^{1,a)}and Patrick Gemünden

^{2}

### Abstract

Surface-induced selective adsorption of homopolymers on a generic level is numerically analyzed for freely jointed chains (with a fixed bond length) whose monomers are attracted by the sites of regular periodic patterns. In particular, the behavior of the specific heat, the gyration tensor, and the bond order tensor are investigated as functions of the temperature. The properties of the transition are related to the interplay of the characteristic lengths. The adsorption proceeds in two steps for certain incommensurabilities of the bond length and the lattice constant. The corresponding adsorption mechanisms are elucidated by looking at the evolution of the inter bond angle distribution upon adsorption. Moreover, the origin of two steps in contrast to adsorption in one step is traced back to entropic restrictions caused by a strongly reduced phase space of the polymer for certain values of the incommensurability.

Fruitful discussions with Friederike Schmid, Shuanhu Qi, Kostas Ch. Daoulas, and Guojie Zhang as well as Thomas Kühne are gratefully acknowledged. This work was partially supported by BMBF grant MORPHEUS (Grant No. FKZ 13N11704).

I. INTRODUCTION

II. DISCRETE EDWARDS MODELS OF POLYMERIC SYSTEMS

III. MODEL SYSTEM FOR POLYMERADSORPTION

IV. SELECTIVE ADSORPTION ON A SQUARE LATTICE

V. ADSORPTION TRANSITION OF A FREELY JOINTED POLYMER

A. Adsorption mechanism

B. Trimer model and two-step adsorption

VI. CONCLUSIONS

### Key Topics

- Polymers
- 105.0
- Adsorption
- 59.0
- Surface patterning
- 26.0
- Tensor methods
- 22.0
- Phase space methods
- 17.0

## Figures

Adsorption energy per monomer (upper panel) and the resulting specific heat (lower panel) of a 26mer as a function of temperature for two different lattice constants of the underlying square lattice: The left-hand side shows l/b = 0.58, the right-hand side l/b = 1. The curves for a real freely jointed chain (black squares) are compared to an ideal chain (black solid curves). (Excluded volume: nearest-grid point scheme, discretization Δ = 0.25, energy parameter v = 0.1.) The shown averages where calculated from 109 Monte Carlo steps.

Adsorption energy per monomer (upper panel) and the resulting specific heat (lower panel) of a 26mer as a function of temperature for two different lattice constants of the underlying square lattice: The left-hand side shows l/b = 0.58, the right-hand side l/b = 1. The curves for a real freely jointed chain (black squares) are compared to an ideal chain (black solid curves). (Excluded volume: nearest-grid point scheme, discretization Δ = 0.25, energy parameter v = 0.1.) The shown averages where calculated from 109 Monte Carlo steps.

Properties of the gyration tensor of a 26mer as a function of temperature for a system with a square lattice site pattern of lattice constant l/b = 1. The ideal freely jointed chain model (left-hand side) is compared to a real discrete Edwards chain (right-hand side). The upper panels show the parallel (black) and the perpendicular (red) components G ∥ = (G xx + G yy )/2 and G ⊥ = G zz of the space-fixed gyration tensor, the middle panels show the three eigenvalues λ1 ⩽ λ2 ⩽ λ3 of the body-fixed gyration tensor. The resulting asphericity (11) and prolateness (12) are displayed in the lower panels by red squares and black squares, respectively.

Properties of the gyration tensor of a 26mer as a function of temperature for a system with a square lattice site pattern of lattice constant l/b = 1. The ideal freely jointed chain model (left-hand side) is compared to a real discrete Edwards chain (right-hand side). The upper panels show the parallel (black) and the perpendicular (red) components G ∥ = (G xx + G yy )/2 and G ⊥ = G zz of the space-fixed gyration tensor, the middle panels show the three eigenvalues λ1 ⩽ λ2 ⩽ λ3 of the body-fixed gyration tensor. The resulting asphericity (11) and prolateness (12) are displayed in the lower panels by red squares and black squares, respectively.

