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Associations leading to formation of reversible networks and gels
83
Associations leading to formation of reversible networks and gels Michael Rubinstein and Andrey V Dobrynin One of the most important properties of associative polymers is their ability to form reversible networks and gels. The issue of whether or not the reversible gelation of associative polymers is a thermodynamic transition has not been settled yet. At the same time the understanding of different mechanisms governing the dynamics of associative polymers has been significantly advanced.
Address Department of Chemistry, University of North Carolina, Chapel Hill, NC 27599-3290, USA Current Opinion in Colloid & Interface Science 1999, 4:83-87 Electronic identifier: 1359-0294-004-00083 0 1999 Published by Elsevier Science Ltd. All rights reserved.
ISSN 1359-0294
Introduction Macromolecules with attractive groups belong to a very interesting and important class of polymeric systems associative polymers [ 1-51. This class includes charged polymers (ionomers, polyelectrolytes and polyampholytes), block copolymers in strongly selective solvents, and polymers with hydrogen bonding. T h e considerable interest in these systems is due to their numerous applications, such as rheology modifiers, adhesives, adsorbents, coatings, biomedical implants, flocculents for waste-water treatment, surfactants and stabilizers for heterogeneous polymerizations, and suspending agents for pharmaceutical delivery systems. T h e associations between attractive groups lead to the formation of physical bonds. T h e difference between chemical covalent bonds and physical bonds is that the latter are reversible. There are two types of physical bonds: weak physical bonds that can be broken and formed on experimental time scales; and strong physical bonds that are stable during the time of the experiment but can be broken by changing experimental conditions (e.g. by heating). Examples of weak physical bonds include hydrogen bonds, multiplets in ionomers, and micelles of block or graft copolymers in selective solvents. T h e barriers to break these weak physical bonds do not usually exceed 10-20 times the thermal energy, kT (where R is the Boltzmann constant and T is the absolute temperature). T h e lifetime of these bonds can vary from microseconds to seconds. Hence, polymers form and break these weak physical bonds many times during the course of an experiment. Examples of strong physical bonds are microcrystals, double and triple helices and other very
strong associations that are effectively permanent under fixed experimental conditions. T h e structures formed in solutions of associative polymers depend on polymer concentration, the number of attractive groups per chain and the strength of physical bonds. In dilute solutions polymers may self-associate by forming physical bonds between the attractive groups of the same chain or by aggregating into a complex containing several molecules (e.g. a micelle). At higher concentrations a network of polymer chains (a physical gel) can be formed. This interchain association into the physical network results in the increase of solution viscosity and the appearance of high elasticity. T h e classification of physical bonds into two types leads to the analogous classification for physical gels - strong and weak physical gels. T h e strong physical gels are soft-solids and have finite elastic moduli on experimental time scales. In the network state their properties depend both on the preparation conditions under which the physical bonds were formed as well as on the test conditions. Strong physical gels are therefore analogous to covalently bonded chemical gels - their bonds are permanent at fixed experimental conditions. Weak physical gels are qualitatively different from chemical gels and are closer to viscoelastic liquids. We limit our discussion below to solutions of associative polymers with weak physical bonds. T h e presence of reversible cross-links in weak physical gels increases the effective attraction between polymers, reducing their affinity to the solvent. There is a delicate balance between association and phase separation in these systems. Increasing the strength of attraction or the number of associative groups per chain leads to stronger aggregation and, in some cases, could result in the separation into phases with different polymer concentrations. We discuss the thermodynamic aspects of gelation and phase separation in the following section. T h e dynamics of associative polymer solutions is reviewed in the third section.
