Excluded volume chain

Tsvetkov VN, Eskin VE, Erenkel SYa (1964) Struktura makromolekul v rastvorakh (Structure of macromolecules in solutions, in Russian). Nauka, Moscow Valleau JP (1996) Distribution of end-to-end length of an excluded-volume chain. J Chem Phys 104(8) 3071-3074... 

In good solvents we thus have B°N = R, where R is either the radius of the mushroom (roughly equal to the radius of gyration In solution) or of the pancake. In the further analysis, we concentrate on the pancakes. From computer simulations abd scaling arguments ) it is known that for a two-dimensional chciin R. This exponent is between that for a three-dimensional chain in a good solvent (R - N ) and a one-dimensional excluded-volume chain (which is a rod... 

The partition coefficient for a real chain (excluded volume chain) has not been obtained except by the scaling theory [6]. The theory gives only a qualitative relationship between K and N for chains sufficiently longer than the pore size Rg Rp) ... 

Modern theories do not solve analytically the excluded volume problem, instead they pursue the same problem from different angles. This is because the excluded volume problem is placed as a category of typical many body problems [36], hence an intractable one. In spite of such a situation, there is an approach to persistently seek closed solutions of the excluded volume chain. Following this trend, some empirical formulations have been put forth for the limiting case of N—>°° [37-41]. The most successful one is that of the des Cloizeaux type equation, written by the form [41] ... 

Fig. 4. Scaled end-to-end distance distribution for an excluded volume chain (solid line), Eq. (28), and the Gaussian chain (shaded area)...

Using the best numerical values of v = 0.588 and y = 1.1619 (hence, 6 = 0.275 and f = 2.427), and with the help of a technique of the pivot algorithm, Valleau [41] could show that the empirical equation remarkably well fits the Monte Carlo simulations on a 3d-lattice, in support of the conjecture for a simple closed form of the excluded volume chain in the asymptotic limit, N— °°. 

In Fig. 4, the conformational behavior of the excluded volume chain (d = 3) represented by Eq. (28) is drawn as a function of the rescaled end-to-end distance to be compared with that of the Gaussian chain. As one can see, the selfavoiding walk does not very much differ from the Gaussian statistics. In the vicinity of r = 1, the difference is only 10%. Thus, replacing real chains (d = 3) with the Gaussian chain is never a poor approximation. However, what is important is that the difference revealed in Fig. 4 between the excluded volume chain and the Gaussian chain does not disappear even in the limit of N— °°. In this real world (d = 3), one can never fit real chains to the Gaussian chain by properly... 

The adsorption of proteins at interfaces is a key step in the stabilization of numerous food and non-food foams and emulsions. Our goal is to improve our understanding of the relationships between the sequence of proteins and their surface properties. A theoretical approach has been developed to model the structure and properties of protein adsorption layers using the analogy between proteins and multiblock copolymers. This model seems to be particularly well suited to /5-casein. However, the exponent relating surface pressure to surface concentration is indicative of a polymer structure intermediate between that of a two-dimensional excluded volume chain and a partially collapsed chain. For the protein structure, this would correspond to attractive interactions between some amino acids (hydrogen bonds, for instance). To test this possibility, guanidine hydrochloride was added to the buffer. A transition in the structure and properties of the layer is noticed for a 1.5 molar concentration of the denaturant. Beyond the transition, the properties of the layer are those of a two-dimensional excluded volume chain, a situation expected when there are no attractive interac-... 

We have also measured y(t) and e(t) during polymer adsorption for a given concentration. In Figure 6, the e-n curve, the equation state of the layer during the adsorption process, is presented. At low surface pressure, one observes a linear increase of the dilational elastic modulus with the surface pressure n. From the slope of the linear part of the e-n curve, a value of 0.66 was found for the excluded volume critical exponent. The same value has been measured elsewhere with another technique.12 This result indicates that, unlike the excluded volume chain behaviour in the bulk, the air water interface is not a good solvent for MeC. At intermediate surface pressures, the modulus levels off and then increases again until the equilibrium surface pressure is reached. 

The results obtained by Clark and Lai for the variation in the net pressure due to free polymer, both with and without excluded volume effects, are presented in Fig. 17.15. According to this model, only the excluded volume polymer chain provides repulsion at larger distances of separation, the non-excluded volume chain generating only attraction at all distances of separation. 

Shaffer s bond fluctuation model (Shaffer, J. S., 1994. Effects of chain topology on polymer dynamics—Bulk melts, J. Chem. Phys., 101 4205 13). Polymers are grown as random walks on a simple cubic lattice, subjected to the excluded volume, chain connectivity, and chain uncrossability constraints described in the text. 

Equations (2.116) and (2.117) give a closed equation for G(R, 0, N). The detailed calculation within this theory is involved, but it predicts many interesting features of the excluded volume chain. For example, the structure factor g(k) at high k region is shown to be... 

