Gaussian conformations

Typically in solution, a polymer molecule adopts a conformation in which segments are located away from the centre of the molecule in an approximately Gaussian distribution. It is perfectly possible for any given polymer molecule to adopt a very non-Gaussian conformation, for example an all-trans extended zig-zag. It is, however, not very likely. The Gaussian set of arrangements are known as random coil conformations. 

The conformation of polymer chains in an ultra-thin film has been an attractive subject in the field of polymer physics. The chain conformation has been extensively discussed theoretically and experimentally [6-11] however, the experimental technique to study an ultra-thin film is limited because it is difficult to obtain a signal from a specimen due to the low sample volume. The conformation of polymer chains in an ultra-thin film has been examined by small angle neutron scattering (SANS), and contradictory results have been reported. With decreasing film thickness, the radius of gyration, Rg, parallel to the film plane increases when the thickness is less than the unperturbed chain dimension in the bulk state [12-14]. On the other hand, Jones et al. reported that a polystyrene chain in an ultra-thin film takes a Gaussian conformation with a similar in-plane Rg to that in the bulk state [15, 16]. 

With respect to the scattering behavior at > 1, the domains of Gaussian conformation and in the opposite case (Q , < 1) the domains of swollen chain conformation are probed. The crossover between both regimes is expected to occur at Q = l/ . 

The polymer we consider here is a semi-flexible chain which has some bending stiffness (Eq. 3). We first estimated the chain conformation in the melt. The calculated mean-square end-to-end distance R2n between atoms n-bond apart has shown that the chains have an ideal Gaussian conformation R2 is a linear function of n (see Fig. 35 given later). The value of R2 for n = 100... 

The expressions 2.7-2.12 which define the Leibler structure factor have been widely used to interpret scattering data from block copolymers (Bates and Fredrickson 1990 Mori et al. 1996 Rosedale et al. 1995 Schwahn et al. 1996 Stiihn et al. 1992 Wolff et al. 1993). The structure factor calculated for a diblock with / = 0.25 is shown in Fig. 2.39 for different degrees of segregation JV. Due to the Gaussian conformation assumed for the chains (Leibler 1980), the domain spacing in the weak segregation limit is expected to scale as d Nm. 

When the block length becomes comparable with Ny distinctions between the behaviors of two copolymers practically disappear. At L > 200, one observes that T Ly with y = 4/3. In this case, the characteristic scale of the microdomain structure behaves as r Ls with <5 = 1/2. This dependence is caused by the fact that flexible chains in the melt have a Gaussian conformation, and the average size of any chain section of n units is proportional to ft1/2 [75]. Hence, for sufficiently large Vs, the spatial scale of microinhomogeneities in the system is determined only by the block size. However, the behavior of the random-block copolymer at L < 102 is more complicated. In particular, r has a minimum at L 10. 

From the vertical shift factor of the master curve, we are able to describe the mass dependence of the zero-shear viscosity in the iso-free volume state which is directly connected to the radius of gyration of the chains. In the molten state, it is generally assumed that the chains exhibit a Gaussian conformation and therefore the viscosity should be proportional to the molecular weight. 

Slater and Noolandi have followed Ltimpkin et al. s approach to the same problem with a similar, semi-quantitative derivation (16). They stress that "non-ideal" electrophoretic behavior stems from the non-gaussian conformation that DNA coils adopt during electrophoresis, as characterized by stretching. As defined in the Lumpkin and Zimm (1) as well as the Lumpkin et al. (9) derivation, the polymer remains in the path tube without any segmental leakage. In terms of a worm in a burrow, this means that none of the worm is able to move in a tube not already defined by the head. For TDP electrophoresis there is no direct experimental evidence for this postulated behavior. On the contrary there may indeed be considerable tube leakeige, which may contribute to the mechanism of separation in TDP electrophoresis. 

Fig. 8.3 Three types of conformations for the same polymer are shown for comparison. The polymer has 1,000 segments of length 1 each. A typical Gaussian conformation is shown in (a). A typical conformation of a polymer that is selfavoiding in real three-dimensional space is shown in (/>). While the self-avoiding chain never crosses itself, its projection on the plane can (and frequently does) have crossings. This is why this figure is very different in spirit and in...

Now assume that the chain distributions are in their most probable (which is assumed to be Gaussian) conformations at the time the crosslinks are introduced. (This is not always so, as in the case of crosslinking in the swollen state as will be discussed below.) The number, Ny, of displacement vectors of type Ry (in the imstretched state) is... 

Fried H, Binder K (1991) Non-gaussian conformational behavior in diblock copolymer melts Is the rpa valid Eur Phys Lett 16(3) 237-242... 

Albeit overly simplistic [1], the analysis leading to Eq. 34 qualitatively captures much of the key phenomena observed experimentally. As indicated in Fig. 5, when the field acting on the chain is weak, then the chain is not strrMigly deformed and retains its Gaussian conformation (with a size proportional to The average... 

In a dense melt, the excluded volume of the monomeric units is screened and chains adopt Gaussian conformations on large length scales. In the following, we shall describe the conformations of a polymer as space curves r(r), where the contoiu" parameter r runs from 0 to 1. The probability distribution P[r] of such a path r(r) is given by the Wiener measure... 

Due to such densely packed molecularly interpenetrated structures, rubbers are incompressible under deformation. Each chain takes a Gaussian conformation following the Flory theorem for screened excluded-volume interaction. On the basis of these characteristics, we can derive the elastic properties of rubbers from a microscopic point of view. 

Fig. 3. Data for the single chain from factor S,(q) in q S,(q). Representation against log qR,i, for the followingsampleofsetI. ,71(t = 10s) +,l8(t" = 00 s) A,49(9000s) and0,50(130000s), Dashed dotted line is for the isotropic Gaussian conformation and dotted lines for the completely affine deformation of that latter in parallel and perpendicular direction...

The first example is the totally affine deformation of the isotropic Gaussian conformation. In that case the deformation is the same on any scale, and thus For a uniaxial deformation X, ... 

Fig. 15a-d. Simulated ESR spectra for radical —CHj— —C(CH3>C00R with Gaussian conformation distributions about the most probable positions 0, =55° and 0j = 65°, with Half-height widths indicated... 

Here we calculate of a chain with a Gaussian conformation. Using the Gaussian distribution given by Eq. 1.34, ( r - for a given m and n is calculated as... 

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