THE JOURNAL OF CHEMICAL PHYSICS 126, 134906 !2007"
Simulation and theory of flexible equilibrium polymers
under poor solvent conditions
LaKedra S. Pam, Larissa L. Spell, and James T. Kindt
Department of Chemistry, Emory University, Atlanta, Georgia 30322
!Received 15 August 2006; accepted 14 February 2007; published online 4 April 2007"
Grand canonical Monte Carlo simulation and simple statistical thermodynamic theory are used to
model the aggregation and phase separation of systems of reversibly polymerizing monomers,
capable of forming chains with or without the ability to cyclize into rings, with isotropic square-well
attractions between nonbonded pairs of monomers. The general trend observed in simulation of
chain-only systems, as predicted in a number of published theoretical works, is that the critical
temperature for phase separation increases and the critical monomer density decreases with rising
polymer bond strength. Introduction of the equilibrium between chains and rings into the theory
lowers the predicted critical temperature and increases the predicted critical density. While the
chain-only theories predict a vanishing critical density in the limit of complete polymerization,
when ring formation is taken into account the predicted critical density in the same limit approaches
the density of the onset of the ring-chain transition. The theoretically predicted effect of cyclization
on chemical potential is in good qualitative agreement with a subset of simulation results in which
chain-only systems were compared with equilibrium mixtures of rings and chains. The influence of
attractions on the aggregation number and radius of gyration of chains and rings observed in
simulations is also discussed. © 2007 American Institute of Physics. #DOI: 10.1063/1.2714945$
I. INTRODUCTION
The phase behavior of equilibrium polymers !EP’s"—
systems featuring reversible formation of chainlike
aggregates—under “poor solvent” conditions of nonspecific
attractions between nonbonded chain segments has been the
topic of numerous experimental and theoretical treatments.
Blankschtein et al. showed that a theory treating selfassembly along with a Flory ! parameter yielded good agreement with the experimental phase diagram of di-octanoyl
phosphatidylcholine, a lipid that forms cylindrical micelles.1
An experimental manifestation of an equilibrium polymer,
short overlapping strands of DNA, has been shown to undergo interesting condensation behavior.2 Jackson et al. applied the theory of associating fluids to make predictions of
coexistence curves for chain-forming systems.3 Douglas and
co-workers have applied lattice mean-field theory to phase
separation of self-assembling chains, and investigated the effects of initiators and activation barriers on phase boundaries
and critical behavior.4,5 Kindt and Gelbart developed theory
for the coupling of phase separation and nematic ordering in
semiflexible equilibrium polymers.6 The interplay between
the chain-forming and condensing properties of the Stockmayer fluid, in which dipolar interactions induce the formation of chain structures, has been investigated by
simulation7,8 and theory.9 Comprehensive simulation studies
of flexible EP’s, with careful attention to excluded volume
effects on chain dimensions and length distributions, were
undertaken by Wittmer et al.,10 while the nematic phase induced by these effects in semiflexible EP’s has been modeled
as well.11,12 Recent simulations by Sayar and Stupp13 explored equilibrium polymerization in the presence of isotropic interchain attractions over a range of chain stiffnesses at
0021-9606/2007/126!13"/134906/10/$23.00
a constant low monomer concentration. Simulations demonstrating phase separation of simple equilibrium polymers under poor solvent or attracting conditions have not, to our
knowledge, been reported in the literature.
The phenomenon of equilibrium polymerization is occasionally complicated by a significant fraction of rings coexisting with chains, as observed, e.g., in some micellar
systems.14 As described in the classic work of Jacobson and
Stockmayer,15 ring formation in reversible linear selfassembly is thermodynamically competitive with chain formation only at low aggregation numbers n. The “J-factor” Jn,
or ratio between cyclization and chain catenation equilibrium
constants, decreases at high n as n−" where " = 5 / 2 for selfassembled noninteracting chains;15 the value of " is greater
by 1 than for cyclization of fixed-length chains because of
the indistinguishability of the n monomers of a selfassembled ring. Within the grand canonical ensemble of noninteracting self-assembled chains and rings, equilibrium concentrations of n-mer chains and rings can be written in terms
of the free monomer concentration c1 and the bond association constant Kassoc,
n
cn1 ,
cn,ring = JnKassoc
!1"
n−1 n
cn,ch = Kassoc
c1 .