As an example, consider the structure factor g(k) of an excluded volume chain. From the dimensional analysis, g(k) must be written as... 

Let (... )o denote the average for the distribution function of the ideal chain, then for the excluded volume chain... 

To calculate Cp. we have to know the distribution of R -R of the excluded volume chain. Since this is not yet known precisely, we assume, for the sake of simplicity, that the distribution of R - R is the same as the distribution of the end-to-end vector of the excluded volume chain with n—m segments, and has the following functional formr f... 

The polymer density profile of ideal chains next to a hard sphere for arbitrary size ratio q was first ealeulated by Taniguchi et al. [125] and later independently by Eisenriegler et al. [126]. Eisenriegler also considered the pair interaction between two colloids for Rg< R [127] and for Rg R [128], as well as the interaction between a sphere and a flat wall due to ideal chains [129]. Depletion of excluded volume polymer chains at a wall and near a sphere was considered by Hanke et al. [130]. One of their results is that the ratio /Rg at a flat plate, which is 1.13 for ideal chains [118, 119], is slightly smaller (1.07) for excluded-volume chains. 

Fig. 4.8 Segment density profiles at a flat wall for non-adsorbing polymers described as ideal chains (filled squares) mean-field chains (open circles) and excluded volume chains (open triangles) compared to the tanh profile (solid curve) of (4.18)...

For other cases (depleted dilute ideal or excluded volume chains, semi-dilute chains with fluctuation effects) density profiles are substantially more involved in comparison with a tanh profile. Surprisingly, the expression (2.55)... 

Figure 1.36 is a sketch of App/kpT. The two terms on the right-hand side have the opposite R dependence. The interaction becomes weaker as the monomer density becomes lower with an increasing R, but the entropy term gains. As a result, there is a minimum in Ach. The R that minimizes Ach is the most probable value of R for a given N. The excluded volume chain will have that dimension. Thus Rp can be obtained from 5(Aeh/kBT)/3R = = 0 as / f = bN, reproducing Fq. 1.62. 

Problem 1.13 We can apply Hory s method to find the Hory exponent for the dimension of a two-dimensional excluded-volume chain. In two dimensions, the entropy term is the same as that of the three-dimensional chain, but the interaction term changes to AUcb/k T = b R N/R ) = b N /R. ... 

Problem 1.16 Nonreverse random walk (NRRW) is often used in lattice simulations. The random walker can choose any direction except for returning to the preceding site. Otherwise, the walker can return to the site it visited earlier. We can expect that the trajectory is between that of an ideal chain and that of an excluded volume chain. For a cubic lattice of lattice unit b, assume that the random walker chooses one out of the five directions at random and answer the following questions. 

The Flory-Huggins theory uses the lattice model to arrange the polymer chains and solvents. We have looked at the lattice chain model in Section 1.4 for an excluded-volume chain. Figure 2.1 shows a two-dimensional version of the lattice model. The system consists of si,e sites. Each site can be occupied by either a monomer of the polymer or a solvent molecule (the monomer and the solvent molecule occupies the same volume). Double occupancy and vacancy are not allowed. A hnear polymer chain occupies N sites on a string of N-l bonds. There is no preference in the direction the next bond takes when a polymer chain is laid onto the lattice sites (flexible). Polymer chains consisting of N monomers are laid onto empty sites one by one until there are a total tip chains. Then, the unoccupied sites are filled with solvent molecules. The volume fraction of the polymer is related to rip by... 

In Section 1.4.2, we derived Rp = for an excluded-volume chain using Flory s method. Here, we use a similar method to derive Rp = for theta chains. The difference in the free energy A<,h of the chain is in the second term of Eq. 1.63. For the theta chains, binary interaction is effectively absent (A2 = 0) and therefore the leading term in the polymer-polymer interaction is b R (N/R y = b N /R, which is due to the ternary interactions. Then, Ajh is given as... 

The linear dimension of the chain in the slit is different from the counterpart in the cylindrical pore. Because the confined chain follows the conformation of two-dimensional excluded-volume chain,... 

Problem 2.33 An excluded-volume chain of N monomers of size b has a dimension of Rp = bN . Grouping monomers into one big monomef makes a chain of N/rii big monomers with excluded volume. Within each big monomer, the chain is stiU an excluded-volume chain of monomer size b. Show that this coarse-grained chain has the same dimension as that of the original chain. 

Table 3.1 compares various measures of the dimension for chains with an ideal-chain conformation (or at the theta condition), excluded-volume chains, and rodlike molecules. The latter will be considered in Section 3.5. 

Wormlike chain of contour length L and persistence length 1 Excluded volume chain of contour length L Infinitely thin rod of length L... 

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