Semiflexible polymer in a strip

Abstract

We study the thermodynamic properties of a semiexible polymer confined inside strips of widths L <= 9 defined on a square lattice. The polymer is modeled as a self-avoiding walk and a short-range interaction between the monomers and the walls is included through an energy <IMG SRC="/img/fbpe/bjp/v32n4/a19img04.gif">associated with each monomer placed on one of the walls. Also, an energy <IMG SRC="/img/fbpe/bjp/v32n4/a19img04.gif">b is associated with each elementary bend of the walk. The free energy of the model is obtained exactly through a transfer matrix formalism. The profile of monomer density and the force on the walls are obtained. We notice that as <IMG SRC="/img/fbpe/bjp/v32n4/a19img04.gif">b is decreased, the range of values of <IMG SRC="/img/fbpe/bjp/v32n4/a19img04.gif">which the density profi le is neither convex nor concave increases, and for sufficiently attracting walls (<IMG SRC="/img/fbpe/bjp/v32n4/a19img04.gif"> < 0) we find that in general the attractive force is maximum for <IMG SRC="/img/fbpe/bjp/v32n4/a19img04.gif">b < 0, that is, for situations where the bends are favored.

Semiflexible Polymer in a Strip

Jürgen F. Stilck

Instituto de Física

Universidade Federal Fluminense

Av. Litorânea, s/n, 24210-340, Niterói, RJ, Brazil

Received on 17 June, 2002

We study the thermodynamic properties of a semiexible polymer confined inside strips of widths L £ 9 defined on a square lattice. The polymer is modeled as a self-avoiding walk and a short-range interaction between the monomers and the walls is included through an energy associated with each monomer placed on one of the walls. Also, an energy _b is associated with each elementary bend of the walk. The free energy of the model is obtained exactly through a transfer matrix formalism. The profile of monomer density and the force on the walls are obtained. We notice that as _b is decreased, the range of values of which the density profi le is neither convex nor concave increases, and for sufficiently attracting walls ( < 0) we find that in general the attractive force is maximum for _b < 0, that is, for situations where the bends are favored.

I Introduction

Polymers are often modeled as self- and mutually avoiding walks placed on a lattice, and much has been learned about their thermodynamic properties through such models [1,2]. Grand-canonical models of this kind, where the number of monomers incorporated into the chain is allowed to fluctuate controlled by an activity z = exp(m/k_BT), display a phase transition at some value of the activity (for infinite chains, that is, in the polymer limit). This transition is discontinuous in the one dimensional case d = 1 [3], and continuous for d > 2. Rather precise estimates of the critical value of z were found in two dimensions through transfer matrix calculations [4] and series expansions [5]. Also, exact values for the critical exponents are available in this case [6].

More recently, properties of such models on lattices limited by walls have attracted much interest [7], following studies of magnetic models in the same situation [8]. The short range interaction between the wall and the polymer may be introduced by associating a Boltzmann factor = exp(–/k_BT) with each monomer placed on the wall, so that < 0 corresponds to attracting walls while repulsive walls are described by > 0. The grand-canonical partition function for a model of a single chain is

where N is the number of monomers in the chain, the sum is over all configurations of the chain with the initial monomer placed on the wall, and N £ N stands for the number of monomers located on the wall. Such a model shows interesting features, and even in the limit where the self avoidance constraint is neglected (the so-called ideal chains) one finds that for sufficiently large values of > ₀ the surface polymerization transition will occur at a lower value of the activity z than the one in the bulk [9]. The point (z₀,₀) in the phase diagram where the bulk (also called ordinary) transition line meets the surface transition line is called the adsorption transition point. In two dimensions, such models have been studied through transfer matrix calculations [10] and series expansions [11]. Additional walls may be added, confining the polymer inside a strip, slab or pore [7], and the force applied on the walls in such situations is of interest even from the point of view of applications of polymers as adhesives [12]. The model of ideal chains confined in a slab has been studied in the past [12,13], and it was found that the force on the walls is attractive if exceeds the adsorption value ₀. In the case of self-avoiding chains confined in a strip on the square lattice, transfer matrix calculations show that attractive forces appear for below ₀ [13]. Also, the profile of the monomer density inside the strip was obtained [14], and, unlike what happens for ideal chains, for a self-avoiding chain profiles which are neither convex nor concave are found for a range of values of . Finally, similar techniques have also been applied to shed light on the scaling behavior of such models for strips with periodic boundary conditions (cylinders) as the width of the strips becomes large [15].

