Pressure–Volume Properties of Endlinked Hard-Chain Polymer Networks N

JOURNAL OF CHEMICAL PHYSICS VOLUME 110, NUMBER 15 15 APRIL 1999

Pressure–volume properties of endlinked hard-chain polymer networks N. R. Kenkare, C. K. Hall,a) and S. A. Khan Department of Chemical Engineering, Box 7905, North Carolina State University, Raleigh, North Carolina 27695-7905 ͑Received 5 November 1998; accepted 22 January 1999͒ Equilibrium molecular dynamics simulations are used to obtain the pressure and conﬁgurational chain properties of near-perfect, off-lattice, trifunctional hard-chain networks of chain lengths 20, 35, 50, and 100, and of tetrafunctional hard-chain networks of chain lengths 20, 35, and 50 over a range of packing fractions. Our simulation results show that the variation of network pressure with density is similar to that of uncrosslinked chain systems of the same chain length, except at low densities where the network pressure shows a negative region, as ﬁrst observed by Escobedo and de Pablo. We present a theoretical treatment leading to an analytical expression for the network pressure as the sum of liquid-like and elastic contributions. The liquid-like contribution is obtained by extending the generalized Flory-dimer theory to networks, and the elastic contribution is obtained by treating the network as a set of interpenetrated dendrimers and using an ideal chain-spring analogy to calculate the free energy. The theoretical predictions for network pressure are in good agreement with simulation data. © 1999 American Institute of Physics. ͓S0021-9606͑99͒51615-2͔

I. INTRODUCTION models are successful in predicting the behavior of real networks at small deformations, but are unsuccessful at large The immense technological importance of polymer net- deformations. works and gels in the automobile, plastics, and medical A number of other theories of network behavior have equipment industries is due primarily to their striking elastic been developed in an attempt to overcome some of the short- properties. As a consequence, a great deal of theoretical and comings of the affine and phantom models. These include experimental research has been conducted over the past fifty the constrained junction models of Flory and Erman,5,6 and years in an attempt to understand the nature of the stress– Ronca and Allegra6,8 as well as several localization, slip-link strain relations of polymer networks. and tube models.7–22 These theories partially account for the In this article, we investigate the molecular mechanisms excluded volume interactions in networks ͑which are ne- underlying polymer network behavior by performing discon- glected by the affine and phantom models͒, and have been tinuous molecular dynamics simulations. A comprehensive qualitatively successful in predicting the stress-strain relation study of the pressure–volume properties of trifunctional and of real networks in the small and intermediate deformation tetrafunctional networks is conducted and the resulting data regimes. are used to evaluate the predictions of several existing net- Much of the theoretical work on crosslinked systems ap- work theories. We also propose an analytical mean-field net- plies to ‘‘perfect’’ networks, i.e., networks with constant work equation of state based on a simple network model; this chain length between the junctions and few structural irregu- equation of state yields good agreement with the data from larities ͑dangling chains, loops, or junctions of varying func- our simulations. tionality͒. The validation of network theories is best done The earliest molecularly based theories for crosslinked using ‘‘model’’ near-perfect networks with well- polymers are the affine model of Flory and Rehner,1 and characterized molecular structure. Experimental efforts in Wall,2 and the phantom model of James and Guth.3,4 Both this direction are, however, hampered by the difficulty asso- models are based on mean-field approaches in which the ciated with obtaining precise information about the network analogy between the elastic properties of an ideal, Gaussian structure. In the last decade, computer simulations have chain and a classical elastic spring is used to calculate the proved to be an excellent method for investigating network network free energy. By considering the network to be a behavior since they allow the investigator to construct and collection of such chains, assuming that the chain vectors study near-perfect model networks with well-characterized deform proportionally with macroscopic deformation ͑the af- structure. fine assumption͒, and neglecting excluded volume interac- Most simulation studies of polymer networks to date tions, the affine and phantom models arrive at a simple have focussed on the kinetics of crosslinking or on network stress–strain relationship for network deformation. These dynamics.23–31 Simulation studies of the pressure–volume– temperature ͑P–V–T͒ properties of networks are relatively a͒Author to whom correspondence should be addressed; electronic mail: rare, even though such investigations would yield valuable [email protected] insights into the elastic response of a network undergoing

Downloaded 22 Feb 2008 to 152.1.209.194. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp J. Chem. Phys., Vol. 110, No. 15, 15 April 1999 Kenkare, Hall, and Khan 7557 isotropic deformation, and would enhance our understanding an uncrosslinked system particularly at large deformations; of the molecular mechanisms underlying gel swelling. the difference in pressures was related to the disparity in the To our knowledge, the only investigations of the P–V–T fractal-like properties of the two systems. behavior of networks are those of Escobedo and de Our study of network behavior is motivated partly by the Pablo,32–34 and Sommer.35 Escobedo and de Pablo conducted paucity of simulation data on the P–V–T and structural prop- NPT ensemble ͑constant number of particles, N, pressure, P, erties of crosslinked systems, and partly by the need for an and temperature, T͒ Monte Carlo simulations of short chain analytical network equation of state. The purpose of this ar- ͑chain lengths from 1 to 32͒, tetrafunctional, athermal, and ticle is to investigate the pressure–volume properties of square-well networks having a diamond-like junction con- model athermal trifunctional and tetrafunctional networks nectivity. The networks were created in such a way that the using equilibrium molecular dynamics simulation tech- systems contained no trapped entanglements. Owing to the niques, and to propose an analytical equation of state for limitations of conventional Monte Carlo methods for simu- networks that is valid over a wide range of densities. lating branched and crosslinked systems, Escobedo and de We use discontinuous molecular dynamics simulations Pablo invented a set of complex system-specific Monte Carlo to construct near-perfect, off-lattice, monodisperse, trifunc- moves, the extended continuum configurational bias tional hard-chain networks ͑i.e., networks in which the ͑ECCB͒, ‘‘slab,’’ and ‘‘cluster’’ moves for relaxing the net- chains consist of strings of hard spheres͒ of chain lengths 20, work and for making the volume changes necessary in a 35, 50, and 100, and tetrafunctional hard-chain networks of NPT simulation. They found that the P–V behavior of net- chain lengths 20, 35, and 50. The networks are created from work systems did not differ substantially from that of linear melts of linear tangent hard-sphere chains ͑chains in which or star polymers except at very low packing fractions where successive hard-sphere segments are tangential to each other͒ a region of negative pressure was seen. They also presented by endlinking, i.e., by allowing bond formation between a semitheoretical method to calculate the network free en- chain ends that collide with each other in the course of the ergy from the sum of the free energy of a system of reference simulation, up to a maximum of two bonds per chain end in polymer molecules ͑e.g., octahedral molecules͒ and the free a trifunctional network or three bonds per chain end in a energy associated with network elasticity, thus dividing the tetrafunctional network. Our crosslinking process is similar network energy into liquid-like and elastic components as to the experimental procedure used to develop ‘‘room tem- suggested originally by James and Guth.3 The elastic free perature vulcanizates,’’ as well as the method used by a energy of the network was obtained from a ‘‘computer ex- number of researchers36–39 to synthesize model networks periment’’ in which the entropy of an isolated chain was with a known chain molecular weight distribution between calculated by enumerating all of the self-avoiding walks pos- crosslinks. Even for these model experimental systems, how- sible between the chain ends when they were maintained a ever, the difficulty of obtaining a sufficiently monodisperse fixed distance apart in space. This single chain entropy was precursor system remains. In addition, it is difficult to obtain then used in conjunction with the affine assumption to obtain a precise knowledge of the extent of crosslinking of the sys- the elastic contribution to the network free energy. The tem, and the proportion of irregularities, which is necessary theory showed good agreement with their simulation data in in order to investigate the thermophysical properties of net- the very low packing fraction regime, but overpredicted the works. In comparison, computer simulations offer a greater data at intermediate to high densities. Escobedo and de Pab- control over the crosslinking process, allowing us to create lo’s approach is an improvement over many existing network monodisperse networks ͑with a constant chain length be- theories in that the elastic term clearly accounts for the ex- tween junctions͒, as well as giving us a precise knowledge of cluded volume of the network chains. However, the fact that the network structure. a set of computer simulations must be conducted at each In our systems, the P – V and structural properties of the chain length in order to calculate the elastic contribution to endlinked networks are investigated over a range of packing the network free energy means that network properties can- fractions ͑0.45–0.005͒ by expanding the network from its not be obtained analytically. This precludes the use of this initial packing fraction of 0.45 to successively lower densi- approach for predicting the properties of an arbitrary net- ties. The network expansion is done by using an algorithm work. that utilizes the variable bond-length attribute of the Belle- Sommer35 conducted bond-fluctuation Monte Carlo mans chain to conduct isotropic volume changes. The advan- simulations of a large randomly crosslinked athermal net- tage of using this algorithm is that it allows us to simulate work consisting of approximately 50 000 segments arranged network expansion/contraction without introducing complex, on a cubic lattice. To simulate an isotropic deformation pro- system-specific moves, and without imposing an affine trans- cess, the network was simply allowed to diffuse into a larger formation on the chain ends. lattice, and the chain deformation and the network pressure Highlights of our simulation results are the following. were calculated as a function of the resulting macroscopic We observe that the network pressure follows the same trend deformation of the network. Sommer observed that the net- as that for uncrosslinked chain systems over packing frac- work chain deformation was dependent on molecular weight, tions from ␩ϭ0.45 to about ␩ϭ0.1, i.e., the pressure de- i.e., the shorter chains in the randomly crosslinked network creases continuously with decreasing packing fraction. At were generally less deformed than the longer chains at a low packing fractions (␩Ͻ0.1) the network pressure be- given macroscopic deformation. The pressure of the comes negative, in contrast to the behavior of uncrosslinked crosslinked system was observed to be different from that of chain systems. As the packing fraction is lowered even fur-

