On the surface tension of directed linear polymers

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On the surface tension of directed linear polymers

V.B. Priezzhev, S.A. Terletsky

To cite this version:

V.B. Priezzhev, S.A. Terletsky. On the surface tension of directed linear polymers. Journal de

Physique, 1989, 50 (6), pp.599-608. �10.1051/jphys:01989005006059900�. �jpa-00210940�

599

On the surface tension of directed linear polymers

V. B. Priezzhev and S. A. Terletsky

Joint Institute for Nuclear Research, Laboratory of Theoretical Physics,

Head Post Office, ^{P.O. Box} 79, ¹⁰¹⁰⁰⁰ Moscow, ^U.S.S.R.

(Reçu ^{le 30} juin 1988, révisé le 21 octobre 1988, accepté le 23 novembre 1988)

Résumé.

Nous trouvons la solution d’un modèle de polymères dirigés ^{à deux ou trois}

dimensions dans une approximation de fermions libres. Nous étudions l’énergie ^libre fs dans ^un grand domaine de variation de la densité de polymères ^{p. Nous montrons} que fs ~ 03C12 ^à

d = 2 et fs ~ 03C13/2 à d = 3 pour p petit. ^{La loi} 03C13/2 ^{coïncide avec} celle obtenue pour la tension de surface d’un modèle isotrope par des arguments ^{d’échelle.}

Abstract.

The directed polymer ^{model on a two-} and three-dimensional lattice in the presence of infinite boundary is solved in the free fermion approximation. ^{The excess} surface free energy

fs ^is studied in a wide range of the polymer density ^p. It is shown that fs ~ 03C12 ^for

d ⁼ 2 and fs ~ 03C13/2 for d ^{= 3 when p} is small. The law 03C13/2 coincides with that obtained for the surface tension of an isotropic ^model by scaling arguments.

J. Phys. ^{France 50} (1989) ^{599-608 15 MARS} 1989,

Classification

Physics ^Abstracts

68.45V - 81.60Jw

1. Introduction.

The well-known approach ^to investigations ^of the surface properties of semidilute polymer

solutions [1] gives ^the following dependence ^{of the surface tension on} density for flexible

isotropic ^linear polymers

where p is the volume fraction of monomers entering into macromolecules. The polymer- magnetic analogy ^and scaling arguments ^{used for} deriving ^the expression (1) ^{are efficient} only

when the value of p is sufficiently ^{small and a} solution has the scaling properties [1-3]. ^When

p is large, ^{the surface tension} depends ^{on the} intermolecular interaction at the monomer level

[1]. Therefore, ^the attempts ^of construction of solvable polymer ^models taking ^{into account}

the excluded-volume effects and giving ^the dependence of u (p) ^{in a} wide range of p ^are of great ^interest.

Recently, ^{a model of} polymers ^at surfaces has been proposed [4] ^{in which a} single polymer

chain has been treated as a directed random walk on a semi-infinite lattice. An array of ^two- dimensional directed walks non-intersecting each other and interacting ^{with a wall has been}

considered in [5]. Polymer models of that type ^{with the} excluded volume effects have been

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005006059900

introduced by Nagle [6], who has shown its equivalence ^to dimer models on decorated lattices. The two-dimensional dimer problem ^is solvable because it belongs ^to the class of free- fermion models [7]. Apart ^{from exact solutions in the} two-dimensional case, free-fermion models can be used as reasonable approximations ^for higher dimensions.

In this work we use the free-fermion representation ^{to consider both the} three-dimensional and two-dimensional model of directed polymers being ^{in contact with a} solid wall. Realistic systems ^{which can be described} by ^{that model} may be products ^{of directed} polymerization ^of

bifunctional monomers. Moreover, ^{the model} considered here can apparently describe the behaviour of polymer solutions in the presence of ^a fluid flow [8].

