Flexible cylinder model

Fig. 10 SANS data of PBS-PFP copolymer with Ci2E5 surfactant in D20. The monomer concentration was 5 x 10 4 M and the ratio of surfactant over monomer x = 2.0. Also shown are fits by flexible cylinder model (solid line) and stiff cylinder (dashed line). Reprinted with permission from [48]. (2006) by the American Chemical Society...

With increasing polymer concentration, we may expect that the polymer global motion changes from the fuzzy cylinder model mechanism to the repta-tion model mechanism. The onset of the crossover should depend on the degree in which the lateral motion of a polymer chain is suppressed by entanglement with its surrounding chains, but it is difficult to estimate this degree. There are some disputes over it in the case of flexible polymers [20]. 

Values of B calculated from the ordinate intercepts are shown in Fig. 23 as a plot of B/(2q)3 against the number of the Kuhn segments N. For N<4, the data points for the indicated systems almost fall on the solid curve which is calculated by Eq. (78) along with Eqs. (43), (51), (52), and Cr = 0. A few points around N 1 slightly deviate downward from the curve. Marked deviations of data points from the dotted lines for the thin rod limit, obtained from Eq. (78) with Le = L and de = 0, are due to chain flexibility the effect is appreciable even at N as small as 0.5. The good lit of the solid curve to the data points (at N 4) proves that the effect of chain flexibility on r 0 has been properly taken into account by the fuzzy cylinder model. 

Kp vs PEG moleculor weight. The best results are obtained for the sphere-cylinder model in which the protein and dextran are modeled as spheres and the PEG is modeled as a cylinder. The sphere-sphere model 5uelds mixed results while the sphere-coil model yields trends opposite to those observed experimentally. We have considered the case in which both polymers are modeled as flexible coils and found that the experimental trends are not predicted. 

Figure 13.18. Comparison between model and measurements for clays dispersed in polymers. Viscosities observed as a function of shear rate by Krishnamoorti et al [46] for dispersions of silicate platelets (weight fractions of 0.06 and 0.13) in poly(dimethyl siloxane) at T=301K are indicated with symbols. Calculated results, assuming platelets to be monodisperse flexible cylinders with aspect ratio Af=(thickness/diameter)=0.01, are indicated as lines, (a) Relative viscosity=r)(dispersion)/r (polymer). (b) Dispersion viscosity, r)(dispersion).

Modarres-Sodeghi, Y., Paidoussis, M. R, Semler, C. (2005). A nonlinear model for an extensible slender flexible cylinder subjected to axial flow. Journal of Fluids and Structures, 21,609-627. 

Finally, some rather recent devdopments must be noted. Several years ago, Yamakawa and co-workers [25-27] developed the wormlike continuous cylinder model. This approach models the polymer as a continuous cylinder of hydrodynamic diameter d, contour length L, and persistence length q (or Kuhn length / ). The axis of the cylinder conforms to wormlike chain statistics. More recently, Yamakawa and co-workers [28] have developed the helical wormlike chain model. This is a more complicated and detailed model, which requires a total of five chain parameters to be evaluated as compared to only two, q and L, for the wormlike chain model and three for a wormlike cylinder. Conversely, the helical wormlike chain model allows a more rigorous description of properties, and especially of local dynamics of semi-flexible chains. In large part due to the complexity of this model, it has not yet gained widespread use among experimentalists. Yamakawa and co-workers [29-31] have interpreted experimental data for several polymers in terms of this model. 

Hie cylinder model gave no theoretical lalue of ]i/] for short flexible chains (AL<2.278 and Ad>0.1) owing to the nature of the kernel in the integral equation. To evaluate ]i/] for such chains, Yoshizaki et adopted the touched-bead model, that is, a discrete chain consisting of N beads of diameter db whose centers are located on the continuous KP or HW contour. These authors showed ](/] to be represented approximately by the sum of the solution of the Kirkwood-Riseman integral equation and the contribution of N beads as Einstein spheres. The result for the KP chain may be expressed in the form... 