!2"
15–17
at high bond
As discussed in many previous works,
strength, the difference in scaling between chains and rings
leads to a crossover in system composition from mostly rings
at low concentration to mostly chains at high concentration.
This is evident from Eqs. 1 and 2: when the free monomer
concentration c1 approaches 1 / Kassoc, the total concentration
of monomers in !finite" rings approaches its maximum value,
126, 134906-1
© 2007 American Institute of Physics
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134906-2
J. Chem. Phys. 126, 134906 !2007"
Pam, Spell, and Kindt
$
#r,max = %n=1
nJn, which is finite for " % 2. Meanwhile, in the
same limit, the concentration of monomers in chains diverges as the system undergoes a polymerization transition.
As the formation of rings reduces the concentration of free
monomers, the polymerization transition is pushed to higher
−1
to &#r,max. At
total monomer concentrations, from &Kassoc
high Kassoc, there will therefore be a rapid crossover in system composition from predominantly rings to predominantly
chains as total monomer concentration approaches #r,max.
Several simulation studies of EP systems have demonstrated
this transition and modeled the equilibrium distribution of
rings.16,17
In the present work, we develop the theory of phase
separation in ring-containing equilibrium polymer systems,
and compare our results to simulations of self-assembling
flexible chains of hard sphere monomers with short-ranged
isotropic square-well attractions. We have applied grand canonical Monte Carlo simulation methods to approximately
determine phase behavior near critical points in these systems over two orders of magnitude in Kassoc. The algorithms
used11,18 are designed to achieve equilibrated chain and ring
size distributions even when bond strengths are very large,
conditions under which conventional Monte Carlo !MC" or
molecular dynamics, relying on rare bond breaking events to
change length distributions, might become slow. Ring formation and growth were explicitly treated in the simulations
through an algorithm designed to achieve rapid equilibration
of ring size distribution, and comparisons between behavior
in the presence and absence of rings are made.
II. SIMULATION METHODS
A. Model potential
As in previous studies,11,19,20 we treat each monomer as
a hard sphere of diameter & that can form up to two bonds at
contact with other spheres. An equilibrium association constant Kassoc, with units of &3, defines the strength of this
bond, which is independent of the length of the chain or the
position of the monomer within the chain. Adjacent bonds
are constrained to have an angle greater than ' / 2, but no
other bending potential is imposed. To represent solvent
quality effects, an isotropic square-well potential of depth (
is added to each monomer. The square-well attractive interaction is in effect between all monomer pairs except those
directly bonded to each other. For the present study, the
range of the square-well interaction is set to 21/2&, such that
next neighbors within a chain are outside the influence of the
interaction as well.
B. Monte Carlo sampling
1. Chains
The polydisperse insertion, removal, and resizing21
!PDIRR" algorithm was used to treat chain formation and
growth of chains within the grand canonical ensemble.
Briefly, each MC step is an attempted insertion, removal,
extension, or shortening of a chain through the addition or
removal of one or more monomers. The probability of choosing a length n for a chain is proportional to a weighting
factor,
−1
n−1
wn = Nchain
VKassoc
exp!n)* − )Unb",
!3"
with Nchain the number of chains in the system, V the system
volume, * the monomer chemical potential, ) = !kBT"−1, and
Unb is the potential energy of the chain’s nonbonded interactions. Details about the method for a system of selfassembled chains without square-well attractions are given in
Ref. 11. Adaptation to the present model is straightforward:
for any chain conformation that satisfies the bond distance,
bond angle, and hard-sphere overlap restrictions, a nonbonded potential contribution Unb = −(nsw was used in Eq.
!3", with nsw the number of the chain’s intra- and interchain
square-well bonds. Care was taken to avoid numerical overflow errors for long chains, where exp!)(nsw" can be very
large.
