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A direct renormalization group study of loops in polymers
Volume 92A, number 7
PHYSICS LETTERS
29 November 1982
A DIRECT RENORMALIZATION GROUP STUDY OF LOOPS IN POLYMERS Fereydoon FAMILY Physics Department, Emory University, Atlanta, GA 30322, USA Received 1 June 1982
A direct position space renormalization group approach is developed and used to study the effects of loops on the universal properties of models of ring polymers, linear polymers, branched polymers and gels. It is found that the presence or absence of loops does not change the critical exponents and other universal properties of the models studied.
1. Introduction. Recently there has been considerable interest in theoretical studies of polymers with loops [1—3]in relation to the circular structure of natural macromolecules [41(supercoiled DNA), and the fundamental question of the effects of topological constraints on conformation of polymers [5,6]. Furthermore, the close connection between statistics of polymers and phase transitions [7] has made it possible to apply the renormalization group (RG) technique to polymers [7]. However, while the direct position space RG (PSRG) has been widely applied to various models of linear [8] and branched polymers [9], most of the studies of loops in polymers have been limited to either numerical studies [1,2,5,10] or field-theory techniques [3,11]. The aim of the present work is to develop and apply a direct PSRG to a model of ring polymers and to study the effects of loops on the conformation of models of linear polymers, branched polymers, and gels. The main advantage of the direct PSRG over numerical methods is that in contrast to numerical evidence the RG flows explicitly demonstrate the universality of two models if they are described by the same fixed point [8]. The main advantage of the direct PSRG over c-expansions and field-theory techniques is that it is not only mathematically simple and physically much more intuitive, its results are valid in realistic dimensions (d 2,3, instead of 4 e or 8 e) [3,11]. A direct PSRG for polymers which I call the cell renormalization (CR) method has already been in—
—
—
—
—
troduced in ref. [9], where the reader is referred to for details. In the CR method [9], the lattice is first divided into cells, and polymer configurations are represented by clusters of bonds (and/or sites) on the cell. Each cluster is weighted by K’1, where K is a fugacity for each bond in the cluster. The recursion relation is determined by mapping all configurations which enter a given cell from the origin and extend across the cell to a single bond with a fugacity K’ on a new lattice whose lattice spacing has been rescaled by b. The critical parameters and exponents are obtained from this recursion relation using standard techniques [9]. 2. Ring polymers and linear polymers. Let us dea single parameter CR approach for ring polymers. Consider a linear chain which is represented by a self-avoiding walk (SAW) on a lattice that starts from the origin of a cell and extends across it either horizontally or vertically. If the chain does not return to the cell to self-intersect, it is renormalized to a single step, in the direction in which it extends across the cell. However, if the chain extends across the cell in one direction and returns to the cell from another direction and self-intersects at the origin, then the configuration is a ring polymer. To renormalize these configurations, I map them into two bonds; one bond in the direction of outgoing chain and one bond in the direction of incoming chain. On a cell of side b = 2 on a square lattice the recursion relation is K’2 =K4 + 2K5 +K6 (1) velop
—
—
—
0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
341
Volume 92A, number 7
PHYSICS LETTERS
This recursion relation has a fixed point at K* = 0.62 and gives a correlation length exponent v = 0.79. For b = 3, v = 0.78 and a cell-to-cell transformation [9] from 3 -÷ 2 gives v = 0.75 in remarkably close agreement with the Flory estimate of 0.75 for linear chains in d = 2 [7,8]. Even though this numerical evidence is interesting, it is not sufficient to ensure us that linear polymers and ring polymers have the same exponent v. In order to study the effects of ring formation on a linear chain I develop a two-parameter RG of the form K’ = F1 (r, K) and K ‘r’ = F2(r, K) where K is the fugacity for each step of the linear chain that starts from the origin and gets across the cell horizontally (vertically), and r is the fugacity for each bond in the chainbyjoining which returns the step cell and finally forms ring to thetofirst at the origin. The aflow diagrams for both d = 2, 3 are similar and the case d = 3 on a simple cubic lattice is shown in fig.1. Note that for all r the flow is to the linear chain fixed point at r 0, K = K ~ and therefore ring polymers and linear polymers are described by the same critical fixed point and therefore belong to the same universality class. This result is not entirely unexpected, because in the scaling limit where the number of statistical units in a polymer tends to infinity [7], whether a linear polymer has open ends or a single point of contact should not change its global universal properties.
\ \\
~
~
~.
