Physics Procedia 53 ( 2014 ) 16 – 23

Available online at www.sciencedirect.com

1875-3892 © 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).Peer-review under responsibility of The Organizing Committee of CSP 2013 conference doi: 10.1016/j.phpro.2014.06.019

ScienceDirect

Pattern-recognising polymer adsorption on structured surfaces: Gaussian

polymers vs. freely jointed chains

Hans Behringer

Institut fur Physik, Johannes Gutenberg-Universitat Mainz, Staudinger Weg 7 D – 55128 Mainz, Germany

Abstract

Selective adsorption of homopolymers is exploited as a model for pattern recognition. To this end the strong

adsorption regime of Gaussian polymers adsorbed on a regularly structured surface is investigated for square and

triangular lattices within a discrete Edwards model. The equilibrium behaviour of the specific heat, the gyration

tensor and the (nematic) bond order tensor are analysed and compared to the properties for adsorbed freely jointed

polymer chains.

Keywords: Polymer adsorption, pattern recognition, Monte Carlo simulations

PACS: 82.35.Gh, 68.43.-h, 02.70.Uu

1. Introduction

Generic mesoscopic properties of polymers are often independent of the chemical structure of the molecules and

can be described efficiently in coarse-grained models [1, 2]. Coarse-grained modelling reduces the description to the

most dominant physical ingredients such as the connectivity of the chains, their architecture, topology, and excluded

volume interactions. A phenomenon which can be described satisfactorily in such an approach is the adsorption of

polymers onto surfaces [3, 4, 5]. Particularly interesting systems are those with structured surfaces as the correspond-

ing selective adsorption is related to pattern recognition [6, 7, 8]. Pattern recognition phenomena are at the centre of

biological interaction networks as the selective binding of particular biomolecules is related to the mutual recognition

of the binding sites. Across the binding sites the pattern of the two molecules have to match, i. e. the geometries have

to fit and favourable interaction sites (e. g. hydrogen bonds, slat bridges) have to be in proximity [9].

Aspects of the generic principles of pattern recognition processes have been investigated in a number of works on

polymer adsorption onto structured surfaces [10, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,

28]. Addressed issues comprise among others how the adsorption depends on the interplay of the imprinted structure

on the surface with the sequence of heteropolymers. The structure of the surface can hereby consist of frozen regular

geometries (e. g. stripes, checker boards) or of a statistical distribution of patches with different affinities towards the

different monomer types of the heteropolymer [8, 29, 11, 30, 13, 31, 31, 19, 23, 32, 33]. The recognition itself is

seen as an adsorption where the polymer chain eventually fits snugly to the surface pattern. Often the corresponding

adsorption transition proceeds in two steps with a first unspecific attachment and a second low-temperature transition

that establishes the final ’recognising’ polymer conformations [6, 7, 16, 34, 35, 20, 36, 37].

Email address: behringh@uni-mainz.de (Hans Behringer)

© 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).Peer-review under responsibility of The Organizing Committee of CSP 2013 conference

Hans Behringer / Physics Procedia 53 ( 2014 ) 16 – 23 17

The notion of pattern recognition comprises a second aspect apart from the perfect match of polymer sequences

to surface structures. Biological systems such as the cellular milieu are heterogeneous and crowded [38, 39]. Macro-

molecular recognition processes of proteins and DNA molecules have to work reliably although different interaction

partners are present simultaneously. By specifically binding, one of the biomolecules reads out the specific structure

on the other molecules binding site: the molecule recognises the relevant and designated pattern from an ensemble of

many different patterns. The recognition is related to the formation of a long-living complex whereas the complexes

with the other present biomolecules are only transient. Recently we concentrated on this aspect of pattern recognition

by looking at the strong adsorption regime of polymers on structured surfaces as a generic model system [40, 41]. In

particular, we characterised the polymer conformations for freely jointed polymers on regularly patterned surfaces. In

this system two characteristic lengths show up, the (fixed) bond length of the polymers and the characteristic length

(lattice constant) of the surface structure. These two lengths may be incommensurable which in turn crucially influ-

ences the properties of the adsorbed conformations and of the transition mechanism. In this article the same system

(Sec. 2) is investigated for Gaussian polymers where the bond length is not restricted (Sec. 3).

