Macromolecules fractal dimension

FIGURE 104 The dependence of macromolecule fractal dimension on polymerization degree N for PSD. 

In Fig. 110 the dependence x) D is adduced for the considered den-drimers is adduced, from which file fast decay of relaxation time at macromolecule fractal dimension growth follows. Such shape of the dependence x) D can be explained by two factors influence. The macromolecular coil gyration radius is linked with dimension and polymerization degree A by the Eq. (8). As one can see, the indicated equation supposes very strong (power) dependence of on The estimation according to the Eq. (4) has shown that for the considered dendrimers Devalue varies within the limits of 1.52-2.42. Using reasonable values iV=1000 [227] and 5=0.35 [234], let us obtain variation within the limits of approx. 6-33 nm, that is, in 5.5 times, for file considered dendrimers, that corresponds well to the experimental data [227,230]. It is obvious, that reorientation of small macromolecules in solution occurs much easier than large ones and requires much less duration, which is, (x... 

FIGURE 110 The dependence of fiee relaxation time on macromolecule fractal dimension D, for solutions of dendrimeis PSD in tetrahydrofuran (1) and LPA-3 in chloroform (2). 

The fractal dimension that is determined from Eq. (38) is not in all cases the true fractal dimension of the individual macromolecules. In a polydisperse ensemble one has to take the ensemble average which yields an ensemble fractal dimension df g [7,92] ... 

As already mentioned, we chose three different physicochemical properties for studying the influence of the surface area and fractal dimension in the ability of dendritic macromolecules to interact with neighboring solvent molecules. These properties are (a) the differential chromatographic retention of the diastereoisom-ers of 5 (G = 1) and 6 (G = 1), (b) the dependence on the nature of solvents of the equilibrium constant between the two diastereoisomers of 5 (G = 1), and (c) the tumbling process occurring in solution of the two isomers of 5 (G = 1), as observed by electron spin resonance (ESR) spectroscopy. The most relevant results and conclusions obtained with these three different studies are summarized as follows. 

The interaction between the water and the polymer occurs in the vicinity of the polymer chains, and only the water molecules situated in this interface are affected by the interaction. The space fractal dimension da is now the dimension of the macromolecule chain. If a polymer chain is stretched as a line, then its dimension is 1. In any other conformation, the wrinkled polymer chain has a larger space fractal dimension, which falls into the interval 1 < d( < 2. Thus, it is possible to argue that the value of the fractal dimension is a measure of polymer chain meandering. Straighter (probably more rigid) polymer chains have da values close to 1. More wrinkled polymer (probably more flexible) chains have da values close to 2 (see Table III). 

The choice of dimension D, depends on the value of relation dhldm [9], At dm<0,6dh interaction of diffusant molecules with walls of free volume microvoid is small and transport process is controlled by fractal dimension of structure (structural transport). At dm<0,6dh on transport processes has strong influence interaction of diffusant molecules with walls of free volume microvoid, which are polymeric macromolecules surface with dimension >/(/)/ is the dimension of excess energy localization regions) [10], In this case Dt=Df (molecular transport) [9] is adopted. 

Eq. (5.75) with good solvent fractal dimension 23 = l/v 1.7. Data from M. Daoud etal.. Macromolecules 8, 804(1975). 

Muthukumar M, Winter HH (1986) Fractal dimension of a cross-linking polymer at the gel point. Macromolecules 19 1284-1285... 

The dp value reflects the dimension of the sub-lattice in which the macromolecule is arranged, i.e., it is a fractal dimension of the medium in which the molecule is located rather than of the molecule itself dp does not always coincide with the Euclidean... 

Attempts to resolve this contradiction have been undertaken [68]. Maritan and Stella [69] calculated the fractal dimensions for 50 proteins using two methods. In one method, fractal dimension is a scaling index for the contour length of a macromolecule with respect to the distance between its ends (it is equal to 1.19-1.82), while in the other method, the fractal dimension is a scaling index for the total mass with respect to the length, which is equal to 1.62-2.24. The calculations were performed using X-ray diffraction crystallographic parameters of proteins. 

When D = d = 2, Equations (11.32) and (11.34) are correct only when d = 2. This means that the ranges of variation of the fractal dimension of a macromolecule section between crosslinking points are d,[Pg.314]

Figure 11.7 Fractal dimension (D) of a macromolecule section between chemical crosslinking points versus crosslinking density vf) for epoxy polymers based on epoxy oligomers and amino-containing curing agents. The D values were found from Equation (11.31) for a=A (1), A (2) and 0.25 nm (3).

Figure 11.10 Fractal dimension of the macromolecule section between chemical crosslinking point (D) versus scale of measurement a for EP-1 with v of 1 0.2 X 2 0.6 x lO m 3 1.7 x lO m ...

Figure 11.11 Fractal dimension of a section of macromolecule between chemical crosslinking points (D) versus cross-sectional area of the macromolecule S with of...

