Gaussian chain statistics

Rg)cai2 is the calculated radius of gyration with the parameters for PIP, taken from P.J. Flory (Statistical Mechanics of Chain Molecules, Wiley Intersciences, New York 1969) and the assumption of Gaussian chain statistics. 

Pab is the mutual concentration of reactive groups separated by a chain of v bonds, assuming the chain obeys Gaussian statistics. 

A satisfactory theory of macrocyclisation equilibria for chains obeying Gaussian statistics was presented by Jacobson and Stockmayer (1950) long before the availability of suitable experimental data for the proper testing of the theory. According to this theory, the macrocyclisation equilibrium constant Kx (see p. 10) is related to the density W(0) of the distribution of the end-to-end vector r in the region r = 0 through relation (57), where NA is... 

This result is appHcable to semi-dilute and concentrated solutions [21], and is also useful to check many simulations that do not include HI. For non-draining chains, introducing Gaussian statistics in Eq. (35), and transforming the summations over a large number of units in Eq. (34) into integrals, the translational friction coefficient can finally be written as [ 15]... 

In this model the linear chains become the rays of a star molecule. The rays, consisting of m repeating units, can be considered stiff rods, but in most cases they will be flexible and can be described in a first approximation by Gaussian chain statistics. A star molecule has only one branching unit among/xm units which belong to Hnear chains. Their properties can be expected to show a close... 

Assuming that the chain statistics is an effective Gaussian with the uniform expansion coefficient l, we obtain... 

Cyclic oligomers with x - 2-9 are found to be present in poly(1,3-dioxolane) samples prepared by monomer-polymer-equilibrations using boron trifluoride diethyl etherate as catalyst. The molecular cyclization equilibrium constants 7fx are measured and the values are in agreement with those calculated by the Jacobson-Stockmayer theory, using an RIS model to describe the statistical conformations of the corresponding chains and assuming that the chains obey Gaussian statistics. 

In addition to the above experimental point, one can raise a theoretical objection against the way in which Volkenstein et al. introduce the effect which the structure in a network has on its elastic behaviour. In their theory the Gaussian chain statistics are left unchanged in spite of the fact that the chain molecules run through bundles. Such a decoupling of chain statistics and bundles is unwarranted. In Fig. 29 c a schematic representation of the approach of Volkenstein et al. to a structured network is given. Only a two chain network is drawn, although it should, of course, be remembered that in reality a bundle structure will comprise parts of many molecules. 

As is well-known, this pair of expressions will not be valid for the most general case of a second order fluid, since p22 — tzi must not necessarily vanish for such a fluid. Eq. (2.9) states that the first normal stress difference is equal to twice the free energy stored per unit of volume in steady shear flow. In Section 2.6.2 it will be shown that the simultaneous validity of eqs. (2.9) and (2.10) can probably quite generally be explained as a consequence of the assumption that polymeric liquids consist of statistically coiled chain molecules (Gaussian chains). In this way, the experimental results shown in Figs. 1.7, 1.8 and 1.10, can be understood. 

Also, assuming Gaussian chain statistics for PFPE conformation in the surface normal direction, we can conjecture that the monolayer thickness is proportional to the average molecular size normal to the surface ... 

Figure 15.8 2H NMR spectra obtained in a randomly crosslinked, deuterated PB network. Precursor chain molecular weight 115700 g.mol"1, 1.2% crosslink agent, the average molecular weight between junctions is 27400 g.mol 1 (as determined by swelling experiments) or 11600 g.mol"1 (elastic measurements). The smooth curves are fits with a most probable distribution of chain lengths with number average molecular weight Mc = 11600 and with non-Gaussian chain statistics...

Modulus. Jackson et al. (1) calculate the contribution of amorphous material to the shear modulus of a semicrystalline polymer by assuming that only tie chains (chains whose ends are attached to different crystallites) contribute to the modulus and that these chains follow Gaussian statistics. They assume that the chains deform affinely. The predicted modulus values are lower than the observed values. The... 

The most important fact that you should grasp from this discussion is the entro-pic nature of rubber elasticity. Although the agreement between the simple model described here and experiment is not that great you have to keep in mind that there are both theoretical assumptions (e.g., affine deformation) and mathematical approximations (Gaussian chain statistics) that have... 

With regard to the molecular origin of these hindrances, it should be mentioned that the linear dimension of Aerosil (300 m g ) particles (about 7 nm) is comparable with the mean average distance between primarily filler particles in the PDMS matrix. Since the Gaussian chain statistics might be applied for PDMS chains in the filled rubbers [18], it is easy to show that the chain portions between the primarily filler particles should contain about 30-80 elementary units. This value is in the same range as the apparent number of elementary chain units between topological hindrances as measured by NMR... 

Fig. 20. Experimental molar cycliza-tion equilibrium constants (in mol dm for oligomeric sodium meta- diosidiat (NaPOalj in mdts at 1000 K (demted o) are compared with theoretical values calculated by the JacoNon and Stockmayer theory assuming that all the chains obey Gaussian statistics (dermted by the unbndcen line)...

Ideal chains obey Gaussian statistics in any dimension with R — bN, leading to the overlap volume fraction ... 

Modifications to these classical statistical models can also be made, such as by the incorporation of loose chains (non-load-bearing chains), physical crosslinks (temporary or permanent) and intramolecular crosslinks (loops). At higher deformations (strains) or increasing crosslink densities, it may be necessary to use a non-Gaussian statistical treatment that considers the finite extensibility of the chain. Non-Gaussian models are reviewed extensively by Treloar (1975). 

So far the development of the theory is very general - the fact that we deal with polymers has not yet been used. Now the important point is that the RPA [2,114] allows an explicit calculation of the r s, using gaussian chain statistics. The central assumption that the interactions, Eq. (173), do not affect the coil configurations, will be critically examined in the light of computer simulation results in Sect. 5.4. We shall not give the details of this RPA calculation here, but just quote its most relevant results [43] ... 

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