Thermodynamics of associative polymers One of the simplest theoretical approaches to the problem of associative polymers with weak physical bonds is to model them as molecules which carry several functional groups each. T h e functional groups on the chains are capable of forming weak physical bonds by association. These groups could be functional monomeric units,
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charged groups, hydrophobic side chains etc. T h e weak physical bonds are thermally equilibrated on experimental time scales. As more bonds between molecules are formed, the resulting branched aggregates are larger and the distribution of their sizes is broader. A macroscopic cluster, called a gel, scanning the whole system is formed at a certain fraction of formed bonds, called a gel point. T h e transition from a polydisperse system of branched aggregates (sol) to a mixture of a macroscopic gel and finite branched aggregates is called gelation. Despite of half a century of studies [6-20",21-23",24] including m a n y recent theoretical works [17-20",21-23",24] the question of whether or not the reversible gelation of associative polymers is a thermodynamic transition is still not settled. Below we present our point of view on this controversial issue. T h e modern theory that describes the gelation transition is called the percolation theory [25]. T h e mean field limit of the percolation theory was developed by Flory [6] and Stockmayer [7] as an analytical theory of gelation long before the percolation theory was introduced [26]. This mean field limit of the percolation theory is called in the literature the Flory-Stockmayer theory, but this name is misleading. T h e original Flory theory of gelation [6] and the Stockmayer version of it [71 have very different predictions. In the pre-gel regime the Flory and Stockmayer approaches are the same and assume a tree-like structure of finite clusters of chains (sol). In the post-gel regime these approaches are based on different assumptions. T h e Flory model treats the finite clusters in the post-gel regime in the identical way to the pre-gel regime. For systems further above the gel point the finite clusters get smaller and their distribution gets narrower. T h e gel in the Flory theory is allowed to have cycles. T h e Stockmayer model does not allow cyclization in the gel and assumes that the molecular weight distribution of the sol fraction does not change in the post-gel region and is the same as at the gel point. T h e description of the gelation transition itself is very different in the two theories. T h e Flory theory does not predict a thermodynamic transition at the gel point. In the framework of the Stockmayer approach the weak physical gelation is a third order phase transition with a discontinuity in the derivative of the isothermal osmotic compressibility [181. This disagreement is still not resolved, with some scientific groups accepting Flory approach with no thermodynamic transition at the gel point, while other groups predict third or even second order thermodynamic transitions. T h e Stockmayer approach with a phase transition at the gelation line was adopted by Tanaka and co-workers in their extensive study of the reversible gelation. They have calculated the phase diagrams for a number of polymeric systems. These include the solutions of multi-functional associative polymers [13,141, the mixture of multi-functional A- and B-polymers [15], two-
component physical gels [171 and associative polymers with junctions of variable multiplicity [ 181. We believe that the Stockmayer model is based on the wrong assumptions (tree-like structure of a gel and a constant molecular weight distribution of the sol friction above the gel point) and therefore the results of Tanaka and co-workers are incorrect. T h e misleading comparison between Flory and Stockmayer approaches in the postgel region for multi-functional associative polymers was recently published by Ishida and Tanaka [ZO"]. Another theory predicting gelation to be a thermodynamic transition has been recently proposed by Erukhimovich [24]. In the framework of the field theoretical approach the author shows that the reversible gelation is a second order phase transition. In order to account for the formation of the cycles in the post-gel regime an additional order parameter is introduced. This new description of the gel fraction, however, results in the violation of the physical equilibrium - the chemical potential of monomers in the gel is not properly balanced by the one in the sol. Several detailed considerations [8-11,21-23**] of the post-gel region support the Flory approach and show that the reversible gelation is not a thermodynamic transition at all. T h e free energy of the system and all its derivatives are analytical functions at the gel point. T h e Flory-like model of the physical gelation was applied by Drye and Cates [211 to associations in solutions of rod-like polymers, Gaussian chains and self-avoiding chains, by Nyrkova and co-workers [22] to solutions of telechelic polymers and by Semenov and Rubinstein [23"] to multi-functional associative polymers. T h e mean field theory has been extended to take into account local loops and excluded volume interactions [21,23" 1. We think that the Flory model of gelation [6], in combination with the lattice theory of polymer solutions proposed by Flory and Huggins [27], provide a correct mean field description of associative polymers well above their overlap concentration. T h e gelation is a connective transition but not a thermodynamic one. A typical phase diagram for a solution of associative polymers in terms of polymer concentration c and temperature T is presented in Figure 1. T h e diagram shows a single-phase region bounded by a coexistence curve (solid line in Figure 1) - a dome-shaped miscibility gap with an upper critical solution point. Below the coexistence curve the system is unstable with respect to macrophase separation into a concentrated and a dilute phase. T h e single-phase region itself is divided into two parts by the gelation line (dashed line in Figure 1) which is not a phase boundary. If several competing species are involved in the formation of the physical network the resultant phase diagram can have a reentrant transition and a more complicated topology (closed-loop miscibility gaps at higher temperatures together with or without dome-shaped
Associations leading to formation of reversible networks and gels Rubenstein and Dobrynin
Fiaure 1
T
0
0
/
GEL
I
SOL
I I
,’
One phase
C Current Opinion in Colloid &Interface Science
Schematic phase diagram of the solution of associative polymers. The coexistence curve (thick solid line) is the boundary between one phase and two-phase regions. The gelation threshold is shown as
miscibility gaps with an upper critical solution point). This type of phase diagram is predicted for the gelation of polyelectrolyte solutions in the presence of the multivalent salt ions where the complexes between dissociated salt groups on polymers and multivalent ions act as effective cross-links [28-301 and for the gelation of hydrated polymers [31]. A complementary approach to the solutions of associative polymers explicitly takes the internal structure of aggregates into account (micelles in amphiphilic polymers or multiplets in ionomers) and has its origin in the micellization of block copolymers [32-35.1. T h e micelles connected by bridges form a temporary (reversible) network with micellar aggregates as junction points. T h e formation of these bridges between micelles gives rise to an effective attractive interaction that could result in the phase separation. A detailed review of this approach has been recently presented by Borisov and Halperin [35’1.
