poly(vinyl alcohol), etc., formgels when supercooled.
(5) Complexformation
Gels with zipper-like cross-links of sequential H-bonds, with double or triple helices, with eggbox-shaped complexes involving ions, as shown in Figure3.5(e), may be classified as gels with extended junction zones of complexes. Conformation change of prepolymers
is necessary to formsuch complex junction zones, so that coil–helix transition often takes place before gelation. The gelation of polysaccharides, such as carrageenan and alginate, falls into this category.
(6) Gels with sliding junctions
Recently, new gels have been synthesized by cross-linking ring-shaped cyclodextrins after threading theminto poly(ethylene oxide) chains [8]. The gels have junctions that are mobile along the chains to release the stresses caused by deformation. Gels with such mobile junctions fall into the category of chemical gels in the sense that the number of cross-links is preserved. However, since they can relax the external force by sliding their junctions, their rheological properties are more like physical gels.
(7) Entanglement
Entanglements of long rigid polymers in concentrated solutions and melts often lead to gel-like rheological properties (Figure 3.5(f)) [9]. The entanglements are regarded as delocalized cross-links whose spatial range is difficult to specify. They are created and destroyed by the thermal motion of the polymers or by external force. The number of cross-links is not conserved. The name pseudo-networks is therefore more appropriate for such viscoelastic liquids.