2. Model system for polymer adsorption

The coarse-grained model for the investigation of the universal properties of polymer adsorption is designed to

capture only the most dominant degrees of freedom. On the side of the polymer these characteristic features are the

connectivity and the excluded volume interactions. The connectivity of the chain is modelled by a flexible spring of

averaged length b between two successive monomers at positions �Ri+1 and �Ri leading to the so-called Gaussian chain

model

βHG({�Ri}) = 3

2b2

N−1∑i=1

(�Ri+1 − �Ri)2. (1)

Here β = 1/kBT denotes the inverse temperature of the system and N is the number of monomers of the chain. The

monomer positions �Ri are confined to a simulation volume V with two impenetrable walls along the z direction. Along

the x and y direction periodic boundary conditions are applied.

Additionally to the spring energy one has to account for excluded volume interactions of monomers. This can be

done by recasting the individual pairwise contributions into a functional of the local density. The resulting expression

is expanded into a power series leading to the Edwards Hamiltonian [1, 2]

βHex[ρ�r(�R)] =v2

∫V

d3r ρ2�r (�R) (2)

if one truncates the series after the first non-trivial term. Here, ρ�r gives the local monomer density at position �r in the

volume V of the system. Formally this density is given by the operator

ρ�r (�R) =∑α

∑i

δ(�r − �Rα,i) (3)

acting on the system configuration �R = {�Rα,i} (α is the index for the n chains of the system, i labels the monomers). The

excluded volume parameter v > 0 accounts for the repulsion of monomers and depends in general on the temperature.

For the systems discussed here, however, only a weak temperature dependence is assumed in v for the range of interest

so that βHex is taken to have no explicit temperature dependence.

The total energy of the configuration �R of the polymer system finally is the sum of the two discussed terms and

the attractionHsu with the surface pattern, βH(�R) = βHG(�R)+ βHex(�R)+ βHsu(�R). The attraction with the surface is

due to the interaction of individual monomers with localised sites. This interaction of strength ε > 0 is modelled by a

potential well of range a:

Vsu(r) =

{ −ε r < a0, else,

(4)

18 Hans Behringer / Physics Procedia 53 ( 2014 ) 16 – 23

so that the total surface energy is

Hsu =∑α,i,s

Vsu(|�Rα,i − �ζs|). (5)

The surface sites at positions �ζs are arranged on one of the two hard walls in a regular lattice which is characterised

by a lattice constant l. In this work a square and a triangular lattice pattern are used.

The equilibrium properties of the system are investigated numerically by means of Monte Carlo simulations. To

this end the polymer conformation is updated by local monomer moves and by reptation moves which are accepted

according to the usual Metropolis criterion [42]. For the calculation of the energy change due to a proposed move the

energy of the excluded volume term has to be evaluated. To this end the continuous expression (2) is converted into a

discrete approximation. This is done by introducing a cubic grid with grid nodes A. Each grid node is at the centre of

a cell of linear size Δ and has consequently a volume ΔV = Δ3. The integral in (2) is then approximated by

∫d3rρ2

�r →∑

A

ΔV( cA

ΔV

)2(6)

where cA denotes the total weight of all monomers that are attributed to the cell A. This weight is in the simplest

density assignment scheme just the number of monomers in the cell A. This nearest-grid-point scheme is used in the

following. More generally, the contribution of a given monomer to cA is obtained by using weight functions which

attribute the monomer to the cells in its proximity (for details see [43]). The recast version (6) then defines a discrete

Edwards model for excluded volume interactions which can now be used in Monte Carlo simulations [44, 45, 43].

3. Gaussian chain conformations in the strong adsorption regime

For the investigation of the adsorption properties the excluded volume interaction (i. e. the parameter v and the

discretisation length Δ of the grid) is fixed and the strength of the surface attraction βε is varied. The characteristics of

the specific heat curve is used to identify the transition. As outlined above βHG and βHex are temperature independent

and provide the structure of the phase space, only the surface interaction term βHsu exhibits a direct temperature

dependence. We therefore consider only the specific heat per monomer related to this adsorption energy:

c =1

(N + 1)kBT 2

(〈H2

su〉 − 〈Hsu〉2). (7)

In the following Gaussian chains with a bond length b will be considered in a slit geometry with extension Lz = 5Rg

in z direction; Rg is the radius of gyration of the ideal chain in the bulk. The value of b is fixed to b = 1 (in simulation

units) and the lattice constant l is varied. In contrast to the fixed bond length of a freely jointed chain model the spring

flexibility of the Gaussian chain will allow a varying bond length. This is illustrated in Fig. 1 for two exemplary

values of l/b and a square lattice of attractive sites. (For the incommensurability l/b = 0.58 the adsorption of the

freely jointed chain proceeds in two steps [40].)