The model is formulated in section 2. The calculation method used in this work allows us in

a combinatorial _way to find the partition ^{function of the} semi-infinite system ^{and to obtain the}

excess surface free energy (Sect. 3). Assuming the existence of the stable boundary ^we identity ^{the surface} free energy with the surface tension ^a and derive the p dependence ^of

u for d ⁼ 2, ^3. Equation (1) ^{follows from this} dependence ^{in the} three-dimensional case when p is small. The results are discussed in section 4.

2. Model and calculation method.

For d ⁼ 2 the model of directed linear polymers with excluded volume being ^{in contact with a} nonadsorbing solid wall is constructed as follows. Consider a horizontal one-dimensional

lattice, sites of which are denoted by 0, 1, 2, ^{..., N. We define an} M-step ^{walk as a connected} path along ^{M bonds} (perhaps, ^with repetitions) starting ^and ending ^{at the same} point. ^The

returns to the starting point ^after 2, 4, ..., ^{M - 2} steps ^{are not} prohibited. Further, ^we

introduce a vertical coordinate along ^{which the} positions ^{of a} walking particle ^{are fixed versus}

the number of steps ^or discrete time (Fig. 1). ^{Consider an ensemble of} particles performing M-step walks. Two different particles ^{cannot be} simultaneously ^{found at the same site and}

avoid the origin of coordinates and the site N. The two-dimensional polymer ^model [6] ^arises

from these definitions if one associates time with the spatial coordinates. Indeed, ^the M-step walks may be regarded ^as closed directed polymers rolled around a cylinder. Polymer

chains avoid each other and don’t cross the walls, i. e. the vertical lines going through ^the origin of coordinates and the site N.

Fig. ^1.

^Directed polymers ^{at d = 2. Walks occur on} the one-dimensional array 0, 1, ..., N. M is the number of steps.

In the case d ⁼ 3 the model is defined quite analogously [9-11]. Figure ² illustrâtes the above construction for this case.

We have identified a polymer ^{link with a} step of the walk. Now, ^{ascribe to it a} statistical

weight (fugacity) ^x. To control the hits of walkers on the boundaries, ^{we introduce an}

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Fig. ^2.

^Directed polymers ^{at d = 3. Walks occur on} the two-dimensional plane. ^M is the number of steps.

additional weight y. To that end ^we glue ^the edges ^{of the} cylinder and consider polymer configurations ^{on a torus} (hypertorus). Ascribe the weight y ^to the boundary steps ^{from the} sites 1 to 0 and from N - 1 to N. Then, putting y ⁼ 0 we select the polymer configurations ^on

the cylinder. ^In the case y ^{= x we} get ^{the unbounded} configurations ^{on the torus.}

An M-step walk w which visits boundaries T times has the weight

The weight ^{of a} configuration gn containing ⁿ M-step ^{walks is}

The partition ^{function can be written in the} following ^form

where the summation runs over all possible configurations ^{on the torus.}

According ^{to these} definitions, ^{the excess surface free} energy f, per ^one site of the

boundary with area S is defined as

The mean monomer concentration p (x ) ^{of the unbounded} system may be found by ^the partition ^{function in the} following way

where V is the lattice volume.

Analogously, ^{the mean} concentration of the polymer ^{hits on} the walls is

Therefore, expression (5) ^can be written as

Now, ^{we shall} reformulate the problem ^of determination of the number of all closed

interacting paths ^{as a more} simple ^{task on} the free random walking. ^{When d =} 2, ^{this method} gives ^{an exact} solution. For d ⁼ 3 it results in a free-fermion approximation. ^{A detailed} analysis ^{of this} approximation ^is given ⁱⁿ [9,11]. ^{Here we confine ourselves to a brief accounl}

of the method.