Properties of the gyration tensor of a 26mer as a function of temperature for a system with a square lattice site pattern of lattice constant l/b = 0.58. The ideal freely jointed chain (left-hand side) is compared to a real chain (right-hand side). The upper panels show the parallel (black) and the perpendicular (red) components G ∥ = (G xx + G yy )/2 and G ⊥ = G zz of the space-fixed gyration tensor, the middle panels show the three eigenvalues λ1 ⩽ λ2 ⩽ λ3 of the body-fixed gyration tensor. The averaged asphericity (11) and prolateness (12) are displayed in the lower panels by red squares and black squares, respectively.

Properties of the gyration tensor of a 26mer as a function of temperature for a system with a square lattice site pattern of lattice constant l/b = 0.58. The ideal freely jointed chain (left-hand side) is compared to a real chain (right-hand side). The upper panels show the parallel (black) and the perpendicular (red) components G ∥ = (G xx + G yy )/2 and G ⊥ = G zz of the space-fixed gyration tensor, the middle panels show the three eigenvalues λ1 ⩽ λ2 ⩽ λ3 of the body-fixed gyration tensor. The averaged asphericity (11) and prolateness (12) are displayed in the lower panels by red squares and black squares, respectively.

Diagonal entries of the (averaged) bond order parameter (13) of a 26mer as a function of temperature for two different lattice constants of the underlying square lattice (upper row): left-hand side l = 0.58b, right-hand side l = b. The curves for a real freely jointed chain (squares) are compared to an ideal chain (solid lines). The upper black curves displays the xx components (which are identical to the yy components) the lower red curves the zz components. To point at a slight deficiency of the discrete Edwards model the lower row shows the same curves for a real chain and a triangular lattice where the yy component is shown in addition (light orange curve). The small deviation between the S xx and S yy curves is a numerical artefact of the excluded volume interaction as the cubic grid of (4) does not fit to the triangular lattice.

Diagonal entries of the (averaged) bond order parameter (13) of a 26mer as a function of temperature for two different lattice constants of the underlying square lattice (upper row): left-hand side l = 0.58b, right-hand side l = b. The curves for a real freely jointed chain (squares) are compared to an ideal chain (solid lines). The upper black curves displays the xx components (which are identical to the yy components) the lower red curves the zz components. To point at a slight deficiency of the discrete Edwards model the lower row shows the same curves for a real chain and a triangular lattice where the yy component is shown in addition (light orange curve). The small deviation between the S xx and S yy curves is a numerical artefact of the excluded volume interaction as the cubic grid of (4) does not fit to the triangular lattice.

Possible regions for l/b where a single bond vector can align to (some of) the diagonals in a square (left-hand side) and triangular (right-hand side) lattice of attractive sites with range a = 0.1. (Due to the symmetry of the lattice several bond vectors have the same region, therefore representative ones are only shown.) The dashed lines indicate the two values investigated in this work. Note the region in the triangular case where a complete alignment of a given bond is not possible so that not all monomers of an adsorbed polymer can gain binding energy. The appearance of such regions is determined by the range a of the attractive sites.

Possible regions for l/b where a single bond vector can align to (some of) the diagonals in a square (left-hand side) and triangular (right-hand side) lattice of attractive sites with range a = 0.1. (Due to the symmetry of the lattice several bond vectors have the same region, therefore representative ones are only shown.) The dashed lines indicate the two values investigated in this work. Note the region in the triangular case where a complete alignment of a given bond is not possible so that not all monomers of an adsorbed polymer can gain binding energy. The appearance of such regions is determined by the range a of the attractive sites.

Illustration of the dangling monomer scenario for the adsorption of a trimer as described in the main text. A trimer first binds with its end monomers to the lattice and the monomer in between remains still in solution. In a second step one of the end monomers jumps to a neighboring site (“stretching”) and the trimer can then align and fully bind to the pattern (“alignment”).

