^{1,a)}and Hans Behringer

^{1,b)}

### Abstract

Homopolymer adsorption onto chemically structured periodic surfaces and its potential for pattern recognition is investigated using Monte Carlo simulations. To analyze the surface-induced selective adsorption on a fundamental geometric level polymer chains are represented by freely jointed chains with a fixed bond length whose monomers are attracted by the sites of regular lattice patterns. The structural properties of the adsorbed low-temperature state are comprehensively discussed for different lattices by looking at the radius of gyration and the inter bond angle distributions. These observables show a non-trivial dependence on the commensurability of characteristic lengths given by the lattice constant and by the bond length. Reasons for this behavior are given by exploiting geometric and entropic arguments. The findings are examined in the context of pattern recognition by polymer adsorption. Furthermore, the adsorption transition is discussed briefly. For certain incommensurable situations the adsorption occurs in two steps due to entropic restrictions.

I. INTRODUCTION

II. POLYMERMODEL

III. SIMULATION DETAILS

A. Calculating non-bonded interactions

B. Non-commensurable grid assignment

IV. SELECTIVE ADSORPTION

A. Adsorbed polymer conformations on square lattices

B. Hexagonal lattice

V. PATTERN RECOGNITION

VI. ADSORPTION TRANSITION

VII. CONCLUSION

### Key Topics

- Polymers
- 93.0
- Adsorption
- 64.0
- Surface patterning
- 50.0
- Surface structure
- 26.0
- Pattern recognition
- 15.0

## Figures

Schematical sketch of a polymeric system confined to a slit where one of the confining walls is structured with attractive sites (see main text). Along the *x* and *y* direction periodic boundaries are applied.

Schematical sketch of a polymeric system confined to a slit where one of the confining walls is structured with attractive sites (see main text). Along the *x* and *y* direction periodic boundaries are applied.

Drawbacks of the nearest-grid-point method for studying adsorption processes. Two monomers can be located at a single surface point without being affected by the excluded volume repulsion if the lattice constant and the discretization length of the density mesh are incommensurable (left-hand side). Furthermore, two spatially close monomers at the borders of neighboring cells do not interact via the excluded volume repulsion whereas distant monomers in the same cell do (right-hand side).

Drawbacks of the nearest-grid-point method for studying adsorption processes. Two monomers can be located at a single surface point without being affected by the excluded volume repulsion if the lattice constant and the discretization length of the density mesh are incommensurable (left-hand side). Furthermore, two spatially close monomers at the borders of neighboring cells do not interact via the excluded volume repulsion whereas distant monomers in the same cell do (right-hand side).

Total energy of a system with 10 freely jointed chains of 20 bonds each as a function of inverse temperature β and surface site spacing *d* for the square lattice. For the excluded volume interaction the nearest-grid-point assignment has been applied (*v* = 1, Δ = *b* = 1).

Total energy of a system with 10 freely jointed chains of 20 bonds each as a function of inverse temperature β and surface site spacing *d* for the square lattice. For the excluded volume interaction the nearest-grid-point assignment has been applied (*v* = 1, Δ = *b* = 1).

Specific heat capacity per monomer *c* and fraction of adsorbed monomers *n* _{ads} for different temperatures (interaction range *a* = 0.1). (Left panels) Square lattice and spacing *d* = 1, systems with different number of real chains (cloud-in-cell scheme, *v* = 1, Δ = *b*/3 = 1/3) are shown: *N* _{p} = 1 (black curve), *N* _{p} = 10 (red), and *N* _{p} = 40 (blue). The excluded volume interaction prevents the adsorption of all monomers for *N* _{p} = 40 chains. (Right panels) One chain and spacings *d* = 0.84 (orange triangles) and *d* = 0.58 (blue diamonds). (The shown averages were calculated from 2 × 10^{8} Monte Carlo steps, for *d* = 0.84 averages from 10^{9} Monte Carlo steps are displayed by a black line as well.)

