Displacement length

The Displacement Distance theory suggests that since the stmcture of the flame is only quantitatively correct, the flame height can be obtained through the use of the displacement length or "displacement distance" (35,36) (eq. 12), where h = flame height, m V = volumetric flow rate, m /s and D = diffusion coefficient. 

Eq. (16 ) reverts to (8) for sufficiently small values of the ratio, r/r, of the displacement length to the maximum length. The bracketed quantity in the exponential of Eq. (16 ) may be looked upon as a correction factor on the value of which should have been used in the simpler Eq. (8). For r/vm, less than about one-half, this correction is negligible. The ratio of the root-mean-square length to the maximum length varies inversely as the square root of the maximum length, according to Eq. (13), i.e., / / m = from which it... 

The introduction of vectors of constant displacement length to represent the individual elements, which actually vary in length, is rendered more plausible by inquiry into the effect of incorporating this artifice in the treatment of the freely jointed chain. In this case V = m H. Upon substitution of this expression together with n nlm in Eq. (17), the previous expression for / , Eq. (6), is recovered. Hence the calculated distribution is unaff ected by an arbitrary subdivision of the chain in this manner. We conclude that the value chosen for m in the reduction of the real chain to an equivalent freely jointed chain likewise is inconsequential (within the limits on m stated above). 

In order to arrive at the desired relationship between the average of sf over all chain elements and the mean-square end-to-end distance it will be required to evaluate the sum Si in terms of displacement lengths r. To this end, we proceed with the aid of Eq. (C-l) as follows ... 

For the sake of definiteness, let us assume a freely jointed chain consisting of n bonds each of length Z. Then the mean-square displacement length between elements i and jj by analogy with Eq. (13), will be given by which yields... 

Parameter characterizing the Gaussian distribution of the end-to-end (displacement) length of a polymer chain (Chap. X et seq.). 

Although the disorientation process might be looked upon as a melting process, eq. (9.54) does not give a good estimate of the entropy of fusion of polymers. A more reasonable approach is to introduce the mean square displacement length of the actual chain in its unperturbed state [7], The number of such units would then replace r in eq. (9.54), but one still needs to determine the coordination number z in an independent manner. 

To find the probability of a displacement length between r and r + dr, independent of direction, we integrate over the corresponding volume of space, much as we did in deriving the Maxwell-Boltzmann distribution ... 

Find the most probable displacement length of a linear polymer chain, having 1000 bonds, each of length 0.14nm. 

Using Eqs. (55) and (90), show that the average displacement length of a freely jointed polymer chain with N bonds each of length d is within 10% of -jNd. 

Here k T has its usual meaning and L is the displacement length of the chain considered. The expansion factor a is then a function of a single parameter which may be written... 

Fig. 6. Mean square displacement lengths of two-choice square lattice chains with nearest-neighbor...

For densely crosslinked networks such as epoxies, n could conceivably be less than 5, and the validity of Eq. (8) is therefore questionable. King and Andrews were apparently the first investigators to address this point. The lack of q values for epoxy polymers motivated them to derive an alternate expression for 2J o that did not incorporate q. The main assumption of their theory was that the distance between nearest crosslinks, L, in short-chain networks was equivalent to the mean displacement length, L, of the network chains. They then showed that the number of chains crossing a unit area of crack plane was given by (1/2) and therefore expressed 2 /q as... 

For the discrete stochastic model given by the group of relations (4.58), written considering a unitary and uniform displacement length for the step k ... 

Cowie, P.A. and Scholz, C.H. 1992. Displacement-length scaling relationships for faults data synthesis and discussion. J. Struct. Geol., 14 1149-1156. 

The root-mean-square displacement length may then be calculated from... 

Since solvent power is a function of temperature, there is for any solvent a temperature, 0, at which ideal behavior should be observed—the solvent is then referred to as the theta-solvent. As the solvent power is increased, an isotropic expansion of the polymer molecule occurs, and the root-mean-square displacement length is then given by... 

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