Localised phantom network

Localised phantom network The equilibrium position of all segments of the net chains is determined by suitable boundary conditions without any need for segments or junctions to be fixed. 

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

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. 

In order to serve as a useful model for real networks a phantom network must be defined in which the equilibrium positions of all segments are localised without any segments or junctions being fixed. Such a phantom network will be called a localised phantom network. 

The proporties of localised phantom networks can be derived from James theory for fixed networks by considering networks forming part of a larger network of which a number of junctions is fixed. James has shown that once a number of junctions is fixed, equilibrium positions of all other junctions are settled. By fixing junctions only of the part of the larger network outside the network under consideration, this network can be considered as localised by external forces. 

The configuration function of a localised phantom network is thus... 

Thus, the front factor Eq. (10) or (26) is generally valid for a localised phantom network. However, the value of the front factor cannot be derived from the values of F alone. For a network of given connectivity many geometrical arrangements of equilibrium positions of the junctions are possible. 

We have already argued that a localised phantom network appears to be the most suitable type of phantom network to correspond with a real network. Once the chemical structure of the network is given, the Kirchhoff matrix is known. The phantom network corresponding to a given real network should have the same Kirchhoff matrix... 

Real networks differ from phantom networks in that segments and junctions are constrained by liquid forces whereas in a localised phantom network only network forces are operative. 

By writing the elastic energy of the network as a sum of the elastic energy of the phantom network of 31 degrees of freedom, and the elastic energy stored in the fluctuation of polyfunctional junctions, Flory has introduced a model of the phantom network different from James phantom network with fixed junctions and from the localised phantom network discussed above. 

Since the basic idea of fluctuation domains being transformed in relation to the strain appears fruitful and also because Flory s relations have been confirmed by experiments, we will now see how Flory s assumptions can be introduced into the framework of the theory of localised phantom networks. 

The configuration function of a localised phantom network can, according to Eq. (23), be factorised into Zg and Z and the configuration integral into Ze and (Eq. (24)). Since Z and are independent A of the deformation of the material, the elastic energy is determined by Ze. ... 

To networks of class 2 as defined by Eq. (40b) neither the theory of phantom networks with fixed junctions of James nor the theory of localised phantom networks as presented in Sects. 2 and 3 is applicable since no junctions are fixed and the network is not in equilibrium withjtself. There is no direct and simple relation between the Kirchhoff matrix and th i -configuration. The i reai-values have been displaced from the corresponding R ph-values by liquid or entanglement forces. The exact nature of these forces in not known. 

Vapour-sorption experiments on different polymer plus solvent systems have shown that the elastic component of the solvent chemical potential exhibits a maximum, contrary to the phantom network theories or the Mooney-Rivlin equation. Furthermore, evidence has been found that the localisation and height of the maximum is dependent upon the nature of the diluent. 

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... 

Equation (20) shows that once the equilibrium positions of a phantom network are localised by external forces, the configuration function of the fluctuations around the equilibrium positions is given by Eq. (20) irrespective of where the equilibrium.positions are located and by which external forces they have been settled. 

Clearly, this assumption implies that, once the Kirchhoff matrix and the size of the fluctuation domains ate known, also the reai configuration of the real network is known. It also implies that the ),eai configuration is not in equilibrium with the network forces. Therefore, the theory of localised or fo fixed phantom networks is not applicable. 

---