Comparison of the Adsorption Transition for Grafted and Nongrafted Polymers

We compare the thermodynamic behavior of a finite single nongrafted polymer near an attractive substrate with that of a polymer grafted to that substrate. After we recently found first-order-like signatures in the microcanonical entropy at the adsorption transition in the nongrafted case, and given the fact that many studies on polymer adsorption in the past have been performed for grafted polymers, the question arises, to what extent and in what way does grafting change the nature of the adsorption transition? This question is tackled here using a coarse-grained off-lattice polymer model and covers not only the adsorption transition but also all other transitions a single polymer near an attractive substrate of varying strengths undergoes. Because of the impact of grafting especially on the translational but also on the conformational entropy of desorbed chains, the adsorption transition is affected the strongest. Our results are obtained by a combined canonical and microcanonical analysis of parallel tempering Monte Carlo data.

💡 Research Summary

In this work the authors investigate how grafting a polymer chain to an attractive substrate influences its thermodynamic transitions compared with a free (nongrafted) chain of identical composition. Both systems are modeled by a coarse‑grained off‑lattice bead‑spring polymer consisting of N = 40 monomers. The intrachain energy contains a standard 12‑6 Lennard‑Jones pair potential, a weak bending term, and a surface interaction obtained by integrating a 12‑6 potential over the half‑space below the substrate. The strength of the surface attraction is controlled by a single parameter εₛ. Two scenarios are studied: (a) one end of the chain is permanently grafted to the substrate, and (b) the chain is free to move between the substrate and a hard wall placed at a distance L_z = 60.

The authors perform extensive parallel tempering Monte Carlo simulations with 64–72 replicas covering the temperature range T = 0.001–50. Each replica runs for 5 × 10⁷ sweeps, and configurations are sampled every ten sweeps. Energy histograms from all replicas are combined using a weighted‑histogram reweighting scheme that directly yields the logarithmic derivative ΔS(E)/ΔE, i.e., the microcanonical inverse temperature β(E). From β(E) the microcanonical entropy S(E)=ln g(E) is obtained by integration. Both canonical observables (average energy per monomer h(T), specific heat C_V(T), radius of gyration R_g² and its tensor components parallel and perpendicular to the surface, number of surface contacts, and center‑of‑mass distance to the substrate) and microcanonical quantities (S(E), β(E), curvature ∂β/∂E) are analyzed.

The resulting phase diagram in the (T, εₛ) plane contains four broad structural regimes: expanded (E), globular (G), compact/crystalline (C) and adsorbed (A) or desorbed (D) states, with sub‑phases such as planar globules (AG1) and three‑dimensional adsorbed globules (AG). The authors find that most transitions—coil‑to‑globule collapse and the low‑temperature freezing into compact structures—are essentially unchanged by grafting. Their signatures (peaks in C_V, changes in R_g², etc.) appear at the same temperatures and εₛ values for both grafted and free chains.

In contrast, the adsorption transition is dramatically affected. For weak surface attraction (εₛ≈0) the free and grafted chains have nearly identical energies and entropies. As εₛ increases beyond ≈2, the free chain can desorb into the bulk region far from the surface, gaining a large translational entropy proportional to ln L_z. This manifests in the microcanonical entropy as a constant offset between the free and grafted curves at high energies, and, for sufficiently strong εₛ, as a convex intruder in S(E) that signals a first‑order‑like transition. The corresponding β(E) shows a back‑bending region, and the canonical specific heat exhibits a pronounced, narrow peak. The grafted chain, constrained at one end, lacks this translational freedom; its S(E) curve remains smooth, the convex intruder disappears, and the adsorption transition becomes continuous‑like in the microcanonical sense, with broader canonical peaks.

Thus, grafting primarily reduces the entropy gain associated with desorption, suppressing the first‑order‑like character of the adsorption transition while leaving other transitions essentially intact. The study demonstrates that the presence or absence of a single grafting point can qualitatively change the nature of polymer adsorption, a result that is relevant for interpreting experiments on end‑grafted polymer brushes, thin polymer films, and free polymers in solution near attractive surfaces. The combined canonical/microcanonical approach provides a comprehensive picture, showing that canonical analyses capture the overall location of transitions, whereas microcanonical analysis uniquely identifies the order of the transition through the curvature of S(E). This dual methodology could be applied to other finite‑size systems where surface effects and constraints play a crucial role.