Illustration of the dangling monomer scenario for the adsorption of a trimer as described in the main text. A trimer first binds with its end monomers to the lattice and the monomer in between remains still in solution. In a second step one of the end monomers jumps to a neighboring site (“stretching”) and the trimer can then align and fully bind to the pattern (“alignment”).

Distribution of the angle θ between two consecutive bond vectors of a real freely jointed 26mer for different temperatures with square lattice patterns of l/b = 1 (upper panel (a)) and l/b = 0.58 (lower panel (b)). The gray shaded curve indicates the expected bulk distribution 1/2sin θ. The shaded brown curves represent the fully adsorbed state (with k B T/ε = 0.1 for l/b = 1 and k B T/ε = 0.05 for l/b = 0.58). The other temperatures are indicated in the legends. The insets show the fully adsorbed state again in comparison to the result for an ideal chain (light orange curve). Notice the pronounced peak around θ = 0° for the ideal chain and l/b = 0.58. This peak contains approximately 60% of the total distribution (for comparison the same peak for the real chain contains approximately 26%). The right inset in panel (a) shows the high-temperature distribution (solid black line) of the real chain in comparison to the expected ideal behavior (gray shaded curve).

Distribution of the angle θ between two consecutive bond vectors of a real freely jointed 26mer for different temperatures with square lattice patterns of l/b = 1 (upper panel (a)) and l/b = 0.58 (lower panel (b)). The gray shaded curve indicates the expected bulk distribution 1/2sin θ. The shaded brown curves represent the fully adsorbed state (with k B T/ε = 0.1 for l/b = 1 and k B T/ε = 0.05 for l/b = 0.58). The other temperatures are indicated in the legends. The insets show the fully adsorbed state again in comparison to the result for an ideal chain (light orange curve). Notice the pronounced peak around θ = 0° for the ideal chain and l/b = 0.58. This peak contains approximately 60% of the total distribution (for comparison the same peak for the real chain contains approximately 26%). The right inset in panel (a) shows the high-temperature distribution (solid black line) of the real chain in comparison to the expected ideal behavior (gray shaded curve).

Properties of the model partition function (17) in dependence on the reduced phase space volume γ (see main text). The upper panels show the specific heat c per monomer for three exemplary values of γ, the middle panels show the corresponding adsorption energies e per monomer. The lower panel displays the resulting pseudo phase/state diagram where the lines are determined from the position of the peaks of the specific heat. The desorbed state is characterized by an adsorption energy e close to zero, for the attached phase e ∼ −2/3 (two monomers attached, one dangling) and for the fully adsorbed state e is close to one.

Properties of the model partition function (17) in dependence on the reduced phase space volume γ (see main text). The upper panels show the specific heat c per monomer for three exemplary values of γ, the middle panels show the corresponding adsorption energies e per monomer. The lower panel displays the resulting pseudo phase/state diagram where the lines are determined from the position of the peaks of the specific heat. The desorbed state is characterized by an adsorption energy e close to zero, for the attached phase e ∼ −2/3 (two monomers attached, one dangling) and for the fully adsorbed state e is close to one.

Ideal trimer phase space volume for a square lattice with sites of radius a = 0.1 (black squares in steps 0.1 of l/b). Apart from the absolute value (18) a reduced phase space γ is shown as well (see main text for definition). The possible alignment regions of a single bond are indicated in the lower panel, compare main text and see Fig. 5 . The gray shade represents the region where sites begin to overlap and thus the surface appears to be homogeneous. The filled black squares correspond to values of l/b where two peaks are found in the specific heat (see main text and Fig. 11 ).

Ideal trimer phase space volume for a square lattice with sites of radius a = 0.1 (black squares in steps 0.1 of l/b). Apart from the absolute value (18) a reduced phase space γ is shown as well (see main text for definition). The possible alignment regions of a single bond are indicated in the lower panel, compare main text and see Fig. 5 . The gray shade represents the region where sites begin to overlap and thus the surface appears to be homogeneous. The filled black squares correspond to values of l/b where two peaks are found in the specific heat (see main text and Fig. 11 ).

