Maximum extended length

We may note that function (1.5) has a non-realistic feature that R can be larger than the maximum extended length M of the chain. Though more realistic distribution functions are available (Birshtein and Ptitsyn 1966 Flory 1969), in this monograph, approximation (1.5) is sufficient for our purpose. 

Let /c be the maximum extended length of a lipid molecule. As N decreases we eventually reach a critical radius below which packing constraints become important. This radius R is determined by... 

Thus, as given by Eq. (1.42), the probability distribution function for the end-to-end vector R is Gaussian. The distribution has the unrealistic feature that R can be greater than the maximum extended length Nb of the chain. Although Eq. (1.42) is derived on the freely jointed chain model, it is actually valid for a long chain, where the central limit theorem is applicable, except for the highly extended states. 

FIGURE 15.10. The geometric approach to the evaluation of surfactant aggregation processes is based on three molecular quantities (a) the minimum interfacial area occupied by the head group, oq (b) the volume of the hydrophobic tail (or tails), v and (c) the maximum extended length of the tail(s) in a fluid environment, 4. 

The distribution (2.26) has the unrealistic feature that J can be larger than the maximum extended length Nb of the chain. A more realistic... 

As the kinetics simply scale with the diffusion coefficient the choice of this parameter is unimportant. Hence, all reaction/survival yields are presented in the form D t, which factors out the effect of the diffusion coefficient. The choice of the encounter radius used to model reactions inside the micelle (reported alongside each result) is based on the data given in Ref. [10]. The radius of the micelle (reported alongside each result) was chosen based on the work by Bruce and co-workers [11]. For a spherical micelle the maximum radius R must be the same as the maximum extended length of the hydrophobic chain Ic. Hence, for a micelle to form a spherical structure the following condition must be satisfied... 

Repeaters are very simple devices. They allow a cabling system to extend beyond its maximum allowed length by amplifying the network voltages so... 

Size-exclusion chromatography [9] with a coupled molecular-weight-sensitive detection is a simple convenient method for characterizing dendrimers for which limited sample quantities are available. The polyether dendrimers increase in hydrodynamic radius approximately linearly with generation and have a characteristic maximum in viscosity. These properties distinguish these dendrimers from completely collapsed, globular structures. The experimental data also indicate that these structures are extended to approximately two-thirds of the theoretical, fully extended length. 

Stains are extending down scale for not more than J of the maximum scale length. 

The fully extended length of the chain is equal to nl so that the maximum extensibility of a random coil with r = Zrms is from n l to nl, i.e. a factor of rfi. The value of Znns also gives a measure of the spatial extent of a chain. A second useful measure of this is the radius of gyration, r, which is the RMS distance of the atoms of the chain from the centre of gravity of the chain. Debye showed that, provided that n is very large, Zg = r /s/6. 

A hydrocarbon chain is in a constant thermal motion, and without external force field, the chains fluctuate around the most stable position given by the distribution of possible conformations at the temperature. The action of external forces at the ends of a molecule causes displacements of chains from their equilibrium conformations and evokes retractive forces. For a hydrocarbon chain of M = 14,000, extended length 125.5 nm, and the end-to-end distance r = 1 mn, the maximum exerted force is 10 MPa. The level of forces exerted by the random coil macromolecules are much lower than the theoretical strength of the primary bonds. The presence of strong intermolecular interactions, such as hydrogen bonds in polyamides, affects the retractive force substantially, causing a restriction of the number of possible chain conformations. In addition, the transitions... 

A fully extended chain, without distortion of bond angles or deformation of bonds, is represented in Figure 9(b). In this conformation, the value of the end-to-end distance is nZp, where Zp is the length of the bond vector projected on the chain axis. If the equivalent random chain is now required to have the same end-to-end distance in full extension and also the same mean-square displacement length as the real chain, then equations (33) and (34) result. The maximum extension ratio of unperturbed macromolecules may be defined as the ratio of the fully extended length nZp to the root-... 

---