Flory network model

CALCULATION OF MOLECULAR DEFORMATION AND ORIENTATION IN ELASTOMERS USING THE FLORY NETWORK MODEL... 

The elastic contribution to Eq. (5) is a restraining force which opposes tendencies to swell. This constraint is entropic in nature the number of configurations which can accommodate a given extension are reduced as the extension is increased the minimum entropy state would be a fully extended chain, which has only a single configuration. While this picture of rubber elasticity is well established, the best model for use with swollen gels is not. Perhaps the most familiar model is still Flory s model for a network of freely jointed, random-walk chains, cross-linked in the bulk state by connecting four chains at a point [47] ... 

Therefore, Flory s theory concludes that as the functionality of a network increases, the constraint contribution, fc, should decrease and eventually vanish. Furthermore, in the extreme limit in which junction fluctuations are totally suppressed, the Flory theory reduces to the affine network model ... 

An early model based on crosslinked rubbers put forward by Flory and Rehner (1943) assumed that chain segments deform independently and in the same manner as the whole sample (affine deformation) where crosslinks were fixed in space. James and Guth (1943) then described a phantom-network model that allowed free motion of crosslinks about the average affine deformation. The stress (cr) described from these theories can be described in the following equations ... 

Statistical network models were first developed by Flory (Flory and Rehner, 1943, Flory, 1953) and Stockmayer (1943, 1944), who developed a gelation theory (sometimes referred to as mean-field theory of network formation) that is used to determine the gel-point conversions in systems with relatively low crosslink densities, by the use of probability to determine network parameters. They developed their classical theory of network development by considering the build-up of thermoset networks following this random, percolation theory. 

These two relations result from the phantom network model. Their derivations are given elsewhere (Flory, 1976 Mark and Erman, 2007). 

In recent years, important advances in the theory of rubber elasticity have been made. These include the introduction of the so-called phantom networks by Flory (I) and a two-network model for crosslinks and trapped entanglements by Ferry and coworkers (2,3). 

The results above are only valid for tetrafunctional crosslinking of monodisperse polymer. However, in many thermoreversible systems the crosslinks have functionalities that are much larger than four. Moreover, the polymers used are not monodisperse in general. In order to be able to calculate network parameters the present author [39—44] extended the Flory-Stockmayer model for polydisperse polymer which is crosslinked with f-functional crosslinks. It was possible to calculate network parameters for polymers of various molecular weight distributions (monodisperse polymer with D s M, /r3 = 1, a Schulz-Flory distribution with D = 1.5, a Flory distribution with D = 2, a cumulative... 

Flory s intuitive argument was later confirmed by Fricker [15] with minor modification by a replica calculation. Owing to the rather arbitrary assumption about chain creation and annihilation, the addition-subtraction network model is difficult to apply to real physical gels. 

In recent years, important advances in the theory of rubber elasticity have been made. These include the introduction of the so-called phantom networks by Flory and a two-network model for crosslinks and trapped entanglements by Ferry and co-workers, The latter builds on work by Flory and others on networks crosslinked twice, once in the relaxed state, and then again in the strained state. In other studies, Kramer and Graessley distinguished among the three kinds of physical entanglements as crosslink sites the Bueche-Mullins trap, the Ferry trap, and the Langley trap. 

To make the phantom model more realistic, the corrections for the effects of entanglements resulting from the exclusion of volume of a subchain to others were introduced. They lead to the reduction of the number of configurations available to each chain. According to the constrained jtmction model proposed by Flory, the deviation of a real network from the phantom network model results from constraints affecting the flurtua-tions of junctions, that is, the jimctions are chosen as structural... 

For example, the phantom network model of James and Guth (1,2) gave a recipe for predicting the deformation of a polymer network by an applied stress, and allowed predictions of the change in chain dimensions as a function of network expansion or distortion. In an effort to make the phantom model more realistic, and to fit the model to a variety of experimental results, P.J. Flory and collaborators (3,4,5) proposed that the fluctuation of crosslink junction points calculated by the James-Guth method should be very much restricted by chain entanglements. 

Both the experiments of Alfrey and Lloyd and the studies of Ferry and Kramer are inexplicable by a phantom network model. Incorporation of constraints on junction fluctuations cannot account for the memory of the conditions of crosslinking. Ferry and Kramer use a two network model proposed earlier by Flory (14) to explain their results. 

Other network models based on the inverse Langevin function are the tetrahedral model of Flory and Rehner (1943) subsequently modified by Treloar (1946) and the inverse Langevin approximation (Treloar, 1954). The relative merits of these approaches, which yield similar results have been discussed by Treloar (1975) who points out the overwhelming advantages of the three-chain model in ease of computation. 