The ansatz consists in providing for the walks « fermionic » properties. To this end we

introduce the auxiliary ^function

which is also the weight ^{of the set} of n walks gn, but any walk from gn ^is weighted ^{with the}

« minus » sign. ^{Let P be} any nonperiodic, K-step ^{walk which} gets ^{back at the} origin point ^after

K ⁼ kM steps, where k > 1 is an integer, ^{M as} above is the size of the lattice along ^the

vertical. Let X (P) ^{be the} weight of P in the sense of equation (2) ^{where we have to} put kM instead of M. Then the following ^{formula is true :}

where the I.h.s. contains the product ^{over all} possible ^closed nonperiodic ^{walks. A} simple explanation ^for (10) ^{is as} follows. Each term in the I.h.s. of expression (10) may be associated with a path configuration ^on the lattice. For the configuration ^with intersection at any point

two representations ^are equivalent : ^{in a form of two} paths Pl ^and P2 (Fig. 3(a)) ^{which are} intersecting ^{at this} point ^{and in a form of a} single self-intersecting path (Fig. 3(b)). ^In expansion (10) the first situation is described by weight (-1 ) X (Pl ) (-1 ) X (P2 ) ^{the second}

one by ^the weight (-1 ) X (P1) X (P2) . Contributions from intersecting ^and self-intersecting paths ^are cancelled, ^and only ^{the terms} remain which correspond ^{to all terms of the sum} X, ^{X (g ).}

-

The connection between X (g ) ^and X (g ) is established as follows. In the case

d ⁼ 2 the paths ^on the two-dimensional lattice are rolled around a torus, therefore any path enveloping ^{the lattice more than once in one} direction is surely self-intersecting. Hence in the

sum g X ^{(g ) those paths} remain which envelop the lattice only ^once and, so, they all consist

Fig. ^3.

(a) ^Two intersecting paths Pl ^and P2 ; (b) ^a single self-intersecting path ^P.

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of M steps. ^{Therefore a} simple substitution of variables --+ x exp (i irIM) and ) - y exp (1 w/M) changes ^the sign in any path. ^Then £,, ; (g Xg y (g ) and in the ther-

modynamic ^{limit M -} oo, A (x, y) ^and A(£, Y) ^are coinciding. ^{In the case d =} 3 the above substitution of variables does not improve signs ^of paths ^which envelop ^{the lattice more than}

once. These paths ^{enter into the} partition function : the paths ^which envelop the lattice twice enter with sign ^{« minus} », those with a triple enveloping, ^{with the} sign ^« plus ^{», etc. It is not}

difficult to verify ^{that this} approximation ^{for the} partition ^function exactly corresponds ^{to the}

free-fermion approximation [9].

Let Kp be the number of steps along ^a path ^{P. From} equation (4) ^{with due} regard ^to (10) ^we

get ^the partition ^function of the unbounded system ^{in the form :}

where NK ( ) ^{is the number} of all closed paths ^of length K beginning ^and ending ^{at the lattice} point ^f.

Denote the weighted ^{sum over all} paths ^{on the torus which} begin ^{at the} point f and after K steps ^{end at the} point ^m by WK (f 1 m ; ^x, y ). ^{Then the} product N K(f) ^xxin (11)

is WK(l P ; ^x, x). ^{To obtain the} partition ^function A(x, y) ^{it is} enough ^to substitute in (11)

W K(f 1 f ; ^x, y ) instead of W K(f 1 f ; x, x ). ^Let fo be any point ^{of the lattice} belonging ^{to the}

wall. Then the equality

is valid.

Indeed, ^the differentiation in (12) ^with respect ^to y selects among paths ^{which enter into the} partition ^function only ^{those which at least once fall on} the wall. If the path ^{falls on the wall}

v times, ^its weight contains the factor y ". The action of the operator ya/ay results in the factor

v in this weight. Therefore, in the left-hand side of (12) every path which falls on the

boundary v ^times is taken into account v times. The factor S in the right-hand ^{side of} (12) signifies that every path ^can begin ^at any point ^{of the} boundary. Therefore the path ^{that falls}

on the boundary v times is also taken into account v times in the right-hand ^{side of} (12).