2. Chains and rings
Some simulations treated equilibrium self-assembly of
monomers into rings at equilibrium with chains. Rather than
rely on chain cyclization to give an equilibrium distribution
of rings, a set of MC moves to add, remove, grow, and shrink
rings was implemented in parallel with the chain simulation
moves described above. Equilibration of the size distribution
of rings was performed using the pivot-coupled grand canonical MC !PCGMCMC" method,18 which allows adding or
removing a monomer to a ring in a single move, as opposed
to relying on a sequence of moves to open, grow or shrink,
and reform the ring. The method has been shown to give
quantitatively correct distributions of ring sizes for ideal
freely jointed chains, and to be consistent with other theoretical and simulation results for isolated semiflexible chains
and flexible chains with excluded volume.18 Distributions of
ring sizes in the present simulations reflect interactions with
other rings and chains as well as these intrinsic free energies.
To treat a system of multiple rings interacting with each
other and at equilibrium with chains, MC moves to create
and remove entire rings have been developed. Only rings of
the minimum size !five monomers" were subject to creation
and removal moves. The first step in the creation and insertion of a pentamer ring is the creation of a tetramer. A position r1 for the first monomer in the tetramer is chosen from
the simulation box. The second monomer is placed at a position r2 selected at random on the sphere of radius & centered at r1. The third monomer position is selected such that
bond vector r3 − r2 forms an angle +2 with respect to vector
r2 − r1, where sin +2 is taken at random from a uniform distribution between 0 and 1. The fourth monomer position is
chosen at random on the circle equidistant from monomers
one and three. At any point, if the added monomer overlaps
an existing monomer on a different ring or chain !or for the
fourth monomer, on the same ring" the move fails.
The free energy of the tetramer ring intermediate has
been derived through comparison of its partition function to
that of a tetramer chain. As in previous work,18,19 to evaluate
the equilibrium probabilities of formation of different structures we treat each sticky hard-sphere bond as the limit of a
square-well potential of infinitesimal width ,. The free energy of a given structure can then be related to the volume in
configuration space it occupies, multiplied by a Boltzmann
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134906-3
J. Chem. Phys. 126, 134906 !2007"
Flexible equilibrium polymers
factor exp!−)U" with U its potential energy. To express
the equilibrium partitioning between a tetramer ring
!U = −4Ubond" and a tetramer chain !U = −3Ubond", we compare the volume of the annulus of possible sites for the fourth
monomer in the ring creation scheme above, which can be
shown to equal '&,2 csc! 21 +2" in the limit of low ,, to the
volume 2'&2, of the half-spherical shell of possible sites
for the fourth monomer of a tetramer chain. After including
the fourfold degeneracy of tetramer ring structures compared
to tetramer chains, we conclude that the equilibrium ratio of
tetramer rings to tetramer chains !absent nonbonded interactions" is
P4,ring , exp!)Ubond"
=
.
P4,chain
8& sin 21 +2
!4"
Given Kassoc = 2'&2, exp!)Ubond", this ratio corresponds to
Kassoc / !16' sin! 21 +2"&3".
Except for the singular case of a perfect square, the tetramer will not satisfy bond angle constraints, so its creation
is just an intermediate in the formation of a pentamer. To
generate a pentamer, a pivot-coupled addition move is performed on the tetramer, as described in Ref. 18. In this move,
a bond is broken between adjacent monomers at positions ra
and rb, a new monomer is added bound to position rn, and a
section of the ring from a pivot monomer at r p to rb is
rotated into position to reform the ring including the new
monomer. If the final structure satisfies angle and excluded
volume restrictions, the addition is accepted with probability,
'(
acc = min 1,
-
(
V exp!nsw)("
Nring + 1
)(
4
Kassoc
exp!4)*"
16' sin! 21 +2"
4*rb − r p*
Kassoc exp!)*"
5*rn − r p*
)+
.
)
!5"
The expression in the first set of parentheses accounts for the
translational entropy and attractive interactions of the ring,
Nring represents the number of rings present in the system,
nsw is the number of nonbonded square-well attractive interactions formed by the ring, and V is the simulation box volume. The expression in the second set of parentheses is the
internal partition function of the tetramer, and the expression
of the third set of parentheses represents the addition of the
fifth monomer through the PCGCMC algorithm as derived in
Ref. 18. For ring removal, a ring is selected at random; if it
is a pentamer, then a PCGCMC removal step is attempted
and, if successful in yielding a tetramer without hard-sphere
overlaps then the removal proceeds with probability,
'
acc = min 1,
+
16' sin! 21 +2" 5*rr − r p*
Nring
.
