# Dimensional properties of brush-like polymers with sodium Poly(styrene sulfonate) facet chains

### Appendix

Calculation of the stiffness parameter for the primary chain of brush-like polymers  containing rod-like facet chains with electrostatic interactions

If the Debye–Hückel potential is assumed for the electrostatic interplay between the ith and the jth prices on the completely different facet chains p and q, the potential, wij, between the 2 prices is described as 

$$fracw_ijk_mathrmBT = Q_mathrmBfrac$$

Right here, QB is the Bjerrum size and κ is the reciprocal of the Debye size. The indices i and j run from 1 to (n_^prime) from the (efficient) cost closest to the junction level to the farthest one. The the gap rij between the ith and the jth prices earlier than bending the primary chain is represented by



Right here, the primary chain is assumed to be on the z axis. Within the above equation, h and b point out the spacings of the junction factors and of the fees on the facet chain, respectively. ϕp and ϕq denote the rotating angle of the pth and qth facet chains, respectively, from the xz aircraft (see Determine eight).

When the primary chain is bent within the xz aircraft with the radius of curvature Rc, rij is modified to (r_ij^prime), as given by the next equation,



if we contemplate as much as the order of (R_mathrm^2). Right here, θpq is the angle made by the pth and the qth junction factors on the primary chain across the heart of the curvature and is given by

$$theta _ = (q – p)h/R_mathrm$$

The change within the potential power, Δwij, by bending the primary chain may be calculated as

$$frack_mathrmBT = Q_mathrmBleft e^ – kappa r_/r_ – e^ – kappa r_ij/r_ij proper = frac(q – p)^4h^4left{ proper}e^ – kappa r_ij$$

(eight)

Right here, Δwij was averaged over ϕp and ϕq, assuming that the path of the facet chain round the primary chain is random. By this assumption, rij could also be changed by the averaged worth, (bar r_ij), over ϕp and ϕq:

$$bar r_ij = (i^2 + j^2)b^2 + (q – p)^2h^2$$

The change within the whole free power of the molecule is given by,

$$Delta F = mathop sumlimits_p < q w_ij$$

The efficient stiffness parameter of this molecule is given by,

$$lambda _mathrm^ – 1 = fracfrack_mathrmBT$$

Contemplating the Manning counterion condensation idea, we assumed that the efficient interval of the fees, b, on all sides chain is the same as QB. Since (lambda _mathrm^ – 1) at every Cs trusted the variety of facet chains N, we plotted (lambda _mathrm^ – 1) towards N−1 and extrapolated to N−1 = zero to acquire (lambda _mathrm^ – 1).

Fig. eight Geometry of a brush-like polymer molecule consisting of straight facet chains with charged teams