Size exponent

Units Unit Cost (S) Size Exponent Bare Module Factor ... 

Size exponent 0.85 Costa s eosti (sizej/slzei) ... 

Surface area, sq. ft. Size exponent 0.80 Design pressure, psig. 

For K 7 0, a part of Aqo cancels 1 jft exactly in the non-free-draining limit and the remainder is dependent on the structure factor of the polymer and the size exponent v. For large values of KRg, p becomes... 

In another connection, we showed (Chapter 2, Section 2.3) that the dimensionality of a chain with a size exponent v is given by... 

Very recently, this investigation method has been extended to two dimensions. Improvements have been made concerning the measurement technique of the surface osmotic pressure of polymers spread on a liquid-air interface. Vilanove and Rondelez thus measured, for the first time in 1980, the characteristic size exponent v of chains in a two-dimensional space. 

In 1970, K. Wilson discovered that to describe certain critical properties correctly it was necessary to resort to renormalization techniques borrowed from field theory. In 1972, P.G. de Gennes showed that a long polymer chain is a critical object and that the same techniques are applicable to its study. This new approach proved fruitful. Moreover, it explained the errors contained in the old theories. These errors arise partly from the fact that they take into account only one exponent, namely the size exponent v, overlooking the fact that other exponents may exist. Actually, renormalization theory shows that there are two fundamental exponents v and y. All the older theories implicitly state that y has... 

Thus, the size exponent v characterizing the stretching of the chain is related to i by... 

From a conceptual point of view, this theory seems much better than those that had previously been propounded. However, the method gives a value of the size exponent v which is not Flory s value vF = 3/(d + 2) but Stockmayer s value v = 2jd, which is too large to be realistic. 

In spite of appealing features, the Gaussian approximation has a very serious defect it predicts a size exponent v = 2/d which is too big, in contradiction to the results of computer simulations and experiments for real polymers. 

It is well-known that in a polymer melt as in a semi-dilute solution, the size exponent has the trivial value v = 1/2. However, for a melt, it is also possible to define exponents aM as was done for dilute solutions (see Section 3.3) as before, these exponents are directly related to the partition functions associated with a network [see eqns (12.3.64) and (12.3.65)]. The values of aM in the present case are not obvious. However, the exact values in two-dimensions... 

A critical chain, like the Brownian chain or the Kuhnian chain, is characterized by a size exponent v corresponding to the dimensionality D — 1/v. In this case, the order of magnitude of the distances r(n) — (0) contributing to H(q) is given by the length x defined by... 

The capital cost of the dewar is also determined using a sizing exponent and the ratio of the base cost and size to the actual storage capacity of the dewar... 

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