\ *
\.~
~,
--~~----
~ o r Fig. 1. Flow diagram showing that ring polymers (r > 0) are in the same universality class as Imear polymers (r = 0) because for all r, the flow is to the linear polymer fixed point at r = 0,K = K*. Solid dots are trivial fixed points,
342
29 November 1982
3. Linear polymers with and without loops. The SAW model of linear polymers is a random walk which is constrained never to visit a site or a bond of a lattice more than once [7]. Under these conditions the chain cannot fold back on itself or form closed loops. The question is: which of these two conditions produces the excluded volume effect? First, let us consider SAWs which cannot fold back on themselves by visiting a bond more than once, but can form closed loops by visiting a site twice. There are no such configurations on a 2 X 2 cell on a square lattice. On a 3 X 3 cell the recursion relation for SAWs with loops is 3 + 3K4 + 9K5 + 5K6 + 9K7 + 4K8 K + = 8K9 K + 2K10 (2) ,
which has a fixed point at K* = 0.45 and gives v = 0.72. Now, let us allow the chain to fold on itself by visiting some bonds twice; the recursion relation on a 2 X 2 cell on the square lattice is K’ = K2 + 2K3 + 3K4 + 4K5 + 6K6 + 8K7 + 5K8 (3 This recursion relation has a fixed point at K * = 0.38 and gives v = 0.59. The remarkable difference between the values of the two exponents indicates that allowing for loops does not change the excluded volume effect and the chain is expanded, but by folding the chain its dimension is considerably reduced to its value at the 0-point [13] where two.body interactions vanish. Similar results are obtained in d = 3 where by allowing loops on a simple cubic lattice, ~ retains its excluded volume value (~0.59[12,141), whereas by allowing foldings it decreases to 0.5; its 0-point value ind=3 [7]. To show that linear chains with and without loops are in the same universality class I construct a twoparameter RG. As before,K is the fugacity for each step of the chain, and a second fugacity 1 is chosen to be the fugacity for a 1oop. The renormalized weight 1’ for loops contains only the weights of those configurations that span the cell and have a loop. The two coupled recursion relations for K’ and 1’ may be solved for the fixed points and the global RG flows. The results ford = 2, 3 on the square and simple cubic lat.
.
tices, respectively, look qualitatively the same; the case ofd = 3 is shown in fig. 2. There is only one crit-
Volume 92A, number 7
PHYSICS LETTERS ~
tion [16—18](the q = 1 limit of the q.state Potts model) and I am not concerned about the statistics of a correlated percolation model in which loops are not allowed. Such a model, commonly referred to as tree percolation [19], belongs to a different universality class than percolation [191 (q = 0 limit of the q-state Potts model) and is known to have different critical exponents [19]. The CR study of loops in random animals and percolation clusters follows the same line of reasoning as
~
>,,
_______ __
~ i~•’:i i:_ ~
~
for linearthe chains: theofrenormalized loop fugacity 1’ contains weight only those spanning configurations which have a loop. Thus, each cluster of size N
_~.—‘
.~—
29 November 1982
withL loops contributes a weight KN1L to 1’. Renormalization of bond fugacityK follows the usual pro-
~—
Loop fugacity 2 Fig. 2. Flow diagram showing that polymers with loops (1> 0) are in the same universality class as polymers without loops (I = 0), because for all 1, the flow is to the! = 0, K = K* fixed point. Solid dots are trivial fixed points,
ical fixed point at 1= 0 and for all 1 the flow is to this fixed point. Thus, loops are irrelevant, v is unchanged, and SAW with and without 1oops are in the same universality class.
cedure [9,12,15,17]. In this way I obtain two-coupled recursion relations for 1’ and K’ which I solve for the fixed points and the RG flow diagram. The resulting flow diagram in both d = 2, 3 and for random animals and percolation clusters are similar to fig. 2. In all cases I have found only a single critical fixed point at 1= 0 and K = K ~ corresponding to clusters without loops (self-avoiding trees embedded on a lattice [9]). This fixed point controls the entire critical behavior for clusters with loops. Thus, in random animals and percolation clusters ioop fugacity is a non-critical parameter (i.e., it only changes the magnitude of the excluded volume and not the exponents), and clusters with and without loops are in the same universality class. The universality of random animals with and without loops is in agreement with the same conclusion reached by Lubensky and Isaacson [11] who applied a field-theory formalism to study the random animal model in d = 8 c dimensions.
