Binary Blend

Two-component RPA model with a flat background.

DEFINITION
This model calculates the scattering from a two component polymer blend using the Random Phase Approximation (RPA).

The two polymers are assumed to be monodisperse.

The scattered intensity $I(q)$ is calculated as[1]

$$
\frac {(\rho_A - \rho_B)^2}{N_A I(q)} = \text{scale} \cdot
[\frac {1}{\phi_A M_A v_A P_A(q)} + \frac {1}{\phi_B M_B v_B P_B(q)} -
\frac {2 \chi}{N_A V_0}] + \text{background}
$$

where

$$
P_i(q) = \frac{2 [\exp(-Z) + Z - 1]}{Z^2}
$$
$$
Z = (q \cdot Rg_i)^2
$$

Here, $\phi_i$, is the volume fraction of polymer i (and $\phi_A + \phi_B = 1)$, $M_i$ is the molecular weight of polymer i, $v_i$ is the specific volume of monomer i, $Rg_i$ is the radius of gyration of polymer i, $\rho_i$ is the sld of polymer i, $N_A$ is Avogadro's Number, and $V_0$ is the reference volume (taken to be 1 $cm^3$). $P_i(q)$ is the Debye Gaussian coil form factor.

$\chi$ is the Flory-Huggins interaction parameter expressed per unit volume and not per monomer as is more usual. However, the influence of the third term on the RHS of the first equation is small.

This model works best when as few parameters as possible are allowed to optimise such as, for example, when the actual blend composition is well-known and $M_i$, $\phi_i$, and $\rho_i$ can be fixed.

REFERENCES
1. Lapp, Picot & Benoit, Macromolecules, (1985), 18, 2437-2441 (Appendix)