Density monomers

The mean monomer density decreases with the increasing degree of polymerization. 

In the collapse phase the monomer density p = N/R is constant (for large N). Thus, the only confonnation dependent tenn in (C2.5.A1) comes from the random two-body tenn. Because this tenn is a linear combination of Gaussian variables we expect that its distribution is also Gaussian and, hence, can be specified by the two moments. Let us calculate the correlation i,) / between the energies and E2 of two confonnations rj ]and ry jof the chain in the collapsed state. The mean square of E is... 

Theoretical predictions [73,74] concern also the monomer density profile p(z), which is predicted to scale as... 

Figure 4. Monomer density profile Pn,(y) normalized such that p ,(y) = 1 if the monomers were distributed homogeneously in the simulation box plotted vs. y at various values of s/ksT from e/ksT = 0.1 (circles to s/kfiT = 0.019 (erosses). Data are for a symmetrical AB polymer mixture with Na = Nb = N = 32, the interface occurring at y = 0. From Miiller et al." ...

Finally, the 4-monomer density, Eq. (45), can be rewritten in terms of the end-segment functions ... 

Factors which influence properties chain length, branching vs. linear, nature of the monomer, density, interchain bonds, hydrophobic and hydrophilic interactions. 

Using and S, the translational friction coefficient and the electrophoretic mobility can be calculated by simply replacing G by in the respective formulas. Before we proceed to do this, we present the collective dynamics of monomer density and counterions. 

The brush thickness and the axial distance per side chain are interrelated due to the condition of fixed monomer density r in the collapsed brush as R = Vn/ 11 h. The axial tension is given by... 

Figure 6 shows the behavior of the reduced monomer density p z)Rp/Np at increasing anchoring density. The stretching of the chains with increasing surface coverage, which is due to the repulsion between monomers, is evident. This plot has to be compared with Fig. 3b, where the same type of rescaling has been used. However, note that at this point, direct and quantitative comparison is not possible, since it is a priori not clear which value of the interaction parameter /3 in the self-consistent calculation corresponds to which set of simulation parameters ct, N, pa. 

N21<, is the number of bound dimers in the volume v and N2 — N(N —1)/2 is the number of pairs (bound or free) in the volume v. Note that the right-hand side of this expression is roughly proportional to the reciprocal volume, /v. We may, therefore, write the expression to the left in terms of dimer and monomer densities, n2 = N2/V and nj = N/v, respectively [27],... 

Here a is the Kuhn or segment length. The delta functional selects out only those configurations satisfying the incompressibility constraint. The Flory-Huggins parameter, measures the incompatibility between A and B monomers. The dimensionless A monomer-density operator is given by... 

This is a large area of research, and significant progress in this direction has already been made by Schweizer [199] and by Freed and coworkers [200], However, these two schemes are entirely different. While the approach of Freed pays special attention to the coupling between different Rouse modes, the formulation of Schweizer is more in line with the mode coupling theory of liquids with monomer density as the slow collective mode. The main... 

Figure 7 displays the root mean square end-to-end distance as a function of the monomer density for various chain lengths. As expected, the chains are expanded in dilute solution and they contract with increasing density. The dots mark results from computer simulations [27, 49]. As is obvious from the figure, the above approach yields an excellent agreement with simulation results not only qualitatively but also quantitatively. Hence, we expect to find reliable results also for other quantities by our approach. 

M. Manciu, E. Ruckenstein On the Monomer Density of Grafted Polyelectrolyte Brushes and Their Interactions, LANGMUIR 20 (2004) 8155-8164. 

The simple Alexander—de Gennes theory, which assumed a steplike monomer density in the brush, captured the dependence of the interaction on the physical parameters (length of the polymer, density of grafting, quality of the solvent) and provided a satisfactory approximation for the calculation of the steric repulsion. However, new applications of grafted polymers on surfaces, such as the control of the catalytic selectivities of some chemical reactions by varying the thickness of a brush,4 the prevention of the adsorption of proteins on surfaces (a condition required for biocompatibility),5 or the control of... 

---