Collapse of a polymer : evidence for tricritical behaviour in two dimensions

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Collapse of a polymer : evidence for tricritical behaviour in two dimensions

A. Baumgärtner

To cite this version:

A. Baumgärtner. Collapse of a polymer : evidence for tricritical behaviour in two dimensions. Journal

de Physique, 1982, 43 (9), pp.1407-1411. �10.1051/jphys:019820043090140700�. �jpa-00209521�

Collapse ^{of a} polymer : evidence for tricritical behaviour in two dimensions

A. Baumgärtner (*)

IBM Research Laboratory, ^San Jose, California 95193, ^U.S.A.

(Reçu ^{le ler}

^mars

1982, révisé le 9 mai, accepté ^{le 3 mai} 1982)

Résumé.

²⁰¹⁴

Le paramètre ^{d’ordre et} la chaleur spécifique ^des polymères

^{sur un}

réseau carré sont calculés par

une

méthode de Monte Carlo et analysés

^en

^{termes de} loi d’échelle de taille finie. Près de la transition

vers un

état

globulaire, ^ils présentent ^les caractéristiques ^d’un point tricritique. ^Les exposants critiques correspondants 03B1t, 03C5t et 03A6t, ^{déduits de cette} analyse ^sont

^en

bon accord

avec

les prévisions ^obtenues d’après ^le développement

^en

^série

de

03B5.

Des corrections du 1er ordre sont calculées pour le comportement asymptotique.

Abstract.

²⁰¹⁴

Finite-size scaling of Monte Carlo generated ^order parameter (intrinsic density ^of monomers) ^and specific ^heat of square lattice polymers

^near

^the collapse transition exhibit characteristics of

a

tricritical point.

The estimated corresponding ^critical exponents 03B1t, 03BDt and 03A6t

^are

ⁱⁿ good agreement ^with predictions from 03B5-expan- sions. First-order corrections to the asymptotic ^behaviour

^are

calculated.

Classification

Physics ^Abstracts

61.40K - 64.70

^-

68.60

1. Introduction.

^-

A long-standing problem ^{in the} theory ^of polymer ^{chains in solution} is the effect of intramolecular forces on the shape ^{and size of an}

isolated chain. The intramolecular forces are usually

assumed to be of van der Waals’ type, consisting ^of strong, short-range repulsive ^and weak, long-range

attractive interactions. By ^suitable changes ⁱⁿ tempe-

rature or in solvent composition, ^{the chain} crosses over

from an extended coil to a collapsed ^dense globule. ^This phenomena has attracted a great ^{deal of} attention, ^both

theoretical [1-20] ^and experimental [21-25] possibly

because of its connection with protein folding [26, 27],

and it is also of theoretical interest as a first step towards the understanding of micellar structures.

Our present understanding ^{of the} collapse ^transition

within the context of critical phenomena ^{is still} incomplete ^and controversial. Both analytical [1-14]

and numerical attempts [15-20] ^{have been made ‘in}

order to elucidate this problem.

Recently ^{de Gennes} proposed [4] ^{that the} collapse

transition should exhibit characteristics of a tricritical

point [28]. Although ^{several works} [5,.6, 8,13,14] ^have

been devoted to elaborate the theory, ^{no evidence for} tricriticality ^{of the} collapse has been demonstrated so

far, ^neither by experiment ^nor by computer simulation.

In order to verify ^the theory, ^{one is} confronted with

an important difficulty : ^{at the} marginal dimensiona-

lity ^d

⁼

3, ^{the critical} properties of the transition are

governed by logarithmic singularities, ^{which are hard}

to analyse conclusively. A detailed Monte Carlo

investigation ^{for d}

⁼

^{3 will be} published [29].

In order to avoid this difficulty ^{and to extend our} understanding ^{of the} collapse transition to two-dimen- sional systems, ^the present ^work reports ^{on a finite-} size scaling analysis of Monte Carlo generated ^thermo- dynamic functions of square-lattice polymers.

