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Selective homopolymer adsorption on structured surfaces as a model for pattern recognition
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10.1063/1.4773470
/content/aip/journal/jcp/138/2/10.1063/1.4773470
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/2/10.1063/1.4773470

Figures

Image of FIG. 1.
FIG. 1.

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.

Image of FIG. 2.
FIG. 2.

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).

Image of FIG. 3.
FIG. 3.

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).

Image of FIG. 4.
FIG. 4.

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 × 108 Monte Carlo steps, for d = 0.84 averages from 109 Monte Carlo steps are displayed by a black line as well.)

Image of FIG. 5.
FIG. 5.

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.

Image of FIG. 6.
FIG. 6.

An adsorbed bond of a polymer can compensate the misfit between surface site spacing and bond length within a range of ±2a by rearranging. For spacing d = b − 2a, 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 + 2a. 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 |db|, see main text.

Image of FIG. 7.
FIG. 7.

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).

Image of FIG. 8.
FIG. 8.

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 Δd = 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 109 Monte Carlo steps for one chain and from 3 × 108 steps for 10 polymers.)

Image of FIG. 9.
FIG. 9.

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

Image of FIG. 10.
FIG. 10.

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 .)

Image of FIG. 11.
FIG. 11.

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.

Image of FIG. 12.
FIG. 12.

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.

Image of FIG. 13.
FIG. 13.

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.

Image of FIG. 14.
FIG. 14.

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 ).

Image of FIG. 15.
FIG. 15.

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.

Image of FIG. 16.
FIG. 16.

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.

Image of FIG. 17.
FIG. 17.

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.

Image of FIG. 18.
FIG. 18.

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

Generic image for table
Table I.

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.

Generic image for table
Table II.

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.

Generic image for table
Table III.

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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/content/aip/journal/jcp/138/2/10.1063/1.4773470
2013-01-10
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Selective homopolymer adsorption on structured surfaces as a model for pattern recognition
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/2/10.1063/1.4773470
10.1063/1.4773470
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