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Homopolymer Adsorption on Hexagonal Surfaces: A Replica-Exchange Monte Carlo Study

The adsorption behavior and thermodynamic properties of a coarse-grained flexible homopolymer, grafted on a hexagonal patterned surface, are investigated by means of parallel-tempering replica-exchange Monte Carlo simulations. In this study, the strength of the polymer-surface interaction, which is based on a standard Lennard-Jones potential, is changed systematically, mimicking different honeycomb-structured substrate materials. Utilizing specific order parameters, measured over a range of various temperatures and surface adsorption strengths, different structural phases are discriminated into classes of expanded, globular, droplet, and compact conformations. Finally, we construct the hyperphase diagram for a polymer with 55 monomers and discuss representative polymer structures.


Introduction
Recent research on two-dimensional nanomaterials revealed interesting surface properties, which are expected to lead to new applications in emerging technologies such as energy storage and novel electronic materials for integrated circuits. The most prominent example in this regard is graphene with outstanding mechanical, electrical, and thermal properties [1].
Since graphene as a semi-metal does not exhibit a band gap, which is required for many advanced applications like photovoltaics or field-effect transistors (FETs), structurally similar classes of materials have been investigated viz. transition metal dichalcogenides (TMX 2 ) and metal monochalcogenides (MX). Examples for these materials are MoS 2 or GaSe, exhibiting also hexagonal surface structures. These topological insulators and semiconductors allow for band gap engineering and thus show interesting optical properties such as photoluminescence. However, in contrast to pure graphene sheets, these materials are, in general, not chemically inert. Therefore, a protective coating is required to prevent reactions with the environment. Polymer coats are explored to serve this purpose [2,3]. One key aspect of our study is the systematic investigation of polymer adsorption properties on generic hexagonal surfaces at different adsorption strengths, representing an entire class of miscellaneous substrates.
Graphene, as a potential substrate for drug delivery [1], may also be relevant for health care applications. Important in this context is the adhesion enhancement with polymers [4], as well as the compatibility with a biological environment [5,6]. Therefore, it is necessary to understand the structural and thermodynamic behavior of such finite and comparatively small systems under thermal conditions. Since finite-size effects are essential for structure formation on mesoscopic scales, qualitative changes of macrostate ensembles of such systems are not phase transitions in the conventional sense, because the thermodynamic limit is out of reach and scaling properties are typically not of interest. Key towards understanding these small interacting systems is the systematic investigation of the cooperative polymer adsorption behavior, which is currently only accessible by means of computer simulations. The systeminherent complexity generally entails a rugged free-energy landscape in the space of relevant degrees of freedom, requiring advanced generalized-ensemble Monte Carlo sampling methods [7]. In order to unravel the general adsorption behavior of polymer chains on surfaces, we employ a generic hybrid coarse-grained polymer-substrate model, in which the monomers in the linear polymer chain, representing entire chemical groups, are attracted by the interaction sites of the patterned substrate. The polymer is grafted with one end to the surface [8][9][10].
Here, we present first insights into the adsorption behavior of a flexible polymer at hexagonal surfaces for different adsorption strengths and temperatures. Dominant structural conformations are identified in this parameter space and enable the construction of a structural hyperphase diagram.