5
V exp!nsw)(" Kassoc
exp!5)*" 4*rb − r p*
!6"
To check that the ring-chain equilibrium is treated correctly in simulations, the equilibrium ratio of pentamer rings
to chains obtained in a dilute grand canonical system was
compared to the ring closure probability of an independent
set of 108 randomly generated pentamer chains with the
angle and excluded volume restrictions of the current model.
The ring closure probability for pentamer chains within a
tolerance , was 9.56- 10−3, / &. The cyclization constant
for fixed-length chains is therefore 9.56- 10−3, / &
-exp!)Ubond", but reduced by a factor of 5 for the degeneracy
of
self-assembled
chains
to
Kcyc,5 = 1.91
and
substituting
Kassoc
- 10−3, / & exp!)Ubond",
= 2'&2, exp!)Ubond" gives a ratio of 3.04- 10−4&−3Kassoc.
The full simulation gives a ratio of 3.1- 10−4&−3Kassoc, the
good agreement between the two values confirming that the
treatment of pentamer formation, insertion, and removal is
consistent with the true ring-chain equilibrium for this
model.
The size distribution of rings is sampled using singleparticle PCGCMC addition and removal moves, which were
attempted with 15% probability at each MC step. The
PCGCMC method has been previously shown to give accurate equilibrium size distributions of rings, compared with
analytical expressions and other simulation methods.18 To
confirm the implementation of the method, we generated 108
hexamer chain structures and found the fraction cyclizing to
within a tolerance , to be 7.77- 10−3, / &, giving an expected ratio of self-assembled hexamer rings to chains of
!7.77- 10−3, / &"Kassoc / !6 - 2'&2," = 2.06- 10−4&−3Kassoc,
consistent with observations from full simulations that
#ring,6 / #chain,6 , 2.0- 10−4&−3Kassoc.
Finally, in simulations incorporating rings, chain addition and removal/resizing moves were attempted with 17.5%
probability at each Monte Carlo step through the PDIRR
method as described previously.
3. General
To aid in sampling of chain interior and ring conformations, after every chain or monomer addition/removal move,
five independent crankshaft moves were attempted. Four
were attempted rotations of one interior monomer of a chain
or ring about the axis of its two neighbors. One crankshaft
move was also attempted on a randomly selected ring; in this
case, a segment of up to n − 3 monomers, with n the number
of monomers in the rings, was rotated by a randomly selected angle about the axis of rotation of its end points. This
large rotation move was included to allow for topological
changes in ring arrangements, such as nontrivial knotting or
catenation, which are nearly impossible to achieve through
single-monomer steps. Acceptance of trial structures generated through rotation moves was subject to excluded
volume restrictions and an acceptance probability of
min#1 , exp!,nsw)("$, where ,nsw is the change in the number of square-well contacts during the move.
For each set of simulation conditions, histograms of the
total numbers of monomers, chain and ring aggregates by
size, and nonbonded interactions, along with mean sizedependent radius of gyration were collected over a 107-step
simulation run following equilibration. A ring size limit of
100 particles was used; at Kassoc = 500 000&3, equilibrium
ring concentrations were negligible compared with chain
concentrations for aggregates of this maximum size.
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134906-4
J. Chem. Phys. 126, 134906 !2007"
Pam, Spell, and Kindt
4. Efficiency considerations
The PDIRR method suffers in the present simulation
context, even when measured against simulations from the
last decade,10,16 for several reasons. Calculation of the grand
canonical weight factor for a chain, which is trivial when
only excluded volume interactions are operating, is significantly more complex when the number of square-well interactions needs to be taken into account. While a bond list was
implemented to avoid the need to calculate i - N distances to
obtain the weight of an i-mer in a system of N monomers, a
calculation of order i2 was still necessary to appropriately
count intrachain interactions for removal and resizing. The
advantage to biasing moves in general is less important for
flexible chain systems than for semirigid systems where
monomers face large entropic barriers to bond, and where the
PDIRR method performed very well. For future work on
flexible systems, an end-bridging algorithm22 would aid in
achieving equilibrium length distributions.