4. Branched polymers and gels with and without loops. As in refs. [7,15], I use random animals as a model for randomly branched polymers with 1oops and self-avoiding trees as a model for branched polymers without loops in the dilute limit in good solvents, I use random percolation as a model for gelation[7,16]. In both cases I am interested in the scaling limit where the polymer size N (i.e., the number ofbonds in the cluster) becomes very large. The problem we are concerned with is whether the critical exponents, such as I would like to thank members of the Center for the exponent v for the cluster radius R (which is dePolymer Studies, Boston University, for discussions fined byR ~Nv, and should not be confused with and comments. the exponent v~which is defined by ~ (p Pc)~~)’ are the same for clusters with and without loops or References they are different. Since for i’
-‘
—
—~
343
Volume 92A, number 7
PHYSICS LETTERS
[5] J. des Cloizeaux and M.L. Mehta, J. de Phys. 40 (1979) 665. [6] J. des Cloizeaux, J. de Phys. Lett. 42(1981) L-433. [7] P.G. de Gennes, Scaling concepts in polymer physics (Cornell Univ. Press, Ithaca, NY, 1979). [8] For a review of the applications of PSRG to polymers see: H.E. Stanley, PJ. Reynolds, S. Redner and F. Family, in: Real-space renormalization, eds. T. Burkhardt and J.M.J. van Leeuwen (Springer, Berlin, 1982) Ch. 7, p. 171, and references therein. [9] F. Family,J. Phys. Al 3 (1980) L325. [10] D.C. Rapaport, 1. Phys. A8 (1975) 1328, and references therein. [11] T.C. Lubensky and J. Isaacson, Phys. Rev. A20 (1979) 2130.
344
29 November 1982
[12] F. Family, J. de Phys. 42(1981)189. [13] J.A. Marqusee and J.M. Deutch, J. Chem. Phys. 75 (1981) 5179. [14] See, e.g., A. Baumgartner, J. Phys. A13 (1980) L39; K. Kremer, A. Baumgartner and K. Binder, Z. Phys. B40 (1981) 331. [15] F. Family and A. Coniglio, J. Phys. A13 (1980) L403. [161 D. Stauffer, Physica 106A (1981) 177. [17] F. Family and P.J. Reynolds, Z. Phys. B45 (1981) 123. [18] D. Stauffer, in: Intern. Conf. on Disordered systems and localization, ed. C. Di Castro (Springer, Berlin, 1982), to be published. [19] MJ. Stephen, Phys. Lett. 56A (1976) 149.
PHYSICS LETTERS
29 November 1982
A DIRECT RENORMALIZATION GROUP STUDY OF LOOPS IN POLYMERS Fereydoon FAMILY Physics Department, Emory University, Atlanta, GA 30322, USA Received 1 June 1982
A direct position space renormalization group approach is developed and used to study the effects of loops on the universal properties of models of ring polymers, linear polymers, branched polymers and gels. It is found that the presence or absence of loops does not change the critical exponents and other universal properties of the models studied.
1. Introduction. Recently there has been considerable interest in theoretical studies of polymers with loops [1—3]in relation to the circular structure of natural macromolecules [41(supercoiled DNA), and the fundamental question of the effects of topological constraints on conformation of polymers [5,6]. Furthermore, the close connection between statistics of polymers and phase transitions [7] has made it possible to apply the renormalization group (RG) technique to polymers [7]. However, while the direct position space RG (PSRG) has been widely applied to various models of linear [8] and branched polymers [9], most of the studies of loops in polymers have been limited to either numerical studies [1,2,5,10] or field-theory techniques [3,11]. The aim of the present work is to develop and apply a direct PSRG to a model of ring polymers and to study the effects of loops on the conformation of models of linear polymers, branched polymers, and gels. The main advantage of the direct PSRG over numerical methods is that in contrast to numerical evidence the RG flows explicitly demonstrate the universality of two models if they are described by the same fixed point [8]. The main advantage of the direct PSRG over c-expansions and field-theory techniques is that it is not only mathematically simple and physically much more intuitive, its results are valid in realistic dimensions (d 2,3, instead of 4 e or 8 e) [3,11]. A direct PSRG for polymers which I call the cell renormalization (CR) method has already been in—
—
—
—
—