Downloaded 22 Feb 2008 to 152.1.209.194. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 7558 J. Chem. Phys., Vol. 110, No. 15, 15 April 1999 Kenkare, Hall, and Khan ther (␩Ͻ0.01), there is a sharp downturn in the network sphere chain. The potential energy of interaction between pressure, that is, the pressure drops suddenly towards the any two beads on the same or neighboring chains in the highly negative region. These observations are in agreement network is given by with the results of Escobedo and de Pablo.33,34 We find that 0, rϾ␴ the sharp downturn in network pressure observed at very low U͑r͒ϭ ͑1͒ packing fractions occurs when the average end-to-end dis- ͭ ϱ, rр␴ , tance of the network chains is between 50% and 70% of its where r is the bead-bead distance and ␴ is the bead diameter. maximum extended length. We also observe that the P – V The treatment of a system of tangent hard sphere chains properties and chain dimensions of tetrafunctional networks using standard molecular dynamics techniques is difficult do not differ appreciably from those of trifunctional net- due to the tangency requirement and flexible nature of such works. chains. To circumvent this problem, Rapaport47,48 and Belle- We present a model based in statistical thermodynamics mans, Orban, and Belle49 suggested an approach in which that predicts the pressure–volume behavior of athermal poly- the bond length, l, between successive segments of the chain mer networks. The network is modeled as a set of interpen- is not kept constant, but is instead allowed to vary freely etrated dendrimers, whose branches are the network chains. over a range, ␴(1Ϫ␦)ϽlϽ␴(1ϩ␦) where ␦Ӷ1. The chain 3 Following James and Guth, the network pressure at a given then effectively becomes a collection of segments connected packing fraction is taken to be the sum of a liquid-like term by sliding links ͑the Bellemans chain͒.As␦ 0, the Belle- which arises from the repulsive part of the intermolecular mans chain approximates the tangent hard-sphere→ chain interaction, and an elastic term which arises from the retrac- model. The relaxation of the constant bond length require- tive force exerted by the chains opposing the network defor- ment allows the chain segment trajectories to be partially mation. The liquid-like term is obtained from the generalized decoupled, resulting in linear trajectories between collisions. 40–42 Flory–Dimer equation of state by considering each den- The system dynamics can then be treated using the discon- drimer to be made up of a number of connected star-like tinuous molecular dynamics techniques developed by Alder molecules. The elastic term is obtained by calculating the and Wainwright50–52 for hard-sphere systems. free energy of a dendrimer in terms of the free energies of its Molecular dynamics simulations of systems with discon- component chains using the simple ideal-chain/spring anal- tinuous potentials, such as the hard-sphere potential, take ogy. The dendrimer chains are treated in two ways: ͑1͒ as advantage of the fact that the equations of motion for these having a Gaussian chain vector distribution ͑the Gaussian systems can be solved analytically at successive ‘‘colli- dendrimer model͒, and ͑2͒ as having a non-Gaussian chain sions.’’ A collision ͑or event͒ occurs when the distance be- vector distribution of the form suggested by Kuhn and tween any two segments becomes equal to a point of discon- 43,44 Grun ͑the non-Gaussian dendrimer model͒. We find that tinuity in the potential. Thus the dynamics of a system of while the Gaussian dendrimer model predicts the network molecules governed by the hard-sphere potential evolves on pressures fairly well over a wide range of packing fractions, a collision-by-collision basis. For a Bellemans chain, colli- it is unable to predict the sudden drop in pressure that occurs sions can be of two types: core collisions and bond stretches. at very low densities. In comparison, the non-Gaussian den- Core collisions can occur between any pair of segments, i drimer model predicts the network pressures over the entire and j, in the system when the distance between two bonded range of packing fractions and is remarkably successful in segments becomes equal to ␴(1Ϫ␦), or when the distance the extremely low density region. between two nonbonded segments becomes equal to ␴. The organization of our article is as follows: Section II Bond stretches occur between pairs of bonded segments describes the molecular model and the general simulation when the distance between the segments becomes equal to technique, as well as the simulation methods used for net- the maximum stretch of the Bellemans bond, ␴(1ϩ␦). The work construction and for network volume changes. In Sec. simulation proceeds by locating the next event, advancing all III, our theoretical model is outlined, and in Sec. IV, our the segments until the event occurs, and calculating collision simulation and theoretical results are discussed. A brief sum- dynamics for the colliding pair of segments. This process is mary and some further discussion are given in Sec. V. performed repeatedly. A number of time-saving techniques are used to improve the algorithm efficiency based on the II. MOLECULAR MODEL AND SIMULATION work of Smith, Hall, and Freeman,46 including neighbor lists, TECHNIQUE linked lists, binary trees, and delayed position updates. In this article, the discontinuous molecular dynamics In order to construct trifunctional or tetrafunctional net- technique that we have used previously to treat polymer works, we endlinked melts of linear Bellemans chains. The melts45,46 has been adapted to the case in which linear melt endlinking was done at an initial system packing fraction of chains are endlinked to form a polymer network. We begin ␩ϭ0.3 after which the packing fraction was increased to ␩ this section by reviewing the discontinuous molecular dy- ϭ0.45, where ␩ is defined by namics technique as it is applied to a polymer melt. We then 3 ␲ Nc n␴ outline our method for constructing and relaxing polymer ␩ϭ , ͑2͒ 6 V networks, and finally we describe our technique for conducting network volume change moves. with Nc equal to the number of network chains, n equal to The basic unit of the polymer network is a polymer the chain length, and V equal to the volume of the simulation chain which we represent as a freely jointed tangent hard- cell. We adopted this procedure ͑see our previous

TABLE I. The chain length, n, number of chains, Nc and the total number TABLE II. The network elastic fraction, %el , the number of trifunctional of segments, Nt for each simulation system. and tetrafunctional junctions, N3 and N4 , the number of unsaturated junctions, Nunsat , the number of dangling ends, Nend , the number of loops, Trifunctional Tetrafunctional Nloop , and the number of free chains, Nfree in the trifunctional and tetrafunctional network systems. n 20 35 50 100 20 35 50 Nc 90 90 90 90 120 120 120 Trifunctional Tetrafunctional Nt 1800 3150 4500 9000 2400 4200 6000 N 20 35 50 100 20 35 50

%el 0.99 0.96 0.90 0.84 0.96 0.94 0.93 N3 60595857347 30 publication for details͒ because it is relatively easy to N4 ¯¯¯¯57 56 52 achieve near-complete extents of reaction at lower densities, Nunsat 0123013 N 0223325 particularly for the longer chains. This crosslinking proce- end Nloop 1135641 dure is found to impact network elastic properties, causing Nfree 0020000 the elastic behavior of networks that are crosslinked at low densities and then compressed to the ͑high͒ density of interest to be somewhat different from those of networks that are network.54,55 It is calculated in Table II by subtracting the crosslinked at high densities.30 This is attributed to topologi- number of unsaturated junctions, dangling chains, and loops cal differences between these networks; for instance, net- from the total number of network chains, and dividing by the works that are crosslinked at low initial densities are less total number of network chains. Since the elastically active interpenetrated and contain fewer trapped entanglements fraction is close to 1 for all the networks considered, and the than networks crosslinked at high densities. Also the chains number of dangling chains, loops and unsaturated junctions in the networks crosslinked at low densities are somewhat is low, we consider these networks to be near-perfect sys- collapsed relative to the chains in the networks crosslinked at tems. high densities, even after the system density is increased. Once the networks attained their final packing fraction of The decreased chain dimensions and changed entanglement 0.45, they were relaxed for 500 million collisions. The com- topology associated with networks crosslinked at low densi- pressibility factors were then evaluated from the Clausius ties are expected to affect the elastic properties of these net- virial theorem in the following form: works so that they have ‘‘superelastic’’ properties ͑see Ob- hukov, Rubinstein, and Colby53͒. That is, networks PV ms͚collrij–⌬vij crosslinked at low initial density ͑which are akin to solution- Znetϵ ϭN nϪ , ͑3͒ net c 3k Tt crosslinked experimental systems from which the solvent is N kBT B e extracted͒ can be elongated to much larger deformations be- where P is the network pressure, Nnet is the number of ‘‘net- fore they rupture than networks crosslinked at a high initial work molecules’’ (Nnetϭ1 since the network is considered density ͑which are akin to melt-crosslinked networks͒. Al- to be one large molecule͒, te is the elapsed simulation time though we have chosen a fairly high initial density for over which the sum is calculated, kB is the Boltzmann con- crosslinking (␩ϭ0.3), in order to ensure that the network stant, T is the temperature, ms is the mass of a segment, rij is topology is not greatly affected by our crosslinking proce- the vector between segment centers at a collision, and ⌬vij is dure, we expect that the network’s elastic properties will be the velocity change for the colliding pair. In this article, the influenced to some extent. simulation results are reported mostly in terms of the reduced 3 net net In this article, four trifunctional networks of chain pressure, P*ϵP␴ /kBTϭZ N 6␩/(␲ Ncn). We also lengths 20, 35, 50, and 100 and three tetrafunctional net- calculated the mean-squared end-to-end distance of the networks of chain lengths 20, 35, and 50 are studied. As shown work chains, ͗R2͘, which is defined by in Table I, each trifunctional network contains 90 chains, and R2 ϭ r Ϫr ͒2 , ͑4͒ each tetrafunctional network contains 120 chains, the num- ͗ ͘ ͗͑ 1 n ͘ bers being chosen so that the number of junctions, N f , in all where r1 and rn are the coordinates of the first and last chain of the networks is approximately the same (N f ϭ2Nc / f segments, respectively, and the ensemble average is taken ϳ60, where f is the network functionality͒. The chain seg- over all the chains in the network. ment diameter, ␴ϭ1, the Bellemans bond variation factor, In order to obtain the properties of networks at different ␦ϭ0.05, and the box length varies with the packing fraction. packing fractions, we devised a novel algorithm to isotropi- The structural characteristics of the networks, displayed in cally expand or contract the network from its initial, equili- Table II, are described in terms of the elastically active frac- brated state at ␩ϭ0.45 to a final deformed state at any other tion, %elastic , the number of trifunctional junctions, N3 , the packing fraction. This algorithm takes advantage of the vari- number of tetrafunctional junctions, N4 , the number of bi- able bond length of the Bellemans chain and allows us to functional ͑unsaturated͒ junctions, Nunsat , the number of dan- conduct network volume changes without having to break gling ends and loops, Nend and Nloop, and the number of free and regrow the chains, and without causing segment over- chains, Nfree . The elastically active fraction is an important laps. The procedure for network expansion is as follows: measure of the ‘‘perfectness’’ of our networks, and is defined When the system undergoes a core collision, we conduct a to be the fraction of chains that are connected at both ends to search among all the bonded pairs of segments in the system junctions with at least three independent paths to the for the pair of segments, i and j, that have the largest inter-

