Mark-Kuhn equations

The dependences of the intrinsic viscosities [Tl] on the molecular weight are shown in Fig. 3.2, and the parameters K, Kp in Mark-Kuhn equation (3.12) for a series of mesogenic comb-shaped polymers are reported in Tables 3.2 and 3.3. The nature of these dependences (a-b = 0.5) for polymers 2-5 in Table 3.2 and 1-3 in Table 3.3 are in agreement with the constancy of the experimental values of the hydrodynamic invariant Aq = TiQD([T ]M/100) r with respect to the fractions [14]. 

With such a large hydrodynamic section of the molecules (d = 30-40 A), the degree of stretching of the molecules p = Lid of the fractions studied is low from 7-5.5 to 1-0.7. In this region of p, the dependences of [nl and D on M for rod-shaped particles in the form of Mark-Kuhn equations are characterized by values of exponents a and b of less than 0.5. For this reason, Eqs. (3.17)-(3.19) for approximation of the dependences of [t]], D, and Sq on M (cf. Fig. 3.4, lines 1, 2, 3) should only be considered a first approximation. Lines 1, 2, and 3, plotted by the method of least squares with slopes of less than 0.5, more accurately convey the features of the hydrodynamic properties of short-chain molecules and quantitatively more precisely correspond to the experimental data. The relations... 

Preservation of the value of the exponent in the Mark-Kuhn equations for [q] in different solvents is a characteristic feature of the reaction of polymers of this type with the solvent. This is seen from Fig. 3.2 lines log [q] = jT(M) are parallel for polymer 5 (Table 3.2). 

Since the intrinsic viscosity depends not only on the size of the macromolecule but also on its shape, on the solvent, and on the temperature, there is no simple relationship for the direct calculation of molecular weights from viscosity measurements. However, the Mark-Houwink-Kuhn equation gives a general description of how the molecular weight can be calculated from the intrinsic viscosity ... 

The thermodynamic quality of a solvent for a polymer can be also estimated from Kuhn-Mark-Houwink-Sakurada viscosity law (often called Mark-Houwink equation) ... 

The polymers were fractioned into 8-14 components by coacervate extraction from the benzene - methanol system. For fractions and nonfractioned polymers, characteristic viscosities [t ], were me-asured. Because that was the first example of studying conformations of macromolecules of this ty-pe in diluted solutions, authors of the work [56] paid much attention to selection of an equation, which would adequately describe hydrodynamic behavior of polymeric chains. Figure 10 shows de-pendencies of [q] on molecular mass (MM), represented in double logarithmic coordinates. Parame-ters of the Mark-Kuhn-Hauvink equation for toluene medium at 25°C were determined from the slo-pe and disposition of the straight lines. 

The dependence of // and Mco values in lg[//] - lg Mo coordinates, shown in Figure 15, indicates that the following Mark-Kuhn-Hauvink equations are corresponded to silarylene carboorganosilo-xane copolymer ... 

Values of parameter a in the Mark-Kuhn-Hauvink equation, close to 0.5, which were obtained for silarylenecarboorganosiloxane fractions, allow correspondence of the polymer molecules to the coil- like type and toluene at 20 and 25°C to 0-solvents. As a consequence, extrapolation data of the Mark-Kuhn-Hauvink equation, silarylenecarboorganosiloxane molecules display the coil-like conformation, which is slightly disturbed in toluene at 20 and 25 °C by interaction with the solvent. 

Table 6 shows experimental values of conformational parameters for the Kuhn segment, A= /nl0 (where 10 is the structural unit length), hindrance factor of rotation around the single bond, cr=(/)l/2, and constants of the Mark-Kuhn-Hauvink equation, K(f [ri]/Mm, at different temperatures. 

In the Mark-Kuhn-Hauvink equation, parameter a for copolymer 6 solution in toluene at 25°C equ-als 0.30, which is typical of branched macromolecules. For cyclolinear polymer 3 under the same conditions, this parameter equals 0.62 (Figure 7). Molecular masses vary within the range from 4xl03 to 565xl03. 

Equations (4) and (5) show that when the parameter x = 2 L/A changes from 0 to the hydrodynamic properties of a worm-like chain change from those of a thin straight tod to those of an undrained Gaussian coil. In accordance with this the dependence of intrinsic viscosity (nl and diffusion coefficient D on molecular weight M of a rigid-chain polymer cannot be described by the usual Mark-Kuhn dependence... 

U = Mw/Mn is the generally used parameter of polymer polydlspersity and a the exponent in the Mark-Kuhn relation (Eq. (10), p. 13). Equation (65) assumes that in Eq. (62) the weight average molecular weight Mw is used as the experimental value of molecular weight. 

Table 2.2 Parameters of Mark-Kuhn-Houwink equation for different lignin preparations ...

