Free phantom network

Free phantom network Network without any constraints, which consequently collapses,... 

Three types of phantom networks can be distinguished free phantom networks, fixed phantom networks, and localised phantom networks The first type is without any constraints and will consequently collapse. The second type is a phantom network with some junctions fixed in space. As a result, it is subjected to contraints that do not really exist. The most natural phantom network model is the last variant, in which the equilibrium positions of all segments are determined by suitable boundary conditions without any need for segments or junctions being fixed. 

However, for a network with given Kirchhoff matrix three types of phantom networks can be defined free phantom networks, fixed phantom networks and localised phantom networks. 

A free phantom network is a phantom network without any constraints. Such a network will collapse to a microscopic volume and is therefore not a useful model for a... 

The properties of such a localised expanded network can be derived from James theory in Flory s version. Consider a large phantom network with variable Kirchhoff matrix T. The configuration function of the free phantom network is Zfree = exp[- J T J ] where R represents the set of N vectors (3N components) R, Rj of the junctions i, j. James and Flory have shown that for a fixed phantom network with a junctions fixed and r junctions free, the configuration function can be written... 

Comparison of the expressions for the elastic free energies for the affine and phantom network models shows that they differ only in the front factor. Expressions for the elastic free energy of more realistic models than the affine and phantom network models are given in the following section. 

The elastic free energy of the constrained-junction model, similar to that of the slip-link model, is the sum of the phantom network free energy and that due to the constraints. Both the slip-link and the constrained-junction model free energies reduce to that of the phantom network model when the effect of entanglements diminishes to zero. One important difference between the two models, however, is that the constrained-junction model free energy equates to that of the affine network model in the limit of infinitely strong constraints, whereas the slip-link model free energy may exceed that for an affine deformation, as may be observed from Equation (41). 

Early theories of Guth, Kuhn, Wall and others proceeded on the assumption that the microscopic distribution of end-to-end vectors of the chains should reflect the macroscopic dimensions of the specimen, i.e., that the chain vectors should be affine in the strain. The pivotal theory of James and Guth (1947), put forward subsequently, addressed a network of Gaussian chains free of all interactions with one another, the integrity of the chains which precludes one from the space occupied by another being deliberately left out of account. Hypothetical networks of this kind came to be known later as phantom networks (Flory, 1964,... 

A subsequent theory [6] allowed for movement of the crosslink junctions through rearrangement of the chains and also accounted for the presence of terminal chains in the network structure. Terminal chains are those that are bound at one end by a crosslink but the other end is free. These terminal chains will not contribute to the elastic recovery of the network. This phantom network theory describes the shear modulus as... 

Each monomer is constrained to stay fairly close to the primitive path, but fluctuations driven by the thermal energy kT are allowed. Strand excursions in the quadratic potential are not likely to have free energies much more than kT above the minimum. Strand excursions that have free energy kT above the minimum at the primitive path define the width of the confining tube, called the tube diameter a (Fig. 7.10). In the classical affine -and phantom network models, the amplitude of the fluctuations of a... 

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 ... 

Free-fluctuation limit (localised phantom network),... 

The basic assumption behind the second hmit, the affine limit of mbberlike elasticity (sometimes also called the affine hmit of a phantom network, Hermans-Flory-Wall (Wall 1942, 1943, 1951 Hermans 1947)), is that the cross-links are firmly embedded in their surroundings and, therefore, they do not fluctuate. Their position is transformed affinely with the macroscopic strain. The elastic part of free energy is given by... 

In case of the free-fluctuating limit of a phantom network, the fluctuations are not hindered. K (2) is equal to zero, leading to (20). 

It should be noted that in the case of the phantom network (see Sect. 3) the free energy of any typical representative with a given number of chemical crosslinks shows the same behaviour. 

Equation (22) holds for phantom networks of any functionality, irrespective of their structural imperfections. In case b), fluctuations of junctions are assumed to be suppressed fully. The junctions themselves are considered to be firmly embedded in the medium and their position is transformed affinely with the macroscopic strain. This leads to the free energy expression for an f-functional network possibly containing free chain ends... 

The shortcomings of the phantom network concepts have stimulated a number of attempts to find theoretically a more satisfactory elastic free energy function to describe the properties of elastomers at different states of deformation. The efforts to explain the real network behaviour by a special non-topological mechanism can be divided into four types. The first group considered intra- and intermolecular effects los-ni) wjtii these assumptions it is hard to explain values of the Mooney-Rivlin parameter which are of the order of the corresponding Cj parameter. 

The so-called domain of constraints is assumed to be cubical or spherical The degree of constraint on fluctuations is affected by the degree of deformation. All centres of domains are distributed with respect to the mean positions of the junctions in the phantom network and the mean positions of the domains are assumed to transform affinely with the macroscopic strain. Further, the shapes of domains are assumed to transform non-affinely due to some relaxation of the constraints. The theory results in an elastic free energy change... 

A very interesting way to explain deviations from the phantom network theory is the approach proposed by Ball et al. who modelled the topological constraints by sliding links which make contacts between the network chains. Formally, this approach is based on the Deam-Edwards concept. Assuming that the chemical junctions are free to fluctuate about their mean positions the calculation of the elastic free energy change leads to the expression... 

An old point of controversy in rubber elasticity theory deals with the value of the so-called front factor g = Ap which was introduced first in the phantom chain models to connect the number of elastically effective network chains per unit volume and the shear modulus by G = Ar kTv. We use the notation of Rehage who clearly distinguishes between A andp. The factor A is often called the microstructure factor. One obtains A = 1 in the case of affine networks and A = 1 — 2/f (f = functionality) in the opposite case of free-fluctuation networks. The quantity is called the memory factor and is equal to the ratio of the mean square end-to-end distance of chains in the undeformed network to the same quantity for the system with junction points removed. The concept of the memory factor permits proper allowance for changes of the modulus caused by changes of experimental conditions (e.g. temperature, solvent) and the reduction of the modulus to a reference state However, in a number of cases a clear distinction between the two contributions to the front factor is not unambiguous. Contradictory results were obtained even in the classical studies. 

For uniaxially stretched networks, the molecular deformations are characterised by the radii of gyration R u an respectively parallel and perpendicular to the stretching direction. From the small-angle scattering function, the molecular deformation of a stretched elastic phantom chain has been calculated for three cases 1) Free-fluctuation phantom network... 

The eKperimental situation can be summarised as follows Crosslinked and swollen polystyrene gels exhibit a chain deformation which is less than that given by all phantom network models Early experiments by Clough et al, on radiation crosslinked polystyrene networks had been interpreted to be consistent with the free-fluctuation phantom scattering law. Most of the scattering experiments which were performed on uniaxially stretched end-linked siloxane networks showed much smaller molecular deformations than would be predicted even from Eq. (78) To... 

The tube model presented here yields values of the front factor of the crosslink contribution close to the front factor of the free-fluctuating phantom network. It is felt that the stronger constraints acting on the crosslinks have to be simulated by tube dimensions that depend on the distance from the crosslinks. In this way, the crossover from the free-fluctuating to the fixed phantom network value of the front factor, characteristic for the model of restricted junction fluctuation, can also be reproduced by tube models. 

Using Eq. (5.53) for the elastic free energy, we obtain the following expression for the free energy of the phantom network... 

In real polymer network the effects of excluded volume and chain entanglements should be taken into account. In 1977 Hory [26] formulated the constrained junction model of real networks. According to this theory fluctuations of junctiOTs are affected by chains interpenetration, and as the result the elastic free energy is a sum of the elastic free energy of the phantom network AAph (given by Eq. (5.78)) and the free energy of constraints AA ... 

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