Another generalization of the original polymer model is to introduce an energy associated with the operation of bending the chain. On hypercubic lattices, the elementary bends will always be at right angles, and an energy

_b may be associated with each of them. This semiflexible polymer problem (also called persistent or biased walks), has been studied some time ago [16,17,18]. Recently, this model has attracted renewed interest , since it may describe some relevant aspects in the protein folding problem [19]. The thermodynamic properties of the model have been studied on the Bethe lattice [20] and, equivalently, in the Bethe approximation [21]. The end-to-end distance of semi-flexible chains on Bethe and Husimi lattices was obtained [22]. In this paper we study the thermodynamic behavior of a semiflexible polymer confined inside a strip. The partition function of the model may be written as

where N_b is the number of elementary bends in the configuration and _b = exp(–_b/k_BT) is the Boltzmann factor associated with each elementary bend and the sum is over all configurations of the chain. We define a transfer matrix for the model and obtain the grand-canonical partition function in the thermodynamic limit, determined by the largest eigenvalue of this matrix. In order to be able to obtain the distribution of monomers in the transverse section of the strip, we define a position dependent activity for the monomers. Also, the force applied by the polymer on the walls is calculated, as a function of the width of the strip, and _b. All thermodynamic properties are calculated at the polymerization transition, that is, at the value of the activity of a monomer for which the number of monomers incorporated into the polymer diverges.

In section the model is defined in detail and its solution is presented. The results we find for the thermodynamic behavior of the model are shown in section as well as final conclusions and discussions.

II Definition of the model and its solution

The self avoiding chain is constrained inside a strip of width m defined on the square lattice in the (x,y) plane, so that 0 £ x £ m. The chain runs through the whole strip, from y ® – ¥ to y ® +¥. We may define a transfer matrix for the problem following a prescription proposed by Derrida, in a way to take into account the self-avoidance constraint exactly [4]. The connectivity properties of all vertical bonds of the chain arriving at a line y₀ from below are specified through the indication of

1. The (unique) bond connected to the initial monomer of the chain (placed in y ® –¥) through a path lying entirely below the line y₀ (passing though sites with y < y₀);

2. The pairs of bonds connected to each other through a path lying entirely below the line y₀.

In Fig. 1 the five configurations for the case m = 2 are depicted, and a portion of the chain placed inside the strip is shown with each configuration indicated. Configurations 1 and 3, as well as 4 and 5, are related to each other by reflection symmetry. We define column dependent activities z_i, i = 1, 2, ...n_a(m) according to this reflection symmetry, as indicated also in Fig. 1. The number of activities n_a(m) is equal to m/2+1 for even values of m and (m+1)/2 for odd values of m.

For a fixed connectivity configuration of a set of m+1 vertical bonds arriving at the line y₀, the possible configurations of the set of vertical bonds arriving at y₀+1 may be obtained, as well as the contribution to the partition function from the sites comprised between both sets of vertical bonds. This contribution will be given by

where Ni,y₀ is the number of monomers with activity z_i in line y₀ and Nb,y₀ is the number of elementary bends in this line. These contributions, shown for a particular example in Fig. 2, define a line of the transfer matrix. For m = 2 the transfer matrix will be given by

The grand canonical partition function of the model,considering periodic boundary conditions in the y direction, will be given by

where N_y is the total length of the strip in the y direction, so that the total number of sites is given by N_s = N_y(m+1). The number of monomers with activity z_i is

and the total number of monomers in the chain will be

The fraction of monomers placed at column x is

where x = 0,1,¼,n_a(m) –1 and r(x) = r(m–x). The Kronecker delta in the denominator contributes only for even values of m.

In the thermodynamic limit N_y® ¥ the partition function 4 is dominated by the largest eigenvalue of the transfer matrix ₁, so that X ~ and in this limit

The first order polymerization transition in the finite strip will take place when the thermodynamic potential

is equal to the one for the empty lattice y₀ = 0. Thus, since in the thermodynamic limit we have

the polymerized phase will coexist with the non-polymerized phase for

₁ = 1. Therefore, all thermodynamic quantities below will be calculated for z_i = z, i = 1,¼,n_a(m)–1 and z(n_a(m)) = z, where the activity z is then fixed at the coexistence value z_c so that, for a given value of _b, we have ₁ = 1.