Polymer-substrate model and evaluation of the surface potential
In this adsorption study, the interactions of a homopolymer with N = 55 monomers are modeled by a potential U NB describing the van-der-Waals polarization effects between nonbonded monomers of the polymer, by a bond potential U B between subsequent polymer beads, and by the interaction U S of each bead with the vertices of a hexagonally patterned substrate.
The non-bonded interaction is given by a shifted and truncated 12-6 Lennard-Jones (LJ) potential, with where r is the monomer-monomer distance and σ parametrizes the van-der-Waals radius of a monomer. The potential shift U shift = U LJ (r c ) is subtracted to avoid a discontinuity at the cutoff distance r c . In our simulations, we fix the minimum location of the potential to r 0 = 2 1/6 σ ≡ 1, the cutoff distance to r c = 2.5σ, and set the energy scale ϵ to unity. The bonded potential between adjacent monomers is based on the finitely extensible nonlinear elastic (FENE) potential [11][12][13]. A Lennard-Jones potential is added to account for the asymmetry of the bond due to volume exclusion and polarization, The FENE potential restricts the bond extension to the interval r ∈ [r 0 − R, r 0 + R], with parameters fixed to the typical values R ≡ 3/7 and k ≡ 90/4. The parameter η is set to η ≡ 0.1. The constant shift leads to the same minimum value for U B as for U NB . 1 One terminal monomer of the homopolymer chain is grafted to a vertex of the hexagonal surface. While the positions of the remaining (N − 1) monomers are sampled, locations of surface sites are kept fixed in the z = 0 plane, allowing for a three-fold rotational symmetry and two linearly independent translational symmetries. The interaction between a monomer and a lattice site at the distance r is given by where the parameter ϵ S represents the surface adsorption strength. For simplicity, we choose the same cutoff radius r c as for the non-bonded potential. In this study, the attraction strength of the surface was varied in the interval ϵ S ∈ [0.0, 4.5], effectively representing different materials or solvent properties. The total energy of a conformation X = (⃗ r 1 , . . . , ⃗ r N ) with the monomermonomer distance r ij = |⃗ r i − ⃗ r j | and the distance r ia = |⃗ r i − ⃗ s a | of a monomer i from a surface site a eventually reads, For the evaluation of the last term in Eq. (5), it is computationally favorable to store the ten vertex positions ⃗ s 1 , . . . , ⃗ s 10 of the basis cell, next-nearest cells, etc., in one 4π/3-sector (see Fig. 1) by utilizing the following algorithm. First, a monomer ⃗ r i is shifted by integer multiples of the lattice vectors towards the fundamental cell, where we assign to it the new position ⃗ r i ′ . Second, two copies ⃗ r i ′′ and ⃗ r i ′′′ are created by rotation about the z-axis by 2π/3 and 4π/3, respectively. The calculation of the distances r ia between monomers and surface sites is performed in sets of Fig. 1, right), where the positions ⃗ s 1 , . . . , ⃗ s 10 are sorted by increasing distance with respect to the origin of the coordinate system. The evaluation can be terminated as soon as all contributions from one set vanish. Finally, this procedure is applied to all monomers close to the surface, within LJ cutoff distance z < r c .
In Fig. 2(top left), the effective potential U S (x, y, z) is plotted for ϵ S = 1 as a function of the lateral coordinates x, y in the distance z = 0.81 above the surface, probing the monomer positions in a triangular overlayer. This overlayer is observed for adsorbed compact phases, where monomers in the surface plane are located inside the centers of hexagons, i.e., on the vertices of the Wigner-Seitz cells depicted in Fig. 1(left). A small lateral translation of a polymer bead from this favored position results in a steep increase in energy compared to a vertical displacement (Fig. 2, top right). Vertical cuts of the surface potential along y = 1 and x = √ 3 at z = 0.81 are shown in Fig. 2  with increasing vertical distance the effective potential is smeared out. Thus, pattern effects are expected for polymer beads below z = 1.5 for this particular lattice type and parametrization of the potential.  Whereas in each simulation the positions of surface sites were kept fixed, the conformational space of monomer positions was sampled at a given temperature T m according to the Metropolis rule,