Fig. 7.2 The a subunit has an 8-fold a/fS barrel tertiary fold. /3-Strands are shown as flattened arrows with arrowheads at their C-termini. -Helices are represented as cylinders and are labeled on their N-termini. In addition to the eight strands and helices found in a canonical a/fl barrel, the a subunit contains three other helices labeled 0,2 and 8. N and C mark the N- and C-termini. P marks the site of proteolysis at Arg-188 that occurs in the darkened, flexible loop (residues 179-192) following strand 6. A darkened, disordered loop following strand 2 contains a catalytic residue, Asp-60, and makes important contact with the a active site and with the /3 subunit. A substrate analog, indole-3-propanol phosphate, binds at the active site as indicated by the ball-and-stick model. (See also color plate). (Reproduced with permission from Hyde et al J. Biol. Chem., 263, 17857 (1988)).

The dendritic core adopts an extended conformation which is possible due to the great flexibility (conformational freedom) of the PAMAM and PPI skeletons. In this way, the molecular model of thick disks proposed for the LC dendrimers with two-terminal chain mesogenic units is transformed into a model consisting of a long cylinder, which is the result of the axial elonga-... 

Different ways of depicting the conformation of proteins convey different types of information. The simplest way to represent three-dimensional structure is to trace the course of the backbone atoms with a solid line (Figure 3-5a) the most complex model shows every atom (Figure 3-5b). The former, a Co, trace, shows the overall organization of the polypeptide chain without consideration of the amino acid side chains the latter, a ball-and-stlck model, details the interactions between side-chain atoms, which stabilize the protein s conformation, as well as the atoms of the backbone. Even though both views are useful, the elements of secondary structure are not easily discerned in them. Another type of representation uses common shorthand symbols for depicting secondary structure—for example, colled ribbons or solid cylinders for a helices, flat ribbons or arrows for (3 strands, and flexible... 

The most important of recent theoretical studies on semi-flexible polymers is probably the formulation of Yamakawa and Fuji [2,3] for the steady transport coefficients of the wormlike cylinder. This hydrodynamic model, depicted in Figure 5-2, is a smooth cylinder whose centroid obeys the statistics of wormlike chains. In the figure, r denotes the normal radius vector drawn from a contour... 

This EVB model was employed to the systematic study of model pores or channels. Taking a simphstic view, the polymer was regarded as a rigid framework in which slab or cylinder pores of constant thickness or radius, respectively, are formed. Within this approach, proton transport in pores has been studied as a function of a variety of generic structural and dynamical features of the polymer and operational parameters of the working fuel cell (such as temperature and humidity). These studies revealed a number of factors determining the proton mobihty, such as the width of the channel, distance between the sidechains, and their flexibility. The main lessons of the simulations and the theoretical analysis were ... 

The cylinder representations were particularly useful for comparing 3D models from the various SRP RNAs. First we superimposed cylinders from the M. jannaschii structure and the atom coordinates of the human SRP RNA to improve the human model. We then used the cylinder representations of the human S RNA to adjust the atom coordinates of M. jannaschii SRP RNA. These two sequences illustrate well how comparative modeling is mutually beneficial. For example, a coaxial orientation of helices 5b, 5c, and 5d in the human SRP RNA model is supported by the extended helix (5bcd) of the M. jannaschii SRP RNA. On the other hand, the M. jannaschii sequence inserts extra nucleotides in the loop of helix 4 and between helix 2 and helix 5 a however, the high degree of potential flexibility that would normally result from these insertions, is contained by comparison with the human model. 

A wormlike chain is specified by the persistence length A and the contour length Lp. However, it does not have a thickness. We need to give it a diameter b for the chain to have a finite diffusion coefficient. The model is called a wormlike cylinder (Fig. 3.62). The expressions for the center-of-mass diffusion coefficient and the intrinsic viscosity were derived by Yamakawa et al. in the rigid-rod asymptote and the flexible-chain asymptote in a series of h/A and A/A-... 

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