Network theories

There is a growing consensus of scientists who consider the present approach to pore size determination to be unreliable and prefer a different approach. They consider that pore size determination from the desorption branch of the isotherm gives misleading data, particularly if the hysteresis loop is broad. 

Network theory describes isotherms in terms of pore connectivity and pore size distribution. At the end of adsorption, when a high relative pressure has been reached and the adsorption isotherm has formed a plateau, all the accessible pores have been filled. On redudng the pressure, liquid will evaporate from the larger open pores but will be prevented from evaporating from equally large pores that are connected to the surface via narrower channels. Desorption more closely reflects the distribution of channels rather than the distribution of pores. 

Network theory defines the resulting hysteresis between the adsorption and desorption branches of the isotherm in terms of pore interconnectivity. As the pressure is reduced, a liquid filled cavity cannot convert to the gas phase until at least one of the channels to the outside has evaporated. If the radii of all the channels are less than the equivalent radius of the cavity, the emptying is governed by the largest channel radius, rather than the cavity radius, which will take place at a reduced pressure. Once the liquid in the channel evaporates, the liquid in the cavity can also evaporate and the cavity empties. This is what causes hysteresis adsorbate in the small channel must evaporate before the adsorbate in the cavity can evaporate. As a result the geometry of the network determines the shape of the desorption branch of the isotherm. 

Adsorption provides more accurate pore size information since adsorption ttdces place through the porous network. This means that even though smdl cavities fill first they do not block off what is happening in the internal big cavities. Adsorption is a continuous 

To illustrate, suppose an increase in external gas pressure causes liquid to condense into a small channel. If the other end is connected to a bigger cavity the liquid would simply evaporate from that end of the channel into the bigger cavity until equilibrium was reached. Adsorption filling is determined by the size of the pore correlated to relative pressure whereas desorption is determined by the branching interconnectivity of the porous network [79]. 

---

Phan-Thien, N. and Tanner, R.T., 1977. A new constitutive equation derived from network theory, Non-Newtonian Fluid Mech. 2, 353-365. 

Wagner, M.H., 1979. Towards a network theory for polymer melts. Rheol. Acta. 18, 33 - 50. 

The important point we wish to re-emphasize here is that a random process is specified or defined by giving the values of certain averages such as a distribution function. This is completely different from the way in which a time function is specified i.e., by giving the value the time function assumes at various instants or by giving a differential equation and boundary conditions the time function must satisfy, etc. The theory of random processes enables us to calculate certain averages in terms of other averages (known from measurements or by some indirect means), just as, for example, network theory enables us to calculate the output of a network as a function of time from a knowledge of its input as a function of time. In either case some information external to the theory must be known or at least assumed to exist before the theory can be put to use. 

One approach to extend such theories to more complex media is network theory. This approach utihzes solutions for transport in single pores, usually in one dimension, and couples these solutions through a network of nodes to mimic the general structure of the porous media [341], The complete set of equations for aU pores and nodes is then solved to determine overall transport behavior. Such models are computationally intense and are somewhat heuristic in nature. 

Jeme N.K. (1974) Towards a network theory of the immune system. Ann Immunol (Instil Pasteur), 125C, 373-389. 

Allanic AL, Jezequel JY, Andre JC (1992) Application of neural networks theory to identify two-dimensional fluorescence spectra. Anal Chem 64 2618... 

In this review, we have given our attention to Gaussian network theories by which chain deformation and elastic forces can be related to macroscopic deformation directly. The results depend on crosslink junction fluctuations. In these models, chain deformation is greatest when crosslinks do not move and least in the phantom network model where junction fluctuations are largest. Much of the experimental data is consistent with these theories, but in some cases, (19,20) chain deformation is less than any of the above predictions. The recognition that a rearrangement of network junctions can take place in which chain extension is less than calculated from an affine model provides an explanation for some of these experiments, but leaves many questions unanswered. 

The remaining question is, how the deviations from phantom network theory at high branching densities can be explained. 

Figure 7. Ratio of experimentally observed and theoretically calculated modulus, using phantom network theory with f2 and v2, versus branching density z-...

The two-network theory for a composite network of Gaussian chains was originally developed by Berry, Scanlan, and Watson (18) and then further developed by Flory ( 9). The composite network is made by introducing chemical cross-links in the isotropic and subsequently in a strained state. The Helmholtz elastic free energy of a composite network of Gaussian chains with affine motion of the junction points is given by the following expression ... 

Figure 1. Effective first network modulus, Gle, after complete removal of first network cross-links plotted against second network modulus, Gi. Calculated from the composite network theory of Flory (19J for G, — 0.75 MPa.

Figure 2. The principle of the two-network method for cross-linking in a state of simple extension. First network with modulus Gy is entirely due to chain entangling. Second network with modulus Gx is formed by cross-linking in the strained state. Both Gy and Gx can be calculated from the two-network theory.

The two-network method has been carefully examined. All the previous two-network results were obtained in simple extension for which the Gaussian composite network theory was found to be inadequate. Results obtained on composite networks of 1,2-polybutadiene for three different types of strain, namely equibiaxial extension, pure shear, and simple extension, are discussed in the present paper. The Gaussian composite network elastic free energy relation is found to be adequate in equibiaxial extension and possibly pure shear. Extrapolation to zero strain gives the same result for all three types of strain The contribution from chain entangling at elastic equilibrium is found to be approximately equal to the pseudo-equilibrium rubber plateau modulus and about three times larger than the contribution from chemical cross-links. 

The first theory of the structure of glass to become widely accepted was that of Zachariasen (1932), called the random network theory [now commonly referred to as the continuous random network (CRN) theory]. This arose... 

---