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segment distance, Rmax . Since the maximum allowable dis- configurational entropy of the network chains is lower than tance between a pair of bonded segments is ␴(1ϩ␦), the that of the uncrosslinked system chains, again at the same ␩. bond length lij between that pair, i and j, could be increased For all of these reasons the network is considered to be un- until lijϭ␴(1ϩ␦). Accordingly we scale the positions of all deformed at the initial packing fraction of ␩0ϭ0.45. The the segments in the box, as well as the box length by the deformation ratio of a network at any packing fraction ␩ is defined as factor, ␬ϭ␴(1ϩ␦)/Rmaxу1. This does not cause bond р␩0 breakage or overlaps, since i and j are the bonded pair that 1/3 1/3 1/3 ␩0 0.45 V L are the furthest apart in the system. The procedure for net- ␭ϭ␭ ϭ␭ ϭ␭ ϭ ϭ ϭ ϭ , ͑5͒ x y z ␩ ␩ V L work contraction is similar: When the system undergoes a ͫ ͬ ͫ ͬ ͫ 0ͬ 0 bond stretch, we conduct a search among all bonded and where ␩0 , V0 , L0 , and ␩, V, L, are the packing fractions, unbonded pairs of segments in the system for the pair of simulation cell volumes, and lengths for the undeformed and segments, i and j, that have the smallest intersegment dis- deformed networks, respectively. Once the network has at- tance, Rmin . Since the minimum allowable distance between tained its final packing fraction, ␩, it is relaxed for 500 mil- a pair of segments is either ␴(1Ϫ␦)or␴, depending on lion collisions, and the compressibility factor and chain di- whether the segments are bonded or nonbonded, the distance mensions are calculated. lij between i and j could be decreased until lijϭ␴(1Ϫ␦) We also conducted simulations of uncrosslinked chain ͑bonded pair͒ or lijϭ␴ ͑unbonded pair͒. Accordingly we systems containing 20 chains each of chain lengths 20, 35, scale the positions of all the segments in the box, as well as 50, and 100, over a packing fraction range of ␩ the box length by the factor, ␬ϭ␴(1Ϫ␦)/Rmin ,ifiand j are ϭ0.45– 0.01. Our purpose in conducting these simulations bonded, or by ␬ϭ␴/Rmin if i and j are nonbonded, without was to obtain accurate data for the pressure–volume behav- causing segment overlaps. After the segment positions and ior and the chain dimensions of uncrosslinked systems which box length are scaled, the system neighbor lists and time lists could be compared to our network data. The 20-mer uncrosslinked chain systems were relaxed for 100 million col- are regenerated. This procedure is repeated every Ncoll colli- lisions each, and all the other systems were relaxed for 500 sions ͑for e.g., Ncollϭ1000), until the required packing fraction ͑or volume͒ is achieved. million collisions each. The compressibility factors were The use of the algorithm described above to expand/ evaluated using the Clausius virial theorem in the following contract the network to the required packing fraction repre- form: sents a considerable computational saving over the alterna- PV ms ͚coll rij –⌬vij tives of either constructing a network at each required Zchainϵ ϭn Ϫ . ͑6͒ N k T 3N k Tt density or using complex chain breakage and regrowth c B c B e moves to maintain the network topology during volume The chain dimensions were evaluated using Eq. ͑4͒. changes. This method also offers us the advantage that the network structural characteristics are the same at each den- III. THEORY sity, allowing us to ignore variations in the network structure in our analysis of network static properties. The disadvantage In this section, we propose a new equation of state for of this algorithm is that it is inefficient when applied to large polymer networks which is based on modeling the network ͑Ͼ10 000 segments͒ systems, the reason being that all of the as a set of interpenetrated dendritic structures. The network segment positions and collision times have to be recalculated free energy is taken to be the sum of a liquid-like contribu- after a volume change move is carried out, causing a com- tion and an elastic contribution. The liquid-like contribution plete rescheduling of events. is obtained from the generalized Flory-dimer ͑GFD͒ theory, We have used this algorithm to expand the network from and the elastic contribution is obtained by extending the for- an initial packing fraction of ␩ϭ0.45 to successively lower malism of the affine model to the dendrimer-network model. packing fractions of ␩ϭ0.35,0.25,...,0.003. Since the net- Although the assumption that the network free energy can be work at ␩ϭ0.45 is the starting point for all expansions of the divided into liquid-like and elastic contributions has been the system, the network is considered to be undeformed at this subject of debate ͑Gee, Herbert, and Roberts,56 Eichinger packing fraction. At any lower packing fraction (␩Ͻ0.45), and Neuburger57͒, we have used it in this work because the the network is considered to be deformed. An additional cri- simplicity of the assumption allows us to obtain an analytical terion on which to base the description of the ␩ϭ0.45 net- expression for the network free energy. work as being undeformed is that the mean-squared end-to- To help us explain the physical basis for our model and end distance of the network chains at ␩ϭ0.45 is equal to the to point out its similarities and differences with existing mean-squared end-to-end distance of uncrosslinked system theories of network elasticity, we begin Sec. III with a brief chains at the same packing fraction ͑this will be shown in description of the affine1,2 and phantom3,4 theories. Next, we Sec. IV͒. Hence the configurational entropy of the network outline the network theories of Duiser and Staverman,58 and chains at a packing fraction of 0.45 is approximately the of Graessley59 who modeled the polymer network as an en- same as that of the equivalent uncrosslinked system chains at semble of micronetworks, and obtained results that deviate the same packing fraction. At any lower packing fraction somewhat from the affine and phantom predictions. Last, we (␩Ͻ0.45), the mean-squared end-to-end distance of the net- describe our theory in detail. work chains is greater than the mean-squared end-to-end dis- In the affine and phantom theories, the polymer network tance of uncrosslinked system chains at this same ␩, and the is modeled as a system of ideal chains whose elastic proper-

ties are similar to those of classical elastic springs. Thus the kchain A ͑R͒ϩCЈϭ R2. ͑11͒ forces exerted by the network are the same as would be chain 2 produced if each chain were replaced by an elastic spring.

This means that each pair of chain ends in the affine or If the network is assumed to consist of Nc similar, parallel, phantom network is under the action of a force tending to and indistinguishable chains which exert equal restoring pull the chain ends together, and that, in the absence of other forces, then the total ͑elastic͒ free energy of the athermal 44 forces, the network would collapse to a zero volume. In a network is simply the sum of the individual chain energies, real rubber-like network, this tendency to collapse is bal- anced by the mutual repulsive forces between molecules. 2 net 3NckBT͗R ͘ Thus, in a real rubber, two different forces can be distin- AaffineϩCЉϭ , ͑12͒ 2 R2 guished ͑James and Guth3͒, the first being the elastic force ͗ 0͘ that is related to the configurational entropy of the chains and where ͗R2͘ is the mean-squared end-to-end distance of the the second being the internal ‘‘hydrostatic’’ or liquid-like network chains and CЉ is a constant. force that is associated with the interatomic forces between When the network is deformed, the chain vectors are the network segments. The latter force is the same as that assumed to transform affinely with macroscopic deforma- which exists in an ordinary liquid. tion, i.e., We now briefly review the steps taken in the affine theory to calculate the free energy of a deformed ideal chain 2 2 2 ͗R ͘ϭ␭ ͗R0͘, ͑13͒ network when the network undergoes an isotropic expansion/ 2 contraction from an initial volume, V0 , to a final volume, where ͗R ͘ is the mean of the squared end-to-end vectors of 3 Vϭ␭ V0 . The theory rests on a number of premises. Each the deformed network chains, averaged over all the chains. network chain is assumed to be ideal, which means that the The assumption underlying Eq. ͑13͒ is that the chain vector probability distribution associated with its end-to-end vectors transformation with deformation is independent of the pres- is Gaussian at all deformations. The probability distribution, ence of its neighbors. The free energy of the deformed ather- p(R), of the chain end-to-end vectors, R, is expressed as the mal network in the affine model can be obtained by substi- product of the probability distributions of the components of tuting Eq. ͑13͒ in Eq. ͑12͒, the chain end-to-end vectors along each of the coordinate 2 axes, 3NckBT␭ Anet ϩCЉϭ . ͑14͒ affine 2 pchain͑R͒ϭpchain͑X͒pchain͑Y ͒pchain͑Z͒ 3R2 In addition to the free energy resulting from the loss of chain ϭC exp Ϫ , ͑7͒ configurational entropy, Flory61 suggested that another free ͩ 2͗R2ͪ͘ 0 energy term should be added to Eq. ͑14͒ which accounts for 2 where C is a constant and ͗R0͘ is the mean squared end-to- the change in entropy of the crosslinks due to the increased end distance of the undeformed chain, which is taken to be volume available to them in an expanded network. The total equal to the mean squared end-to-end distance of a chain in network free energy is then61 the equivalent melt, i.e., ͗R2͘ϵn␴2. The chain entropy, 0 3N k T 2 S(R) is given by net c B ␭ 3 A ͑␭͒ϩCЉϭ ϪN k T log͑␭ ͒ , ͑15͒ affine 2 f B 2 3kBR Schain͑R͒ϭkB ln pchain͑R͒ϭkB ln CϪ , ͑8͒ 2͗R2͘ where N f is the number of crosslinks of type f ( f ϭ3,4, etc.͒. 0 This equation can be rewritten in terms of the network pack- and the chain Helmholtz free energy is given by 3 ing fraction using the relation ͓cf. Eq. ͑5͔͒, ␭ ϭ␩0 /␩,

Achain͑R͒ϭUchain͑R͒ϪTSchain͑R͒ 2 3 3NckBT ␩0 ␩0 2 net 3R Aaffine͑␩͒ϩCЉϭ ϪN f kBT log . ͑16͒ ϭϪ ϩ 2 ͩ ␩ ͪ ͩ ␩ ͪ kBT ln C kBT 2 , ͑9͒ 2͗R0͘ The pressure of the deformed network in the affine model is where Uchain(R) is the chain internal energy ͑equal to zero given by for the ideal chain͒. Equation ͑9͒ can be rewritten as Anet N k T ␩ 1/3 N k T ␩ץ 2 net affine c B f B 3R Paffine͑␩͒ϭϪ ͉TϭϪ ϩ . V V0 ͩ␩0ͪ V0 ͩ␩0ͪץ ϩ ϭ Achain͑R͒ CЈ kBT 2 , ͑10͒ 2͗R0͘ ͑17͒ where CЈϭkBT ln C, and is a constant. By comparing Eq. We can express Nc /V0 and N f /V0 in the pressure equation ͑10͒ to the equation for the elastic energy of a classical above in terms of the network packing fraction using the 2 3 spring of length R, AspringϭkspringR /2, where kspring is the packing fraction definition, ␩0ϭ␲ (Nc n)␴ /6V0 , and the classical spring constant, we can recover the equivalent relation, N f ϭ2 Nc / f , between the number of crosslinks, N f , 60 2 ‘‘spring constant’’ of the ideal chain, kchainϭ3kBT/͗R0͘. and the number of chains, Nc , in a perfect network. Then, net The chain Helmholtz free energy can then be written as Paffine(␩) can be rewritten as

Downloaded 22 Feb 2008 to 152.1.209.194. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 7562 J. Chem. Phys., Vol. 110, No. 15, 15 April 1999 Kenkare, Hall, and Khan

of Fig. 1͒. The total number of concentric layers is called the rank of the dendrimer. The ﬁrst ͑innermost͒ layer of the dendrimer contains f branches or chains, the second layer contains f ( f Ϫ1) branches or chains, the third layer consists of f ( f Ϫ1)2 branches or chains, and so on. The total number of chains in a dendrimer of rank m is given by

