The theory of polymer dynamics

612

The theory of polymer dynamics Glenn H Fredrickson Recent

years have brought

to understanding

under nonequilibrium diffusion-controlled molecular

insights theories

aid in the design

to reactive relating

facilitate

the design

rheological

blending agents

systems in

are bringing

technologies,

to associating

theories

polymers

and should

and coatings.

Recent

of flow and deformation

of branched

properties

advances

Developments

of polymers

of thickening

in molecular

theoretical

of macromolecular

conditions. reactions

improved progress

exciting

the behavior

polymers

and improved

may

with tailored

adhesives.

Addresses Departments of Chemical Engineering and Materials, University of California, Santa Barbara, CA 93106,

USA; e-mail:

[email protected] Current

Opinion

in Solid State & Materials Science 1996,

1:612-616 0 Current Chemistry Ltd ISSN 1359-0266 Abbreviations cmc

By incorporating functional groups into the chains of the engineering polymers, complementary groups from the two phases can meet in the interfacial regions and react to form a graft copolymer. This interfacial reaction is generally carried out in parallel with the blending operation. A significant advantage of reactive blending is that the copolymer is delivered at the interface, precisely where it is needed, avoiding the task of dispersing it from a third phase by mechanical agitation.

critical micelle concentration

Introduction The science of synthetic polymers has grown to be an extremely rich and active field over the past few decades. Theoretical notions have proved to be invaluable in the formulation of new plastic parts, coatings, and adhesives, and in the design and operation of the plants necessary for the commercial-scale production of such materials. Perhaps in no other branch of materials science is theory so inextricably linked with both experiment and industrial practice. Because of the present high level of activity in theoretical polymer science, it is difficult to provide a comprehensive survey of even the most recent developments. Thus, I have restricted the scope of this review to nonequilibrium phenomena, which, in my opinion, is where the most exciting advances in polymer theory are occurring today. Unfortunately, space limitations do not permit an exhaustive coverage of this subfield: thus, I have simply tried to highlight a few of the most interesting theoretical developments. I apologize in advance for not being able to discuss many other important pieces of work dealing with related issues.

Reacting and associating Reactive

formation of plastics with large inclusions, possessing poor optical properties and often poor mechanical properties arising from weak interfaces. As a result, polymer ‘compatibilizers’ are added (usually some type of copolymer) in order to strengthen the interfaces between the alloy’s components and reduce the average particle size. While a few commercial polymer blends are produced by melt-mixing a preformed compatibilizer with two or more engineering polymers, most preparations actually involve the reactive formation the compatibilizer in situ.

polymer systems

blending

Reactive processing has become an important technology for control of morphology in polymer blends. As the miscibility of engineered polymer alloys is generally very limited, mechanical mixing alone usually results in the

In spite of the importance of this technology, very little is understood concerning the basic transport phenomena and kinetics that dictate morphology development and ultimate materials properties. Successful process models must evidently grapple with simultaneous non-Newtonian and two-phase flow, non-Fickian diffusion, and interfacial reaction. Recent theories have begun to address some of these issues. Fredrickson and Leibler [l’] generalized earlier theoretical studies by Doi and de Gennes to include the effects of convection on homogeneous polymer melts undergoing an intermolecular diffusion-controlled reaction. For a homogeneous melt containing a low concentration of chains functionalized at one end, these earlier studies had shown that the long time decay of the reactive chain population can be described by a bimolecular rate coefficient R. For high molecular weight chains and in the absence of flow, it was argued that R satisfies the Smoluchowski-like expression R, aDoR, where Do is the center-of-mass diffusivity of a labeled chain and Rg is the radius of gyration. Fredrickson and Leibler [l’] showed that application of a steady linear flow in effect multiplies this formula by a dimensionless function of the Deborah number, De=-, where K is the shear (or extension) rate and T is the longest internal relaxation time of a reactive polymer. For weak flows, that is, when De<< 1, the first flow corrections are nonanalytic: R=R, [ 1 + a De’iz], where a is a positive numerical coefficient that depends on the flow type. For strong flows, convection was shown to affect much more significant enhancements in the reaction rate: k=k,DelI3 for NcN,, and k=k,(De/ln De) for N>N, (where N, is the entanglement degree of polymerization).