Linear dynamics of associative polymers T h e rheology of the solution of associative polymers changes dramatically near the gelation threshold. There are four rheological regimes with qualitatively different relaxation mechanisms [36’1. T h e first regime is the chemical gelation regime [36*’,37,38]. Far below the gel point only small clusters of chains are formed by reversible bonds. T h e relaxation time, rrefaS, of these small clusters (branched
85
polymers) is shorter than their lifetime, T ~ Therefore, ~ ~ ~ rheology in this regime is similar to that for chemical gelation with branched polymers formed by permanent bonds. T h e dynamics of these branched clusters is partially draining (Rouse-Zimm-like dynamics) [39,401, with diffusion coefficients inversely proportional to their molecular weights [37,381. T h e second regime is that of dynamical clusters [36*’,41,42]. As polymer concentration increases further approaching gelation concentration cg, clusters of chains grow very large and their relaxation time, T , , ~ ~ in~ , creases dramatically. However, the lifetime, T f i f e , of these clusters decreases because larger clusters do not survive as long as smaller ones. Indeed, it is not enough to break any ‘crucial’ bond to separate these clusters into two large parts. T h e system behaves as a polydisperse solution of clusters up to the crossover size. T h e relaxation time of these crossover dynamical clusters is equal to their lifetime, where T~~~~= T,,/,,. These dynamical clusters dominate the rheology of associative polymers at the gelation transition (and in its vicinity). T h e system does not ‘feel’ that it gets closer to the gel point because the giant clusters break before they begin to participate in a coherent motion. This regime of dynamical clusters continues above the gelation threshold until the elastic strands of the temporary network become smaller than these crossover dynamical clusters. T h e viscosity in this dynamical cluster regime is almost independent of the extent of reaction both below and above the gel point. T h e third regime is the bond breaking regime. When the elastic strands of the reversible network are smaller than the crossover dynamical clusters, but still larger than individual chains, the relaxation of solution is controlled by breaking ‘crucial’ bonds along these elastic strands. T h e viscosity in this regime grows as a relatively high power of the difference of the extent of the reaction from the gel point. T h e final regime is the sticky motion regime. At higher concentrations the network is well-developed and consists of multiple connected chains. Each chain (on average) is connected to the network by more than one reversible bond. In the bond-breaking regime the cutting of a single ‘crucial’ bond leads to the relaxation of the system, while in the sticky motion regime several bonds need to be broken (and reformed many times) for the stress to relax. These bonds become effective friction centers, as they typically dominate the energy dissipation (friction) of the reversible network. T h e dynamics of the unentangled reversible gels far above the gelation transition are described by the ‘stickyRouse’ model [43-481. Above the entanglement threshold the sticky motion of the chains has to be modified to allow polymers to reptate [49] along the confining tubes formed by their neighbors (sticky-reptation model [50,51]). Another important feature in the dynamics of reversible networks is the transformation of intrachain
.
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Acknowledgements
Figure 2
T h e authors are grateful to L Bromberg for providing them with the experimental data for Figure 2 and to S Panyukov for illuminating discussions.
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References and recommended reading Papers of particular interest, published within the annual period of review, have been highlighted as:
-0
Dynamical clusters 0.01
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0.1
1.o
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(C-CJ/C*
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Pineri M, Eisenberg A (Eds): Structure and Propertiesof lonomers. Dordrecht: D Reidel Publishing; 1987. [NATO AS1 Series vol 198.1
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Calbo LJ (Ed): Handbook of Coating Adhesives, vol 2. New York: Marcel Dekker; 1993.
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Goethals EJ (Ed): Telecbelic Polymers:Syntbesis and Applications. Dallas: CRC Press; 1989.
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Schlick S (Ed): lonomers: Characterization, Theory and Applications. CRS Press; 1996.