A crosslinked polymer is formally one giant macromolecule (fractal cluster) therefore, the models considered above provide the dimension d( of exactly this cluster. The Vilgis concept [61, 62] assumes the existence of several clusters of this type and linear macromolecules between them. In practice, parameters used more widely are the density of the network of nodes (chemical crosslinks) or the molecular mass of the macromolecular section between the crosslinks M, which are related to each other in the following way ... 

Below we consider practical aspects of the estimation of the fractal dimension D of a macromolecule section between chemical crosslinking points with allowance for these statements as well as for those considered in the preceding Sections. This approach differs fundamentally from that of the Cates [56] and Vilgis models [61, 62]. All the foregoing is valid not only for a network of chemical bonds but also for a network of macromolecular entanglements in linear polymers. 

Now we attempt to estimate D within the framework of the theoretical views on polymer fractals [22, 36, 56, 61-64]. It is assumed [68] that the fractal dimensions d, d, and D are related to one another via an expression similar to Equation (11.8). The relationship between the parameters d and D has been considered [6]. It is believed that d is a dynamic value which responds to a change in the conditions of the interaction of a macromolecule with its surrounding and D is a static parameter. However, in our opinion, the opposite situation occurs in reality this is indicated by the following facts, some of which have been noted previously. It has been shown experimentally [72] that a two-fold increase in the crosslinking density does not change d, while, according to the plot shown in Figure 11.12b, has an appreciable influence on D. Moreover, the monomer and the crosslinked epoxy polymer have nearly identical d values (see Table 11.4). Helman... 

Thus, the fractal dimension of a section of the macromolecule between chemical crosslinking points in network polymers varies over limits close to those predicted for a fractal broken line, i.e., 1molecular mobility and is related to the fractal dimension of the cluster structure of epoxy polymers... 

Figure 11.15 Dielectric loss (tangent tan 8) versus fractal dimension (D) of the macromolecule section for copolyethersulfone formals with the content of formal blocks of 1 0, 2 5, 3 10, 4 30, 5 50 and 6 70 mol%. Measurement frequency 1 kHz.

Fractals are macromolecular coils in good solvents and in Euclidean space with dimension d = 3 their fractal dimension is df = 1.65 [5]. To demonstrate this situation or the fractality of a polymer at a molecular level the following will be considered as an example. In the review [6], the data for the variation of radius of a macromolecule s gyration Rg and molecular weight are presented for polymethyl methacrylate (PMMA) and... 

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]. 

The authors [15-17] measured the fractal dimension of cross-linked polydimethylsiloxane (PDMS) at the gelation point with the aid of rheological measurements and obtained the value D =2. By the confession of the authors themselves, this result is imexpected, since the indicated dimension corresponds to a linear macromolecule in 0-solvent. The similar measurements of the branched chains structure of the epoxy polymer (EP) with the aid of X-raying gave the value D =2.17 0.03 [16]. In connection with these measurements several comments should be made. 

FIGURE 14 The dependence of interaction parameter e on macromolecular coil fractal dimension Dj. A hnear macromolecule characteristic states according to the classification [10] are indicated by points. 

In Fig. 40, the dependences of on testing temperature T have been shown for F-1 solutions in tetrachloroethane and N, N-dimethylfomiamide (c=0.5 mass %). As one can see, Z) monotonous increasing at Tgrowth is observed, that is, macromolecular coil compactness degree enhancement is realized. The plots of Fig. 40 extrapolation to D=2.0 [10] allows one to estimate 0-temperature values for the indicated solvents. Let us note, that the achievement of 0-conditions does not mean any critical state of the macromolecular coil. Such state forthe coil in diluted solution (practically isolated macromolecule) can be reached at the fractal dimension critical value F /, determined by the Eq. (4) of Charter 1. As it follows from tiie indicated relationship, 2.285. Let us note, that approximately at this Devalue the dependence [ti](2) for polyarylate F-1 in tetrahydrofiuan (Fig. 38) becomes parallel to abscissa axis, that is, this Devalue is a critical one. 

With the Eqs. (4) and (141) using the following results were received for PDMDAAC in water solution Z) =1.44 and in NaCl solution Z) =1.65. The indicated values define macromolecular coil conformation for linear polymers [10]. The first from the indicated values corresponds to permeable coil fractal dimension, the second one— to Devalue for a macromolecular coil in good solvent. D decreasing means more unfolded conformation of polymer chain (Z) =1.0 corresponds to completely stretched chain and to transition to Euclidean behavior [182]). Thus, the main influence of medium on polymer macromolecule is its conformation change (and, hence, Z) value) at variation of solvent quality in respect to polymer [56]. 

The Eq. (182) at the condition /g=const=0.270 nm and C =const=10 is reduced to a purely fractal form, that is, to the Eq. (8) with 5=0.349. Let us note essential distinctions of the Eqs. (180) and (8). Firstly, if the first from the indicated equations takes into accoimt object mass only, then the second one uses elements number N of macromolecule, that is, takes into account dynamics of molecular structure change. Secondly, the Eq. (8) takes into account real structural state of macromolecule with the aid of its fractal dimension The indicated above factors appreciation defines correct description by the equation (8) the dependence of macromolecular coil gyration radius R on molecular weight MIT of polynner [235]. 

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