Theory of polymer dynamics Fredrickson

Another interesting development was the recent publication of theories for diffusion-controlled coupling of polymer chains across a polymer-polymer melt interface. Fredrickson [2*] and O’Shaughnessy and Sawhney [3’] independently arrived at expressions for the rate coefficient describing the initial stages of diffusion-controlled coupling at a symmetric A-B, polymer-polymer interface. For A and B bulk phases that initially contain equal and uniform number densities pc of reactive chains, the initial growth rate of the number of copolymers (reaction product) per unit area of interface, Q(t), is given by daldt=k,p#. For reactive polymers of comparable size, the interfacial reaction rate coefficient, k;, was shown to scale as k;- R$/(~ln~e), which implies that ki- l/in N for NN,. This rate law holds for times less than the characteristic time for reactants to be depleted near the interface, ~,-,==D~(k~po)~. At longer times, center-of-mass diffusion of the reactive polymers to the interface is rate controlling, and at very long times the chemical potential barrier presented by the copolymer brush built up at the interface dictates the coupling rate. In addition to these studies that addressed the kinetics of the copolymer formation, an important theoretical paper by Milner and Xi [4**] has recently shed light on the role of copolymer once it has been formed and localized at the interface of droplets. It has been known for a long time that even small amounts of copolymer can reduce the mean inclusion (drop) size and limit the drop size polydispersity during reactive blending. Both of these effects are beneficial because they improve the homogeneity and mechanical properties of the polymeric composite. Until very recently, it was believed that these improvements arose from the copolymer’s ability to reduce the interfacial tension between the two engineering polymer phases. Milner and Xi. [4**] argue that the reduction in interfacial tension at the surface coverages employed (typically of the order of one copolymer chain per Rg squared of interfacial area) is far too small to explain the observations from existing blending experiments. Instead they suggested that the beneficial action of the copolymer was primarily derived from its ability to suppress coalescence between drops. Although the presence of copolymer led to static repulsive forces, arising from the deformation of the copolymer layer when two drops were held fixed at a separation less than Rg, Milner and Xi argued that the repulsive hydrodynamic lubrication force between two approaching drops was even more dramatically enhanced by the presence of copolymer. In the absence of copolymer, the fluid mechanics of two closely approaching drops involved both a recirculating fountain flow within the drops and a ‘squirting’ lubrication flow within the gap between them. Once copolymer is added to the interface, the fountain flow is suppressed because such flow would entrain and compress the two-dimensional ‘gas’ of copolymer at the interface. Milner and Xi estimated that the work involved in this compression would be substantial, that is, many thousands

813

of thermal energy kBT, even for very low copolymer coverage. With the fountain flow suppressed, the drops act more like ‘solid’ particles and large repulsive forces arise from the squirting lubrication flow in the gap. Milner and Xi proceeded to estimate these forces and further build a phenomenological model of drop breakup and coalescence under shear. Numerical solutions of the master equation governing the droplet dynamics yielded trends with droplet volume fraction that closely mirrored trends observed in recent experiments. Associating

polymer

rheology

Associating polymers, such as telechelics, polysoaps, and ionomers, are typically composed of hydrophilic backbones with two more more ‘associating’ hydrophobic groups (‘stickers’) per chain. These macromolecules have very rich linear and nonlinear rheological properties that derive from the propensity of the associating groups to cluster, forming transient networks at sufficiently high polymer concentrations. The resulting dramatic increase in solution viscosity and elasticity makes such materials invaluable as thickening agents and rheology modifiers. A complete theoretical description of the equilibrium and nonequilibrium properties of associating polymers is not in hand. Some progress with regard to equilibrium clustering and self-assembly has been achieved in recent years by a combination of analytical theory and computer simulation methods. A few scaling results are also now available on the linear viscoelasticity of simple model associating polymers, such as telechelics, under restricted concentration regimes. In contrast, very little is understood about the nonlinear viscoelastic regime, where for example, nonmonotonic shear rate dependence of the viscosity is frequently observed. In view of the reactive blending problem discussed in the previous section, this is perhaps not surprising. Very few theoretical tools are available to deal with the behavior of strongly heterogeneous polymeric systems that are subjected to flows strong enough to break, extend, or otherwise perturb self-assembled or other localized structures. The analogy with reactive processing can be taken even further if one views the association process as a reversible chemical reaction, in which the breakdown or build-up of associating polymer aggregates under shear can then be interpreted as flow-induced shifts of chemical reaction equilibria. While little progress has been made on the nonlinear rheology of true associating polymers with two or more stickers per chain, a recent theoretical study by Jones, Marques, and Joanny [SO] addressed the role of steady shear on the micellization of diblock copolymers in a selective solvent. In such a system, which is analogous to a solution of associating polymers with a single sticker, the associations are primarily intermolecular. Jones et a/. [S’] considered the change in the free energy of a spherical micelle under steady shear and used their results to predict