FIG. 1. !Color online" Inverse of Flory ! parameter !proportional to temperature at fixed nonbonded attraction energy" and volume fraction at the
critical point as functions of association constant Kassoc, with and without
ring formation, from theory.
$
) f = % #ci!ln ci − 1" − !i − 1"ci ln Kassoc$
i=1
$
+ % #c j!ln c j − 1" − jc j ln Kassoc − c jkB−1,Scyc
III. THEORY FOR PHASE SEPARATION OF
EQUILIBRIUM POLYMER CHAINS AND CHAIN-RING
MIXTURES
The modified Flory-Huggins lattice theory for phase
separation of equilibrium polymers as given in Ref. 6 has
been extended to include ring formation. The free energy of
a system of monomers occupying a fraction . of lattice sites,
capable of forming two chain-type bonds per monomer with
association constant Kassoc !with dimensionless units of the
lattice site volume", mean-field excluded volume interactions, and isotropic intermonomer attraction energy proportional to the Flory parameter ! = )n( / 2, with ) = !kBT"−1 and
n the number of nearest neighbor sites on the lattice, is given
by
$
) f = % #ci!ln ci − 1" − !i − 1"ln Kassoc$
i=1
+ !1 − ."#ln!1 − ." − 1$ − !.2 .
!7"
Minimizing f with respect to the distribution of i-mer concentrations ci gives a mean chain length M & !Kassoc."1/2. To
simplify the free energy and obtain scaling behavior at high
Kassoc, we follow Tlusty and Safran23 in associating a free
energy of −kBT to each equilibrium defect in the system,
giving
)f , −
( )
.
Kassoc
1/2
− . ln Kassoc + !1 − ."ln!1 − ." − !.2 ,
j=5
+ c j2.5 ln j$ + !1 − ."#ln!1 − ." − 1$ − !.2 .
!9"
For comparison with the present simulations, we have assumed a minimum ring size of 5 and a power law contribution of j−5/2 !as for ideal rings", added to a size-independent
cyclization entropy ,Scyc = −4.09kB, selected to give the appropriate ratio of pentamer rings and chains as determined
for the simulation model !3.0- 10−4&−3Kassoc, determined in
Sec. II B 3 above". In the terms of Eq. !1", we are defining
Jn = exp!−4.09"n−2.5; the appropriate magnitude and length
dependence of the J function will depend, in general, on the
chain flexibility and intrachain interactions. Figure 1 shows
critical values of . and !, determined numerically as a function of Kassoc for both models, by solving for d3 f / d.3c = 0 and
d2 f / d.2c = 0 using Eqs. !7" and !9" for the chain-only and
ring-containing systems, respectively. Consistent with the
−1/5
in the
scaling predictions obtained using Eq. !8", .c & Kassoc
long-chain limit in the absence of rings. Ring formation
leads to a significant increase in .c and a slight increase in !c
for values of Kassoc % 105. When rings are taken into account,
−1/5
but
in the limit of high Kassoc, .c does not vanish as Kassoc
approaches the onset concentration of the ring-chain transition. A proof of this result is given in the Appendix. Phase
coexistence boundaries, shown in Fig. 2 at several values of
Kassoc with and without rings, were determined by numerical
search for pairs of volume fractions .A and .B equal in
chemical potential *!." = df / d. and in pressure p!."
= -0.*!.!"d.!.
IV. SIMULATION RESULTS AND DISCUSSION
!8"
which, when solving for the critical condition d3 f / d.3c = 0,
−1/5
.
gives .c & Kassoc
Making no distinction between monomers in rings and
chains in determining contributions to attractive and repulsive interactions, we can include ring structures,
A. Chain-only systems
Twelve series of chain-only grand canonical MC simulations were performed: at Kassoc = 5000, 50 000, and
500 000&3, simulations covering a range of concentrations
were made at three values of the isotropic square-well potential energy ( near the critical point and at ( = 0. For certain
combinations of *, Kassoc, and ( the mean total monomer
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134906-5
J. Chem. Phys. 126, 134906 !2007"
Flexible equilibrium polymers
TABLE I. Parameters for nonideal contribution to * vs # fit: *nonbonded = a
+ b# + c#2. For interpolation purposes, the formulas b = −68&3!)( − 0.395"2
and c = 0.48&6 + 83&6!)( − 0.395"2 were used.