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Stockmayer WH: Theory of molecular size distribution and gel formation in branched-chain polymers. J Cbem Pbys 1943, 11:45-55.
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Coniglio A, Stanley EH, Klein W: Site-bond correlated-percolation problem: a statistical mechanical model of polymer gelation. PbysRev Lett 1979,42:518-522.
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Current ODinion in Calloid & InterfaceScience
The viscosity h of the thermoreversible hydrogeis of poly(ethy1ene oxide)-b-poly(propy1ene oxide)-b-poly(ethy1ene oxide)-g-poly(acry1ic acid) as a function of the relative distance to the gel.
bonds into interchain ones. T h e dependence of the solution viscosity on concentration in the part of the sticky motion regime where the intrachain bonds are transformed into interchain ones is extremely strong. Most of the above predictions were recently confirmed by Bromberg [52**] for the reversible gelation of polyacrylic acid grafted with pluronic triblocks (see Figure 2) and by English et al. [53] for solutions of a hydrophobically-modified alkali-soluble associative polymer.
10.
Daoud M, Coniglio A: Singular behavior of the free energy in the gel-sol transition. J PbysA 1981, 14:L301-L305.
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Panyukov SV: Replica technique in the theory of equilibrium polymer systems. Sov PbysJETPl985, 61:1065-1072.
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Kuchanov SI, Korolev SV, Panyukov SV: Graphs in chemical physics of polymers. Adv PolymSci 1987, 72:115-324.
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Tanaka F, Matsuyma A: Tricriticality in thermoreversible gels. Pbys Rev Lett 1989, 62:2759-2762.
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Tanaka F: Theory of thermoreversiblegelation. Macromolecules 1989, 22:1988-1994.
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Tanaka F: Thermodynamictheory of network-forming polymer solutions. Macromolecules 1990, 23:3784-3795.
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Stockmayer W: Thermoreversible gelation via multichain junctions. Macromolecules 1991, 24:6367-6368.
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Tanaka F, lshida M: Phase formation of two-component physical gels. PbysicaA 1994, 204:660-672.
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Tanaka F, Stockmayer WH: Thermoreversiblegelation with junctions of variable multiplicity. Macromolecules 1994, 27:3943-3954.
19.
Tanaka F: Phase formation of associative polymers: gelation, phase separation and microphase separation. Adv Colloid Interface Sci 1996, 63:23-32.
Conclusions Recently there has been a renewed interest in the fascinating properties of associative polymers. T h e works reviewed above illustrate a wide spectrum in opinions on the nature of gelation transition. We believe that this controversy will soon be resolved by a combination of careful experiments, computer simulations and theory. Several interesting ideas have been proposed on linear dynamics of the solutions of associative polymers near the gelation threshold and well above it. These ideas have to be tested experimentally and numerically before they can be extended to non-linear dynamics of these solutions. We are certain that this exciting area of polymer science will witness a number of major breakthroughs in the near future.
of special interest of outstanding interest
lshida M, Tanaka F: Theoretical study of the post-gel regime in thermoreversiblegelation. Macromolecules 1997, 30:3900-3909. The paper compares the Flory and Stockmayer models of reversible gelation. Unfortunately the authors have used a wrong expression for the free energy of the Flory model in the post-gel regime. Therefore, the phase diagrams obtained in the paper for the Flory model are, in our opinion, incorrect. 20.
0.
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Drye TJ, Cates ME: Living networks 2 the role of cross-links in entangled surfactant solutions. J Chem Phys1992,96:1367-1375.
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Semenov AN, Nyrkova IA, Cates ME: Phase equilibria in solutions of associating telechelic polymers: rings versus reversible network. Macromolecules 1995, 28:7879-7885.
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Semenov AN, Rubinstein M: Thermoreversible gelation in solutions of associating polymers: 1. Statics. Macromolecules 1998, 31 :1373-1385. In this paper the mean-field Flory model is rederived and generalized to take into account local intrachain loops as well as excluded volume interactions.
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Cates ME: Brownian dynamics of self-similar macromolecules. J PhysFrance 1985, 46:l 059-1 077.
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Muthukumar M: Dynamics of polymeric fractals. J Chem PhySl985,83:3161-3168.
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Rubinstein M, Colby RH, Gillmor JR: Dynamic scaling for polymer gelation. In Space-time Organizations in Macromolecular fluids. Tanaka F, Doi M, Ohta T (Eds). Berlin: Springer-Verlag; 1989. [Springer Series in Chemical Physics, vol 51 .I
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Colby RH, Gillmor JR, Rubinstein M: Dynamics of near critical polymer gels. Phys Rev E 1993, 48:3712-3716.