8 14

Polymers

how the association equilibria, for example, critical micelle concentration (cmc) and average micelle size, would be shifted by flow. Due to the fact that the distortion energy per chain induced by shear decreases as the aggregate size increases, Jones et a/. [5*] found that flow stabilizes larger micelles and thereby lowers the cmc. Although this quasi-thermodynamic treatment is suggestive, it neglects an important feature of convection: two aggregates on different streamlines are set in relative motion. Thus, coalescence events are promoted by flow. Such events cannot be captured by a purely thermodynamic approach, but require kinetic descriptions similar to those employed in theories of reactive blending [1*,4’]. Another interesting paper, by Semenov, Joanny, and Khokhlov [6*], dealt with the equilibrium and linear viscoelastic properties of telechelic polymers in the limit where the aggregation number p (half the number of hydrophobic ends in a cluster) is large. These authors argue that, at low polymer concentrations, telechelics tend to form ‘flower-like’ micelles with p >> 1 (typically S-50). They further argue that the micelles attract each other so that bridges can be formed, resulting in an increase of conformational entropy. Since the binding energy of two micelles scales as p”.3 ksT, they suggest that at large p and sufficiently high polymer concentration the micelle ‘gas’ will condense to become a macrophase of close-packed ‘flowers’. On laboratory timescales, this phase corresponds to a physical gel, rather than a liquid, due to the presence of bridges between flowers. At the point of condensation, the viscosity is predicted to rise dramatically over that of the sol phase. On further raising the concentration, interestingly enough, the viscosity is predicted to actually decrease over a restricted range. Reaction-induced

structuring

or phase

separation

A related topic that has received some recent theoretical attention concerns chemical reactions, such as polymerizations, with simultaneous structure formation or phase separation. Such situations frequently arise during copolymerization of monomers (particularly macromonomers) of limited solubility, or during precipitation or other heterogeneous polymerizations. Besides the motivation to develop a quantitative description of such technologically relevant situations, theoretical studies may identify ways in which chemical reactions could be used to regulate phase separation processes and thereby provide a route to novel or better-controlled heterogeneous polymeric structures. A recent letter by Glotzer, Di Marzio, and Muthukumar [7*] explicitly discussed this possibility and performed a linearized analysis and numerical study of spinodal decomposition in a simple model of a binary A + B mixture that could undergo the reversible reaction of A = B. While this model is not particular to polymers, nor is the reaction appropriate for any of the situations described

above, the results do indicate an interesting coupling between chemical reaction and phase separation kinetics. In particular, the reaction was found to suppress the instability of long-wavelength concentration modes and to stabilize intermediate-scale structures, even in the late stages of phase separation. Further studies on different classes of reaction models should prove quite interesting. Another recent study by Fredrickson and Leibler [8*] examined the situation in which a bulk copolymerization was carried out between two monomers that are initially homogeneously mixed, but very close to a critical consolute point. Such a situation could be achieved, for example, by copolymerizing two macromonomers whose lengths were carefully tuned. Prior to commencing the copolymerization, the reaction-bath in such a near-critical system has long-ranged and long-lived composition fluctuations that are thermally excited. As the reaction is started, ‘blocky’ random copolymers grow, because the active radicals experience a correlated heterogeneous environment. In turn, these quenched chemical correlations in the copolymer backbones are predicted to destabilize the bath to yield phase separation. At some point, a spinodal decomposition begins to occur simultaneously with the reaction. An interesting feature of this decomposition in the early stages is that the wavelength characterizing the bicontinuous structure is predicted to decrease with time as -t-‘/z, a consequence of the continuous progress of the reaction that further reduces the thermodynamic compatibility of the components in the multicomponent bath. A third example of structuring during chemical reaction is given in the work of Wittmer et a/. [9*]. The authors considered a situation in which monomers are brought into contact with a surface containing a high coverage of fixed initiator sites. On commencing the polymerization reaction, propagation initially occurs at the surface, but chain growth moves the propagation front outwards along the surface normal. The ‘polymer brush’ grown this way is expected to be polydisperse, but much higher densities of grafted chains can in principle be achieved at the early stages of the polymerization. Wittmer eta/. [9’] constructed a simple model of the growth process: involving three coupled continuum equations for the density of unreacted monomer; the density of reactive chain ends; and the density of monomers incorporated into polymer. Working in the adiabatic limit of an infinitesimal flux of incoming monomers, they showed that a constitutive swelling law could be combined with a classical diffusion-reaction model to produce a rather complete scaling description of the growth. Specifically, they predict the formation of a brush with a (polymerized) monomer density varying with distance, z, away from the wall as $(z)-z-213 and with a characteristic height (h) that grows according to /i(t)- t3. Such layers may prove useful as polymeric coatings for colloids, or in impact modification, toughening or bonding.