Experimentally the two-dimensional collapse ^can

be realized if the polymer ^is strongly ^{adsorbed on a flat}

surface [30] ^or if the solution is confined in a slit.

2. Model and simulation technique.

^-

^{The chain is} represented by ^a random walk (of ^N steps), ^{which at}

each step ^{moves from one} point of the square lattice

to any of the four nearest-neighbour sites. The effect of the excluded volume is represented by preventing ^the

walk from visiting any lattice point ^{more than once.}

The length ^{of one} step ^{will be called}

^a.

Following

Orr [31], the mutual interactions ^of polymer ^and

solvent can be included in the above model by ^intro- ducing

^a

^Boltzmann factor, exp(- slkb T), for every

nearest-neighbour ^contact of the chain with itself. The energy 8 ^measures the _energy of

a

polymer-polymer

(*) ^Permanent Address : Institut fiir Festkorperforschung

der Kernforschungsanlage Jflich, ^Postfach 1913, ^D-5170 Jflich, ^West Germany.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019820043090140700

1408

contact relative to the energies ^of polymer-solvent ^and

solvent-solvent contacts. Thus,

⁸

^{0 describes}

^a

polymer ⁱⁿ

^a

poor solvent, ^{which means that} polymer- polymer ^{contacts are} favoured, leading ^to

^a

collapse

in the chain dimensions at low temperatures.

The total interaction _energy, E(X)

⁼

El (X) ⁺ E2(X), ^for

^a

^chain configuration ^{X is thus} given by E I (X)

⁼

s tj (X), ^where q(X) is the number of nearest-

neighbour corltacts, ^and E2(X) ^{= oo, if any lattice site}

is occupied twice. An ensemble of configurations ^dis-

tributed according ^to the canonical distribution

,P(X) ^{oc exp( - E(X)/kB T) was} generated by

^{a «}

rep- tation » Monte Carlo dynamics [32]. Starting ^from

^a

state X, ^{there one} first selects one of the ends of the chain at random and then removes this end link of the chain and adds it at the other end, specifying ^ran- domly the orientation of the link. The resulting ^state

X’ is accepted ^{as a new} configuration, ^if according ^to

the Metropolis ^criterion [33]

where ( ^{is a} random number 0 C 1; ^otherwise

state X’ is rejected and the old configuration ^{X is}

counted once

more

in the averaging. ^This mechanism,

which corresponds ^{to a movement} of the chain along itself, produces ^an approach ^towards equilibrium.

Denoting ^ensemble averages by angular brackets,

energy U, specific heat C and intrinsic density p are then obtained by

where

is the mean-square radius of gyration, ^and rk ^are the lattice sites occupied by the chain. The unperturbed (e

⁼

0) ^{chain dimension for} N >> ^{1 is} given by [34]

3. Numerical results.

^-

The temperature ^variation of intrinsic density, internal energy and specific ^{heat is}

shown in figure ¹ for the entire range of chains studied.

Identifying kB T,(N)IB ^{with the} positions ^{of the} spe- cific-heat maxima, ^{we examine the size} dependence ^of

the transition temperature T, ⁱⁿ figure ^{2. The} asymp- totic behaviour seems described by

where kB TB(oo)/s

⁼

1.31 ± 0.06 and 0, ^{the crossover}

Fig. ^1.

^-

Raw data of monomer-density p (upper part),

internal energy U (middle part) ^and specific ^{heat C} (lower part) plotted

^versus

temperature ^{for chain} lengths ^N

⁼

20, 40, 80, ^{160. The} suggested asymptotic (N - oo ) ^behaviour

is indicated qualitatively ^for p and U.

Fig. ^2.