Ideal trimer phase space volume for a triangular lattice for a radius a = 0.1 of the lattice sites (black squares, l/b in steps of 0.1). Apart from the absolute value Γ from (18) a reduced phase space γ is shown as well (see main text for definition). The possible alignment regions of a single bond are indicated in the lower panel, compare main text and also Fig. 5 . The gray shade represents the region where sites begin to overlap and thus the surface appears to be homogeneous. The filled black squares indicate values of l/b where two peaks are found in the specific heat (see main text and Fig. 12 ).

Ideal trimer phase space volume for a triangular lattice for a radius a = 0.1 of the lattice sites (black squares, l/b in steps of 0.1). Apart from the absolute value Γ from (18) a reduced phase space γ is shown as well (see main text for definition). The possible alignment regions of a single bond are indicated in the lower panel, compare main text and also Fig. 5 . The gray shade represents the region where sites begin to overlap and thus the surface appears to be homogeneous. The filled black squares indicate values of l/b where two peaks are found in the specific heat (see main text and Fig. 12 ).

Pseudo phase/state diagram for the adsorption of a real freely jointed chain of N = 25 bonds onto a square lattice with commensurabilities l/b and site radius a = 0.1. The error in determining the border line between attached and fully adsorbed states is approximately twice the size of the symbols.

Pseudo phase/state diagram for the adsorption of a real freely jointed chain of N = 25 bonds onto a square lattice with commensurabilities l/b and site radius a = 0.1. The error in determining the border line between attached and fully adsorbed states is approximately twice the size of the symbols.

Pseudo phase/state diagram for the adsorption of a real freely jointed chain of N = 25 bonds onto a triangular lattice with commensurabilities l/b and radius a = 0.1 for the attractive sites.

Pseudo phase/state diagram for the adsorption of a real freely jointed chain of N = 25 bonds onto a triangular lattice with commensurabilities l/b and radius a = 0.1 for the attractive sites.

Adsorption energy per monomer of a real freely jointed chain for some values of l/b for a triangular lattice pattern (see main text and compare Fig. 12 for the expected states).

Adsorption energy per monomer of a real freely jointed chain for some values of l/b for a triangular lattice pattern (see main text and compare Fig. 12 for the expected states).

Smallest eigenvalue of the gyration tensor averaged in the body fixed frame of reference of a real freely jointed chain for some values of l/b for a triangular lattice pattern (squares). In addition, the dashed lines show the averaged zz component of the gyration tensor in the laboratory frame.

Smallest eigenvalue of the gyration tensor averaged in the body fixed frame of reference of a real freely jointed chain for some values of l/b for a triangular lattice pattern (squares). In addition, the dashed lines show the averaged zz component of the gyration tensor in the laboratory frame.

## Tables

Possible inter bond angles of the dangling monomer scenario for l/b = 0.58 with the neighboring monomers attached to the indicated diagonals (m, n). The angle is computed using the law of cosines: .

Possible inter bond angles of the dangling monomer scenario for l/b = 0.58 with the neighboring monomers attached to the indicated diagonals (m, n). The angle is computed using the law of cosines: .

Phase space (18) of trimer alignments onto a square lattice with l/b = 0.58 as a function of the inter bond angles θ (normalized to the total phase space volume Γ). Only those angles are shown, for which the value is non-zero. The values give an impression for the expected angles and their weights at very low temperatures. Compare also Fig. 7(b) for the actual distribution of angles for a long real and ideal chain at finite temperatures.

Phase space (18) of trimer alignments onto a square lattice with l/b = 0.58 as a function of the inter bond angles θ (normalized to the total phase space volume Γ). Only those angles are shown, for which the value is non-zero. The values give an impression for the expected angles and their weights at very low temperatures. Compare also Fig. 7(b) for the actual distribution of angles for a long real and ideal chain at finite temperatures.

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