Theory of polymer dynamics Fredrickson

Rheology, slip, and deformation Homogeneous polymer melts

With the advent of commercial synthetic methods that offer a combination of unprecedented control of molecular architecture and attractive economics (e.g. metallocene catalysts), there is much current interest in the range of rheological and mechanical properties that can be achieved by purposeful manipulation of chain architecture. Among other options, such manipulation may involve the introduction of long chain branches into the macromolecules. While considerable theoretical attention has been given in the past to molecular models of stress relaxation and dynamics of star and ring polymers, surprisingly little work has focused on more practical architectures such as combs and trees. Bick and McLeish [lo*] recently developed theoretical estimates of the damping function L(y), which describes the nonlinear strain softening behavior, for homogeneous polymer melts with arbitrary long branch content and chain topology. Employing the tube model of entanglement constraints popularized by Doi-Edwards and de Gennes, they argue that a(y) is dictated primarily by the distribution of chain ends appropriate for a given architecture. Each free end is associated with an entropic tension; the retraction of a chain segment within a tube after a large applied strain is sensitive to the forces transmitted by segments on either side of it and, ultimately, to the entropic tensile forces. Bick and McLeish [lo*] devised a scheme for prioritizing the response of chain segments of arbitrary position along the macromolecular backbone and the summing of these responses to compute the strain softening behavior. In addition to a general formula for’the damping function, they provide numerical calculations for model comb, tree, and percolation clusters. The comb architecture is theorized to provide the weakest strain softening behavior; in contrast, the Cayley tree model gives an expression for b(y) that most closely matches the experimentally determined damping function for commercial low-density polyethylene. Even for polymer melts with a strictly linear architecture, the best industrial-scale synthetic approaches yield materials with considerable chain length dispersity. So-called ‘double-reptation’ theories have been proposed that relate the molecular weight distribution to the dynamic (linear) viscosity. While these theories have proved quite successful at reproducing experimental measurements on idealized polydisperse blends, their underlying molecular basis has been unclear. Recent work by Milner [l 1.1 has shown how the microscopic constraint release model of Viovy, Rubinstein, and Colby reduces to the simpler double-reptation model, subject to certain conditions on the molecular weight distribution. For realistic distributions, for example in commercial grades of polyethylene, the double-repcation description is argued to be appropriate, and may prove quite

815

useful in deconvolving information about the distribution of chain lengths from linear viscoelastic data. Milner further provided a heuristic theoretical argument for the Cox-Merz empirical rule, which relates the frequency dependent linear viscosity to the shear rate dependent nonlinear viscosity. Rheology of interfaces and heterogeneous