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Erukhimovich IYA: Statistical theory of sollgel transition in weak gels. Sov PhysJ€TPl995,81:553-566.
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Cates ME: Gelation and associating polymers. NATO AS/ Series 1989, 21 1:319-329.
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Stauffer D, Aharony A: Introduction to Percolation Theory. London: Taylor and Frances; 1992.
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Essam JW: Percolationand cluster size. In PhaseTransition and Critical Phenomena. Domb C, Green MC (Eds). New York: Academic Press; 1973.
Cates ME, McLeish TCB, Rubinstein M: Living trees: dynamics at a reversible classical gel point. J PhysCond Matter 1990, 2:749-754.
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Green MS, Tobolsky AV: New approach to the theory of relaxing polymer media. J Chem Phys1946, 14:80-92.
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Flory PJ: Principles of Polymer Chemistry. London: Cornell University Press; 1953.
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Keita G, Ricard A, Audebert R, Pezron E, Leibelr L: The poly(viny1alcohol)-borate systems: influence of polyelectrolyte effects on phase diagram. Polymer 1995, 36:49-54.
Tobolsky AV: Propetfiesand Structure of Polymers.New York: Wiley; 1960.
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Lodge AS: Elastic Liquids. London: Academic Press; 1964.
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Baxandal LG, Edwards SF: Deformation-dependent properties of polymer networks constructed by addition of cross-links under strain. Macromolecules1988, 21:1763-1772.
47.
Jenkins RD, Silebi CA, El-Aasser MS: Polymers as rheology modifiers. Washington: American Chemical Society; 1991:222-233. [ACS Symposium Series vol 462.1
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Tanaka F, Edwards SF: Viscoelastic properties of physically cross-linked networks. Transient network theory. Macromolecules 1992, 251 51 61 1523.
49.
Doi M, Edwards SF: The Theory of PolymerDynamics.Oxford: Clarendon Press; 1985.
50.
Leibler L, Rubinstein M, Colby RH: Dynamics of reversible networks. Macromolecules 1991, 24:4701-4712.
51.
Leibler L, Rubinstein M, Colby RH: Dynamics of telechelics ionomers. Can polymers diffuse large distances without relaxing stress? J Physlll993, 3:1581-1590.
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Wittmer J, Johner A, Joanny J-F: Precipitation of polyelectrolytes in the presence of multivalent salts. J PhySlll995, 5~635-654.
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Olvera de La Cruz M, Belloni L, Delsanti M, Dalbiez JP, Spalla 0, Drifford M: Precipitation of highly charged polyelectrolyte solutions in the presence of multivalent salts. J Chem Phys1995, 95:5781-5791.
31.
Tanaka F, lshida M: Thermoreversible gelation of hydrated polymers. J Chem SOCFaraday Trans 1995,91:2366-2670.
32.
Semenov AN, Joanny J-F, Khokhlov AR: Associating Dolvmers: eauilibrium and linear viscoelasticitv. Ma&-omolecdes 1995, 28:1066-1075.
33.
Borisov OV, Halperin A: Micelles of polysoaps. Langmuir 1995, 11:2911-2920.
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Borisov OV, Halperin A: Micelles of polysoaps: the role of bridging interactions. Macromolecules1996,29:2612-2618.
Borisov OV, Halperin A: Self assembly of polysoaps. Curr Opin Colloid lntehwe Sci 1998, 3:415-421. The paper presents an overview of the recent theoretical models of polysoaps. 35.
Rubinstein M, Semenov AN: Thermoreversible gelation in solutions of associating polymers. 2. Linear dynamics. Macromolecules 1998, 31 :1386-1397. The paper presents a detailed analysis of the Rouse-Zimm dynamics of associative polymers below and above the gel point. It was shown that multiple dissociations and recombinations between the same pair of associative groups lead to the increase in the apparent activation energy and slow down the relaxation process. 36.
--
Bromberg L: Scaling of rheological properties of hydrogels from associating polymers. Macromolecules 1998, 31 16148-6156. Rheological properties of thermoreversible hydrogels are demonstrated to be in good agreement with the Rouse-Zimm model of associative polymers [36**]. 52.
53. English RJ, Gulati HS, Jenkins RD, Khan SA: Solution rheology of a hydrophobically modified alkalisoluble associative polymers. J Rheoll997, 41 :427-444.