2 ͑mϪ1͒ ndenϭ f ϩ f ͑ f Ϫ1͒ϩ f ͑ f Ϫ1͒ ϩ¯ϩ f ͑ f Ϫ1͒ ͑ f Ϫ1͒mϪ1 ϭ f , ͑21͒ ͫ ͑ f Ϫ2͒ ͬ

where f is the functionality of the dendrimer ( f ϭ3,4...). The spatial dimension or diameter of the dendrimer of rank FIG. 1. Model of a single dendrimer in the network. m is denoted by D, and is given by the relation, D2 ϭ2m͗R2͘, i.e., D2 is the square of the distance covered by a random walk beginning at any point on the circumference of the dendrimer, passing through each layer and through the net 3 1/3 Paffine͑␩͒␴ 6␩0 ␩ 2 ␩ origin of the dendrimer, and ending at another point on its ϭ Ϫ ϩ . ͑18͒ k T ␲ n ͫ ͩ ␩ ͪ fͩ␩ ͪͬ circumference. B 0 0 The elastic free energy contribution for our dendrimer The restoring force per unit initial area ͑stress͒ is given by model is first calculated for the case in which the network chains are modeled as ideal Gaussian chains ͑the Gaussian net -Aaffine 3NckBT␭ 3N f kBT dendrimer network͒. The same approach is subsequently exץ 1 ␶ϭ ϭ Ϫ L ͯ V ␭V tended to the case in which the network chains are modeledץ 2 L 0 0 0 T as non-Gaussian chains ͑the non-Gaussian dendrimer net- 3 1/3 1/3 ͑19͒ ␶␴ 6␩0 ␩0 6 ␩ work͒. ϭ 3 Ϫ . k T ␲n ͫ ͩ ␩ ͪ f ͩ␩ ͪ ͬ We now derive the free energy for a Gaussian dendrimer B 0 network. The free energy of a network of interpenetrated A bridge between the affine and phantom chain models, dendrimers can be obtained by replacing each dendrimer and the dendrimer-network model proposed in this article, is ͑composed of nden ideal gaussian chains of ‘‘spring con- 58 2 provided by the work of Duiser and Staverman and stant,’’ kchainϭ3kBT/͗R0͘) by a single equivalent chain, of 59 Graessley who both modeled crosslinked polymers as mi- spring constant, kdendrimer . Using a simple spring-resistor cronetworks of ideal chains. These micronetworks are en- analogy, kdendrimer can be obtained by treating the dendrimer sembles of small networks having mobile inner junctions as a set of springs arranged in series and in parallel confor- ͑central junctions͒ and fixed outer junctions. A statistical me- mations, as described in Appendix A. For a dendrimer of chanical approach based on the affine and the phantom mod- rank m, containing nden chains, the spring constant is given els is employed in conjunction with the affine assumption by ͓Eq. ͑13͔͒ to obtain the restoring force per unit initial area to be ͑ f Ϫ1͒͑mϪ1͒ 1/3 kdendrimerϭkchain f ͑ f Ϫ2͒ ␶ ͑ f Ϫ2͒ 3Nc␭ ͑ f Ϫ2͒ 18␩0 ␩0 ͫ ͑ f Ϫ1͒mϪ1 ͬ ϭ ϭ , ͑20͒ k T f V f 3ͩ ␩ ͪ B 0 ␲n␴ ͑mϪ1͒ 3kBT ͑ f Ϫ1͒ ϭ f ͑ f Ϫ2͒ . ͑22͒ which has the same form as the first term of Eq. ͑20͒ with the ͗R2͘ ͫ ͑ f Ϫ1͒mϪ1 ͬ 0 exception of the prefactor, ( f Ϫ2)/f. We now describe our approach to calculating network properties. Following James and Guth, we assume that the The Helmholtz free energy of the dendrimer, network free energy, and hence the pressure, is the sum of an Adendrimer(␩), can then be obtained in a form similar to Eq. elastic contribution and a liquid-like contribution. The elastic ͑10͒ by replacing the term kchain in Eq. ͑11͒ with kdendrimer , contribution is calculated by modeling the network as a set and by replacing the squared end-to-end distance, R2, of the of interpenetrating dendritic structures, each of whose chain, with the squared spatial dimension, D2ϭ2m͗R2͘,of branches is either an ideal, gaussian chain, or a non-Gaussian the equivalent dendrimer spring. The term, ͗R2͘, represents chain ͑i.e., a chain having a non-Gaussian end-to-end vector the mean-squared end-to-end distance of a constituent chain probability distribution͒. Figure 1 is an example of a single within the dendrimer, averaged over all the nden dendrimer such dendritic structure. It consists of concentric layers of chains. When the network deforms we assume that the den- 2 2 2 2 branches, where each branch represents a network chain of drimer also deforms affinely, that is, D ϭ␭ D0, where D0 2 1/2 2 end-to-end distance ͗R ͘ ͑as shown by the right hand side ϭ2m͗R0͘ denotes the mean-squared spatial dimension of

Downloaded 22 Feb 2008 to 152.1.209.194. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp J. Chem. Phys., Vol. 110, No. 15, 15 April 1999 Kenkare, Hall, and Khan 7563 the undeformed dendrimer. The Helmholtz free energy of the 3 R 2 9 R 4 p ͑R͒ϭC exp Ϫn ϩ deformed dendrimer of rank, m, can be written as chain 1 ͭ ͫ2ͩ n␴ͪ 20ͩ n␴ͪ 6 Adendrimer͑␩͒ϩCЈ 99 R ϩ ϩ , ͑27͒ 350ͩ n␴ͪ ¯ͬͮ kdendrimer ϭ D2 2 where C1 is a constant, and n is the chain length. As in the case of the Gaussian distribution, this non-Gaussian distribu- kdendrimer tion is also based on the assumption that the chains have no ϭ D2␭2 2 0 excluded volume ͑i.e., that the chains can pass through each other͒. The non-Gaussian distribution reduces to the Gauss- 3k T f Ϫ1 ͑mϪ1͒ 4 B ͑ ͒ 2 2 ian form ͓Eq. ͑7͔͒ when the higher order terms, (R/n␴) and ϭ f ͑ f Ϫ2͒ 2m͗R0͘ ␭ 6 2͗R2͘ ͫ ͑ f Ϫ1͒mϪ1 ͬ (R/n␴) , in the above equation are small enough to be ne- 0 2 2 glected, and when n␴ is taken to be equal to ͗R0͘ ͑the ͑mϪ1͒ 2/3 ͑fϪ1͒ ␩0 relation between the end-to-end distance of an ideal, Gauss- ϭ3kBTmf͑fϪ2͒ . ͑23͒ ian chain and its chain length͒. Deviations in p(R) from the ͫ͑fϪ1͒mϪ1ͬͩ ␩ ͪ Gaussian are about 8% when the end-to-end distance of the Thus far we have focused on the free energy of a single chain is 0.5 times its fully extended length, and about 20% dendritic structure in the network. Since the network consists when the end-to-end distance of the chain is 0.7 times its of identical dendritic structures, the free energy of the net- fully extended length. work is simply taken to be the sum of the free energies of We now outline the steps by which the network free these individual dendritic structures. The free energy of a energy can be obtained for the dendrimer-network model with a non-Gaussian chain vector distribution. The general network composed of ND dendrimers is then given by method is based on the approach described by Treloar44 in net which many of the assumptions of the affine model ͑with Adendrimer͑␩͒ϩCЉϭND ͓Adendrimer͑␩͒ϩCЈ͔ Gaussian chains͒ are applied to the non-Gaussian chain case. ͑fϪ1͒͑mϪ1͒ Our method for the non-Gaussian dendrimer network is simi- ϭND 3kBTmf͑fϪ2͒ lar to our approach in the Gaussian dendrimer-network case, ͫ͑fϪ1͒mϪ1ͬ and involves making similar assumptions, including the ne- 2/3 ␩0 glect of the chain excluded volume, and the use of the affine ϫ , ͑24͒ ͩ ␩ ͪ assumption. We begin by obtaining the free energy of a single non-Gaussian chain in analogy with the method de- where ND is obtained from the undeformed network packing scribed earlier for the single Gaussian chain ͓cf. Eqs. ͑7͒– fraction, ␩0 , using the relation, ͑11͔͒ A ͑R͒ϭϪTS ͑R͒ 3 chain chain ␩0ϵ␲ND ndenn␴ /6V0 . ͑25͒ ϭϪkBT log pchain͑R͒ The quantity, n is given by Eq. ͑21͒. The reduced pressure den 3 R 2 9 R 4 of a network composed of dendrimers of rank m is then ϭϪkBT logC1ϩkBTn ϩ given by ͫ2ͩn␴ͪ 20ͩ n␴ͪ 6 net 3 net 3 99 R Adendrimer͑␩͒ ␴ ϩ ϩ . ͑28͒ץ Pdendrimer͑␩͒␴ ϭϪ 350ͩ n␴ͪ ¯ͬ V kBTץ kBT By comparing this to the equation for the elastic free energy 12␩0 of a classical spring, the spring constant for the non- ϭϪ m͑ f Ϫ2͒2 ␲n Gaussian chain can be written as 2 4 ͑mϪ1͒ 1/3 3 9 R 99 R ͑ f Ϫ1͒ ␩ k ϭk Tn ϩ ϩ ϩ . ͑29͒ ϫ . ͑26͒ chain B 2 4 6 ¯ m 2 ͫ͑n␴͒ 10 ͑n␴͒ 175 ͑n␴͒ ͬ ͫ ͑͑fϪ1͒ Ϫ1͒ ͬͩ␩0ͪ Thus the spring constant of a non-Gaussian chain depends on Next, we derive the free energy for a non-Gaussian den- the chain end-to-end distance, unlike that of a Gaussian drimer network. Although the probability distribution for the chain. end-to-end distances of the network chains is adequately de- The calculation of the elastic free energy of a network of scribed by the Gaussian distribution ͓Eq. ͑7͔͒ for small de- non-Gaussian chains is based on the assumption44 that the formations, at large deformations approaching the maximum non-Gaussian probability distribution ͓Eq. ͑27͔͒ can be extensibility of the chains, this distribution is expected to treated as the product of its components along the x, y, and z show significant deviations from the Gaussian form. Kuhn axes, i.e., pchain(R)ϭpchain(X)pchain(Y)pchain(Z). This as- and Grun43 suggested the following non-Gaussian form for sumption is strictly valid for the Gaussian case ͓cf. Eq. ͑7͔͒ the probability distribution of the chains, based on the con- but is commonly extended to the non-Gaussian case.44 In this cept that a chain has finite maximum extensibility: instance, this assumption allows us to calculate the free en-

Downloaded 22 Feb 2008 to 152.1.209.194. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 7564 J. Chem. Phys., Vol. 110, No. 15, 15 April 1999 Kenkare, Hall, and Khan ergy of the non-Gaussian dendrimer network in the same This equation reduces to the Gaussian dendrimer-network way as for the Gaussian dendrimer network. The Helmholtz pressure equation ͓Eq. ͑26͔͒ when the second and third terms free energy of the non-Gaussian dendrimer network is ob- in the square bracket on the right side are negligible, i.e., at tained by evaluating the free energy of a single non-Gaussian small deformations of the network. dendrimer, Adendrimer(␩), in analogy with the treatment of the Finally we calculate the liquid-like contribution to the Gaussian dendrimer network in Eq. ͑23͒. To do this, the free energy of a network. Since our networks are athermal, effective spring constant for each dendritic structure is again this term arises from the intersegment repulsions between the (mϪ1) m taken to be kdendrimerϭkchainf ( f Ϫ2)͓(fϪ1) /(fϪ1) network monomers. It can be obtained by using an approach Ϫ1͔,asinEq.͑22͒. Again, as in the Gaussian dendrimer- based on the GFD theory of Dickman, Honnell, and network case, we average the chain dimensions over all the Hall,40–42 which has been successfully used to evaluate the network chains, i.e., we replace the chain end-to-end distance thermodynamic properties of a variety of hard-chain sys- R2 with ͗R2͘, and R4 with ͗R4͘. Using the affine assump- tems. We begin by briefly reviewing some pertinent results 2 2 2 4 tion, ͗R ͘ϭ␭ ͗R0͘, and assuming further that ͗R ͘ of the GFD theory for uncrosslinked, linear hard-sphere 4 4 ϭ␭ ͗R0͘, the Helmholtz free energy of a deformed non- chains. Gaussian dendrimer can be written in a form analogous to The GFD prediction for the compressibility factor of a Eq. ͑23͒ as fluid of hard-chain n-mers is given by

kdendrimer ve͑n͒Ϫve͑1͒ A ͑␩͒ϩCЈϭ D2 Z ͑␩͒ϭ Z ͑␩͒ dendrimer 1 2 GFD ͫv ͑2͒Ϫv ͑1͒ͬ 2 e e