Specific heat capacity per monomer *c* and fraction of adsorbed monomers *n* _{ads} for different temperatures (interaction range *a* = 0.1). (Left panels) Square lattice and spacing *d* = 1, systems with different number of real chains (cloud-in-cell scheme, *v* = 1, Δ = *b*/3 = 1/3) are shown: *N* _{p} = 1 (black curve), *N* _{p} = 10 (red), and *N* _{p} = 40 (blue). The excluded volume interaction prevents the adsorption of all monomers for *N* _{p} = 40 chains. (Right panels) One chain and spacings *d* = 0.84 (orange triangles) and *d* = 0.58 (blue diamonds). (The shown averages were calculated from 2 × 10^{8} Monte Carlo steps, for *d* = 0.84 averages from 10^{9} Monte Carlo steps are displayed by a black line as well.)

Snapshot top view on adsorbed 21mer chains on a square lattice with spacing *d* = 1 and β = 25. A system with 10 chains is shown. Surface sites are visualized schematically, which means that their size is not connected to the interaction range.

Snapshot top view on adsorbed 21mer chains on a square lattice with spacing *d* = 1 and β = 25. A system with 10 chains is shown. Surface sites are visualized schematically, which means that their size is not connected to the interaction range.

An adsorbed bond of a polymer can compensate the misfit between surface site spacing and bond length within a range of ±2*a* by rearranging. For spacing *d* = *b* − 2*a*, a chain bond can find exactly one alignment where both monomers form contacts with surface points (upper situation). The second example shows the only possible conformation for a chain bond to form contacts with two surface points with spacing *d* = *b* + 2*a*. For values of *d* in between many different alignments are possible for a single bond. Due to bond connectivity the number of possible rearranged contacts in a row within a polymer shows a strong dependence on |*d* − *b*|, see main text.

An adsorbed bond of a polymer can compensate the misfit between surface site spacing and bond length within a range of ±2*a* by rearranging. For spacing *d* = *b* − 2*a*, a chain bond can find exactly one alignment where both monomers form contacts with surface points (upper situation). The second example shows the only possible conformation for a chain bond to form contacts with two surface points with spacing *d* = *b* + 2*a*. For values of *d* in between many different alignments are possible for a single bond. Due to bond connectivity the number of possible rearranged contacts in a row within a polymer shows a strong dependence on |*d* − *b*|, see main text.

Ranges for *d* where a single bond can bind to the site pair (*m*, *n*) of a square (left-hand side) and a hexagonal lattice (right-hand side) of attractive sites with range *a* = 0.1 (black intervals). Due to symmetry of the lattices several site pairs have the same range. For instance, (1, 0) is equivalent to (− 1, 0) and (0, ±1) for the square lattice. Thus only representative pairs are shown. The two considered hexagonal lattices, the triangular and honeycomb pattern, show the same ranges. Note the gap where no pairs exist for bond alignments in this case. The grey shade represents the region where sites begin to overlap (resembling an homogeneous attractive surface).

Ranges for *d* where a single bond can bind to the site pair (*m*, *n*) of a square (left-hand side) and a hexagonal lattice (right-hand side) of attractive sites with range *a* = 0.1 (black intervals). Due to symmetry of the lattices several site pairs have the same range. For instance, (1, 0) is equivalent to (− 1, 0) and (0, ±1) for the square lattice. Thus only representative pairs are shown. The two considered hexagonal lattices, the triangular and honeycomb pattern, show the same ranges. Note the gap where no pairs exist for bond alignments in this case. The grey shade represents the region where sites begin to overlap (resembling an homogeneous attractive surface).

Radius of gyration *G* _{∥} parallel and *G* _{⊥} perpendicular to the surface as functions of the lattice spacing *d* of a square pattern for a fully adsorbed polymer (black curves) at β = 25. The spacing *d* is varied in steps of Δ*d* = 0.1 (squares, solid lines as guide to the eye). The curves for a system with 10 polymers are also displayed in red. ^{ 56 } In addition, the alignment regions to (*m*, *n*) diagonals from Fig. 7 are indicated again in the lower panel. (The data is generated from 10^{9} Monte Carlo steps for one chain and from 3 × 10^{8} steps for 10 polymers.)

