Equilibrium polymerization in two dimensions

Volume 95A, number 2

PHYSICS LETTERS

18 April 1983

EQUILIBRIUM POLYMERIZATION IN TWO DIMENSIONS Marko V. JARIC 1 and K.H. BENNEMANN Institute for Theoretical Physics, Freie Universitdt Berlin, D-1000 Berlin 33, West Germany Received 18 January 1983

A mean-field theory of equilibrium polymerization on a square lattice is presented. A novel feature of the model is the treatment of the whole range of rigidity of polymers (from flexible to stiff) as well as a simultaneous account of both polymer rings and excluded volume. As a result a line of tricritical points is found.

The general features of equilibrium polymerization have recently been studied in several models. Most attention has been devoted to polymerization of flexible polymers [ 1,2] for which it was found that the criticality associated with polymerization is the same as for an n-vector model: an exclusion of ring polymers led to n = 0 [ 1 ] while a suppression of the excluded volume led to n = 1 [2]. The opposite extreme of stiff polymers on a square lattice has also been investigated [3] with an emphasis on the general features of the phase diagram rather than on the nature of criticalities. In this paper a mean-field calculation which encompasses the whole range between flexible and stiff polymers on a square lattice is presented. The model also includes both formation of ring polymers and the excluded-volume effect. Although the model represents a simplification of reality it gives some insight into peculiarities of polymerization in two dimensions. Its purpose is not to give a correct description of polymerization but merely to show a possibility of ordering polymerization in two dimensions. Furthermore, the model, which is a type o f classical "lattice gas", is amendable to investigation by other techniques and there is a possibility that the mean-field results presented below can be considerably refined and improved. We consider a square lattice. At each site of the lattice we place either an inactive monomer or an 1 Present address: Department of Physics, Montana State University, Bozeman, MT 59717, USA.

activated one. Due to bifunctionality an activated monomer has two "arms" which extend from the site half-way along different lattice bonds. Activation energy is denoted by e > 0 (all energies will include the Boltzmann factor 1/kT). Out o f the six configurations of an activated monomer the four with arms forming an angle of 90 ° are assigned an additional bending energy 5 (stiff, flexible and coiled polymers correspond to 5 = 0% 0 and _0% respectively). Since activated monomers have "sticky", attractive ends, there is a release o f an energy v > O, whenever two arms of two different monomers meet at a lattice bond. The hamiltonian for this seven-state model with anisotropic nearest-neighbour interactions can be written in a straightforward fashion

H= ~ e(a)n~(R) R,a

_ 1 ~ v(e;ct, fl)na(R)n~(R + e ) , (1) 2R,e,a,fl where na(R) is the occupation number for the monomer state a at the lattice site R, e(a) is the energy o f the state a (i.e. zero for an inactive monomer, e for a straight monomer and e + 5 for a bent monomer), v(e, or,fl) is the interaction energy between two monomers, one in the state a and the other, a nearest neighbour at the relative lattice site e, in the state/3 (i.e. zero unless ct and/3 are activated m o n o m e r states with arms meeting along e in which case the energy is v).

0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland

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The ground states for the hamiltonian (1) are the following. For/5 > 0 and e < v the ground state is a doublet: infinite (endless)parallel polymers either along one or the other axis o f the square lattice. For (5 > 0 and e = v the ground state is infinitely degenerate: any configuration of endless straight polymers and inactive monomers. For/5 < 0 and e < v the ground state is infirfitely degenerate: any configuration o f endless (infinite or ring) polymers with no straight portions (coiled polymers). For 15 < 0 and e = v the ground state is infinitely degenerate: any configuration of endless coiled polymers and inactive monomers. For/5 = 0 and e < v the ground state is infinitely degenerate: any configuration o f endless polymers. For/5 = 0 and e = v the ground state is infinitely degenerate: any configuration of endless polymers and inactive monomers. For all/5 and e > v the ground state is a singlet: all monomers are inactive. A mean-field approximation can be formulated starting from ground states for e < o in a standard way. Choosing a particular ground-state configuration, we denote by A 0 the concentration of inactive monomers, by A 1 the concentration o f (activated) monomers which are in the same configuration as in the ground state, by A 2 the concentration o f activated monomers with precisely one arm in the ground-state configuration, and by A 3 the concentration of activated monomers with no arms in the ground-state configuration. In the ground state A 1 = 1, A 0 = A 2 = A 3 = 0. The mean-field (variational) [4] free energy is then 3 ¢ ( A ) = A o l n A 0+i=~l.= AilnA i + A ' E - ~ A . V . A ,

(2)

where 3

A = [A1,A2,A3] ,

A O= l - ~ Ai, i=1

E = [e', e ' - ln(~ - 2 ) , e'] (e' = e and K = 2 + 4 exp(-/5) for/5 > 0, e' = e +/5 and K = 4 + 2 exp(/5) for 6 < 0), and

V=v

1 1

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.

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18 April 1983

The free energy, eq. (2), which should be minhnized with respect to A, leads immediately to the concentrations of all activated monomers Q = Oq~/ae, of open polymers nop = aq~/ae + O¢/ao, and o f bends n b = o¢/~8.