^-

Variation of the critical temperature T ^{with chain} length ^N.

exponent described below. It should be noted that the

reliability ^{of the} present Monte Carlo calculations at low temperatures ^is supported by ^{that fact, that our}

estimates of internal energies ^{seem to coincide} (by extrapolating ^{the data to T -} 0) ^{with the} ground-

state energies, ^{which are indicated} by ^{short horizontal}

lines in the middle part ^of figure ^1. Obviously ^the ground-state ^{for an} infinite chain is highly degenerated,

but for finite N, ^we found that the

«

spiral » configu-

ration has the highest ^{number of} contacts il :

where I

4. Finite-size scaling analysis.

^-

^The expected ^ther- modynamic ^{behaviour of} finite systems ^of interacting particles ^{has been discussed} by ^Fisher [35] ^{in terms of a} scaling theory involving the critical exponents ^{of the} corresponding infinite system. ^An assumption ^of generalized scaling ^at

^a

tricritical point [36, 37] ^{for an}

isolated polymer [4] then leads to a density ^{of the form}

where vt ^and Ot ^are the correlation length exponent ^at the transition temperature T, (-r =- T - Tt I/Tt) ^and

the crossover exponent, respectively. ^{For the consi-}

dered case d

⁼

2, approximate ^values of vt ^and Ot

have been calculated using s-expansion techniques [6, 38], ^{carried to order S2 and} a, respectively :

The scaling ^function /(x) ^should asymptotically reproduce ^for large

^x

^{(i.e., T 1, but N -} 00) ^the

infinite-chain critical behaviour [4]

where

v

and _vc are the well-known correlation length exponents ^{in the} coil and in the globule ^state, respecti- vely :

Thus one obtains finally ^the asymptotic behaviour :

where B t, ^{B I are} the critical (nonuniversal) ampli-

tudes for an infmite chain. A similar scaling ^ansatz

for the specific ^{heat is} given by

where the tricritical exponent a, has been calculated

[38] ^to first order in

c ⁼

3 - d

The scaling ^function g(x) asymptotically yields ^the

infinite-chain critical behaviour, g(x) -+ A t x-11,

Using ^{the data} presented ⁱⁿ figure 1, ^{we can now test} the finite-size scaling ^relations equations (4)-(11).

In figure ^{3 we} examine the chain length dependence

of _p for various temperatures. ^Well above and below the critical temperature kB T,/e -- 1.31, ^{the data are}

in good agreement ^with p - fi (Eq. (8a)) ^and

p

⁼

^const (Eq. (8c)), respectively. ^{In order to distin-} guish properly between the behaviour just ^{at the critical} point p ’" N - 0.011 (Eq. (8b») and in the globule phase

p

⁼

const, it would be _necessary to extend our investi-

gations ^{to much} larger ^values of N, ^{which is} reasonably unapproachable ^{with the} present generation ^of compu- ters.

Fig. ^3.

^-

Variation of the monomer-density p with chain

length ^N for various temperatures.

However, ^we adopt ^the given ^value (5a) of v, - ^{0.505 5}

in order to examine the scaling ^form of p (Eq. (4)) ; ^this

shown in figure ^{4. In} good agreement ^{with the} scaling

ansatz, ^the scaled data all lie on two smooth curves, for T > Tt ^{and T} T t, respectively. ^{The two solid}

lines correspond ^{to the} asymptotic ^forms given by equations (6)-(8) ^{with B + =-} 2.4, B - - 1.2 and the

predicted values., p + -- - 0. 7 5, p - - ^0.031. However,

Fig. ^4.

^-

Finite-size scaling plot ^{of the}

^monomer

density according ^to equation (4) for various N at temperatures

T

>

7t (lower part) ^{and T} T, (upper part).

1410

from the (N It i)

^-

region ^covered by the Monte Carlo

data, it is rather suggestive ^{to infer an} asymptotic approach of the data to their solid lines. The reason

behind this slow convergence is that corrections to the

asymptotic ^form (Eq. (6)) ^{for x 10 are} important.

This is shown in figure ^{5. Now our data are well-}

described for

x

> 1 by

with B’ - - 2.4, B- ^{0.8 and} JJi - ^1.4, JJl - 0.1 (compare also table I). ^In figure ^5, Asp ^is

defined as Asp ^N2Vt-l

⁼

^{Bt xu’ -} p(x) ^{N2vt-1. The}

uncertainties in the critical amplitudes and critical exponents ^{are about} 30% ^and 10%, respectively.