Radius of gyration *G* _{∥} parallel and *G* _{⊥} perpendicular to the surface as functions of the lattice spacing *d* of a square pattern for a fully adsorbed polymer (black curves) at β = 25. The spacing *d* is varied in steps of Δ*d* = 0.1 (squares, solid lines as guide to the eye). The curves for a system with 10 polymers are also displayed in red. ^{ 56 } In addition, the alignment regions to (*m*, *n*) diagonals from Fig. 7 are indicated again in the lower panel. (The data is generated from 10^{9} Monte Carlo steps for one chain and from 3 × 10^{8} steps for 10 polymers.)

Snapshot top view on adsorbed 21mer chains for a square lattice and commensurability parameter *d* = 0.84 and temperature β = 25.

Snapshot top view on adsorbed 21mer chains for a square lattice and commensurability parameter *d* = 0.84 and temperature β = 25.

Integrated inter bond angle distribution *D* as a function of the angle θ for different values of the commensurability parameter around *d* = 0.84 for a freely jointed 21mer at β = 25 (black solid lines). (*d* = 0.81 and *d* = 0.85 have been chosen as they have approximately the same *G* _{∥}.)

Integrated inter bond angle distribution *D* as a function of the angle θ for different values of the commensurability parameter around *d* = 0.84 for a freely jointed 21mer at β = 25 (black solid lines). (*d* = 0.81 and *d* = 0.85 have been chosen as they have approximately the same *G* _{∥}.)

Radius of gyration *G* _{∥} parallel to the surface as functions of the lattice spacing *d* of hexagonal lattices for an adsorbed polymer at β = 25. The *G* _{∥} curve for a system with 10 polymers is also displayed in red. ^{ 56 } The possible alignment regions to (*m*, *n*) pairs as shown in Fig. 7 are repeated in the lower panel. Little arrows indicate values for which conformations are discussed in more detail in the main text.

Radius of gyration *G* _{∥} parallel to the surface as functions of the lattice spacing *d* of hexagonal lattices for an adsorbed polymer at β = 25. The *G* _{∥} curve for a system with 10 polymers is also displayed in red. ^{ 56 } The possible alignment regions to (*m*, *n*) pairs as shown in Fig. 7 are repeated in the lower panel. Little arrows indicate values for which conformations are discussed in more detail in the main text.

Radius of gyration *G* _{⊥} perpendicular to the surface as functions of the lattice spacing *d* of hexagonal lattices for an adsorbed polymer (black) at β = 25. The results for a system with 10 chains are shown in red as well. ^{ 56 } The lower panel shows possible alignment regions to (*m*, *n*) diagonals, see Fig. 7 . Little arrows indicate the value *d* = 1.13 for which conformations are discussed in more detail in the main text.

Radius of gyration *G* _{⊥} perpendicular to the surface as functions of the lattice spacing *d* of hexagonal lattices for an adsorbed polymer (black) at β = 25. The results for a system with 10 chains are shown in red as well. ^{ 56 } The lower panel shows possible alignment regions to (*m*, *n*) diagonals, see Fig. 7 . Little arrows indicate the value *d* = 1.13 for which conformations are discussed in more detail in the main text.

Snapshot top view on adsorbed chains on a triangular lattice with spacing *d* = 1.13 and β = 25. Typical conformations show zigzag structures, most of them being straight and some exhibiting a bend.

Snapshot top view on adsorbed chains on a triangular lattice with spacing *d* = 1.13 and β = 25. Typical conformations show zigzag structures, most of them being straight and some exhibiting a bend.

Schematic depiction of the origin of asymmetric zigzag shapes of adsorbed chains on a triangular lattice structure with spacing *d* = 1.13 (see also Fig. 13 ).

Schematic depiction of the origin of asymmetric zigzag shapes of adsorbed chains on a triangular lattice structure with spacing *d* = 1.13 (see also Fig. 13 ).

Distribution *P* of the inter bond angle θ for adsorbed polymers on different lattices with *d* = 1 (square lattice: black; triangular: red; and honeycomb: blue). See Table III for angles where peaks are expected. The lower panel shows the integrated bond angle distribution *D* for the hexagonal lattices.

Distribution *P* of the inter bond angle θ for adsorbed polymers on different lattices with *d* = 1 (square lattice: black; triangular: red; and honeycomb: blue). See Table III for angles where peaks are expected. The lower panel shows the integrated bond angle distribution *D* for the hexagonal lattices.

