Polymer-pair correlation function

Integral equations can also be used to treat nonuniform fluids, such as fluids at surfaces. One starts with a binary mixture of spheres and polymers and takes the limit as the spheres become infinitely dilute and infinitely large [92-94]. The sphere polymer pair correlation function is then simply related to the density profile of the fluid. 

This function represents how the states of polymers o and r at a given instant are correlated, and it is called the polymer-pair correlation function. 

In the case of coherent scattering, which observes the pair-correlation function, interference from scattering waves emanating from various segments complicates the scattering function. Here, we shall explicitly calculate S(Q,t) for the Rouse model for the limiting cases (1) QRe -4 1 and (2) QRe > 1 where R2 = /2N is the end-to-end distance of the polymer chain. 

Rouse motion has been best documented for PDMS [38-44], a polymer with little entanglement constraints, high flexibility and low monomeric friction. For this polymer NSE experiments were carried out at T = 100 °C to study both the self- and pair-correlation function. 

The pair-correlation function for the segmental dynamics of a chain is observed if some protonated chains are dissolved in a deuterated matrix. The scattering experiment then observes the result of the interfering partial waves originating from the different monomers of the same chain. The lower part of Fig. 4 displays results of the pair-correlation function on a PDMS melt (Mw = 1.5 x 105, Mw/Mw = 1.1) containing 12% protonated polymers of the same molecular weight. Again, the data are plotted versus the Rouse variable. 

Chapter 4 deals with the local dynamics of polymer melts and the glass transition. NSE results on the self- and the pair correlation function relating to the primary and secondary relaxation will be discussed. We will show that the macroscopic flow manifests itself on the nearest neighbour scale and relate the secondary relaxations to intrachain dynamics. The question of the spatial heterogeneity of the a-process will be another important issue. NSE observations demonstrate a subhnear diffusion regime underlying the atomic motions during the structural a-relaxation. 

Figure 5.24 shows that this approach fails not only quantitatively but also qualitatively. Neither is the strong increase of the collective times relative to the self-motion in the peak region of Spair(Q) explained (this is the quantitative failure) nor is the low Q plateau of tpair(Q) predicted (this is the quaUtative shortcoming). We note that for systems hke polymers an intrinsic problem arises when comparing the experimentally accessible timescales for self- and collective motions the pair correlation function involves correlations between all the nuclei in the deuterated sample and the self-correlation function relates only to the self-motion of the protons. As the self-motion of carbons is experimentally inaccessible (their incoherent cross section is 0), the self counterpart of the collective motion can never be measured. For PIB we observe that the self-correlation function from the protonated sample decays much faster than the pair... 

The residual Helmholtz energy due to the dissociation of polymer chains in pure state and the association of polymer chains in mixture state can be calculate by Equation (5). The pair correlation functions of component i in the corresponding Ising lattice system are calculated by gf = 1 / fyfij (Liu al., 2007). The residual... 

The PRISM (Polymer-Reference-Interaction-Site model) theory is an extension of the Ornstein-Zernike equation to molecular systems [20-22]. It connects the total correlation function h(r)=g(r) 1, where g(r) is the pair correlation function, with the direct correlation function c(r) and intramolecular correlation functions (co r)). For a primitive model of a polyelectrolyte solution with polymer chains and counterions only, there are three different relevant correlation functions the monomer-monomer, the counterion-counterion, and the monomer-counterion correlation function [23, 24]. Neglecting chain end effects and considering all monomers as equivalent, we obtain the following three PRISM equations for a homogeneous and isotropic system in Fourier space ... 

This description is a manifestation of the self-similarity (fractal nature) of polymers, discussed in Section 1.4. The fractal nature of ideal chains leads to the power law dependence of the pair correlation function g(r) on distance r. This treatment for the ideal chain can be easily generalized to a linear chain with any fractal dimension V. The number of monomers... 

Tiencc, Eq. (2.121) is a special case of this result, with P — 2 for an ideal chain. The fractal dimension of a rod polymer is — 1 and the pair correlation function is g(r) ... 

The intramolecular pair correlation function g(r) is the number of monomers per unit volume of a section of chain with section size r inside its pervaded volume r, plotted in Fig. 6.31. The linear subsections of the randomly branched polymer have approximately Aq monomers connected in a linear chain (P = 2). The intramolecular pair correlation function g r)... 

Intramolecular pair correlation function e( r I of directly connected monomers within a sphere of radius r of a given monomer, for randomly branched polymers with N monomers made from ulcanizing linear chains with degree of polymerization Nq. Both axes have logarithmic scales. 

In this section, we describe a theory for calculating observables resulting from incoherent excited state transport among chromophores randomly distributed in low concentration on isolated, flexible polymer chains. The pair correlation function used to describe the distribution of the chromophores is based on a Gaussian chain model. The method for calculating the excitation transfer dynamics is an extension to finite, inhomogeneous systems of a truncated cumulant expansion method developed by Huber for infinite, homogeneous systems (25.26). 

In the site-site representation, the virial pressure involves higher spherical harmonic coefficients of the site-site pair correlation functions. The difficulties involved in the standard virial route to the pressure in polymer systems have been well illustrated in recent work. ... 

The above derivation can also be applied to colloidal or polymer-based liquids and is then used to calculate the so-called form factors of soft matter samples. The major difference between a monoatomic liquid and a polymer chain in the melt or in solution is that the total structure factor consists of two parts. The first is the inter-particle structure factor and the second the intra-particle structure factor. This second part is also often called the particle form factor P(q). Using Eq. (2.38) it is straightforward to calculate P(q) for a given soft matter sample. A good example is the form factor of a single polymer coil in a melt [88, 92]. The pair correlation function of such a coil is given by... 

For beads belonging to different polymers a and r we may define the interchain two-body bead distribution function /2(ri(interchain bead-pair correlation function Qiic ri( ),rk(T)) as follows ... 

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