The early molecular-based statistical mechanics theory was developed by Wall (1942) and Flory and Rehner (1943), with the simple assumption that chain segments of the network deform independently and on a microscopic scale in the same way as the whole sample (affine deformation). The crosslinks are assumed to be fixed in space at positions exactly defined by the specimen deformation ratio. James and Guth (1943) allowed in their phantom network model a certain free motion (fluctuation) of the crosslinks about their average affine deformation positions. These two theories are in a sense limiting cases, with the affine network model giving an upper... 

Small-angle neutron scattering (SANS) of labelled (deuterated) amorphous samples and rubber samples detects the size of the coiled molecules and the response of individual molecules to macroscopic deformation and swelling. It has been shown that uncrosslinked bulk amorphous polymers consist of molecules with dimensions similar to those of theta solvents in accordance with the Flory theorem (Chapter 2). Fernandez et al (1984) showed that chemical crosslinking does not appreciably change the dimensions of the molecules. Data on various deformed network polymers indicate that the individual chain segments deform much less than the affine network model predicts and that most of the data are in accordance with the phantom network model. However, defmite SANS data that will tell which of the affine network model and the phantom network model is correct are still not available. 

Swelling equilibrium in a network polymer can be predicted by the Flory—Rehner equation, derived on the basis of the affine network model and the Flory—Huggins expression for polymer solutions. 

The above picture of the network structure of vulcanized rubber is supported by -the success of the kinetic theory of rubberlike elasticity (see part 4, page 14) calculations based on this model agree well with experimental measurements of stress-strain curves and other properties (James and Guth, 1943 Flory, 1944). Excellent evidence that the swollen gel contains the same network as the unswollen rubber has been presented by Flory (1944, 1946), based on studies of butyl rubber. Using the network model, the number of cross-links in the structure can be calculated in three ways (o) from measurements of the proportions of insoluble (network) and soluble (unattached) material in samples of different initial molecular lengths (b) from the elastic modulus of the unswollen rubber (c) from the maximum amount of liquid imbibed by the gel when swollen in equilibrium with pure solvent. The results of these three calculations for butyl rubber samples were in good agreement. 

The Affine Network Model. One of the earlier assumptions regarding microscopic deformation in networks is that the junction points in the networks move affinely with macroscopic deformation. The affine network model was developed by Kuhn, Wall, and Flory (1,175,176). According to the model, chain end-to-end vectors deform affinely, which gives... 

The Phantom Network Model. The theory of James and Guth, which has subsequently been termed the phantom network theory, was first outlined in two papers (186,187), followed by a mathematically more rigorous treatment (188-190). More recent work has been carried out by Duiser and Staverman (191), Eichinger (192), Graessley (193,194), Flory (82), Pearson (195), and Kloczkowski and co-workers (196,197). The most important physical feature is the occurrence of junction fluctuations, which occur asymmetrically in an elongated network in such a manner that the network chains sense less of a deformation than that imposed macroscopically. As a result, the modulus predicted in this theory is substantially less than that predicted in the affine theory. 

This analysis shows that Flory s model implies much more detailed assumptions about the liquid forces than only the assumed deformation of fluctuation domains. The effect of that assumption alone is estimated by Eq. (50) for real networks in equilibrium. Flory s additional assumption (ii) that C, the centre of the fluctuation domain undergoes displacement affine with the strain together with assumption (iv) about the distribution of fluctuations around 0, leads to non-affine displacement of 0, the equilibrium position in the phantom network. An alternative assumption could have been that the displacement of 0 is affine with the strain while that of C is not. This is perhaps even more probable because with increasing strain the chain forces will increase in the direction of strain while the liquid forces will not. Also the additional assumption (iv) about the distribution of fluctuations around 0 implies a detailed model of the / p, -configuration and the / reai-configuration. 

Recently the polymeric network (gel) has become a very attractive research area combining at the same time fundamental and applied topics of great interest. Since the physical properties of polymeric networks strongly depend on the polymerization kinetics, an understanding of the kinetics of network formation is indispensable for designing network structure. Various models have been proposed for the kinetics of network formation since the pioneering work of Flory (1 ) and Stockmayer (2), but their predictions are, quite often unsatisfactory, especially for a free radical polymerization system. These systems are of significant conmercial interest. In order to account for the specific reaction scheme of free radical polymerization, it will be necessary to consider all of the important elementary reactions. 

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