systems

There is currently much interest in interfacial rheology and ‘slip’ phenomena in polymer melts as they relate to instabilities observed in industrial-scale processing flows and to the microscopic physics of adhesion and fracture. A related phenomenon in bulk systems is ‘shear-banding’, where a material spontaneously stratifies into layers of different shear rate, usually in response to high levels of stress. Spenley, Yuan, and Cates [12”] recently examined shear-banding behavior in the context of a class of viscoelastic constitutive equations in which the shear stress is a nonmonotonic function of the shear rate. They used numerical simulations to solve for the structure of plane Couette flows in the case of an integral constitutive equation appropriate for entangled worm-like micelle solutions, and compared the results with predictions from a toy differential constitutive model. Shear-banding was explicitly observed and it was proposed that stress gradient terms need to be included in constitutive equations to stabilize the interfaces between shear bands. These results provide considerable insight into the mathematical structure of conscitutive laws leading to shear banding, irrespective of the type of material. In another recent study, Semenov [13*] examined the slip behavior of molten polymer brushes (dense assemblies of chains end-grafted to a surface) when sheared against each other. By assuming that the microscopic Rouse model is appropriate in the region of interpenetration between two brushes, Semenov reproduced earlier theoretical predictions for the linear viscosity. More interesting, however, is his scaling prediction for the shear thinning behavior at large shear rate. He finds that the nonlinear viscosity rl scales with chain length A? shear rate IC, and layer height h as ~~-fV/(~rc)‘/r. Similar shear thinning behavior has recently been observed in experiments using polymer melts confined in the surface forces apparatus. Finally, I mention the work of Brochard-Wyart, Gay and de Gennes [14*]. These authors considered the slippage of a polymer melt of degree of poiymeri~tion P against a surface with end-grafted chains of length N. They examined the friction associated with shearing the melted chains past the tethered chains and elucidate the extent to which slippage can be suppressed by properly designing such tethered surfaces. At low grafting densities o, they find that the ability of the N-chains to suppress slippage increases linearly with v. This effect is predicted to saturate above a coverage vc, however, at which point all

616

Polymers

melted chains in the grafting layer are trapped. As the rate of shear is increased at fixed v, the authors predicted a further cascade of transitions in which the tethered chains become increasingly stretched and disentangled from the surrounding melted chains. Experiments to systematically test these predictions on the model surfaces would be quite interesting.

Conclusions While the above work constitutes only a small subset of the theoretical activities in polymer science over the past two years, it does provide a sense of the continuing vitality of the field. A constant flux of new polymer architectures, new synthetic approaches, new processing strategies, and new applications provides impetus for basic experimental and theoretical advances in years to come. One of the greatest theoretical challenges for the field remains the development of tools capable of describing the response of solid and liquid polymers to large external forces, particularly when heterogeneous structures are involved. Our ability to predict failure mechanisms and structure-property relations in systems with complex morphologies, such as blends, copolymers, and semicrystalline polymers, remains very limited. Nevertheless, more attention is being focused on these far from equilibrium situations and our analytical and numerical tool-chests continue to expand, so I am optimistic that the next few years will prove to be yet another exciting period for polymer theory!

Acknowledgement This work was supported by the National Science Foundation under grant NSF-DI\IR 9505599.

References

and recommended

readings

Papers of particular interest, published within the annual period of review, have been highlighted as: . l

*

of special interest of outstanding interest

1. Fredrickson GH, Leibler: Theory of diffusion-controlled reactions . in polymers under flow. Macromolecules 1996, 29:2674-2665. A theoretical study of the effect of linear flow fields on the rate of intermolecular diffusion-controlled reactions of polymers. Shear and extensional flows are found to enhance the rate of irreversible reactions between functional groups on different chains. Explicit calculations are presented for the reaction rate coefficient of unentangled chains (Rouse model) and entangled chains (reptation model) subjected to steady linear flows of arbitrary strength. 2. .

Fredrickson GH: Diffusion-controlled reactions at polymer-polymer interfaces. Phys Rev Let? 1996, 763440-3443. A theoretical study of the rate of diffusion-controlled coupling of end-functionalized polymers at an A-B interface. Expressions are derived for the interfacial rate coefficient, k;, which controls the rate of copolymer formation at times less than the time zp discussed in this review. See also 13.1 3.

O’Shaughnessy B, Sawhney U: Polymer reaction kinetics at interfaces. Phys Rev Let! 1996, 76:3444-3447. in independent theoretical study of the problem described in 12’1. Similar scaling results for kr with the degree of polymerization are found, although the approach is different. The authors also discuss the situation at late stages, where a dense brush of copolymer is formed at the interface and presents a barrier to further reaction. 4. ..

Milner ST, Xi H: How copolymers promote mixing of immiscible homopolymers. I Rheo/l996,40:663-697.

A theoretical study of the mechanisms that influence particle size distribution during reactive blending. Contrary to the popular notion that copolymers are effective at reducing particle size as a result of their ability to lower interfacial tension, the authors argue that the principal role of copolymer during reactive blending is to enhance lubrication forces between drops, thereby lowering the rate of coalescence. 5.