3k T ͑ f Ϫ1͒͑mϪ1͒ ve͑n͒Ϫve͑2͒ B Ϫ Z ͑␩͒, ͑34͒ ϭ f ͑ f Ϫ2͒ v ͑2͒Ϫv ͑1͒ 1 2 ͫ ͑ f Ϫ1͒mϪ1 ͬ ͫ e e ͬ where Z ( ) and Z ( ) are the compressibility factors of 2 2 1 ␩ 2 ␩ 1 3 ͗R0͘ ␭ ϫ ϩ fluids composed of hard sphere monomers and dimers, re- ͫ n␴2 10 n͑n␴2͒2 spectively, at the same packing fraction as the n-mer fluid, ve(1),ve(2) are the monomer and dimer excluded volumes, 33 R4 ␭4 ͗ 0͘ 2 2 and ve(n) is the volume excluded by an n-mer to a mono- ϩ ϩ¯ 2m͗R0͘␭ . ͑30͒ 175 n2͑n␴2͒3 ͬ mer. The excluded volumes of a monomer, dimer, and trimer can be evaluated exactly, and are given by ve(1) 2 2 3 3 3 We use the relation, ͗R0͘ϵn␴ ͑since the undeformed net- ϭ(4␲/3)␴ , ve(2)ϭ(9␲/4)␴ , and ve(3)ϭ9.826 05␴ . 4 2 2 work chains are Gaussian͒, the approximation, ͗R0͘Ϸ͗R0͘ The excluded volume of an n-mer linear chain is difficult to ϵ(n␴2)2, to recast the equation in the following form: evaluate exactly; the following approximate form was used by Dickman and Hall40 ͑fϪ1͒͑mϪ1͒ Adendrimer͑␩͒ϩC1Јϭ3kBTm f͑fϪ2͒ v ͑n͒ϭv ͑3͒ϩ͑nϪ3͒ ͓v ͑3͒Ϫv ͑2͔͒. ͑35͒ ͫ͑fϪ1͒mϪ1ͬ e e e e The compressibility factor for the monomer fluid, Z1(␩), is 3 4 33 6 62 2 ␭ ␭ obtained from the Carnahan-Starling equation of state ϫ ␭ ϩ ϩ ¯ . ͑31͒ ͫ 10 n 175 n2 ͬ 1ϩ␩ϩ␩2Ϫ␩3 The free energy of the non-Gaussian dendrimer network is Z1͑␩͒ϭ , ͑36͒ ͑1Ϫ␩͒3 obtained by summing the free energies of ND dendrimers ͓as done in Eq. ͑24͔͒, and is given by and the compressibility factor for the dimer fluid, Z2(␩), is 63 net obtained from the Tildesley–Streett equation of state Adendrimer͑␩͒ϩC1ЉϭND͑Adendrimer͑␩͒ϩC1Ј͒, ͑32͒ 1ϩ2.456 96␩ϩ4.103 86␩2Ϫ3.755 03␩3 where ND is related to ␩0 as shown in Eq. ͑25͒. The reduced Z2͑␩͒ϭ . ͑37͒ pressure of the network composed of non-Gaussian dendrim- ͑1Ϫ␩͒3 ers of rank m is then Thus, from the GFD equations of state, we can obtain the net 3 net 3 pressure of a chain fluid at any packing fraction based on Adendrimer ␴ץ Pdendrimer͑␩͒␴ ϭϪ knowledge of the hard-monomer and dimer compressibility V k Tץ k T B B factors and the excluded volume of the chain molecule. ͑mϪ1͒ To evaluate the liquid-like contribution to the network 6␩0 ͑fϪ1͒ ϭϪ m͑fϪ2͒2 pressure, Pnet , we postulate that Pnet is equal to the pressure ␲n m 2 liq liq ͫ͑͑fϪ1͒ Ϫ1͒ ͬ of a reference polymer having the same excluded volume as the network. For the dendrimer-network model described ␩ 1/3 6 ␩ Ϫ1/3 ϫ 2 ϩ earlier in Sec. III, the reference polymer is simply a fluid of ͫ ͩ␩0ͪ 5n ͩ␩0ͪ f -functional dendrimers of rank m. To calculate the compressibility factor of this dendrimer fluid using the GFD 198 ␩ Ϫ1 ϩ ϩ . ͑33͒ equations, we need only evaluate the excluded volume of a 2 ¯ 175n ͩ ␩0 ͪ ͬ single dendrimer. To do so, we note that a dendrimer is es-

chains ͓cf. Eq. ͑21͔͒ of which f ( f Ϫ1)(mϪ1) chains are in the last layer of the network. The total dendrimer excluded volume will be

͑ f Ϫ1͒͑mϪ1͒Ϫ1 v ͑n,m, f ͒ϭ 1ϩ f v ͑ f ͒ dendrimer ͫ ͑ f Ϫ2͒ ͬ center f ͑ f Ϫ1͒mϪ1 ϩ͑nϪ4͒ ͫ ͑ f Ϫ2͒ ͬ ͑mϪ1͒ ϫ ͓ve͑3͒Ϫve͑2͔͒ϩ2 f ͑ f Ϫ1͒

FIG. 2. ͑a͒ Portion of a trifunctional dendrimer consisting of three stars. The ϫ͓ve͑3͒Ϫve͑2͔͒. ͑39͒ dendrimer shown contains four star centers ͑dark lines͒ with connecting chains ͑light lines͒. ͑b͒ One of the constituent stars is enlarged to show that The excluded volumes, vcenter( f ), of the trifunctional and the star center consists of 2 f segments ͑dark beads͒ and that the connecting tetrafunctional star centers are calculated using the Monte chains consist of (nϪ4) segments each ͑light beads͒. Carlo algorithm mentioned above.64,65 For a trifunctional network ( f ϭ3), the star center contains six segments and vcenter(3)ϭ18.01Ϯ0.05; for a tetrafunctional network ( f sentially a set of connected f-functional star molecules, as ϭ4), the star center contains eight segments and vcenter(4) illustrated in Fig. 2. To calculate the excluded volume of the ϭ23.06Ϯ0.03. dendrimer we need to evaluate the total excluded volume of The compressibility factor of the fluid of dendrimers, this set of connected star molecules. In this context it is each of excluded volume, vdendrimer(n,m, f ), and rank, m, useful to consider the expression developed by Yethiraj and can then be calculated from Hall64 for the excluded volume of a star in a fluid of uncon- vdendrimer͑n,m, f ͒Ϫve͑1͒ nected star molecules Znet ͑␩, f ͒ϭ Z ͑␩͒ GFD ͫ v ͑2͒Ϫv ͑1͒ ͬ 2 e e nstarϪ1 v ͑n , f ͒ϭv ͑ f ͒ϩ f Ϫ2 ͓v ͑3͒Ϫv ͑2͔͒, v ͑n,m, f ͒Ϫv ͑2͒ e star center ͩ f ͪ e e dendrimer e Ϫ Z1͑␩͒. ͑40͒ ͑38͒ ͫ ve͑2͒Ϫve͑1͒ ͬ where nstar is the number of segments in the star, and The liquid-like contribution to the network pressure is ob- net net vcenter( f ) is the excluded volume of the star center. In Eq. tained from the relation, PGFD(␩, f )/kBTϵZGFD ND /V net 3 ͑39͒, the star center consists of 2 f ϩ1 segments — two seg- ϭZGFD(␩, f )6␩/␲␴ (ndenn) where (ndenn) is the number of ments from each arm of the star and one central bead. The segments in a dendrimer, and nden is obtained from Eq. ͑21͒. excluded volume of the star center is not obtained analyti- The reduced liquid-like contribution to the network pressure cally, but is calculated from a Monte Carlo simulation in is given by which a monomer is inserted successively at many randomly net 3 chosen points in a box containing random configurations of PGFD␴ 6␩ ͑ f Ϫ2͒ 65 ϭZnet ͑␩, f ͒ . ͑41͒ the star center ͑Alejandre and Chapela, Yethiraj and k T GFD n m 64 B ␲ ͭ f ͑͑fϪ1͒ Ϫ1͒ͮ Hall ͒. We can employ an equation similar to Eq. ͑38͒ to calcu- The total reduced pressure of the network is the sum of late the excluded volume of a dendrimer of rank m.Todoso the liquid-like contribution ͓Eq. ͑41͔͒ and the elastic contri- we subdivide the dendrimer into a set of star centers con- bution. The elastic contribution to the pressure has been cal- nected by arms which are simply linear chains. Figure 2 culated for the cases, ͑1͒ a dendrimer network of Gaussian shows a portion of a dendrimer consisting of three connected chains ͓Eq. ͑26͔͒, and ͑2͒ a dendrimer network of non- star molecules ͑the left side of Fig. 2͒ and a single star mol- Gaussian chains ͑chains that are Gaussian at low deforma- ecule enlarged to show the constituent beads ͑the right side tions and non-Gaussian at large deformations͓͒Eq. ͑33͔͒. of Fig. 2͒. For each star molecule on the left side, the dark net 3 The total reduced network pressure, P ␴ /kBT lines denote the star centers and the lighter lines denote the net 3 net 3 ϭPGFD␴ /kBTϩPdendrimer␴ /kBT is given for the Gaussian chains connecting them, while for the enlarged star molecule dendrimer network by on the right side, the dark beads denote the star centers and the light beads denote the chain segments connecting them. Pnet␴3 6␩ ͑ f Ϫ2͒ Each star center is composed of 2 f segments, i.e., two seg- ϭZnet ͑␩, f ͒ k T GFD ␲n m ments from each arm of the star ͑there is no central bead B ͭ f ͓͑fϪ1͒ Ϫ1͔ͮ since the chains are endlinked͒. Each connecting chain is ͑mϪ1͒ 1/3 12␩0 ͑ f Ϫ1͒ ␩ composed of (nϪ4) segments, where n is the length of the Ϫ m͑ f Ϫ2͒2 , m 2 network chains. The chains in the last layer of the network, ␲n ͭ ͓͑fϪ1͒ Ϫ1͔ ͮͩ␩0ͪ being connected to star centers only at one end, are com- ͑42͒ posed of (nϪ2) segments each. A dendrimer of rank m will mϪ1 contain 1ϩ f ͓( f Ϫ1) Ϫ1͔/(fϪ2) star centers and nden and for the non-Gaussian dendrimer network by

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3 TABLE III. Simulation results for the reduced pressure, P*ϭP␴ /kBT,as a function of packing fraction, ␩, for trifunctional networks at chain lengths, nϭ20, 35, 50, and 100.

␩ 20 35 50 100

0.45 5.129 5.149 5.155 5.166 0.40 3.021 3.042 3.048 ¯ 0.35 1.736 1.755 1.758 1.769 0.30 0.959 0.973 0.975 ¯ 0.25 0.495 0.507 0.509 0.516 0.15 0.087 0.095 0.094 0.099 0.10 0.018 0.025 0.024 0.028 0.05 Ϫ0.0065 Ϫ0.00098 Ϫ0.0017 Ϫ0.0014 0.03 Ϫ0.0107 Ϫ0.0045 Ϫ0.0049 Ϫ0.0017 0.01 Ϫ0.0224 Ϫ0.0074 Ϫ0.0075 Ϫ0.0027 0.005 ¯ Ϫ0.012 Ϫ0.0118 ¯ 0.0028 ¯¯¯Ϫ0.0028 FIG. 3. Reduced pressure, P*, multiplied by the network chain length, n,vs the packing fraction, ␩, for trifunctional networks of chain lengths, 20, 35, 50, and 100. Lines are added solely to guide the reader’s eye.