( !kBT"
0.432
0.451
0.470
0.495
0.513
M = 21 +
FIG. 2. !Color online" Coexistence curves from theory showing phase separation of equilibrium polymer systems !chain only: solid black curves;
chains+ rings: dotted red curves" as a function of inverse Flory ! parameter
for labeled association constants Kassoc. The lines of critical points are also
shown; for chain-only system, the line terminates at . = 0, !−1 = 2, while for
ring-containing system, the line terminates at . = 0.016, !−1 = 1.97.
concentration # exhibited hysteresis, staying near one of two
values throughout the simulation depending on the initial #.
This hysteresis was taken as evidence of a phase separation.
Data are shown in Fig. 3, along with estimates of the best-fit
curves and of the coexisting phase densities. Contributions to
the chemical potential can be approximately separated into
an ideal-chain term that incorporates the bonding and translational entropy of monomers and density can be approximately separated into an ideal-chain contribution,
)*ideal!#" = M −1!#"ln!#/M 2"
+ !1 − M −1!#""#ln!1 − M −1!#"" − ln!Kassoc"$,
!10"
where M represents the mean number of monomers per chain
in the ideal system,
FIG. 3. !Color online" Grand canonical Monte Carlo !GCMC" simulation
results !symbols" and fits to data for chain-only simulations. Circles: Kassoc
= 500 000&3; squares, Kassoc = 50 &3; diamonds, Kassoc = 5000&3. Solid curve
!violet": ( = 0.513kBT; dash curve !blue": ( = 0.495kBT; dot-dash curve
!black": ( = 0.470kBT; dot-dot-dash curve !green": ( = 0.451kBT; dotted curve
!red": ( = 0.432kBT. Fit parameters are given in Table I. The gray horizontal
lines indicate coexistence boundaries calculated from fits.
1
2
a
b ! & 3"
c ! & 6"
−0.013
−0.019
−0.022
−0.024
−0.026
−0.095
−0.222
−0.385
−0.675
−0.950
0.600
0.755
0.925
1.30
1.65
.1 + 4Kassoc# ,
!11"
added to a quadratic polynomial with adjustable coefficients
to account for nonbonded interactions. Best-fit coefficients
are shown in Table I; we found that one set of polynomial
coefficients at a given isotropic attraction strength ( would
give a reasonably good fit to data obtained for two or more
values of the chaining constant Kassoc. The linear and quadratic terms in the fit to *!#", corresponding to second and
third terms in the virial expansion of the free energy, are
included to account for the strength of attractions and of the
three-body effects that counteract these attractions at high
density. The constant term, independent of # but dependent
on (, cannot be a true constant because in the limit of very
low density !where free monomers predominate" it must vanish to allow recovery of the ideal chemical potential of Eq.
!10". This pseudoconstant term can be understood as arising
from the average intrachain attraction for a monomer added
to the end of a growing chain, contributing !in the dilute
case" to the effective chain association free energy in a manner that is only weakly dependent on chain length. Consistent with this interpretation, as shown in Fig. 4, a significant
fraction of monomers have nonbonded attractions even at the
lowest densities investigated. Phase boundaries corresponding to the fits were also obtained. In all cases, the coexisting
phase concentrations fell near the low end of the range of
hysteresis observed in simulation. These phase boundaries
are placed on a phase diagram !analogous to Fig. 2" in Fig. 5,
with coexistence curves drawn as a guide to the eye obtained
from interpolation of the parameters of Table I.
