Macromolecular coils in solution

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. 

In this chapter a method is proposed for finding the fractal dimension (D) of a macromolecular coil in solution, which uses only the characteristic viscosity of the polymer in an arbitrary solvent and a 0 solvent. Several examples are given to illustrate the applicahility of the proposed method to biopolymers of different classes. From D, one can determine a number of other important parameters characterising the behaviour of polymers in dilute solutions. 

Linear polymer macromolecules are known to occur in various conformational and/or phase states, depending on their molecular weight, the quality of the solvent, temperature, concentration, and other factors [1]. The most trivial of these states are a random coil in an ideal (0) solvent, an impermeable coil in a good solvent, and a permeable coil. In each of these states, a macromolecular coil in solution is a fractal, i.e., a self-similar object described by the so-called fractal (Hausdorff) dimension D, which is generally unequal to its topological dimension df. The fractal dimension D of a macromolecular coil characterises the spatial distribution of its constituent elements [2]. 

All these methods require quite complex and laborious measurements [3-5]. The simplest of these methods, which requires no sophisticated instrumentation, is measurement of [q], which can be performed in virtually any laboratory. Therefore, in this work, we propose a simple rapid method of estimating the fractal dimension (D) of macromolecular coils in solutions, which is based on the same principles as applied in deriving Equations (16.4)-(16.6). The coefficient of swelling of a macromolecular coil is known to be defined as [6] ... 

THE INTERCOMMUNICATION OF MACROMOLECULAR COIL IN SOLUTION STRUCTURE AND CHARACTERISTICS... 

It should be noted, that the considered above correction can be obtained and without fractal analysis application as well, using well-known Flory concept, that follows from the Eq. (1) of Chapter 1. However, obtaining of the anal dical correlation between Flory exponent and structural characteristics of polymers in solid phase is very difficult, if possible at all. At the same time this can be made within the framework of fractal analysis, since both macromolecular coil [25] and solid-phase polymer structure [62] are fractal objects. Hence, the possibility of solid-phase polymers properties quantitative prediction appears in such sequence molecular characteristics (for example, ) structure of macromolecular coil in solution polymer condensed state structure— polymer properties. In Section 2.6, this problem will be considered in detail. These considerations predetermined the choice of fractal analysis in Ref. [59] as a mathematical calculus. 

Thus, the results obtained above confirmed the intercommunication of polymer chain rigidity and macromolecular coil in solution structure. The well-known classification of pol5miers (flexible- and rigid-chain ones) is confirmed also and new criteria of such division ate received. 

As it has been shown above, a macromolecular coil in solution fractal dimension is defined by two groups of factors polymer-solvent interactions and macromolecular coil elements between themselves. As an approximation, the first from the indicated factors can be characterized by the difference of polymer 6 and solvent 6 solubility parameters A8= 6 6j [16]. As to the second group of factors, then a parameters number exists, influencing on value in some way chain rigidity, bulk side groups availability, hydrogen bond and so on. Since at present the strict theory of polymer regular solutions has not still developed, then analytical relationships... 

However, besides the indicated factors, that is, A6 and C, the other parameters can influence on value. In Fig. 32, the dependence Dy(Ad) for polyarylates PD and PF is adduced. As one can see, this dependence breaks down into two curves, in addition one from them includes PAr with rigid para-connections in the main chain (the polyarylates PF-2 and PF-7, Table 5, p. 120 in Ref. [5]) and the other—polyarylates with less rigid metha-connections. The indicated plot demonstrates again the importance of such characteristic as polymer chain rigidity for determination of value of macromolecular coil in solution. 

However, one more specific factors number exists, which are capable to influence on the value in the considered conditions, that confirms again macromolecular coils in solutions behavior complexity. In Ref. [26] an experimental data large set is adduced about the behavior of copolymer stirene-acrylonitril-methacrylate (AN-MMA) in different solvents (see Fig. 3). In Fig. 35, the dependences Z) (A6) are adduced for AN-MMA in two solvents only chloroform and 1,4-dioxane, which form two different dependences for the same copolymers. The authors [68] supposed that two dependences formation was due to 1,4-dioxane (unlike chloroform) ability to form hydrogen bonds polymer-solvent, which results in macromolecular coil compactness enhancement owing to the indicated interaction intensification. Besides, similar kind analysis for copolymers is more complex in general, since solubility of different blocks, making up copolymer, in the same solvent can turn out to be different [26, 57]. 

Thus, summing up the stated above results, the following tendencies of the fractal dimension of macromolecular coil in solution can be pointed out as a function of factors number ... 

Further two variants of choice will be used. The first from them (static) is based on the application of a literary value for PC, equal to 2.4 [69, 70]. The second (dynamical) variant supposes that value depends on the structure of the macromolecular coil in solution, that is, and in this very case value is determined as follows [84] ... 

The change of the fractal dimension D with temperature reflects the corresponding changes in sizes, degree of compactness and asymmetiy of shape of a macromolecular coil in solution [89]. The importance of the temperature dependence of study is determined by strong influence of this parameter on the processes of synthesis [28], catalysis [90], flocculation [91], so forth. At present as far as we know experimental evaluations of the temperature dependence of a macromolecular coil are absent. Theoretical estimations [13] suppose that the temperature enhancement makes the fractal less compact, that is, leads to Devalue reduction. Therefore, the authors [92] performed the experimental study of dependence on temperature for the macromolecular coils of polyarylate F-1 [5] in diluted solutions and evaluation of change influence on s nithesis processes. 