Pnet␴3 6␩ ͑ f Ϫ2͒ ϭZnet ͑␩, f ͒ GFD m kBT ␲nͭ f ͓͑fϪ1͒ Ϫ1͔ͮ would also be the case for uncrosslinked chain systems. Table III shows us that the network pressure becomes nega- ͑mϪ1͒ 1/3 6␩0 ͑fϪ1͒ ␩ tive at very low packing fractions. Although the scale of the Ϫ m͑fϪ2͒2 2 m 2 graph in Fig. 3 does not permit us to observe the low packing ␲n ͓ͭ͑fϪ1͒ Ϫ1͔ ͮ ͫ ͩ␩0ͪ fraction region, this will be more apparent in the later fig- 6 ␩ Ϫ1/3 198 ␩ Ϫ1 ures. ϩ ϩ ϩ . ͑43͒ 5n ͩ␩ ͪ 175nͩ ␩ ͪ ¯ͬ The similarities and differences between the pressure– 0 0 volume behavior of networks and uncrosslinked chain sys- IV. RESULTS AND DISCUSSION tems are clearly illustrated in Fig. 4, which shows simulation results for the compressibility factor of an uncrosslinked 20- In this section, we present our simulation results for the mer chain system, Zchain ͓cf. Eq. ͑6͔͒, and the scaled com- pressure–volume properties of trifunctional and tetrafunc- pressibility factor of a 20-mer trifunctional network, tional networks. The simulation results for the network pres- net Z ND /Nc , as a function of packing fraction, ␩. The curve sure are compared with the predictions of the constrained represents the prediction of the GFD theory40–42 for the com- junction model,5,6 the Obhukov, Rubinstein, and Colby 53 pressibility factor of the uncrosslinked chain system. The model, the affine model, and our Gaussian and non- scaled form of the network compressibility factor, Gaussian dendrimer-network models. Finally we compare net net Z ND /NcϵP V/NckBTϭP*␲n/6␩, is used in order to our results for the pressure and chain dimensions for the facilitate comparison with the uncrosslinked chain system trifunctional networks with those for tetrafunctional net- chain compressibility factor, Z ϵPV/NckBTϭP*␲n/6␩.We works, and with those for uncrosslinked chain systems. see that the variation in the scaled network compressibility Table III shows the simulation results for the reduced 3 pressure, P*ϭP␴ /kBT, for trifunctional networks of chain length 20, 35, 50, and 100, over a packing fraction range of 0.45–ϳ0.003. Each value of the pressure shown in Table III is calculated from simulation runs of between 500 million and a billion collisions ͑depending on the system size͒.It was not possible to repeat these runs for each of the data points because of the large system sizes in this study ͑be- tween 1800 and 9000 segments͒, and because long simulation runs are required for network systems. For selected systems three or more runs were performed in order to estimate the standard deviation for the data. From our sampling of the data, the error associated with the pressure was found to be less than 0.1%. Figure 3 shows our simulation results for the reduced pressure of the trifunctional networks as a function of packing fraction. Since the magnitudes of the pressure for the different chain length networks are similar ͑see Table III͒, FIG. 4. Compressibility factor, ZϵP*n/␩, vs packing fraction, ␩, for a the network pressure is multiplied by the chain length for trifunctional network ͑filled circles͒ and an uncrosslinked chain system ͑open circles͒, both of chain length, nϭ20. The curve represents the GFD clarity. We observe that the pressure shows the expected theory prediction ͑Refs. 40–42͒ for the compressibility factor of the un- continuous increase with increasing packing fraction, which crosslinked chain system.

FIG. 5. Comparison of simulation results and theoretical predictions for the reduced pressure, P*, multiplied by the network chain length, n,vsthe packing fraction, ␩, for trifunctional networks of chain lengths, 20, 35, 50, and 100. Models represented are the afﬁne model ͑dashed line͒, Gaussian dendrimer-network model, and non-Gaussian dendrimer-network model ͑solid lines͒. The predictions of all three models lie on top of each other over the entire packing fraction range shown.

FIG. 6. Comparison of simulation results and theoretical predictions for the reduced pressure, P*, multiplied by the chain length, n, vs the packing factor with packing fraction is similar to that of the un- fraction, ␩, in the region ␩Ͻ0.12, for trifunctional networks of chain crosslinked chain system compressibility factor at high to length: ͑a͒ 20, ͑b͒ 35, ͑c͒ 50, and ͑d͒ 100. The dashed lines represent the moderate packing fractions. The compressibility factor of the affine model prediction ͓Eqs. ͑18͒ and ͑34͔͒, the solid line represents the Gaussian dendrimer-network model ͓Eq. ͑42͔͒, and the bold line represents network is always lower than that of the uncrosslinked chain the non-Gaussian network model ͓Eq. ͑43͔͒. system due to the elastic retractive force exerted by the network chains as they oppose network deformation. At very low densities (␩Ͻ0.1), the uncrosslinked chain compress- The predictions of the three models lie on top of each other ibility factor tends to a value of 1, while the network com- except in the high packing fraction region for the 20-mer, pressibility factor becomes negative. This negative region in 35-mer, and 50-mer networks where there is a slight discrep- the network compressibility factor was first observed by Es- ancy between the affine and dendrimer model predictions. cobedo and de Pablo.33 The reason for its existence is that at The three theories are in good agreement with our simulation these low densities, the liquid-like contribution to the total results for all chain lengths at high to intermediate densities. pressure of the network is negligible, but the ͑negative͒ elas- Figure 6 shows the simulation results for the scaled net- tic contribution is relatively large. work pressure, nP*, versus packing fraction, for the four We now compare our simulation results with the theo- trifunctional networks in the low packing fraction region retical predictions for network pressure described in Sec. III. (␩Ͻ0.1) along with the prediction of the affine model The elastic component of the pressure is obtained from the ͑dashed line͒, the Gaussian dendrimer model ͑solid line͒, and following models: ͑1͒ the affine model ͓Eq. ͑18͔͒, ͑2͒ the the non-Gaussian dendrimer model ͑bold solid line͒. At these Gaussian dendrimer-network model ͓Eq. ͑26͔͒, and ͑3͒ the low packing fractions, the network pressure becomes nega- non-Gaussian dendrimer-network model ͓Eq. ͑33͔͒. The tive, and the network models have different degrees of suc- liquid-like component of the pressure is obtained from the cess. We first consider the 20-mer trifunctional network GFD theory for chains ͓Eq. ͑34͔͒ for the affine model, and shown in Fig. 6͑a͒. As the packing fraction decreases, a from the GFD theory for networks ͓Eq. ͑41͔͒ for the Gauss- downturn in the network pressure is evidenced, which is not ian and non-Gaussian dendrimer-network models. For all predicted by either the affine or the Gaussian dendrimer- three models, the packing fraction of the undeformed net- network models. In fact these two models predict a slight work, ␩0 , is taken to be equal to 0.45. For the dendrimer- increase in the pressure at very low densities. However the network models ͑Gaussian and non-Gaussian͒ the value of non-Gaussian dendrimer-network model is strikingly suc- the network rank, m, is chosen so as to give the best agree- cessful in predicting the downturn in pressure, as well as in ment with simulation results. The value of m is taken to be 2 providing generally good agreement with our simulation re- for all the networks considered in this article. sults in the low packing fraction region. Since the theoretical Figure 5 shows the simulation results ͑symbols͒ for the basis for the non-Gaussian dendrimer-network model is the reduced network pressure scaled by the chain length, nP*, deviation in the network chain probability distribution from versus the packing fraction for the four trifunctional net- the Gaussian form at large network deformations, the success works, (nϭ20, 35, 50, and 100͒ in the packing fraction re- of this theory in predicting the network pressure at very low gion, ␩ϭ0.1– 0.45. The lines in Fig. 5 show the predictions packing fraction ͑i.e., large deformations͒ and the failure of of the affine model ͑dashed line͒, and the Gaussian den- the affine and Gaussian dendrimer-network theories drimer and the non-Gaussian dendrimer models ͑solid lines͒. strengthen the supposition that non-Gaussian chain extension

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TABLE IV. Simulation results for the chain mean-squared end-to-end dis- creases. This is related to the inverse dependence of the 2 2 tance, ͗R ͘/␴ , at each packing fraction, ␩, and corresponding deformation chain extensibilities on chain length, i.e., at a constant pack- ratio, ␭, for trifunctional networks at chain lengths nϭ 20, 35, 50, and 100. ing fraction, the chain extensibility decreases with increasing ␩ ␭ 20 35 50 100 chain molecular weight. This inverse dependence arises from the fact that for all ideal or real chains, the chain end-to-end 0.45 1.0 28.54 54.50 79.40 166.58 distance, R, is related to the chain length as, Rϰna, where 0.4 1.04 31.92 58.14 87.29 ¯ 0.35 1.09 34.25 65.23 96.22 190.66 aϽ1. The chain extensibility, R/n␴, then takes the form, (1Ϫa) 0.3 1.14 37.35 69.10 103.92 ¯ R/n␴ϰ1/n ; thus the extensibility decreases with in- 0.25 1.22 41.16 75.55 114.24 231.85 creasing chain length, and a 20-mer chain at a given packing 0.15 1.44 52.42 100.67 156.97 314.42 fraction is extended to a higher fraction of its fully extended 0.05 2.08 82.73 168.81 280.84 597.40 length than a 100-mer chain at the same packing fraction. 0.03 2.47 212.94 361.97 770.87 ¯ 0.02 2.82 123.44 ¯¯ ¯Hence the 20-mer chain shows departures from Gaussian sta- 0.01 3.56 181.94 382.46 667.25 1389.77 tistics at higher packing fractions ͑smaller deformations͒ 0.005 4.48 ¯ 592.41 1033.05 ¯ than the 100-mer chain, and as a result, the Gaussian net- 0.0028 5.44 ¯¯ ¯2960.19 work theories will be successful over an increasing range of packing fractions as the network chain length rises. Thus for the 100-mer network, both the affine and Gaussian- effects are significant in highly deformed networks. dendrimer models predict the network pressure fairly well The non-Gaussian statistical theory of Kuhn and Grun43 even at low densities, while for the shorter chain length net- ͓Eq. ͑33͔͒ which forms the basis for our non-Gaussian works, the Gaussian theories work well only at high and dendrimer-network model, predicts that for a freely jointed intermediate densities, and fail at low densities. chain, departures from the Gaussian statistical theory first The success of the non-Gaussian theory in predicting become important when the chain extensibility, defined to be network behavior over such a wide range of packing frac- the ratio of the chain end-to-end distance, R, and the fully tions is intriguing because of two assumptions that were extended chain length, n␴, is between one third and one made in arriving at Eq. ͑33͒: the neglect of excluded volume half. In light of this prediction, it is of interest to calculate interactions in the elastic term, and the assumption that the the chain extensibilities at the packing fractions at which the mean end-to-end distances of the chains deform affinely with pressure downturn shown in Fig. 6 occurs. We observe that macroscopic strain. These two assumptions are related in the the downturn in the pressure occurs at a packing fraction of sense that the neglect of excluded volume allows the affine approximately 0.01 for the 20-mer network ͓Fig. 6͑a͔͒, and assumption to be made, since chains that do not exclude at packing fractions of approximately 0.005 for the 35-mer volume could deform freely by passing through each other and 50-mer networks ͓Figs. 6͑b͒ and 6͑c͔͒. The downturn in ͑being ‘‘phantoms’’͒, and their motion would not be con- pressure does not occur at all in the 100-mer network ͓Fig. strained by neighboring chains or crosslinks. Here we ex- 6͑d͔͒ over the packing fraction range studied ͑down to ␩ plore the implications of the neglect of excluded volume by ϭ0.002 8). To obtain the chain extensibilities at these pack- comparing our simulation results to theories developed by ing fractions, we calculate the mean-squared end-to-end dis- other researchers5,6,53 in an attempt to account for excluded tances of the network chains, ͗R2͘/␴2 from our simulations, volume effects. and then set the chain extensibility to be R/nϭ͗R2͘1/2/n. Although the complex nature of real chains makes it The simulation results for the network chain mean-squared difficult to incorporate excluded volume interactions into the end-to-end distance ͗R2͘/␴2 are shown in Table IV. From elastic free energy expression in a simple manner, several the results in Table IV we see that for the 20-mer network, theoretical approaches have been proposed that account for the packing fraction of 0.01, which is where the pressure specific consequences of the monomer–monomer repulsion. downturn occurs, corresponds to a chain extensibility of We focus on two such models: ͑1͒ the constrained junction ϳ67%, i.e., the average end-to-end distance of the 20-mer model of Flory and Erman,5,6 in which the damping effects chains is approximately 67% of its fully extended length. For on junction motion due to the presence of the neighboring the 35-mer and 50-mer networks, the packing fraction of network chains are concentrated in a domain of constraint, 0.005, which is where the pressure downturn occurs, corre- and ͑2͒ the work of Obhukov, Rubinstein, and Colby53 in sponds to chain extensibilities of ϳ70% and ϳ65%, respec- which stress–strain relations are proposed for deswollen net- tively. For the 100-mer network, there is no downturn in works based on the Pincus blob model67 for real chains. Both pressure and the chain extensibility is only 54% at the lowest models assume that the crosslink mean positions ͑and hence packing fraction of 0.002 8. The chain extensibilities at the chain vectors͒ deform affinely with strain. The theoretical which the downturns in pressure are observed are larger than details of these models are given elsewhere;5,6,8,53 here we those suggested by Kuhn and Grun43 ͑between one third and only show the final expressions for the network elastic free two thirds͒, but are in excellent agreement with the experi- energy. mental results of Andrady, Llorente, and Mark,66 who ob- The constrained junction model ͑Flory and Erman,5 served non-Gaussian effects to begin at approximately 60%– Ronca and Allegra68͒ was proposed to incorporate the damp- 70% of the maximum chain extensibility. ing effect of chain entanglements on the spatial fluctuations From Fig. 6, we also see that the downturn in pressure of the junctions. In this model, the free motion of the junc- occurs at lower packing fractions as the chain length in- tions in a phantom domain ͑i.e., in a purely phantom net-

