Macromolecular coils

The equilibrium and non-equilibrium characteristics of the macromolecular coil are calculated conveniently in terms of new co-ordinates, so-called normal co-ordinates, defined by 

It can readily be seen that the determinant of the matrix given by (1.8) is zero, so that one of the eigenvalues, say Ao, is always zero. The normal co-ordinate corresponding to the zeroth eigenvalue 

It is convenient to describe the behaviour of a macromolecule in a co-ordinate frame with the origin at the centre of the mass of the system. Thus p° = 0 and there are only N normal co-ordinates, numbered from 1 to N. 

The transformation matrix Q can be chosen in a variety of ways, which allow us to put extra conditions on it. Usually, it is assumed orthogonal and normalised. In this case, it can be demonstrated (see, for example, Dean 1967) that the components of the transformation matrix and the eigenvalues are defined as 

For large N and small values of a, the eigenvalues are then given by 

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Chapter 13 - It was shown, that limiting conversion (in the given case - imidization) degree is defined by purely structural parameter - macromolecular coil fraction, subjected evolution (transformation) in chemical reaction course. This fraction can be correctly estimated within the framework of fractal analysis. For this purpose were offered two methods of macromolecular coil fractal dimension calculation, which gave coordinated results. 

It is known [3], that macromolecular coil in various polymer s states (solution, melt, solid phase) represents fractal object characterized by fractal (Hausdorff) dimension Df. Specific feature of fractal objects is distribution of their mass in the space the density p of such object changes at its radius R variation as follows [4] ... 

From the equation (1) p decrease at Df reduction follows, as always Djinner regions and results to the fuller chemical transformations, i.e., to conversion degree Q increase. Besides, it is known [4], that at macromolecular coil formation by irreversible aggregation mechanisms in its central part... 

Let s remind, that according [7] the value v characterizes system states fraction, un changing in its evolution process. In case of chemical reactions generally and imidization process particularly this assumes, that the value v characterizes macromolecular coil part inaccessible for chemical transformations. Then the accessible for such transformations coil part P is determined as follows [8] ... 

Figure 2. The comparison of macromolecular coil fractal and calculated according to the...

The hypothesis can be checked by the mathematical experiment, if one takes into consideration the probability of the intemolecular cross-linking. Let s assume that sol fraction contains only intramolecularly cross-linked chains while the formation of even one intermole-cular cross-linkage leads to the sol-gel transition. Because only the properties of sol fraction are of our interest we don t need to follow the intermolecularly cross-linked chains. It is rather natural to assume that the probability of the transition into the gel is proportional to the dimensions of the macromolecular coil. 

Description of polymerization kinetics in heterogeneous systems is complicated, even more so given that the structure of complex formed is not very well defined. In template polymerization we can expect that local concentration of the monomer (and/or initiator) can be different when compared with polymerization in the blank system. Specific sorption of the monomer by macromolecular coil leads to the increase in the concentration of the monomer inside the coil, changing the rate of polymerization. It is a problem of definition as to whether we can call such a polymerization a template reaction, if monomer is randomly distributed in the coil on the molecular level but not ordered by the template. 

The viscoelastic response of polymer melts, that is, Eq. 3.1-19 or 3.1-20, become nonlinear beyond a level of strain y0, specific to their macromolecular structure and the temperature used. Beyond this strain limit of linear viscoelastic response, if, if, and rj become functions of the applied strain. In other words, although the applied deformations are cyclic, large amplitudes take the macromolecular, coiled, and entangled structure far away from equilibrium. In the linear viscoelastic range, on the other hand, the frequency (and temperature) dependence of if, rf, and rj is indicative of the specific macromolecular structure, responding to only small perturbations away from equilibrium. Thus, these dynamic rheological properties, as well as the commonly used dynamic moduli... 

The temperature dependence of the size of a macromolecular coil is included in the coefficient of stiffness C T) which has the meaning of the ratio of the squared length of a Kuhn segment to the squared length of the chemical bond, and can be calculated from the local chemical architecture of the chain. The results of the calculations were summarised by Birshtein and Ptitsyn (1966) and by Flory (1969). 

To characterise the size and form of the macromolecular coil, one can introduce a function of density of the number of particles of the chain... 

However, if one is not interested in observing the variables r°, rl,...,rN at all, the independent on these parameters free energy can be defined. This quantity can be calculated, starting from expression (1.25) and (1.27), so that it depends on the parameters T, TV, 6, v, whereby the arbitrary quantity TV cannot influence the free energy of the macromolecular coil and the explicit... 

The lateral forces depend on temperature at high temperatures the repulsion interactions between particles prevail on the contrary, at low temperatures the attraction interactions prevail, so that there is a temperature at which the repulsion and attraction effects exactly compensate each other. This is the 0-temperature at which the second virial coefficient is equal to zero. It is convenient to consider the macromolecular coil at 0-temperature to be described by expressions for an ideal chain, those demonstrated in Sections 1.1-1.4. However, the old and more recent investigations (Grassberger and Hegger 1996 Yong et al. 1996) demonstrate that the last statement can only be a very convenient approximation. In fact, the concept of 0-temperature appears to be immensely more complex than the above picture (Flory 1953 Grossberg and Khokhlov 1994). 

The mean distance between the centres of adjacent macromolecular coils d ss n /3 can be compared with the mean squared radius of gyration of the macromolecular coil (S2), which presents the mean dimension of the coil. Taking the definition (1.21) into account, one can see, that a non-dimensional parameter n R2)3/2 is important for characterisation of polymer solutions. The condition... 

The curves illustrate two variants of the concentration dependence of the mean size of a macromolecular coil in solution. The example is taken of a macromolecule in a good solvent, so that at low concentrations the size of the macromolecular coil is larger than the size of ideal coil, R2)/ R2)o > 1. 

From the energy point of view polymer solvent contacts as compared with polymer-polymer contacts are preferred for some solvents called good solvents in this situation. A macromolecular coil swells and enlarges its dimension in a good solvent. On the contrary in a bad solvent, a macromolecular coil decreases in its dimension and can collapse, turning into a condensed globule (Flory 1953 Grossberg and Khokhlov 1994). 

The second virial coefficient of the macromolecular coil B(T) depends not only on temperature but on the nature of the solvent. If one can find a solvent such that B(T) = 0 at a given temperature, then the solvent is called the 0-solvent. In such solvents, roughly speaking, the dimensions of the macromolecular coil are equal to those of an ideal macromolecular coil, that is the coil without particle interactions, so that relations of Sections 1.1—1.4 can be applied to this case, as a simplified description of the phenomenon. 

Experiments due to neutron scattering by the labelled macromolecules allow one to estimate the effective size of macromolecular coils in very concentrated solutions and melts of polymers (Graessley 1974 Maconachie and Richards 1978 Higgins and Benoit 1994) and confirm that the dimensions of macromolecular coils in the very concentrated system are the same as the dimensions of ideal coils. It means, indeed, that the effective interaction between particles of the chain in very concentrated solutions and melts of polymers appears changes due to the presence of other chains in correspondence with the excluded-volume-interaction screening effect. The recent discussion of the problem was given by Wittmer et al. (2007). 

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