Associations leading to formation of reversible networks and gels Michael Rubinstein and Andrey V Dobrynin One of the most important properties of associative polymers is their ability to form reversible networks and gels. The issue of whether or not the reversible gelation of associative polymers is a thermodynamic transition has not been settled yet. At the same time the understanding of different mechanisms governing the dynamics of associative polymers has been significantly advanced.
Address Department of Chemistry, University of North Carolina, Chapel Hill, NC 27599-3290, USA Current Opinion in Colloid & Interface Science 1999, 4:83-87 Electronic identifier: 1359-0294-004-00083 0 1999 Published by Elsevier Science Ltd. All rights reserved.
ISSN 1359-0294
Introduction Macromolecules with attractive groups belong to a very interesting and important class of polymeric systems associative polymers [ 1-51. This class includes charged polymers (ionomers, polyelectrolytes and polyampholytes), block copolymers in strongly selective solvents, and polymers with hydrogen bonding. T h e considerable interest in these systems is due to their numerous applications, such as rheology modifiers, adhesives, adsorbents, coatings, biomedical implants, flocculents for waste-water treatment, surfactants and stabilizers for heterogeneous polymerizations, and suspending agents for pharmaceutical delivery systems. T h e associations between attractive groups lead to the formation of physical bonds. T h e difference between chemical covalent bonds and physical bonds is that the latter are reversible. There are two types of physical bonds: weak physical bonds that can be broken and formed on experimental time scales; and strong physical bonds that are stable during the time of the experiment but can be broken by changing experimental conditions (e.g. by heating). Examples of weak physical bonds include hydrogen bonds, multiplets in ionomers, and micelles of block or graft copolymers in selective solvents. T h e barriers to break these weak physical bonds do not usually exceed 10-20 times the thermal energy, kT (where R is the Boltzmann constant and T is the absolute temperature). T h e lifetime of these bonds can vary from microseconds to seconds. Hence, polymers form and break these weak physical bonds many times during the course of an experiment. Examples of strong physical bonds are microcrystals, double and triple helices and other very
strong associations that are effectively permanent under fixed experimental conditions. T h e structures formed in solutions of associative polymers depend on polymer concentration, the number of attractive groups per chain and the strength of physical bonds. In dilute solutions polymers may self-associate by forming physical bonds between the attractive groups of the same chain or by aggregating into a complex containing several molecules (e.g. a micelle). At higher concentrations a network of polymer chains (a physical gel) can be formed. This interchain association into the physical network results in the increase of solution viscosity and the appearance of high elasticity. T h e classification of physical bonds into two types leads to the analogous classification for physical gels - strong and weak physical gels. T h e strong physical gels are soft-solids and have finite elastic moduli on experimental time scales. In the network state their properties depend both on the preparation conditions under which the physical bonds were formed as well as on the test conditions. Strong physical gels are therefore analogous to covalently bonded chemical gels - their bonds are permanent at fixed experimental conditions. Weak physical gels are qualitatively different from chemical gels and are closer to viscoelastic liquids. We limit our discussion below to solutions of associative polymers with weak physical bonds. T h e presence of reversible cross-links in weak physical gels increases the effective attraction between polymers, reducing their affinity to the solvent. There is a delicate balance between association and phase separation in these systems. Increasing the strength of attraction or the number of associative groups per chain leads to stronger aggregation and, in some cases, could result in the separation into phases with different polymer concentrations. We discuss the thermodynamic aspects of gelation and phase separation in the following section. T h e dynamics of associative polymer solutions is reviewed in the third section.