This fact was also reported in earlier studies [4, 5]. Dioxane lignins, as well as other lignin polymers except for lignosulfonates, are characterised by rather low intrinsic viscosities and exponential coefficients in the Mark-Kuhn-Houwink equations (note that b, is always smaller than b ). The lower limit of b is 0.1, whereas the upper limit is 0.3. As demonstrated by the data on translational diffusion, for most polymers, the values of the exponent lie in a rather narrow range = 0.38 0.05, that is, the characteristics of translational friction of the macromolecule are almost independent of the lignin source and the solvent used. 

Determination of D is the first step in studying macromolecular coils by fractal analysis. D is usually estimated by finding the exponents in the Mark-Kuhn-Houwink type equation, which relate the characteristic viscosity [r ], the translational diffusion coefficient Dq, or the rate sedimentation coefficient Sq) with the molecular weight (M) of polymers [3] ... 

Both e and the fractal dimension D of the coil depends on the exponent b in Mark-Kuhn-Houwink Equation (16.1) [5] e depends on b as ... 

The characteristic viscosity [qjg can be estimated either directly from experiment, or from Equation (16.1) under the condition that b = 0.5, which is valid at the point, if the constant in this equation is known. To test the relationship (16.11), we used the data of Pavlov and co-workers [4] for the polysaccharide rhodexman, for which the Mark-Kuhn-Houwink equation has the form ... 

If, for a polymer, the constants in Equations (16.2) and (16.3) are known, then the proposed rapid method allows estimation of Dq and Sq from the known D values by Equations (16.2), (16.5) and (16.3), (16.6), respectively. Table 16.2 presents the comparison of the experimental values [4] of Dq and Sq with their estimates made by the described method. There is a satisfactory agreement between both sets of the experimental and estimated values, although the experimental values systematically exceed the estimates made. This discrepancy is due to the strong dependence of Dq and Sq on bj and b respectively, which is inherent in power laws, to which the Mark-Kuhn-Houwink equations belong. 

In addition, Askadskii [7] presented the coefficients in the Mark-Kuhn-FIouwink equation (16.1) for the same polymers and also provided the approximate equation ... 

If the six arbitrary values of [h] in the range 0.1-1.2 dl/g are used to evaluate the viscosity-average molecular weight by the Mark-Kuhn-Houwink equation using both experimentally [7] and calculated values obtained the above method the constants K,j and b, can be obtained. Table 16.4 for the polyarylate PP-2I in tetrahydrofuran. Comparison between the values of obtained are presented in Table 16.5, demonstrating a satisfactory agreement between the different approaches. 

Table 16.4 Parameters of the Mark-Kuhn-Houwink equation ...

Note and are estimated by the Mark-Kuhn-h found using K and b determined experimentally [7] an the use ofK and b, calculated by equations (17.13) and outvink equation are d are obtained with (17.4), respectively. 

FRACTAL VARIANT OF THE MARK-KUHN-HOUWINK EQUATION... 

Mark-Kuhn-Houwink equation, derived on the basis of large experimental material analysis, obtained wide spreading for polymers average viscosity molecular weight determination by their solutions intrinsic viscosity [r ] measured values [1]. This equation has a look like ... 

The value B can be determined by the following simple enough method. For polyarylate F-2 solution in 1,1,4,4-tetrachloiethane Mark-Kuhn-Houwink equation has a look like [5] ... 

The Eqs. (5)-(8) combination allows to obtain the relationship between [r ] wdMM, similar to Mark-Kuhn-Houwink equation [6-9] ... 

FIGURE 1 The relation between experimental and calculated according to the Eq. (12) Mark-Kuhn-Houwink equation constants for solutions of polyarylates series [5] in 1,1,4,4-tetrachlorethane (1), tetrahydrofuran (2) and 1,4-dioxane (3). 

Let us consider further simple technique of determination of fractal dimension -D of macromolecular coil in diluted polymer solution, within the framework of which the Eq. (11) was obtained. The determination of value is the first stage of macromolecular coils study within the framework of fractal analysis (see chapter 1) and the similar estimations are performed by measurement of the exponents in Mark-Kuhn-Hou-wink t5q)e equations, linking intrinsic viscosity [r ] (the Eq. (1)), translational diffrisivity or rate sedimentation coefficient with pol5mers molecular weight MM [3] ... 

If for polymer the constants in the Eqs. (16) and (17) are known, then the proposed express-method allows to perform values and values by the known magnitudes according to the Eqs. (16), (18) and (17), (19), respectively. The comparison of the experimentally received [17] and estimated by the indicated method values and is adduced in Table 3. In the given case a satisfactory correspondence of both datasets is obtained, although experimental values exceed systematically estimated ones by the proposed method. This discrepancy is due to the strong dependence of and on and a, respectively, in virtue of power character of Mark-Kuhn-Houwink t)q)e equations [20]. 

It is natural, that the proposed technique, having no specific suppositions, is true not only for biopol uners, but also for other types of macromolecules in solution. In Ref [5] the values for polyarylates number in different solvents are adduced. This allows calculating Devalue for these polymers according to the Eq. (4). Besides, in the same chapter the values of coefficient in Mark-Kuhn-Houwink equations (the Eq. (1)) for the same polymers and approximate equation are adduced [5] ... 

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