Simulation method
In every Monte Carlo sweep, a temperature-dependent displacement update was attempted for each non-grafted polymer bead. During the initial equilibration period, the maximum displacement was adjusted to allow for an acceptance rate of about 50%. Replica exchanges between temperature threads facilitated melting and freezing, i.e., decorrelation of polymer conformations. An exchange between neighboring temperature threads (T m , T k ) was proposed after every 10 3 Monte Carlo sweeps and accepted with the exchange probability, As the initial configuration in each thread we used the lowest-energy state found in a short test run. After thermodynamic equilibrium was reached, each simulation with fixed ϵ S value comprised 10 5 replica exchange attempts per thread. Thermodynamic properties were investigated by analyzing the canonical mean values such as energy ⟨E⟩(T ), components of the radius of gyration parallel and perpendicular to the surface, ⟨r gyr,∥ ⟩(T ) and ⟨r gyr,⊥ ⟩(T ), respectively, the end-to-end distance ⟨l ee ⟩(T ), and number of contacts ⟨n s ⟩(T ) at thread temperatures T m .
Transition temperatures were identified by means of peaks and "shoulders" in fluctuation curves of a quantity O, given by For example, the heat capacity C V = d⟨E⟩/dT accounts for energetic fluctuations. Multiplehistogram reweighting [14] was employed to calculate the density of states g(E) from individual estimates obtained in each temperature thread. The resolution of energy histograms was chosen as dE = 0.05. With the density of states at hand, energetic quantities and fluctuations were calculated as continuous functions of temperature, which enabled us to locate transition signals between the simulation temperatures T m more precisely.
Since the system is of finite size, transition signals identified in different response quantities for the same specific structural transition do not typically coincide. Therefore, this information was accumulated in T -ϵ S parameter space in transition bands rather than transition lines. Phases separated by these bands were characterized by means of the polymer geometry, initially by analyzing similarity features of 10 2 − 10 3 conformations within a single phase. This geometric characterization allowed for the introduction of phase-specific order parameters, which are described in the following section.

Structural hyperphase diagram
A schematic overview of the adsorption behavior of the flexible homopolymer is presented in the T -ϵ S hyperphase diagram shown in Fig. 3. As mentioned earlier, transition signals from different quantities are observed within a broad transition band, rather than at a unique transition temperature. Transition bands in Fig. 3 are indicated by gray lines, separating different structural phases which are annotated by a letter code, similar to the labels introduced in Ref. [9].
At high temperatures T > 1.5 and small adsorption strengths, the polymer resides primarily in the phase of desorbed expanded conformations (DE), which is well known from unconstrained homopolymers. For increased surface adsorption strength ϵ S , these structures are found closer to the surface, within the Lennard-Jones cutoff z < r c . Conformations in this regime are in energetic contact with the surface, but the pattern of the substrate is not yet recognized by the polymer. This polymer behavior defines the adsorbed expanded phase (AE). It is observed that the polymer beads in this phase are primarily located at rather large distances 1.5 < z < r c where the surface potential is smeared out [see Fig. 2(top right)]. Therefore, the structural features in this phase compare well to the adsorption on continuous surfaces [15]. At low temperatures, generally compact polymer conformations govern the solid phases. Typical for a flexible homopolymer with 55 constituents is a crystalline structure with icosahedral shape. An icosahedron is characterized by 20 equally shaped tetrahedra. We find it attached to the surface in phase AID (adsorbed icosahedral droplet) if the surface absorption strength approaches zero. The shortest polymer chain forming such a structure is the 13-mer. Therefore, the 55-mer possesses an icosahedral core of 13 monomers while the remaining 42 monomers form the complete overlayer.
The more interesting and model parameter specific features are found for ϵ S > 0. Already for small surface interactions the symmetry of the icosahedral ground state is broken and adsorbed semi-spherical droplet-like (ASD) conformations dominate. At adsorption strengths ϵ S > 0.8 patterned structures are found. The number of surface contacts is a suitable order parameter that allows to distinguish the ordered phases, in which the surface pattern is recognized by the polymer, from disordered phases. It is defined as the number of monomers below a certain threshold distance z lim to the surface and its fluctuation behavior is particularly useful for the identification of transition lines between these phases. One of the most interesting patterned subphases is HOAC3, which is dominated by highly oriented adsorbed compact 3-layer structures (see Fig. 3).