Thermodynamics of associative polymers One of the simplest theoretical approaches to the problem of associative polymers with weak physical bonds is to model them as molecules which carry several functional groups each. T h e functional groups on the chains are capable of forming weak physical bonds by association. These groups could be functional monomeric units,
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charged groups, hydrophobic side chains etc. T h e weak physical bonds are thermally equilibrated on experimental time scales. As more bonds between molecules are formed, the resulting branched aggregates are larger and the distribution of their sizes is broader. A macroscopic cluster, called a gel, scanning the whole system is formed at a certain fraction of formed bonds, called a gel point. T h e transition from a polydisperse system of branched aggregates (sol) to a mixture of a macroscopic gel and finite branched aggregates is called gelation. Despite of half a century of studies [6-20",21-23",24] including m a n y recent theoretical works [17-20",21-23",24] the question of whether or not the reversible gelation of associative polymers is a thermodynamic transition is still not settled. Below we present our point of view on this controversial issue. T h e modern theory that describes the gelation transition is called the percolation theory [25]. T h e mean field limit of the percolation theory was developed by Flory [6] and Stockmayer [7] as an analytical theory of gelation long before the percolation theory was introduced [26]. This mean field limit of the percolation theory is called in the literature the Flory-Stockmayer theory, but this name is misleading. T h e original Flory theory of gelation [6] and the Stockmayer version of it [71 have very different predictions. In the pre-gel regime the Flory and Stockmayer approaches are the same and assume a tree-like structure of finite clusters of chains (sol). In the post-gel regime these approaches are based on different assumptions. T h e Flory model treats the finite clusters in the post-gel regime in the identical way to the pre-gel regime. For systems further above the gel point the finite clusters get smaller and their distribution gets narrower. T h e gel in the Flory theory is allowed to have cycles. T h e Stockmayer model does not allow cyclization in the gel and assumes that the molecular weight distribution of the sol fraction does not change in the post-gel region and is the same as at the gel point. T h e description of the gelation transition itself is very different in the two theories. T h e Flory theory does not predict a thermodynamic transition at the gel point. In the framework of the Stockmayer approach the weak physical gelation is a third order phase transition with a discontinuity in the derivative of the isothermal osmotic compressibility [181. This disagreement is still not resolved, with some scientific groups accepting Flory approach with no thermodynamic transition at the gel point, while other groups predict third or even second order thermodynamic transitions. T h e Stockmayer approach with a phase transition at the gelation line was adopted by Tanaka and co-workers in their extensive study of the reversible gelation. They have calculated the phase diagrams for a number of polymeric systems. These include the solutions of multi-functional associative polymers [13,141, the mixture of multi-functional A- and B-polymers [15], two-
component physical gels [171 and associative polymers with junctions of variable multiplicity [ 181. We believe that the Stockmayer model is based on the wrong assumptions (tree-like structure of a gel and a constant molecular weight distribution of the sol friction above the gel point) and therefore the results of Tanaka and co-workers are incorrect. T h e misleading comparison between Flory and Stockmayer approaches in the postgel region for multi-functional associative polymers was recently published by Ishida and Tanaka [ZO"]. Another theory predicting gelation to be a thermodynamic transition has been recently proposed by Erukhimovich [24]. In the framework of the field theoretical approach the author shows that the reversible gelation is a second order phase transition. In order to account for the formation of the cycles in the post-gel regime an additional order parameter is introduced. This new description of the gel fraction, however, results in the violation of the physical equilibrium - the chemical potential of monomers in the gel is not properly balanced by the one in the sol. Several detailed considerations [8-11,21-23**] of the post-gel region support the Flory approach and show that the reversible gelation is not a thermodynamic transition at all. T h e free energy of the system and all its derivatives are analytical functions at the gel point. T h e Flory-like model of the physical gelation was applied by Drye and Cates [211 to associations in solutions of rod-like polymers, Gaussian chains and self-avoiding chains, by Nyrkova and co-workers [22] to solutions of telechelic polymers and by Semenov and Rubinstein [23"] to multi-functional associative polymers. T h e mean field theory has been extended to take into account local loops and excluded volume interactions [21,23" 1. We think that the Flory model of gelation [6], in combination with the lattice theory of polymer solutions proposed by Flory and Huggins [27], provide a correct mean field description of associative polymers well above their overlap concentration. T h e gelation is a connective transition but not a thermodynamic one. A typical phase diagram for a solution of associative polymers in terms of polymer concentration c and temperature T is presented in Figure 1. T h e diagram shows a single-phase region bounded by a coexistence curve (solid line in Figure 1) - a dome-shaped miscibility gap with an upper critical solution point. Below the coexistence curve the system is unstable with respect to macrophase separation into a concentrated and a dilute phase. T h e single-phase region itself is divided into two parts by the gelation line (dashed line in Figure 1) which is not a phase boundary. If several competing species are involved in the formation of the physical network the resultant phase diagram can have a reentrant transition and a more complicated topology (closed-loop miscibility gaps at higher temperatures together with or without dome-shaped
Associations leading to formation of reversible networks and gels Rubenstein and Dobrynin
Fiaure 1
T
0
0
/
GEL
I
SOL
I I
,’
One phase
C Current Opinion in Colloid &Interface Science
Schematic phase diagram of the solution of associative polymers. The coexistence curve (thick solid line) is the boundary between one phase and two-phase regions. The gelation threshold is shown as
miscibility gaps with an upper critical solution point). This type of phase diagram is predicted for the gelation of polyelectrolyte solutions in the presence of the multivalent salt ions where the complexes between dissociated salt groups on polymers and multivalent ions act as effective cross-links [28-301 and for the gelation of hydrated polymers [31]. A complementary approach to the solutions of associative polymers explicitly takes the internal structure of aggregates into account (micelles in amphiphilic polymers or multiplets in ionomers) and has its origin in the micellization of block copolymers [32-35.1. T h e micelles connected by bridges form a temporary (reversible) network with micellar aggregates as junction points. T h e formation of these bridges between micelles gives rise to an effective attractive interaction that could result in the phase separation. A detailed review of this approach has been recently presented by Borisov and Halperin [35’1.