Once the adsorption transition is identified by the location of the peak of the specific heat the surface affinity is

fixed to be in the strong adsorption regime and the conformations of the polymers are investigated by analysing the

gyration tensor. Using the centre of the polymer

�RCM =1

N + 1

∑i

�Ri (8)

the tensor of gyration G is expressed as

G =1

N + 1

N∑i=0

(�Ri − �RCM) ⊗ (�Ri − �RCM). (9)

Here ⊗ indicates a tensor product between two vectors. Similar to the specific heat the z-component G⊥ signals the

onset of the transition from a three dimensional polymer coil to a flat two-dimensional adsorbed conformation. The

parallel component G‖ = Gxx +Gyy of (9) contains information about the adsorbed polymer conformations.

Hans Behringer / Physics Procedia 53 ( 2014 ) 16 – 23 19

0.05 0.1 0.15 0.2 0.25

kBT/ε

0

10

20

30

Cad

l = 0.58b

0.05 0.1 0.15 0.2 0.25

kBT/ε

0

10

20

30

l = b

0.05 0.1 0.15 0.2 0.25

kBT/ε

1

1.5

Espring

l = 0.58b

0.05 0.1 0.15 0.2 0.25

kBT/ε

1

1.5

l = b

Figure 1: Specific heat per monomer (upper panel) and the averaged spring energy per bond (lower panel) of a Gaussian 26mer as a function

of temperature for two different lattice constants of an underlying square lattice (range a = 0.1b for attraction well of sites): The left hand side

shows l = 0.58b, the right hand side l = b. The curves for a real chain (squares) are compared to an ideal chain (solid lines). (Excluded volume:

discretisation Δ = 0.25b, energy parameter v = 0.1b3.) The spring energy of the ideal chain in the bulk is expected to be 3/2. However, a small

deviation shows up due to the slit geometry in z direction.

In the fully adsorbed state the properties of the freely jointed polymers strongly depend on the commensurability

of the characteristic bond length b and the lattice constant l [40]. If the two lengths are different the structure of

the adsorbed polymer depends on its ability to explore the finite interaction range a of attractive sites to achieve a

favourable binding’ situation. For a given lattice constant l the bonds with fixed length can only bind to a small

number of certain pairs of sites whose separation fit to the length b. In addition, the binding of a bond to a pair of sites

is related to small phase space in situations where the lengths l and b are very incommensurable. As a consequence

binding can only occur in very special conformations with certain characteristics, as for instance the inter bond angles,

being markedly different for different values of the incommensurability d = l/b. This can be observed in the behaviour

of the parallel gyration tensor which is shown for the example of a square and triangular lattice pattern in Fig. 2. The

effects briefly commented on above and their consequences for the behaviour of the parallel gyration radius G‖ of the

fully adsorbed low-temperature state have been analysed in details elsewhere for the freely jointed polymer model

and different lattice patterns [40]. It turns out, that the characteristics allow the extraction of information on the lattice

patterns. The flexibility of the bond length in Gaussian chains on the other hand will still allow for compensating

frustration effects where the values of the bond length and the lattice constant are not commensurable. The gyration

tensor therefore does not exhibit a characteristic behaviour that allows a discrimination of different lattice patterns as

is illustrated in Fig. 2.

Further information on the properties of the adsorbed polymers is contained in the alignment of the bonds to the

pattern of attractive sites. This alignment can be quantified by a order parameter tensor [46]

S =1

2N

N∑i=1

(3�ui ⊗ �ui − I

)(10)

with

�ui =�Ri − �Ri−1

|�Ri − �Ri−1|(11)

denoting the normalised bond vector between two consecutive monomers, and I is the identity tensor. In the des-

orbed bulk state the bonds have no preferential order and the averaged S vanishes. In the adsorbed state, however,

the averaged bond order tensor does not vanish due to the alignment of the bonds. Furthermore, this alignment is

20 Hans Behringer / Physics Procedia 53 ( 2014 ) 16 – 23

0.4 0.6 0.8 1 1.2d

1

2

3

4

5

6

G||

triangular lattice

0.4 0.6 0.8 1 1.2d

square lattice

Figure 2: Component G‖ of the gyration tensor parallel to the surface for fully adsorbed (ideal) chains and different lattice types in the strong

adsorption regime (kBT/ε = 0.05) for the triangular lattice (left hand side) and square lattice (right hand side) as a function of d = l/b. Shown are

curves for freely jointed chains (red) in comparison with results for Gaussian chains (black).