Jones JL, Marques CM, Joanny J-F: Shear-induced micelliration of diblock copolymers. Macromolecules 1996, 26:136-l 42. ; theoretical investigation of the effect of a weak, steady shear flow on the micellization of diblock copolymers in a selective solvent. The critical micelle concentration is predicted to decrease with increasing shear rate, while the mean aggregation number can either increase or decrease with shear, depending on the compositional asymmetry of the diblock copolymer. 6. .

Semenov AN, Joanny J-F, Khokhlov AR: Associating polymers: equilibrium and linear viscoelasticity. Macromolecules 1995, 26:1066-l 075. A theory for the equilibrium properties and linear viscoelastic properties of telechelic associating polymers in the limit of large aggregation number. The flower-like micelles formed in aqueous solution are predicted to attract each other, leading to condensation and physical gel formations at a certain polymer concentration. At higher concentrations, an interesting regime is predicted in which the viscosity decreases as the solvent content is reduced. 7. .

Glotzer SC, Di Marzio EA, Muthukumar M: Reaction-controlled morphology of phase-separating mixtures. Phys Rev Lett 1995, 74~2034-2037. An analytical and numerical study of spinodal decomposition kinetics in a binary fluid that simultaneously undergoes the reversible reaction A = B. The exchange reaction was found to suppress long-wavelength growth modes and stabilize intermediate-scale structures. 0.

Fredrickson GH, Leibler L: Composition fluctuation effects in chain copolymerization. Macromolecules 1995, 265196-5209. ;\n analytical study of bulk copolymerization when the two comonomers are nearly incompatible. Composition fluctuations in the initial reaction medium are frozen into the nascent copolymers, yielding blocky random copolymers that ultimately destabilize the medium to phase separation. The spinodal decomposition that ensues is predicted to have unusual characteristics. 9. .

Wittmer JP, Cates ME, Johner A, Turner MS: Diffusive growth of a polymer layer by in situ polymerization. Europhys Letf 1996, 33:397-402. A scaling and Monte Carlo study of polymerization initiated by fixed sites on a surface. The diffusion-limited growth of the layer is influenced by excluded volume interactions among the tethered chains. The layer is predicted to be more diffuse and polydisperse than polymer brushes formed by equilibrium self-assembly processes. 10. .

Bick DK, McLeish TCB: Topological contributions to nonlinear elasticity in branched polymers. fhys Rev Lett 1996, 76:2667-2590. The damping function (describing the strain softening behavior of polymer melts) is examined in the context of the Doi-Edwards/de Gennes tube model for chains with arbitrary placement of long branches. The comb architecture is shown to be particularly resistant to shear softening. Milner ST: Relating the shear-thinning curve to the molecular weight distribution in linear polymer melts. J Rheol 1996, 40:303-315. The microscopic theory of constraint release is used as a basis for the argument for a relationship between the molecular weight distribution of a linear polymer melt and the dynamic viscosity of the same form as deduced from the notion of ‘double-reptation’. Heuristic arguments are also given for the validity of the Cor-Men empirical rule relating the (linear) dynamic viscosity to the nonlinear steady shear viscosity. 11. .

Spenley NA, Yuan XF, Cates ME: Nonmonotonic constitutive laws and the formation of shear-banded flows. J Phys /I France 1996, 6:551-571. An analytical/numerical investigation of shear-banding in model constitutive equations designed to mimic the viscoelastic behavior of entangled wormlike micelles. The results shed considerable light on the mathematical structure of constitutive laws needed to produce and stabilize shear-bands. 12. ..

Semenov AN: Rheology of polymer brushes: Rouse model. Langmuir 1996, 11:3560-3564. ; scaling analysis of the shear thinning behavior of melt brushes rubbed against each other at fixed separation. 13.

14.

Brochard-Wyart F, Gay C, de Gennes P-G: Slippage of polymer melts on grafted surfaces. Macromolecules 1996, 29:377-302. ;he ability of tethered chains to suppress slippage of polymer melts sheared across a solid surface is examined by scaling methods. The shear stress and slip behavior is examined as a function of shear rate, grafting density, and relative molecular weights of tethered chains and matrix chains.