From equations (7, 8, 12) ^{we have for the excess} surface free energy

Instead of the summation over the paths ^{with a} length K(K ⁼ kM) ^{it is} conveniently ^to pass to the sum over paths ^{with an} arbitrary ^{number of} steps. ^{This can be made} by adding ^a supplementary ^factor exp (2 7Ti j / M) ^{to the} weight of every step. ^Then omitting arguments

x, y ^we may write :

using ^the identity

where m .1 is an integer. ^The unity ^{in the} right-hand ^{side of} (14) compensates ^{the term}

WO(f ^{1 f)} = 1 absent on the left-hand side. Thus the problem of the calculation of the excess

surface free energy reduces ^to finding ^the generating ^{function of} simple random walks

beginning ^and ending ^{at one} point :

Since we deal with uncorrelated random walks, the defined model represents ^a system ^of flexible polymers. Introducing ^a correlation between successive steps ^{one can} control the statistical weight ^{of trans and} gauche ^{states of} polymers. In this way, ^a model of semi-flexible directed polymers ^can be obtained.

3. Surface free energy.

The expression ^for f, ^derived in section 2 after taking ^the thermodynamic ^{limit can be written}

as follows :

We recall that the walk occurs on the one-dimensional lattice 0, 1, ..., N ^{in the two-}

dimensional model and on the two-dimensional lattice when d ⁼ 3. That is, ^{we set}

fo = 0 ^{when d = 2 and} fo ⁼ (o, 0 ) ^{when d = 3. If one is not} interested in finite volume

effects, ^{one can} neglect ^the effective interaction between two boundaries in the limit N --+ oo. Then the function W (fo f 0) ^{tends to the} generating function of random walks on the infinite lattice in the presence of the alone wall.

To obtain W (0 0 ) ^{we consider the evolution} equation ^{of functions} Wn(£ 1 m ; x, y ) :

where y (f, l ’ ) is the transition matrix from point ^{Q’ to} point f. ^It is convenient to represent

-y (f, f’) in the form :

where p (Q - Q’ ) ^{is the} transition matrix for a walk on the infinite lattice, q (f , f’ ) ^{is a matrix of}

« defects » describing ^{the walk near the wall.}

It is convenient to carry ^out further calculations for d ⁼ 2 and d ⁼ 3 independently.

a) ^{d = 2}

In tbjs ^{case matrices} p (f - Q’ ) ^and q (l, f’) ^are of the form :

where y is the weight ^{of a} lattice link over which the walk falls onto the boundary,

Li = (y - x )/x. Assuming ^that Wo(f 1 m) = 8t,m ^{from (16) we get the equation for}

W(l m) : ¹

605

Let the Fourier-transform of W(l 1 m) ^be

Then from equation (22) ^{we have}

where À (ç ) ^{is the} « structure function » of a walk on the lattice :

Passing ^to the inverse Fourier transformation in (24) gives :

Hence the searched function W(O 10) equals

where the lattice Green function G (0 ) ^{is defined} by ^the equalities

From expressions (17, ^27, 28) it follows that the surface free energy is given by the formula

with the substitutions x - x ei{3 _{and y - y} eifJ in (27, 28).

Using the Jensen formula to calculate the last integral ^we finally ^{arrive at the} simple ^result

where

The value xc ⁼ 0.5 is a « critical ^» point of the model. For X :> Xc polymers ^{arise on the} lattice, when x « xc the lattice is emply.

b) d = 3

In this case matrices p (f - Q’ ) ^and q (f, Q’ ) ^are of the form

Like in the preceding part (a) ^{we can derive the} following relation for the Fourier transforms

of W(fo ¹ fo) :

where , The inverse Fourier trans- formation in (33) ^with respect ^to cp 1 gives

where

When f ⁼ fo ⁼ (o, 0 ) ^the inverse Fourier transformation with respect ^to ’P2 in (34) gives finally ^the expression ^{for the} generating ^function W(O 10) :

For the excess surface free energy from (17, 36) ^{we have}

Using the Jensen formula to calculate the integral ⁱⁿ (37) ^we finally get

Here xc ⁼ 0.25 is a critical point of the model when d ⁼ 3.

4. Results and discussion.

To analyse ^{the obtained} expression of the surface free energy (30) ^and (38) ^{as functions of the}

monomer concentration _p, it in necessary ^to eliminate the fugacity ^x.