Distribution *P* of the inter bond angle θ for adsorbed polymers on different lattices with *d* = 0.5 (square lattice: black; triangular: red; and honeycomb: blue). Angles where peaks are expected from the ideal trimer model (see Table III ) are indicated by arrows. Due to excluded volume and “higher order” connectivity constraints the peaks are slightly shifted and some of them merge.

Distribution *P* of the inter bond angle θ for adsorbed polymers on different lattices with *d* = 0.5 (square lattice: black; triangular: red; and honeycomb: blue). Angles where peaks are expected from the ideal trimer model (see Table III ) are indicated by arrows. Due to excluded volume and “higher order” connectivity constraints the peaks are slightly shifted and some of them merge.

Visualization of typical conformations for *d* = 0.58 during adsorption. Left pictures show a side view on the system, while the right pictures show the top view on the adsorbing surface. The adsorption is a two-step process, where a first, weakly attached state is followed by the fully adsorbed state.

Visualization of typical conformations for *d* = 0.58 during adsorption. Left pictures show a side view on the system, while the right pictures show the top view on the adsorbing surface. The adsorption is a two-step process, where a first, weakly attached state is followed by the fully adsorbed state.

Inter bond angle distribution for a square lattice with *d* = 0.58 and different system temperatures (see also Fig. 17 ): upper panel β = 14; middle panel β = 17.8 (right in between the two peaks in the specific heat); and lower panel β = 24. Note that the characteristic angles where peaks are expected dominate only when passing the second cross-over threshold.

Inter bond angle distribution for a square lattice with *d* = 0.58 and different system temperatures (see also Fig. 17 ): upper panel β = 14; middle panel β = 17.8 (right in between the two peaks in the specific heat); and lower panel β = 24. Note that the characteristic angles where peaks are expected dominate only when passing the second cross-over threshold.

## Tables

Possible alignments of a trimer to the sites {(*m* _{1}, *n* _{1}), (0, 0), (*m* _{2}, *n* _{2})} of a square lattice for some values of *d* (see also Fig. 7 ). Shown are all combinations of (*m* _{1}, *n* _{1}) = (1, 0) and (1, 1) for which the phase space volume (12) is non-zero (tabulated is a numerically calculated λ which is Λ from (12) divided by the corresponding volume for {(1, 0), (0, 0), (1, 0)} of the reference system *d* = 1). In addition, the associated ideal inter bond angle is shown.

Possible alignments of a trimer to the sites {(*m* _{1}, *n* _{1}), (0, 0), (*m* _{2}, *n* _{2})} of a square lattice for some values of *d* (see also Fig. 7 ). Shown are all combinations of (*m* _{1}, *n* _{1}) = (1, 0) and (1, 1) for which the phase space volume (12) is non-zero (tabulated is a numerically calculated λ which is Λ from (12) divided by the corresponding volume for {(1, 0), (0, 0), (1, 0)} of the reference system *d* = 1). In addition, the associated ideal inter bond angle is shown.

Possible alignments of a trimer to the sites {(1, 0), (0, 0), (*m*, *n*)} of hexagonal lattices for *d* = 1.13. Shown are all combinations for which the phase space volume (12) is non-zero (as in Table I λ is Λ divided by the corresponding volume for {(1, 0), (0, 0), (1, 0)} of the reference system *d* = 1). The associated ideal inter bond angle is shown as well.

Possible alignments of a trimer to the sites {(1, 0), (0, 0), (*m*, *n*)} of hexagonal lattices for *d* = 1.13. Shown are all combinations for which the phase space volume (12) is non-zero (as in Table I λ is Λ divided by the corresponding volume for {(1, 0), (0, 0), (1, 0)} of the reference system *d* = 1). The associated ideal inter bond angle is shown as well.

Ideal possible inter bond angles θ of an ideal trimer adsorbed on different lattices with different commensurabilities *d* and non-vanishing phase space volume (12) .

Ideal possible inter bond angles θ of an ideal trimer adsorbed on different lattices with different commensurabilities *d* and non-vanishing phase space volume (12) .

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