0.05 0.1 0.15 0.2 0.25

kBT/ε

-0.4

-0.2

0

0.2

S ii

l = 0.58b

0.05 0.1 0.15 0.2 0.25

kBT/ε

-0.4

-0.2

0

0.2

l = b

Figure 3: Components of the bond order parameter (black: xx (identical to the yy components), red: zz components) of a Gaussian 26mer as a

function of temperature for two different lattice constants of the underlying square lattice: The left hand side shows l = 0.58b, the right hand side

l = b. The curves for a real chain (squares) are compared to an ideal chain (solid lines). (Excluded volume: Δ = 0.25b, v = 0.1b3.) The grey lines

show S = diag(1/4, 1/4,−1/2) which one expects by placing bonds onto the lattice sites.

characterised by the distribution of the normalised vectors �u. The associated information can be extracted by using a

polar representation �u = (sin θ cosϕ, sin θ sinϕ, cos θ) and by looking at the distribution of the azimuthal angle ϕ (in

the adsorbed state θ ≈ π/2).

The behaviour of the (diagonal) entries of the bond order tensor is displayed in Fig. 3 again for two different

lattice constants of a square lattice. The onset of non-zero values signal the adsorption transition in the same way as

the appearance of a peak in the specific heat. By placing single bonds onto the possible set of pairs of attractive sites

one calculates an expected bond order tensor S = diag(1/4, 1/4,−1/2) for the strong adsorption regime [41]. The

behaviour in Fig. 3 deviates slightly from this expectation. This observation will be discussed further at the end of

this section.

As shown in Fig. 4 the azimuthal angle distributions show peaks for alignments to different pairs of sites. A bond

of length b can in general align to a pair (m, n) of sites which are separated by a vector �p = m�e1 + n�e2. Here m and ndenote integers and the vectors �e1 and �e2 span the lattice. For a freely jointed chain alignment is only possible if the

length |�p| fits to the length b (taking the ’size’ a of the sites into account). For a Gaussian chain, however, the stretching

and compressing bonds enable binding to many pairs. For the example l = b bonds align most favourably to pairs

(1, 0) (and equivalent ones (0, 1), (−1, 0) and (0,−1), related to each other by lattice symmetries). By stretching the

bond they can also align to (1, 1) (and equivalent) pairs. The possible alignments of bonds lead to certain preferential

azimuthal angles ϕ, which can then by observed in the distribution P(ϕ). For a square lattice one expects preferential

angles 0◦, ±90◦ and ±180◦ due to bond alignment to the sets of pairs (1, 0). Furthermore, angles ±45◦ and ±145◦ show

up due to alignments to (1, 1) pairs, however, with a reduced weight as the Gaussian springs have to be stretched. For

a system with freely jointed chains the later angles are not possible, see Fig. 4. For smaller values of l/b further

Hans Behringer / Physics Procedia 53 ( 2014 ) 16 – 23 21

-180 -120 -60 0 60 120 180ϕ

0

0.5

1

1.5

2

P(ϕ)

l = b

-180 -120 -60 0 60 120 180ϕ

-180 -120 -60 0 60 120 180ϕ

0

0.2

0.4

0.6

0.8

1P(ϕ)

l = 0.75b

-180 -120 -60 0 60 120 180ϕ

-180 -120 -60 0 60 120 180ϕ

0

0.2

0.4

0.6

0.8

1

P(ϕ)

l = 0.58b

-180 -120 -60 0 60 120 180ϕ

Figure 4: Distribution of the azimuthal bond vector angle ϕ of a fully adsorbed Gaussian chain with 26 monomers at temperature kBT/ε = 0.05 for

the triangular lattice (left hand side) and square lattice (right hand side). Shown are the ratios l/b = 1, l/b = 0.75 and l/b = 0.58. The symmetry

of the lattice is reflected by the different numbers of equivalent angles for bond vector alignment. In contrast to the restricted possibilities for the

freely-jointed chain (compare red curves in the panel for l = b) the flexibility of the Gaussian chain allows alignments to sites separated by a wide

range of different lengths.

possible values for angles show up, for instance, due to alignments to (2, 1) pairs. By investigating the peak-structure

and comparing it with the expectations one can identify the lattice type. However, information on the lattice constant

itself is hardly deducible. In case of a freely jointed polymer system unmatching bond length and lattice constant

have a much more pronounced effect on the peak-structure so that one can extract information on the lattice constant

[40, 41]. In contrast, the flexible bond length of Gaussian chains smears out this information. Information about the

lattice constant can be recovered by looking at the distribution of the (flexible) bond length in the fully adsorbed state.