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AelasticϭAphantomϩAconstraint

2 1 ϭ␰kBT ␭i ϩ 2 N f kBT ͚i

ϫ ͓BiϩDiϪln͑Biϩ1͒Ϫln͑Diϩ1͔͒, ͑44͒ ͚i where the summation is over each coordinate axis, i, and ␰ ϭ(1Ϫ2/f ) Nc , is the cycle rank of the network, i.e., the number of chains that must be cut in order to reduce the FIG. 7. Comparison of simulation results for the reduced pressure, P*, multiplied by the chain length, n, vs packing fraction, ␩, for the 50-mer and network to a tree with no closed cycles. The terms Bi and Di 100-mer trifunctional networks, with theoretical predictions for the network are given by pressure in the region ␩Ͻ0.12. The elastic components of the pressure are taken from the constrained junction model Refs. 5,6 dotted line , the 2 2 ͑ ͒͑ ͒ ͑␭i Ϫ1͒ ␭i Bi model of Obhukov, Rubinstein, and Colby ͑Ref. 53͒͑dashed line͒, and the B ϭ␬2 , D ϭ , ͑45͒ i 2 2 i ␬ non-Gaussian dendrimer-network model ͑solid line͒. The liquid-like compo- ͑␭i ϩ␬͒ nent of the network pressure is taken from the GFD theory ͓Eq. ͑34͔͒ for the constrained junction and Obhukov models and Eq. ͑41͒ for the non- where ␭i is the macroscopic deformation ratio along the co- Gaussian dendrimer-network model͒. ordinate axis, i. The parameter ␬ is a measure of the strength of the constraints, and is directly related to the number of crosslinks in the volume of a single network chain, i.e., ␬ 2 3/2 and our networks therefore contain slightly collapsed chains, ϭI͗R0͘ N f /V0 , where I is a constant, and N f is the number of crosslinks in the f-functional network. In order to com- the work of Obhukov, Rubinstein, and Colby is of special pare the predictions of this theory with our simulation re- relevance to us. The underlying idea in their work is as fol- sults, we calculated the network pressure as the sum of the lows: The chains in the deswelled networks form numerous elastic component from the above relation ͑obtained using temporary entanglements because of the solution crosslink- V), and the GFD liquid-like ing and deswelling procedure. When a chain in a deswelledץ/ Aelasticץthe relation PϭϪ contribution to the pressure ͓Eq. ͑34͔͒. The chain mean- network is stretched, it forms a string of blobs, each blob 2 containing a portion of the chain in a special double-folded squared end-to-end distance ͗R0͘ is taken to be equal to the mean-squared end-to-end distance of the network chains, configuration. When the chains in a melt-crosslinked net- 2 work are stretched, they also form a string of blobs,67,60 but ͗R ͘,at␩0, which again takes the value ␩0ϭ0.45. The values of ͗R2͘/␴2 are obtained from Table IV. The value of each blob contains a portion of the chain in a Gaussian con- the constant I is taken to be 1. ͓The value of the constant figuration. does not materially affect the model performance; this was For deswollen networks, Obhukov, Rubinstein, and checked by trying a range of I values from 0.01 to 1000 in Colby predict a linear stress–strain relation in the low defor- Eq. ͑45͒.͔ mation regime, a nonlinear relation at intermediate deforma- Figure 7 shows the constrained junction model predictions, and a linear relation again at very large deformations. tion for the scaled network pressure, nP*, ͑dotted line͒ ver- The elastic free energy in the low, intermediate, and large sus the packing fraction, along with the predictions of the deformation regimes obeys the following relations non-Gaussian dendrimer model ͑solid line͒ and the Obhukov 2 s s 1/3 1/4 model ͑dashed line͒, for the 50-mer and 100-mer networks in AelasticϳNckBT␭ , ␭͓R0͑␩0͒ ͔Ͻ͑nen͒ the low ␩ region. The symbols represent our simulation re- s s 1/3 4/3 Ϫ1/3 4/3 sults. It can be seen that the constrained junction model does ϳNckBT͓R0͑␩0͒ ͔ ͑nen͒ ␭ , not predict the downturn in the pressure at very low packing fractions. At intermediate to high packing fractions, the per- 1/4 s s 1/3 1/2 ͑nen͒ Ͻ␭͓R0͑␩0͒ ͔Ͻn/ne formance of the constrained junction model prediction for the network pressure is similar to that of the affine model. s 5/12 s s 1/3 1/2 The elastic free energy of networks that are solution ϳNc͑␩0͒ ␭, ␭͓R0͑␩0͒ ͔Ͼn/ne , ͑46͒ crosslinked and then deswollen is obtained by Obhukov, Ru- 53 s s 1/3 binstein, and Colby. As mentioned earlier, the chain con- where the quantity R0(␩0) represents the end-to-end dis- figurations in such networks are somewhat collapsed relative tance of the chain in the deswollen undeformed network, s s to melt-crosslinked networks and contain fewer trapped en- with R0 and ␩0 being the chain end-to-end distance and tanglements. This means that they can undergo very large polymer volume fraction in the network preparation ͑solu- deformations before rupturing. Since our crosslinking proce- tion͒ state, and ne being the entanglement length. We have dure is akin to solution crosslinking followed by deswelling, attempted to make a qualitative comparison of our simula-

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TABLE VI. Simulation results for the reduced pressure, P*, at each packing fraction, ␩, and corresponding deformation ratio, ␭, for tetrafunctional networks at chain lengths nϭ20, 35, and 50.

␩ ␭ 20 35 50

0.45 1.0 5.094 5.127 5.143 0.35 1.09 1.717 1.751 1.753 0.25 1.22 0.488 0.502 0.507 0.15 1.44 0.070 0.091 0.094 0.05 2.08 Ϫ0.0055 Ϫ0.0032 Ϫ0.00055 0.01 3.56 Ϫ0.024 Ϫ0.013 Ϫ0.0063

␭2 increases, the uncrosslinked chain dimensions increase, but much more slowly than the crosslinked chain dimensions; in fact the uncrosslinked chains approximately obey the semidilute solution relation,60 ͗R2͘ϰ␩Ϫ1/4ϰ␭3/4 in the FIG. 8. Simulation results for the mean squared end-to-end distance, region ␭2Ͻ2 ͑approximately͒, while the crosslinked chains 2 2 ͗R ͘/␴ , of trifunctional networks ͑solid symbols͒ and uncrosslinked chain obey the affine relation, ͗R2͘ϰ␭2 in the region ␭2Ͻ2 ͑ap- systems ͑open symbols͒ for chain lengths 20 and 50 vs squared deformation 2 ratio, ␭2, on a log–log scale. Lines are added solely as guides to the reader’s proximately͒. For ␭ Ͼ2(␩less than about 0.15), the un- eye. crosslinked chain dimensions are nearly independent of the deformation, the reason being that in this very dilute regime the chains form swollen coils that do not interact with each tion results with the network pressure obtained from the Ob- other, and are therefore unaffected by the overall change in hukov model, by calculating the latter as the sum of the system dimensions. The crosslinked chain dimensions con- elastic component obtained from the above relation, and the tinue to increase with ␭ in the ␭2Ͼ2 regime, with a non- GFD liquid-like component ͓Eq. ͑34͔͒. The parameter ne in affine scaling of ͗R2͘ϰ␭2g where 0ϽgϽ1. The exact value our calculations is taken to be equal to the chain entangle- of g is dependent on the chain length, for instance, g ment length in an equivalent melt at a packing fraction of ϭ0.682 for the 20-mer network, gϭ0.775 for the 35-mer 45 0.45; based on the article of Smith, Hall, and Freeman, on network, gϭ0.826 for the 50-mer network, and gϭ0.841 for the dynamics of melts, neϭ29. Figure 7 shows the Obhukov the 100-mer network. Thus the chain dimensions of un- model prediction for the scaled pressure, nP*, versus the crosslinked chain systems show a much weaker scaling with packing fraction, ␩, in the low ␩ region. It can be seen that the deformation ratio than the network chains. the Obhukov model does not predict the pressure downturn We have also investigated the effect of network func- at very low packing fractions. The Obhukov model is as tionality on the network pressure and chain end-to-end dis- successful as the affine model in predicting the network pres- tances by comparing the behavior of trifunctional and tet- sure at high to intermediate densities. rafunctional networks. Tables VI and VII show the reduced It is also of interest to compare the chain dimensions of pressure and chain dimensions of three tetrafunctional net- network chains with those of uncrosslinked linear chains at works of chain length 20, 35, and 50. The pressure variation the same packing fraction. Figure 8 shows the mean-squared with packing fraction of the tetrafunctional and trifunctional end-to-end distance of 20-mer and 50-mer network chains networks is shown in Fig. 9. The inset is a magnification of ͑solid symbols͒ and uncrosslinked chains ͑open symbols͒ the low ␩ region. The open symbols represent the trifunc- 2 versus the squared deformation ratio, ␭ on a log–log scale. tional network results and the filled symbols represent the The uncrosslinked system chain dimensions are given in tetrafunctional network results. We see that the pressure– 2 Table V. At ␭ ϭ1.0, i.e., at ␩ϭ0.45, the chain dimensions volume behavior of tetrafunctional networks is identical to in the crosslinked and uncrosslinked systems are nearly iden- that of the trifunctional networks. Hence it appears that the tical; evidently at these high packing fractions, the chain di- pressure–volume behavior of these athermal networks is af- mensions are virtually unaffected by the system topology. As fected very little by the network functionality. Our theory too predicts a weak dependence on the functionality, both in the TABLE V. Simulation results for the uncrosslinked chain mean-squared liquid-like term and in the elastic term ͓Eqs. ͑42͒ and ͑43͔͒. end-to-end distance, ͗R2͘/␴2, at each packing fraction, ␩, and correspond- Figure 10 shows the simulation results ͑symbols͒ and non- ing deformation ratio, ␭, at chain lengths nϭ20, 35, 50, and 100. Gaussian dendrimer-network theory predictions for the scaled pressure, nP*, versus the packing fraction, ␩, for a ␩ ␭ 20 35 50 100 trifunctional and tetrafunctional network, both of chain 0.45 1.0 30.43 55.77 76.38 160.68 length 20. Our theory predicts that the tetrafunctional net- 0.35 1.09 32.07 60.07 85.16 168.53 work pressure follows the same trend as the trifunctional 0.25 1.22 ¯ 66.96 99.14 217.14 network pressure, which is in agreement with our simulation 0.15 1.44 39.84 75.25 113.80 233.15 0.05 2.08 45.08 82.47 137.86 305.33 results. 0.01 3.56 47.56 97.46 150.92 349.44 Figure 11 shows the variation in the mean-squared end- to-end distances of the tetrafunctional and trifunctional net-