Linear dynamics of associative polymers T h e rheology of the solution of associative polymers changes dramatically near the gelation threshold. There are four rheological regimes with qualitatively different relaxation mechanisms [36’1. T h e first regime is the chemical gelation regime [36*’,37,38]. Far below the gel point only small clusters of chains are formed by reversible bonds. T h e relaxation time, rrefaS, of these small clusters (branched
85
polymers) is shorter than their lifetime, T ~ Therefore, ~ ~ ~ rheology in this regime is similar to that for chemical gelation with branched polymers formed by permanent bonds. T h e dynamics of these branched clusters is partially draining (Rouse-Zimm-like dynamics) [39,401, with diffusion coefficients inversely proportional to their molecular weights [37,381. T h e second regime is that of dynamical clusters [36*’,41,42]. As polymer concentration increases further approaching gelation concentration cg, clusters of chains grow very large and their relaxation time, T , , ~ ~ in~ , creases dramatically. However, the lifetime, T f i f e , of these clusters decreases because larger clusters do not survive as long as smaller ones. Indeed, it is not enough to break any ‘crucial’ bond to separate these clusters into two large parts. T h e system behaves as a polydisperse solution of clusters up to the crossover size. T h e relaxation time of these crossover dynamical clusters is equal to their lifetime, where T~~~~= T,,/,,. These dynamical clusters dominate the rheology of associative polymers at the gelation transition (and in its vicinity). T h e system does not ‘feel’ that it gets closer to the gel point because the giant clusters break before they begin to participate in a coherent motion. This regime of dynamical clusters continues above the gelation threshold until the elastic strands of the temporary network become smaller than these crossover dynamical clusters. T h e viscosity in this dynamical cluster regime is almost independent of the extent of reaction both below and above the gel point. T h e third regime is the bond breaking regime. When the elastic strands of the reversible network are smaller than the crossover dynamical clusters, but still larger than individual chains, the relaxation of solution is controlled by breaking ‘crucial’ bonds along these elastic strands. T h e viscosity in this regime grows as a relatively high power of the difference of the extent of the reaction from the gel point. T h e final regime is the sticky motion regime. At higher concentrations the network is well-developed and consists of multiple connected chains. Each chain (on average) is connected to the network by more than one reversible bond. In the bond-breaking regime the cutting of a single ‘crucial’ bond leads to the relaxation of the system, while in the sticky motion regime several bonds need to be broken (and reformed many times) for the stress to relax. These bonds become effective friction centers, as they typically dominate the energy dissipation (friction) of the reversible network. T h e dynamics of the unentangled reversible gels far above the gelation transition are described by the ‘stickyRouse’ model [43-481. Above the entanglement threshold the sticky motion of the chains has to be modified to allow polymers to reptate [49] along the confining tubes formed by their neighbors (sticky-reptation model [50,51]). Another important feature in the dynamics of reversible networks is the transformation of intrachain
.
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Self-assembly
Acknowledgements
Figure 2
T h e authors are grateful to L Bromberg for providing them with the experimental data for Figure 2 and to S Panyukov for illuminating discussions.
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References and recommended reading Papers of particular interest, published within the annual period of review, have been highlighted as:
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Dynamical clusters 0.01
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Conclusions Recently there has been a renewed interest in the fascinating properties of associative polymers. T h e works reviewed above illustrate a wide spectrum in opinions on the nature of gelation transition. We believe that this controversy will soon be resolved by a combination of careful experiments, computer simulations and theory. Several interesting ideas have been proposed on linear dynamics of the solutions of associative polymers near the gelation threshold and well above it. These ideas have to be tested experimentally and numerically before they can be extended to non-linear dynamics of these solutions. We are certain that this exciting area of polymer science will witness a number of major breakthroughs in the near future.
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