Finally, the force applied on the walls is given by

where a is the lattice parameter and positive values correspond to attractive forces. An adimensional force per monomer at the coexistence may be then defined as

Since our results correspond to integer values of m, the force f was estimated making the discrete approximation

A simple calculation may be performed in the limit of rigid rods

_b = 0, where bends are not allowed. In this limit the transfer matrix is diagonal of size (m+1) ×(m+1). For > 1 all monomers are on the walls, for = 1 they are uniformly distributed, whereas for < 1 an uniform distribution in the sites away from the walls is found. As expected, in this limit the force on the walls vanishes. In the numerical results below we consider _b³ 1.

III Numerical results and conclusion

The transfer matrices for the model were obtained exactly for strips of widths ranging from 3 to 9. After using the reflection symmetry, the sizes of the matrices are, respectively, equal to 6, 16, 38, 100, 256, 681, and 1805. We obtained the density profile at the coexistence condition for values of and _b mostly between 1 and 3.

For neutral walls ( = 1) the density profile is always concave, with a higher density in the center. This may be understood since the region away from the walls is favored entropically. As becomes larger, monomers on the walls are energetically favored, so for sufficiently large values of a convex density profile is expected. For ideal flexible chains (_b = 1), at the adsorption value = 4/3 the density profile is flat for all sites which are not on the walls [12,14]. Such a flat transition profile is not observed for self-avoiding chains, where convex profiles are separated from concave ones by an interval of values of , located well below the adsorption transition value, where the profile is neither convex nor concave. This interval is quite narrow for flexible chains [14]. In Fig. 3 the densities are plotted as functions of for two values of _b. It is clear that as _b is increased, favoring bends, the interval of values of with a profile without well defined convexity grows.

The values of below which the density profile is concave (₁)and those above which the profile is convex (₂) are plotted in Fig. 4 as functions of _b for two values of m. As is apparent the range with no defined convexity grows with _b in a nearly linear way. As a general rule, we found that as _b is increased from 1, the first pair of densities to cross, destroying concavity, is always correspondent to the columns which are first and second neighbors to the walls. Also, the last crossing, which turns the profile convex, is between densities corresponding to the columns at the wall and the neighbor columns. In the limit _b® ¥ we found that ₂® ¥.

Finally, we calculated the force per monomer on the walls for pair of widths m, m+1, using expression 13. As already found for flexible self-avoiding chains [13], attractive forces appear for sufficiently large values of . Fig. 5 shows results for the force as a function of for some values of _b. The origin of attractive forces in the system are portions of the chain limited by monomers adsorbed on opposite walls. Thus, as expected, the curves f × display a maximum, because the force vanishes as ® ¥, since in this limit such "bridge" segments are absent.

In Fig. 6 the maximum force per monomer (with respect to ) is plotted as a function of _b. It is apparent that the maximum of these curves is located at _b > 1, and thus we conclude that polymers for which bends are somewhat favored give rise, in general, to larger attractive forces than flexible ones. As _b is increased, the maximum attractive force occurs at higher values of

The force per monomer as a function of the width of the strip m, for above a threshold, shows a stable equilibrium point at very low separation m₁ and an unstable equilibrium point at a larger separation m₂ [13]. The force is attractive in the interval m₁ < x < m₂.

The stable equilibrium point m₁(, _b) is found in the range 0.5 £ m₁£ 1.5, showing little variation as a function of the Boltzmann factors, as long as is larger than the threshold value. The unstable equilibrium point m₂(,_b) shows a rather strong monotonic dependence on both variables, being an increasing function of and a decreasing function of _b.

In conclusion, we may summarize the behavior of semiflexible chains confined inside strips observing that as the presence of elementary bends in the chains is favored, the range of values of for which the density profile has no defined convexity grows. Also, the largest attractive forces are found for _b > 1, that is, when the bending of the chains is favored. It should be noticed that the nonmonotonic behavior observed for the tension as a function of the width of the strip, with two equilibrium points, happens in a regime where the finite size scaling behavior has not yet been reached [13].

Acknowledgements

Partial financial support by the brazilian agencies CNPq and FAPERJ is gratefully acknowledged.

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Publication Dates

• Publication in this collection
  13 Mar 2003
• Date of issue
  Dec 2002

History

• Received
  17 June 2002