For this aim we shall need an expression ^for p (x ). ^This dependence ^can easily ^{be found}

with the help ^{of the} expression ^for p ^on the lattice without boundaries (y ⁼ x ) :

following ^from (6, 11) ^and (14-16).

Using expressions (27, 28) ^{we obtain the} following ^{result for d = 2 :}

Therefore from equalities (30) ^and (40) ^we find for afl 0 _pu 1 :

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The low and high density asymptotics ^are

When d ⁼ 3, ^the fugacity ^can only be eliminated from (39) ^{when p --+ 0 and p - 1 by} using

the results of work [12] :

With these expressions, the low and high density asymptotics ^{as can be seen from} (38, 44, 45) ^are given by the formulae

The latter formula implicitly ^{defines the excess surface} free energy in the concentrated- solution limit on the lattice.

In conclusion, ^{we have} proposed the lattice model of directed linear polymers ^which

contact with the nonadsorbed wall. Flexible polymer ^{chains are not} intersecting ^{with each}

other owing ^{to the} short-range repulsion. The enumeration of polymer configurations ^is

carried out by the combinatorial method which gives ^{an exact} result when d ⁼ 2. For d ⁼ 3, ^{it leads to a} free-fermion approximation. The difference of free energies for the model with nonadsorbed wall and the model without boundary ^can be related to the surface tension

cr. The analysis of obtained results for d ⁼ 2 shows that a - p 2 in the semidilute limit and a - - In (1 - p ) ^{in the} concentrated-solution limit. When d ⁼ 3 the free-fermion approxi-

mation gives the known result [1] u ’" ^p 3/2, when p is small. This dependence for the surface tension coincides with the analogous law for the semidilute solution regime ^{of an} isotropic polymer solution which is obtained from the polymer-magnetic analogy ^and scaling arguments

[1].

In spite of the coincidence of the law p 3/2 for three-dimensional isotropic and directed

polymer models, correlation properties ^{of these model are} drastically different. The effective correlation length e in the semidilute limit for directed polymers ^can be estimated from the

osmotic-pressure dependence. ^{In our} model when d ⁼ 3, it has the asymptotic [9] :

From (48) ^and (44) ^we find that for small p :

Using the known estimate for the semidilute solution [1] :

we get

Therefore effective correlations of directed linear polymers with the excluded volume are

very different from those in an isotropic system where the effective correlation length ^is

References

[1] ^{DE GENNES P.} G., Scaling Concepts ⁱⁿ Polymer Physics, (Cornell University ^{Press, Ithaca and} London) ^1979.

[2] ^{BINDER K.,} Phase Transitions and Critical Phenomena Ed. C. Domb, J. L. Lebowitz (Academic Press, London) pp. 1-144 (1983).

[3] ^{DES CLOIZEAUX J., J.} Phys. ^{France 36} (1975) ^289-291.

[4] ^PRIVMAN V., ^FORGACS G., ^{FRICH H.} L., Phys. ^{Rev. 37B} (1988) pp. 9897-9900.

[5] ^{BURKHARDT T. W.,} SCHLOTTMANN P., ^Z. Phys. ^54B (1984) ^155-158.

[6] ^{NAGLE J. F.,} Proc. R. Soc. Lond. Ser. A 337 (1974) ^569-589.

[7] ^FAN C., ^{WU F.} Y., Phys. ^{Rev. 2B} (1970) ^723-733.

[8] ^{LEE J. J., FULLER G. G., J.} Colloid Int. Sci., ¹⁰³ (1984) ^569-577.

[9] KORNILOV E. I., PRIEZZHEV V. B., ^Z. Phys. ^54B (1984) ^351-356.

[10] BHATTACHARJEE S. M., ^{NAGLE J.} F., HUSE D. A., ^{FISHER M.} E., ^{J. Stat.} Phys. ³² (1983) ^361-

374.

[11] ^{IZUYAMA T., AKUTSU Y., J.} Phys. ^Soc. Jpn. ⁵¹ (1982) ^50-58.

[12] ^{LITVIN A.} A., PRIEZZHEV V. B., ^JINR communications, ^P17-87-4 (1987).