Excluded-volume variable

The excluded-volume variable introduced in eq 1.11 has been defined for the Edwards continuous chain with length L and excluded-volume strength V. Since all the theoretical expressions presented above for or and as arc concerned with such chains, the variable appearing in them is the one so defined. However, though identical notation was used, the z that appeared in... 

In the framework of the two-parameter theory this dimensionless quantity becomes a universal function of the excluded-volume variable 2. Yamakawa s book [2] summarizes some representative theories of gf( ) presented by the end of the 1960s. Unfortunately, no substantial progress in the theory of g has since evolved. 

The radius expansion factor as(r) for perturbed spring-bead ring chains was calculated to first order in the excluded-volume variable 2 by Casassa [77], who obtained... 

In these equations, / is the excluded-volume strength, a the bond length (equal to the diameter of one bead), and z an excluded-volume variable, which reduces to z defined previously when 2q is set equal to a. The coefficient K nn) is essentially zero for ni< < 1 and increases to the familiar coil value 4/3 with... 

As expected, in equation (37) is not the same as ao in equation (38), but the most significant feature of these relations is that the expansion factors are functions of the single excluded volume variable z. From the second form on the right hand side of equation (33), we see that z can be expressed in terms of two groups of statistical quantities nb and n p. As mentioned above, the first group is simply related to an observable property and the second is obtainable from thermodynamic measurements (see below. Section 13.4). Thus z is a function of two physically meaningful quantities, and the theory of the expansion factor outlined here is a two-parameter theory in the sense proposed by Stockmayer. ... 

Now the excluded volume variable z can be written in terms of experimental quantities in the form... 

Another interesting version of the MM model considers a variable excluded-volume interaction between same species particles [92]. In the absence of interactions the system is mapped on the standard MM model which has a first-order IPT between A- and B-saturated phases. On increasing the strength of the interaction the first-order transition line, observed for weak interactions, terminates at a tricritical point where two second-order transitions meet. These transitions, which separate the A-saturated, reactive, and B-saturated phases, belong to the same universality class as directed percolation, as follows from the value of critical exponents calculated by means of time-dependent Monte Carlo simulations and series expansions [92]. 

Some GPC analysts use totally excluded, rather than totally permeated, flow markers to make flow rate corrections. Most of the previously mentioned requirements for totally permeated flow marker selection still are requirements for a totally excluded flow marker. Coelution effects can often be avoided in this approach. It must be pointed out that species eluting at the excluded volume of a column set are not immune to adsorption problems and may even have variability issues arising from viscosity effects of these necessarily higher molecular weight species from the column. 

When a dispersed phase is passed through a nozzle immersed in an immiscible continuous phase, the most important variables influencing the resultant drop size are the velocity of the dispersed phase, viscosity and density of continuous phase, and the density of the dispersed phase (G2, HI, H5, M3, Nl, P5, R3, S5). In general, an increase in continuous-phase viscosity, nozzle diameter, and interfacial tension increases the drop volume, whereas the increase in density difference results in its decrease. However, Null and Johnson (N4) do not find the influence of continuous-phase viscosity significant and exclude this variable from their analysis. Contradictory findings... 

Random walks on square lattices with two or more dimensions are somewhat more complicated than in one dimension, but not essentially more difficult. One easily finds, for instance, that the mean square distance after r steps is again proportional to r. However, in several dimensions it is also possible to formulate the excluded volume problem, which is the random walk with the additional stipulation that no lattice point can be occupied more than once. This model is used as a simplified description of a polymer each carbon atom can have any position in space, given only the fixed length of the links and the fact that no two carbon atoms can overlap. This problem has been the subject of extensive approximate, numerical, and asymptotic studies. They indicate that the mean square distance between the end points of a polymer of r links is proportional to r6/5 for large r. A fully satisfactory solution of the problem, however, has not been found. The difficulty is that the model is essentially non-Markovian the probability distribution of the position of the next carbon atom depends not only on the previous one or two, but on all previous positions. It can formally be treated as a Markov process by adding an infinity of variables to take the whole history into account, but that does not help in solving the problem. 

Universality and two-parameter scaling in the general case of finite excluded volume, Be comes about by the much more sophisticated mechanism of renormalization. As will be discussed in later chapters (see Chap. 11, in particular) both the discrete chain model and the continuous chain model can be mapped on the same renormalized theory. The renormalized results superficially look similar to expressions like Eq. (7.13), but the definition of the scaling variables iie, z is more com plica led. Indeed, it is in the definition of R ) and z in terms of the parameters of the original unrenormalized theory, that the difference in microstructure of the continuous or discrete chain models is absorbed. 

Kim, Y.H., Stites, W.E. Effects of excluded volume upon protein stability in covalently cross-linked proteins with variable linker lengths. Biochemistry 2008, 47, 8804-14. 

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