TABLE VII. Simulation results for the chain mean squared end-to-end distance, ͗R2͘/␴2, at each packing fraction, ␩, and corresponding deformation ratio, ␭, for tetrafunctional networks at chain lengths nϭ 20, 35, and 50.

␩ ␭ 20 35 50

0.45 1.00 26.11 56.31 81.24 0.35 1.09 30.69 63.02 96.31 0.25 1.22 36.72 78.46 116.80 0.15 1.44 47.21 104.47 162.03 0.05 2.08 74.46 183.61 285.19 0.01 3.56 177.88 ¯ 686.06

V. CONCLUSION Discontinuous molecular dynamics simulations were conducted in order to investigate the pressure–volume properties of near-perfect trifunctional and tetrafunctional networks. A simple analytical equation of state is presented in which the network chains are treated in two ways; ͑1͒ as having a Gaussian end-to-end vector distribution ͑the Gauss- FIG. 9. Simulation results for the reduced pressure, P*, multiplied by the chain length, n, vs the packing fraction, ␩, for trifunctional ͑larger open ian dendrimer-network model͒, and ͑2͒ as having a non- circles͒ and tetrafunctional networks ͑smaller filled circles͒, for chain Gaussian end-to-end vector distribution ͑the non-Gaussian lengths 20, 35, and 50. The inset shows the pressure in the low density dendrimer-network model͒. The predictions of the Gaussian region. Lines are added solely as guides to the reader’s eye. dendrimer-network model for the network pressure are similar to those of the Flory affine model; both models accurately predict the network pressure over the range of packing frac- work chains with deformation ratio on a log–log scale. As tions from ␩ϭ0.45 to ␩ϳ0.1, and fail at low packing frac- before, the closed symbols represent the tetrafunctional net- tions (␩Ͻ0.1). The non-Gaussian dendrimer-network model work results and the open symbols represent the trifunctional is able to predict the network pressure over the entire range network results. The dimensions of the tetrafunctional net- of packing fractions studied, and successfully predicts the work chains appear to be approximately the same as the di- observed downturn in network pressure at very low packing mensions of the trifunctional network chains at all deforma- fractions, ␩ϳ0.003. tion ratios, with the exception that the 20-mer tetrafunctional A novel algorithm that utilizes the variable bond-length network chains are slightly collapsed relative to the corre- feature of the Bellemans chain was used to perform network sponding trifunctional network chains at ␭2 close to 1.0 volume changes. The efficiency of this algorithm coupled ͑high packing fractions͒. Thus it appears that the network with its applicability to different system topologies allowed chain dimensions are almost unaffected by the functionality us to investigate the pressure–volume behavior of trifunc- at all packing fractions. tional and tetrafunctional networks over a wide range of network packing fractions. The simplicity of the algorithm also makes it an attractive tool for NPT or Gibbs ensemble simu-

FIG. 11. Simulation results for the mean-squared end-to-end distance, FIG. 10. Simulation results ͑symbols͒ and non-Gaussian dendrimer-network ͗R2͘/␴2, of trifunctional ͑larger open circles͒ and tetrafunctional ͑smaller model theoretical predictions ͑lines͒ for the reduced pressure, P*, multi- closed circles͒ networks for chain lengths 20, 35, and 50, vs squared defor- plied by the chain length, n, vs the packing fraction, ␩, for a trifunctional mation ratio, ␭2, on a log–log scale. Lines are added solely to guide the network and a tetrafunctional network, both of chain length 20. reader’s eye.

Downloaded 22 Feb 2008 to 152.1.209.194. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 7572 J. Chem. Phys., Vol. 110, No. 15, 15 April 1999 Kenkare, Hall, and Khan lations of systems with complex molecular structure, for the conductance of an electrical resistor network.10,60 For an which conventional volume change moves cannot be used. ideal chain, the magnitude of the elastic restoring force ex- As mentioned above, we observed a downturn in the erted by the chain is given by FchainϭϪkchain R. The sum of network pressure as the network is expanded to very low the forces in any direction i exerted on a given junction in packing fractions. This occurs when the network chains are the network by the chains attached to that junction must be extended to between 60% and 70% of their maximum ex- zero, i.e., tended length. At these large extensions, the probability distribution of the chain end-to-end vectors is not expected to i ͚ Fchainϭ0. ͑A2͒ follow the Gaussian form. In fact, we found that the non- i Gaussian dendrimer-network equation of state based on the This is analogous10,60 to the Kirchoff law for electrical cir- non-Gaussian distribution suggested by Kuhn and Grun43 cuits, which states that the net current entering a given node ͓Eq. ͑33͔͒ was remarkably successful in predicting this drop in the circuit must be zero. Comparing the ideal chain rela- in pressure. The success of Eq. ͑33͒ does not necessarily tion, F ϭϪk R, and the electrical resistance relation imply that the non-Gaussian probability distribution of Kuhn chain chain ͑Ohm’s law͒, iϭ⌬V/R, where i is the current, R is the re- and Grun is accurate, but it does suggest that network equa- sistance and ⌬V is the voltage drop across the resistor, we tions of state that account for the finite extensibility of poly- see that analogies can be drawn between: ͑1͒ the spring con- mer chains can successfully predict network pressure– stant, k and the conductance, 1/R, ͑2͒ the magnitude of volume behavior at low packing fractions. chain the force, F and the current, i, and ͑3͒ the chain end-to- Comparison of the chain end-to-end vectors in trifunc- chain end distance, R, and the voltage drop, ⌬V. tional networks and uncrosslinked chain systems shows that We now want to use simple electrical circuit laws to at very high packing fractions, the network chain dimensions obtain the effective resistance, R , of the dendritic are approximately equal to the uncrosslinked chain dimen- dendrimer network shown in Fig. 1, when each branch ͑chain͒ of the sions. This is due to the fact that at very high packing frac- dendrimer is replaced by a resistance, Rϭ1/k . As de- tions, the chains are so densely packed together that their chain scribed in Sec. III, the f functional dendrimer network in Fig. dimensions are unaffected by the specific topology of the 1 consists of concentric layers of branches or resistors, with system. As the packing fraction decreases, the chain dimen- the first layer containing f branches or resistors, the second sions begin to be influenced by the system topology and the layer containing f ( f Ϫ1) branches or resistors, the third layer network chain dimensions are observed to increase rapidly containing f ( f Ϫ1)2 branches or resistors, and so on. Now relative to the uncrosslinked system chain dimensions. We consider that the outermost nodes of the dendrimer are also investigated the pressure-volume behavior and the chain grounded, and that a current i enters the network at O. ͑This dimensions of tetrafunctional networks, and found that net- is analogous to applying a force F across the network͒. Then work pressure and chain dimensions are minimally affected the sum of the voltage drops over any path taken by the by the functionality. current between the point O and the grounded nodes is equal to the overall voltage drop, ⌬V, between O and ground. The ACKNOWLEDGMENTS effective resistance of the network, Rdendrimer is then simply This work was supported by the GAANN Computational given by Rdendrimerϭ⌬V/i. Due to the symmetry of the net- Sciences Fellowship of the U.S. Department of Education, work, all branches in the same layer will have the same and the Office of Energy Research, Basic Sciences, Chemical voltage drop across them and the same current flowing Science Division of the U.S. Department of Energy under through them. Hence by simple symmetry arguments, the Contract No. DE-FG05-91ER14181. Acknowledgement is resistors in the first layer of the network will have a current made to the Donors of the Petroleum Research Fund admin- i/ f flowing through them, the resistors in the second layer istered by the American Chemical Society for partial support will have a current i/ f ( f Ϫ1) flowing through them, and so of this work. The authors thank the North Carolina Super- on. The voltage drop across any one of the resistors in the computing Center for grants of CPU time on the Cray Y-MP first layer of the network is then iR/f, the voltage drop and T90, and on the workstation cluster. across any one of the resistors in the second layer of the network is iR/f(fϪ1), and so on. By summing the voltage APPENDIX drops over all the concentric layers of the network, we can calculate the total voltage drop across the network Here we show that the effective spring constant of the dendritic structure displayed in Fig. 1 can be obtained in i i i terms of the spring constant of a single chain ͑branch͒ of the ⌬VϭiR ϭ Rϩ Rϩ Rϩ dendrimer f f f Ϫ1 2 ¯ structure, in the following form: ͑ ͒ f ͑ f Ϫ1͒

͑ f Ϫ1͒͑mϪ1͒ i ϩ R, ͑A3͒ kdendrimerϭkchain f ͑ f Ϫ2͒ , ͑A1͒ mϪ1 ͫ ͑ f Ϫ1͒mϪ1 ͬ f ͑ f Ϫ1͒ where m is the rank of the dendrimer. If the dendrimer chains where m represents the rank or the number of layers in the 2 are ideal and Gaussian, kchainϭ3kBT/͗R0͘. network. The right hand side of the above equation is a geo- To show how Eq. ͑A1͒ is obtained, we invoke the anal- metric series in 1/( f Ϫ1), and can be summed in order to ogy between the elasticity of a perfect polymer network and obtain the effective resistance of the dendrimer network

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