### Quick Start

A polymer adsorbs onto a surface in a manner that depends on N, the number of polymer segments, its χ parameter with respect to the solvent, and the relative χs parameter of solvent and polymer onto the surface. Using Prof Frans Leermakers' implementation of SF theory1, we can readily find out

Chain Length N
N Layers
Bulk φ
χ
χsegment
Converged?
H
%Trains

The polymer has N segments (typically a monomer unit, typically with a unit MWt~80). If N=100 and MWt=80 then we are talking of a small MWt polymer of 8000, typical of dispersants rather than large polymers. The χ parameter to the solvent is the conventional one varying from 0 to 0.5, after which the solvent is "bad". The χs parameter is a measure of whether the surface prefers the solvent (nearer to 0), the polymer (large) or is neutral (low). The simulation assumes a number of layers, each one unit thick into which the polymer units might extend. You will quickly find that something like N/4 is a good starting value - higher values make little difference but slow the calculations.

The graph shows, for reasonably large values of χs the local volume fraction of the polymer chains, φ Polymer, dropping off towards the global concentration at thickness H. The function φ Solvent is the complement - the volume fraction of solvent molecules. The local concentration on the surface, and how fast it falls off, depend on the interplay of the two χ parameter.

Above a value of, say, N=80, the chain length makes surprisingly little difference. The plot shows the polymer in the first layer, the "trains", which are strongly stuck to the surface. Then in the log(φ) plot there is a near-linear fall off in the "power law" region, with whisps of polymer in the distal region (you need the log plot to see them). The end of the power law region is at √N, as you can easily confirm visually. So a polymer of N=100 extends to z=10 and a polymer of N=200 only extends to z=14, not such a big change.

By sliding the Bulk φ slider from a very low value you can get an idea of the adsorption isotherm - how much is attached at a given volume fraction of polymer. For any polymer with N>50 and a reasonable balance of χ parameters, there is little to see. The "Henry" portion of the isotherm, where you go from zero, is at super-low concentrations, not worth including in the slider.

At first it seemed to me rather disappointing that apart from the χ effects (which are super-important, and which fit nicely into my solubility world view via HSP), the app shows that changes with polymer length or concentration are rather small. But that is good news. In terms of providing, say, a steric barrier for particle stability, you don't need a super-large polymer as the barrier size isn't greatly affected. Instead, you need to focus on the relative χ values - get those wrong and other details about the polymer are irrelevant.

Prof Leermakers' "simple" algorithm is not as robust as a full SF algorithm, but then we would not be able to run the full algorithm in an app. We give up if the number of iterations exceeds a reasonable limit where it is likely that the algorithm will fail anyway. Success shows the number of iterations, failure shows "No" with a red background.

1SF Theory is a Self-Consistent Field theory of polymers (and other things) in solution and at interfaces developed by Scheutjens and Fleer at Wageningen U in the late 1970s and developed further at Wageningen and other universities.