The Eq. (95) demonstrates clearly the genetic intercommunication between structures of reaction products (macromolecular coil in solution) and polymers condensed state. 

Let us note, that the value of a macromolecular coil in solution defines the polymer structure in condensed state (see Section 2.6). For instance, higher values D fox PAr, received by high-temperature polyeon-densation, define higher fractal dimension of solid-phase PAr [114] and, henee, larger values of deformation up to failure [143], that is eonfirmed experimentally [5]. 

Hence, the results stated above demonstrated that the cluster model of polymers amorphous state stmcture and fractal analysis allowed quantitative prediction of mechanical properties for pol5miers film samples, prepared from different solvents. Let us note, that the properties prediction over the entire length of the diagram a- was performed within the framework of one approach and with precision, sufficient for practical applications. This approach is based on strict physical substantiation of the analytical intercommunication between structures of a macromolecular coil in solution and pol5miers condensed state [201]. 

The exponent value defines usually a macromolecular coil in solution conformation for linear polymers its value makes up 0.7-0.8, for dendrimers it is practieally equal to zero and superbranched polymers the value a 0.2-0.4 [228]. a value for dendrimers allows to suppose eompac-tization of dendrimers maeromolecules shape in comparison with classical macromolecular coils. Proceeding from these considerations the authors [229] performed theoretical description of intrinsic viscosity [T ] change as... 

Kozlov, G. V Dolbin, I. V Express method of fractal dimension estimation of biopolymers macromolecular coils in solution. Biophysics, 2001, 46(2), 216-219. 

Kozlov, G. V. Afaunov, V. V Temiraev, K. B. Fractal dimension of biopolymers macromolecular coil in solution as a bulk interactions measure. Manuscript deposited to VINIU RAS, Moscow, 08.01.199SfP-V98. 

Kozlov, G. V Burya, A. L Shustov, G. B. Temiraev, K. B. Shogenov, V. N. The influence of main characteristics of polymer and solvent on fractal dimension of a macromolecular coils in solution. Bulletin of University of Dnepropetrovsk. Chemistry, 2000, 5, 96-102. 

Kozlov, G. V Temiraev, K. B. Shustov, G. B. The intercommunication of stmcture of macromolecular coil in solution with stmcture and properties of linear polyarylates condensed state. Proceedings of Higher Educational Institutions, North-Caucasus region, natural sciences, 1999,3, 77-81. 

Burya,A. 1. Kozlov, G. V. Temiraev, K. B. Malamatov, A. Kh. Thebranchirrg influence on fractal dimension of macromolecular coil in solutions. Problems of Chemistry and Chemical Technology, 1999,3, 26-28. 

The results of the numerical calculatirms based on (1.49), (1.50), and (1.66)-(l. 70) depicted in Figs. 1.24 and 1.25 reveal the evolution in time of a jet segment corresponding to one wavelength of perturbatirMi. The relevant dimensionless groups for viscoelastic jets are the volume fraction of the macromolecular coils in solution in... 

Since the introduction in analysis of macromolecular coil stmcture, characterized by its fractal dimension Df, is the key moment of polycondensation process fractal physics, then the value Df determination methods are necessary for practical application of polycondensation fractal analysis for solutions. This parameter for macromolecular coil in solution is defined by two groups of interactions interactions polymer-solvent and interactions of coil elements among them [6]. At... 

Hence, the proposed above assumption about intercommunication of Df and Xj corresponds to the experimental data Floiy-Huggins interaction parameter describes exactly enough interactions system for macromolecular coil in solution, controlling its fractal dimension value. The main problem at the Eq. (12) usage with the purpose of Dj. predicting is the empirical parameter xs calculation method absence [17]. 

As it has been shown above, polymers macromolecular coils in solution are fractal objects, i.e., self-similar objects, having dimension, which differs from their topological dimension. The coil fractal dimension Dp characterizing its structure (a coil elements distribution in space), can be determined according to the Eq. (4). The exponent ax values for polyarylate Ph-2 solutions in three solvents (tetrachloroethane, tetrahydrofuran and 1,4-dioxane) are adduced in [36]. The values ar] for the same polyarylate are also given in paper [37]. This allows to use the Eq. (4) for the macromolecular coil of Ph-2 Devalue estimation in the indicated solvents. The estimations showed D variation from 1.55 in tetrachloroethane (good solvent for Ph-2) up to 1.78 in chloroform. As it is known [38],... 

Let us note, that value of macromolecular coil in solution defines polymer stmcture in condensed state [74], For example, higher values for PAr, of the obtained high-temperature polycondensation, define higher fractal dimension of solid-state PAr [74] and, hence, higher values of fracture strain [75], that is confirmed experimentally [53],... 

As it has been noted above, the value D for macromolecular coil in solution is defined by the interactions of two groups interactions among elements of coil itself and interactions polymer-solvent [24]. Therefore the value can be changed by the solvent variation and, hence, copolymer type, synthesized from the same monomers (oligomers). Increasing means coil compactness enhancement, p reduction and Kh raising. 

Hence, the stated above results have shown, that fractal analysis and irreversible aggregation models application allows to obtain the clear physical interpretation of copolycondensation process and estimate its quantitative characteristics. The fractal dimension D. of macromolecular coil in solution is the main characteristic, controlling this process [142]. 

And at last, further the fractal dimension D of macromolecular coil in solution can be determined, using the eqnation for linear polymers [154] ... 

THE INTERCONNECTION OF MACROMOLECULAR COIL IN SOLUTION AND POLYMER CONDENSED STATE STRUCTURES... 

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