Thermodynamic and structural order parameters
One of the key challenges in this study is that surface-specific phases are found at very low temperatures, where parallel tempering generally suffers from large correlations that cause an effective slowing-down of the simulation. Moreover, structures in these phases depend sensitively on values of the surface adsorption strength ϵ S . Consequently, we expect more subphases at low temperatures in the regime of droplet-like and patterned conformations [16]. Since signals in this regime are much weaker than for more prominent transitions, i.e., the freezing transition or coil-globule transitions, we suppose that subphases in the regime of droplet conformations are separated by steep (vertical) transition bands. In Fig. 3, these transitions are sketched in the interval ϵ S ∈ [0.1, 1.1]. With parallel tempering, they can only be identified if the ϵ S -parameter space is scanned with high resolution and high accuracy of suitable order parameters. A useful parameter that describes the effective distance of the homopolymer from the surface is the zcomponent of the center of mass. In a similar way, fluctuations about the mean of the sum over all surface interactions, which is the last term in Eq. (5), signal these transitions as well.
More specific is the average number of surface contacts, i.e., the number of monomers located near the surface below z lim . In particular in the phase HOAC3 (cf. Fig. 3), the monomers that are arranged in three layers are well separated by the threshold values z lim ∈ {0.3, 1.0, 1.5}. The latter value, for instance, is used to signal the freezing transition from the adsorbed globule (AG) into the highly oriented adsorbed compact (HOAC3) phase, since the number of surface contacts n s,1.5 discriminates monomers that reside above the ordered layers which do not belong to the crystalline conformation.
In addition, we consider the threshold for an energetic contact with the surface given by the LJ cutoff r c ≈ 2.23. For small surface adsorption strengths, the fluctuation signals of n s,0.3 compare well with the peaks in the heat capacity C V . Furthermore, fluctuations of n s,1.0 suggest vertical separations of droplet and globular subphases. For increasing surface adsorption strengths, polymer conformations become flatter (wetting). The divergence of the FENE potential at maximum bond length prevents the occupation of favorable lattice sites by two adjacent monomers of the homopolymer. Thus, at least one additional monomer is required to link monomers residing in the centers of neighboring hexagons. In the simplest case, three monomers form a zig-zag bridge, with the central monomer located above the surface. However, the repulsive character of the surface potential near the interaction sites of the substrate, as it is felt by the central monomer (Fig. 2, bottom row), requires that either monomers in the surface layer slightly lift off from the hexagonal plane or that bonds are stretched. Therefore, the formation of zig-zag bridges is strongly dependent on the surface adsorption strength. In the AC2 phase, zig-zag bridges can occur at the edge of the film-like structure.
Towards the center of typical AC2 conformations ("bulk"), beads in the second layer (z ≈ 0.81) form a regular triangular lattice. The bond length of bonded monomers located in two different layers typically corresponds to the equilibrium distance r 0 . Obviously, the formation of the triangular overlayer requires more beads than are necessary to fill favorable lattice sites in the underlying surface layer. In this phase, we observe a ratio (⟨n s, 1 higher surface adsorption strengths, ϵ S > 2, the formation of a triangular overlayer is reduced in favor of zig-zag bridges, leading to a larger number of surface contacts n s,0.3 .

Conclusion
In this work, we have studied the cooperative adsorption behavior of a generic polymer model interacting with a hexagonally patterned substrate by means of parallel-tempering replicaexchange Monte Carlo simulations. The interaction strength between the homopolymer and the surface was varied systematically.
To locate structural phase transition bands, we extracted canonical and structural quantities from sampled configurations and applied canonical statistical analysis. Different structural phases were characterized, allowing for the identification of pattern recognition effects by the homopolymer. We summarized these results by constructing a complete hyperphase diagram, parametrized by the temperature T and the surface adsorption strength ϵ S . Depending on the adsorption strength, we found for low temperatures T < 0.5 specific wetting and freezing transitions into phases dominated by adsorbed droplet-like and crystal-like conformations. For high temperatures, T > 1, the entropic enhancement prevents pattern recognition by the polymer. In the phases of globular and extended structures, most monomers reside in a distance from the surface where patterns in the potential are averaged out.
The hyperphase diagram of adsorption for a flexible polymer near a hexagonal substrate exhibits a large number of characteristic and unique features. Depending on the surface adsorption strength and temperature, we identified different adsorption scenarios for surface wetting/dewetting and pattern recognition. Prospectively, this allows for both, the classification of surfaces with respect to polymer adsorption and it helps understand how different polymers adsorb on a certain substrate of interest. These questions are apparently of interest for the design of hybrid materials composed of soft and solid matter.