The position of the peaks is directly related to the length of the vector m�e1 + n�e2 between the two sites to which bonds

align. Together with the information about the symmetry of the lattice the lattice constant l can thus be determined.

Note that the spring energy is not subject to thermal fluctuations, see (1). Therefore, the apparently ’high-energy’

peaks in the corresponding distribution will not disappear upon further decreasing the temperature.

To conclude this section let us come back to the adsorption properties of the Gaussian chain, in particular to the

behaviour of the Gaussian energy (1) per bond which is shown in Fig. 1. This energy itself is only subject to thermal

fluctuations in an indirect way, however, its behaviour signals the onset of the transition. For large temperatures one

expects to observe the bulk value 3/2. Due to the energy gain upon binding the possible conformations of bonds are

restricted to a ’quasi-discrete’ set on the two-dimensional surface, compare Fig. 5. The excluded volume interaction

leads to a stretching of the Gaussian spring so that the observed values are above the ones for the ideal chain. The

flexibility allows the alignment of the bonds to many different site pairs of different separation which leads to the rather

broad transition region. The distribution function of the bond length, Fig. 5, shows a peak around the length 0.1b. The

monomers of such a bond are both bound to the same site (recall that the attraction range a = 0.1b and note that the

repulsion of monomers in the discrete Edwards model is rather soft). Such a bond has full two-dimensional rotational

freedom and is basically bound to the adsorption plane although the phase space is still related to the three space

dimensions. Calculating the bond order tensor related to such a (freely rotating) bond gives S = diag(1/8, 1/8,−1/4).

(Here, the integration measure for the calculation is basically 14π

dϕdθ sin θδ(θ − π/2) where the delta function is due

22 Hans Behringer / Physics Procedia 53 ( 2014 ) 16 – 23

0 1 2r

0

0.5

1

1.5

2

2.5

D(r

)

l = 0.58b

0 1 2r

0

1

2

3

4

5 l = b

Figure 5: Distribution function of the bond length r of a Gaussian in the low temperature phase kBT/ε = 0.05 for the two different lattice constants

of the square lattice: left l = 0.58b, right l = b. The black solid curve represents the real chain, the dashed red curve the ideal chain.

to the restriction to the plane as a consequence of the energy gain upon adsorption.) The total bond order tensor of

all bonds in the strong adsorption regime thus contains small contributions from diag(1/8, 1/8,−1/4) related to freely

rotating bonds of small length in addition to the contributions diag(1/4, 1/4,−1/2) associated with alignments to a

pair of sites. This explains the observation of Fig. 3 concerning the slight deviations of the components of the bond

order parameter from the low temperature values expected at first sight by placing bonds with each monomer on a

different site.

4. Concluding remarks

In this work the strong adsorption regime of Gaussian polymers on an attractive regularly patterned surface has

been investigated. In particular, the question whether or not selective adsorption can be used as a means of pattern

recognition has been addressed. Hereby, pattern recognition is understood to be the ability to selectively bind to a

certain surface pattern from a collection of many different structures and thereby reading out information about the

surface. A crucial requirement of pattern recognition phenomena is the presence of several ’competitive’ degrees of

freedom so that different recognition phenomena are related to different degrees of freedom dominating the physical

behaviour. Often this competition aspect is introduced by having heteropolymers with different monomer types that

have different affinities towards different patches of the surface (’multifunctionality’) [11]. Recently we considered a

model for pattern recognition with homopolymers, which, however, have different characteristic lengths in form of a

fixed bond length. This bond length can have incommensurable values with respect to the typical length of the surface

structuring thus leading to a characteristic physical behaviour depending on the incommensurability [40, 41].

In the present work the condition of having a fixed bond length (freely jointed chain) has been relaxed by allowing

a flexible bond length (Gaussian chain). The lacking possibility of having incommensurable lengths with its conse-

quence on entropic restrictions crucially alters the pattern recognition mechanism compared to freely jointed chains.

Nevertheless, the combination of information from the distribution of the azimuthal angles of the adsorbed bonds

together with information about the spring energy of the Gaussian chains allows one to read out information about the

underlying structure of attractive sites. This work concentrated only on the strong adsorption regime and did not touch

the question about the thermodynamics of the transition itself (see, for instance, [47] for an analytical investigation).

Numerical finite-size scaling investigations of the desorption-adsorption transition are left to future studies.

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