Analytical ^{work is} required ^to check the accuracy of

Bi ^and pl.

Table I.

^-

Estimates of critical exponents and critical

amplitudes of the square-lattice polymer ^chain according

to equations (12) ^and (13).

Fig. ^5.

^-

Finite-size scaling plot for the correction to the

asymptotic behaviour of the monomer-density according ^to (12) for various N at temperatures ^T

>

T, (upper part) ^and

T T, (lower part).

Following ^{the above} procedure, ^the scaling analysis

of the specific ^{heat is} presented ⁱⁿ figure ^{6 and} figure ^7.

Since the specific ^heat diverges ^so weakly (see Fig. 1),

the divergent portion ^{does not} completely ^dominate

the

«

background » except ^{very close to} T,. Analog ^to

Fig. ^6.

^-

Finite-size scaling plot ^{for the} singular part ^{of the} specific ^heat according ^to (9) for various N at temperatures T

>

T, (upper part) ^{and T} T, (lower part), ^{and for}

various

^«

background

^»

portions ^bt.

Fig. ^7.

^-

Finite-size scaling plot ^{for the} correction to the

asymptotic ^{behaviour of the} specific ^heat according ^{to (13)}

for various N at temperatures ^T

>

T, (upper part) ^and

T T, (lower part).

numerical investigations ^on Ising models, [39, 40] ^we approximate ^the nonsingular part by ^{the constants} b’. As shown in figure 6, ^we find that the singular portion ^{of the} high-temperature specific ^heat (C ⁺ b+ )

scales quite ^well according ^to equation (9) using

b+ -- 0.05. For T Tt, however, scaling ^{is not} quite convincing ^{for x 8.} The solid lines in figure ^{6 corres-} pond ^{to the} predicted asymptotic power law (11).

Similar to the density ^discussed above, corrections to the asymptotic ^form (11) ^{of the} specific ^{heat are} impor-

tant, which is demonstrated in figure ^{7. The data are in} good agreement ^for

^x

> 1 with

The amplitudes ^and exponents ^are summarized in table I. AC is defined as åC N -rztQ>t

₌

A * x-at -

(C ⁺ b+)N-’t"t. Again, ^the low-temperature specific

heat do not scale accurately : ^the large-N ^{data lie} systematically above the small-N data. This behaviour does not seriously ^{affect the} limiting slope a, - - 1.8;

i.e., ^{at each chain} length ^{N, AC exhibit} approximately

the slope a 1. A probable ^reason of the deviations from

scaling is the assumed temperature independence ^of

the nonsingular part bl ^{of the} specific heat, ^which possibly ^{becomes a bad} approximation ^{at low} tempe-

ratures.

5. Summary and conclusion.

^-

By ^the present ^calcu- lations for isolated square-lattice polymer chains, ^it

was

shown using finite-size scaling analysis ^{that the} collapse transitions of this polymer ^{model is related}

to a tricritical point.

The variation of the critical temperature ^with N, T,(oo) - T.(N) ^oc 11NOt, ^is examined and yields kB T,(oo)/&

⁼

1.31 ± 0.06.

The estimated critical exponents ^{for the} asymptotic

behaviour (N - oo ) ^of monomer-density ^and specific

heat are in good ^{agreement with} previous ^estimates

from

s ⁼

expansions. ^{In addition, we} calculated the

corresponding ^critical amplitudes.

We have also calculated amplitudes ^and exponents of the first-order corrections to the asymptotic ^critical

behaviour. The

corrections ^{become important if the}

scaling ^variable N ot t i ^10.

We conclude that the finite-size behaviour of the two-dimensional collapse transition can be well understood within the context of critical phenomena.

We suggest ^{that some of the above} predictions ^could

be tested experimentally by investigations ^{of dilute} polymer ^{solutions confined in a slit.}

Acknowledgments.

^-

^{The author is} grateful ^{to the}

IBM San Jose Research Laboratory ^for hospitality

where the present work has been completed.

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