troduced in ref. [9], where the reader is referred to for details. In the CR method [9], the lattice is first divided into cells, and polymer configurations are represented by clusters of bonds (and/or sites) on the cell. Each cluster is weighted by K’1, where K is a fugacity for each bond in the cluster. The recursion relation is determined by mapping all configurations which enter a given cell from the origin and extend across the cell to a single bond with a fugacity K’ on a new lattice whose lattice spacing has been rescaled by b. The critical parameters and exponents are obtained from this recursion relation using standard techniques [9]. 2. Ring polymers and linear polymers. Let us dea single parameter CR approach for ring polymers. Consider a linear chain which is represented by a self-avoiding walk (SAW) on a lattice that starts from the origin of a cell and extends across it either horizontally or vertically. If the chain does not return to the cell to self-intersect, it is renormalized to a single step, in the direction in which it extends across the cell. However, if the chain extends across the cell in one direction and returns to the cell from another direction and self-intersects at the origin, then the configuration is a ring polymer. To renormalize these configurations, I map them into two bonds; one bond in the direction of outgoing chain and one bond in the direction of incoming chain. On a cell of side b = 2 on a square lattice the recursion relation is K’2 =K4 + 2K5 +K6 (1) velop
—
—
—
0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
341
Volume 92A, number 7
PHYSICS LETTERS
This recursion relation has a fixed point at K* = 0.62 and gives a correlation length exponent v = 0.79. For b = 3, v = 0.78 and a cell-to-cell transformation [9] from 3 -÷ 2 gives v = 0.75 in remarkably close agreement with the Flory estimate of 0.75 for linear chains in d = 2 [7,8]. Even though this numerical evidence is interesting, it is not sufficient to ensure us that linear polymers and ring polymers have the same exponent v. In order to study the effects of ring formation on a linear chain I develop a two-parameter RG of the form K’ = F1 (r, K) and K ‘r’ = F2(r, K) where K is the fugacity for each step of the linear chain that starts from the origin and gets across the cell horizontally (vertically), and r is the fugacity for each bond in the chainbyjoining which returns the step cell and finally forms ring to thetofirst at the origin. The aflow diagrams for both d = 2, 3 are similar and the case d = 3 on a simple cubic lattice is shown in fig.1. Note that for all r the flow is to the linear chain fixed point at r 0, K = K ~ and therefore ring polymers and linear polymers are described by the same critical fixed point and therefore belong to the same universality class. This result is not entirely unexpected, because in the scaling limit where the number of statistical units in a polymer tends to infinity [7], whether a linear polymer has open ends or a single point of contact should not change its global universal properties.
\ \\
~
~
~.
\ *
\.~
~,
--~~----
~ o r Fig. 1. Flow diagram showing that ring polymers (r > 0) are in the same universality class as Imear polymers (r = 0) because for all r, the flow is to the linear polymer fixed point at r = 0,K = K*. Solid dots are trivial fixed points,
342
29 November 1982
3. Linear polymers with and without loops. The SAW model of linear polymers is a random walk which is constrained never to visit a site or a bond of a lattice more than once [7]. Under these conditions the chain cannot fold back on itself or form closed loops. The question is: which of these two conditions produces the excluded volume effect? First, let us consider SAWs which cannot fold back on themselves by visiting a bond more than once, but can form closed loops by visiting a site twice. There are no such configurations on a 2 X 2 cell on a square lattice. On a 3 X 3 cell the recursion relation for SAWs with loops is 3 + 3K4 + 9K5 + 5K6 + 9K7 + 4K8 K + = 8K9 K + 2K10 (2) ,
which has a fixed point at K* = 0.45 and gives v = 0.72. Now, let us allow the chain to fold on itself by visiting some bonds twice; the recursion relation on a 2 X 2 cell on the square lattice is K’ = K2 + 2K3 + 3K4 + 4K5 + 6K6 + 8K7 + 5K8 (3 This recursion relation has a fixed point at K * = 0.38 and gives v = 0.59. The remarkable difference between the values of the two exponents indicates that allowing for loops does not change the excluded volume effect and the chain is expanded, but by folding the chain its dimension is considerably reduced to its value at the 0-point [13] where two.body interactions vanish. Similar results are obtained in d = 3 where by allowing loops on a simple cubic lattice, ~ retains its excluded volume value (~0.59[12,141), whereas by allowing foldings it decreases to 0.5; its 0-point value ind=3 [7]. To show that linear chains with and without loops are in the same universality class I construct a twoparameter RG. As before,K is the fugacity for each step of the chain, and a second fugacity 1 is chosen to be the fugacity for a 1oop. The renormalized weight 1’ for loops contains only the weights of those configurations that span the cell and have a loop. The two coupled recursion relations for K’ and 1’ may be solved for the fixed points and the global RG flows. The results ford = 2, 3 on the square and simple cubic lat.