Although we started from a particular ground state for e < v the result also holds for e > v, where we find the solution A 1 = A 3 = A 2/(K - 2). The critical surface where this symmetric solution bifurcates (.41 4 : A 3 ) and polymerization occurs is found to be ,1 ffc = ½exp (ec),

(4)

which is written in a reduced form, independent of •, t by introducing °c = Uc - K/2 and ec = ec - K/2. Although M = A 1 - A 3 changes at the critical surface like a mean-field order parameter, the polymer weight fraction Q shows only a finite discontinuity in the slope and it is non-zero at both sides o f the transition. Along the critical line (for constant K) Q decreases as v c is increased, Q = K/2v c. However, this behavior cannot persist to v c ~ co. There, we expect instead to approach the ground-state boundary v = e, at which a first-order transition occurs. Secondly, when polymerization develops Q must be large which also indicates that for large v the continuous transition must change into the first-order one. This happens at a line of tricritical points given by ~t = ( 6 -

K)/(8 -

K),

together with eq. (4). fit is non-negative for physically accessible K (K E [2, 6] for (5 > 0, K @ [4, 6],/5 < 0). For flexible polymers,/5 = 0 and K = 6, fit = 0 and the continuous transition is completely suppressed * 2 The first-order transition line (for a given K) meets the critical line smoothly at v t. However, the first-order line must be calculated numerically. The resulting phase diagrams are shown in figs. 1 - 3 . Fig. l shows the phase diagram for the reduced variables and ~"for all ~ between 2 (upper boundary of the shaded region) and 6 (lower boundary of the shaded region). The phase boundaries for different K are too close to be distinguished and we only indicate tricritical points for K = 2, 4 and 5. In figs. 2 and 3 we show ,1 The analysis is analogous to that of the s = 3/2 lsing model for ternary fluid mixtures [5]. ,2 Tetracritical points do not occur for physically accessible K.

Volume 95A, number 2

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2.0

I

18 April 1983

0.5 - 4- . . . . .

I

I. . . . .

--1--____ K=4.0 ~'~%

x=5.0 l/V

1.0

.........

"-,

x--6.0

025 K:2.0 K=4.0 ffff

0.0

_

- 8.q

_

K

0.0

J 0.0

-z,.0

4.0

Fig. 1. Phase diagram in terms of reduced variables ~ and ~. Dashed line corresponds to the critical line. The first-order transition lines join the critical line at tricritical points indicated for K = 2 , 4 and 5. The first-order transition lines for different n go through the shaded region. The upper boundary corresponds to K = 2 (stiff polymers) whereas the lower boundary corresponds to ~ = 6 (flexible polymers).

1.o

4-

I

I

--

"'- ~ , ~ . K= 2.0

\ \ \ \

1/V

% %

K= t,.0

0.5 \

K=6.0

0.0

-5.0

I

I

-2.5

0.0 E/V

Fig. 2. Phase diagram for polymerization with stiff polymer ground state (8 > 0) for K = 2, 4 and 6. The dashed line is the critical line which is joined at the tricritical point by the first-order transition line (full line).

I

I

-1.5

0.0

1.5

3.0

E/V Fig. 3. Phase diagram for polymerization with coiled polymer ground state (6 < 0) for K = 4, 5 and 6. The dashed line is the critical line which is joined at the tricritical point by the firstorder transition line (full line).

the phase diagrams for 5 ~> 0 (K = 2, 4 and 6) and 6 ~< 0 (K = 4, 5 and 6), respectively. Here, we choose more conventional variables 1 / v and e/u (if e and u were microscopic variables 1/u would be proportional to the temperature and e/u would be approximately constant). However, it must be emphasized that due to the simplicity of the model k T e and k T o should be considered as temperature-dependent enthalpy changes (e.g. A H e - T A S e and A H o - T A S o ) . The above results indicate that an account of polymer rigidity as well as of ring polymers may lead to new phenomena on a square lattice. For example, we find that polymerization is associated, for 6 > 0, with an order-disorder transition. For flexible polymers (6 ~< 0) we also find polymerization to occur along a line rather than at a single point as o -+ ~ [ 1,2]. However, our calculation is in agreement with ref. [2] which finds Q v~ 0 on both sides of the transition with an energy-like singularity at the (continuous) transition. The mean-field approximation employed here must be critically evaluated. A problem of general nature, that a mean-field underestimates effects of fluctuations, is particularly emphasized here since for 6 ~< 0, e < u the ground state is infinitely degenerate. In our calculation we start from a particular ground state. However, if we replace at each site an activated 129

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monomer with one having no arms in common with the original monomer we also obtain a ground state. This corresponds to an interchange between A 1 and A 3 under which the free energy (2) is symmetric. These two ground states are the only two ground states for 15 > 0 (e < o) and M = A 1 - A 3 is a real order parameter. On the other hand, for 6 ~< 0 (e < u), although we consider all of the ground states, we take them only two at a time and the order parameter M loses its meaning. Furthermore, for all 5 and e = v not all the ground states are taken into consideration. Therefore, we expect that the mean-field results are valid for sufficiently large 6 > 0 and sufficiently small e < u. A second objection to the mean-field treatment is that it effectively makes the interactions isotropic, whereas anisotropy seems to be essential to polymerization. A more detailed answer to the above questions is currently sought by both exact techniques and by extending the present calculation for flexible

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polymers on a square lattice to a lattice with arbitrary coordination number. MVJ acknowledges several illuminating discussions with Professor D. Mukamel and financial support and hospitality from the Weizmann Institute of Science. We acknowledge partial support from the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 161.

References [ 1] J.C. Wheeler, S.J. Kennedy and P. Pfeuty, Phys. Rev. Lett. 45 (1980) 1748; J.C. Wheeler and P. Pfeuty, Phys. Rev. A23 (1981) 1531 ; Phys. Rev. Lett. 46 (1981) 1409. [2] R. Cordery, Phys. Rev. Left. 47 (1981) 457. [3] M.V. Jarid and K.H. Bennemann, Phys. Rev. A27 (1983), to be published; Lect. Notes Phys. 172 (1982) 250. [4] H.Falk, Am. J. Phys. 38 (1970) 858. [5] S. Krinsky and D. Mukamel, Phys. Rev. Bll (1975) 399.