.
tices, respectively, look qualitatively the same; the case ofd = 3 is shown in fig. 2. There is only one crit-
Volume 92A, number 7
PHYSICS LETTERS ~
tion [16—18](the q = 1 limit of the q.state Potts model) and I am not concerned about the statistics of a correlated percolation model in which loops are not allowed. Such a model, commonly referred to as tree percolation [19], belongs to a different universality class than percolation [191 (q = 0 limit of the q-state Potts model) and is known to have different critical exponents [19]. The CR study of loops in random animals and percolation clusters follows the same line of reasoning as
~
>,,
_______ __
~ i~•’:i i:_ ~
~
for linearthe chains: theofrenormalized loop fugacity 1’ contains weight only those spanning configurations which have a loop. Thus, each cluster of size N
_~.—‘
.~—
29 November 1982
withL loops contributes a weight KN1L to 1’. Renormalization of bond fugacityK follows the usual pro-
~—
Loop fugacity 2 Fig. 2. Flow diagram showing that polymers with loops (1> 0) are in the same universality class as polymers without loops (I = 0), because for all 1, the flow is to the! = 0, K = K* fixed point. Solid dots are trivial fixed points,
ical fixed point at 1= 0 and for all 1 the flow is to this fixed point. Thus, loops are irrelevant, v is unchanged, and SAW with and without 1oops are in the same universality class.
cedure [9,12,15,17]. In this way I obtain two-coupled recursion relations for 1’ and K’ which I solve for the fixed points and the RG flow diagram. The resulting flow diagram in both d = 2, 3 and for random animals and percolation clusters are similar to fig. 2. In all cases I have found only a single critical fixed point at 1= 0 and K = K ~ corresponding to clusters without loops (self-avoiding trees embedded on a lattice [9]). This fixed point controls the entire critical behavior for clusters with loops. Thus, in random animals and percolation clusters ioop fugacity is a non-critical parameter (i.e., it only changes the magnitude of the excluded volume and not the exponents), and clusters with and without loops are in the same universality class. The universality of random animals with and without loops is in agreement with the same conclusion reached by Lubensky and Isaacson [11] who applied a field-theory formalism to study the random animal model in d = 8 c dimensions.
4. Branched polymers and gels with and without loops. As in refs. [7,15], I use random animals as a model for randomly branched polymers with 1oops and self-avoiding trees as a model for branched polymers without loops in the dilute limit in good solvents, I use random percolation as a model for gelation[7,16]. In both cases I am interested in the scaling limit where the polymer size N (i.e., the number ofbonds in the cluster) becomes very large. The problem we are concerned with is whether the critical exponents, such as I would like to thank members of the Center for the exponent v for the cluster radius R (which is dePolymer Studies, Boston University, for discussions fined byR ~Nv, and should not be confused with and comments. the exponent v~which is defined by ~ (p Pc)~~)’ are the same for clusters with and without loops or References they are different. Since for i’
-‘
—
—~
343
Volume 92A, number 7
PHYSICS LETTERS
[5] J. des Cloizeaux and M.L. Mehta, J. de Phys. 40 (1979) 665. [6] J. des Cloizeaux, J. de Phys. Lett. 42(1981) L-433. [7] P.G. de Gennes, Scaling concepts in polymer physics (Cornell Univ. Press, Ithaca, NY, 1979). [8] For a review of the applications of PSRG to polymers see: H.E. Stanley, PJ. Reynolds, S. Redner and F. Family, in: Real-space renormalization, eds. T. Burkhardt and J.M.J. van Leeuwen (Springer, Berlin, 1982) Ch. 7, p. 171, and references therein. [9] F. Family,J. Phys. Al 3 (1980) L325. [10] D.C. Rapaport, 1. Phys. A8 (1975) 1328, and references therein. [11] T.C. Lubensky and J. Isaacson, Phys. Rev. A20 (1979) 2130.
344
29 November 1982
[12] F. Family, J. de Phys. 42(1981)189. [13] J.A. Marqusee and J.M. Deutch, J. Chem. Phys. 75 (1981) 5179. [14] See, e.g., A. Baumgartner, J. Phys. A13 (1980) L39; K. Kremer, A. Baumgartner and K. Binder, Z. Phys. B40 (1981) 331. [15] F. Family and A. Coniglio, J. Phys. A13 (1980) L403. [161 D. Stauffer, Physica 106A (1981) 177. [17] F. Family and P.J. Reynolds, Z. Phys. B45 (1981) 123. [18] D. Stauffer, in: Intern. Conf. on Disordered systems and localization, ed. C. Di Castro (Springer, Berlin, 1982), to be published. [19] MJ. Stephen, Phys. Lett. 56A (1976) 149.