Both phase diagrams in Figs. 2 and 5 treat critical effects
classically, neglecting the strong fluctuation effects that stabilize the single-phase region near the critical point; true co-
FIG. 4. !Color online" Mean number of square-well interactions per monomer at ( = 0 !open symbols" and ( = 0.470kBT !filled symbols". Circles
!black": Kassoc = 500 000&3; squares !blue", Kassoc = 50 000&3; diamonds
!green", Kassoc = 5000&3. X’s are ring-containing simulations at Kassoc
= 500 000&3.
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134906-6
Pam, Spell, and Kindt
FIG. 5. !Color online" Coexistence curves derived from simulation data.
The symbols represent phase boundaries at different Kassoc as in Fig. 3. The
open circles/dotted curve represents Kassoc = 500 000&3 system with chains
+ rings.
existence curves including critical fluctuations would be significantly flatter at the top.24 Although the method used to
obtain phase diagrams is not as reliable as histogram reweighting approaches widely used to determine critical behavior of fixed-length polymer systems,25 the ability of Eq.
!10" to fit the data after the introduction of some adjustable
virial parameters indicates that the general trends are the
same in simulation and theory. In particular, the apparent
critical concentrations are reduced by roughly a factor of 1 / 3
with each power of ten increase in Kassoc, decreasing somewhat less rapidly than the scaling predictions !10−1/5 = 0.63".
The critical temperature likewise increases; extrapolation of
the trend from Kassoc between 5000 and 500 000&3 gives a
critical temperature of &2.4( / kB in the limit of infinite
Kassoc.
As expected, near and below the critical point in dilute
solution chains are significantly more compact than under
purely repulsive conditions. Figure 6 compares the mean
squared radius of gyration as a function of chain length in the
presence and absence of attractions, in dilute solution. The
expected superlinear scaling of the self-avoiding chain system and sublinear scaling in the presence of attractions at (
= 0.43kBT are apparent. With increasing total monomer concentration, these nonideal effects become less important as
interchain interactions become more important. Intrachain
effects also influence the length distribution of chains. Ideal
FIG. 6. !Color online" Mean square radius of gyration vs chain contour
length under dilute conditions !# = 0.004&−3". The black curve, to guide the
eye, is 0.265i6/5.
J. Chem. Phys. 126, 134906 !2007"
FIG. 7. !Color online" Length distribution of chains under dilute conditions.
Symbols are as in Fig. 6; the dashed black line gives ideal exponential
distribution.
equilibrium polymer chains exhibit an exponential dependence of number density on chain length at equilibrium. Deviations from this ideal distribution occur when chain free
energies depend nonlinearly on chain length, either from intrachain or interchain interactions. The effects of these interactions on length distributions of a flexible self-avoiding
model equilibrium polymer have been analyzed in detail by
Wittmer et al.10 In dilute solution, they found an enhancement of short chains relative to the ideal exponential dependence. Both this effect and its converse for chains under poor
solvent conditions are evident in Fig. 7, which shows length
distributions from three simulations at nearly identical total
monomer concentrations from which an ideal length distribution may be calculated. Relative to this ideal !single exponential" distribution, a system of monomers with purely repulsive nonbonded interactions is enriched in shorter chains,
the distribution for chains slightly below the critical attraction threshold for phase separation is close to the ideal prediction, and the distribution for chains under condensing
conditions shows depletion of short chains and enrichment of
long chains. As observed in Ref. 10, we also see !data not
shown" that when concentrations are raised into the semidilute regime, chain length distributions return to a simple exponential function as the local environment of a monomer is
influenced less by its own chain and more by other chains.
The dependence of mean chain-length M !defined as the
ratio of mean total monomer concentration # to mean concentration of chains and free monomers together" on #,
shown in Fig. 8 falls near the ideal system expression of Eq.
!11". Careful inspection reveals some consistent trends in the
small deviations observed. In the dilute regime, the intrachain effects described above lead to a small depression of
mean chain length in the absence of attractions !open circles"
and a small elevation of mean chain length in the presence of
attractions. These effects are most notable for long chains,
and contrast with observations on semiflexible polymers,
where nearly perfect adherence to Eq. !11" is seen in dilute
solution.11 At higher concentrations, in the absence of attractions a steric end effect raises the observed M above the ideal
system prediction, as seen for semiflexible chains,11 but this
effect is suppressed when attractions are important. With or
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134906-7
Flexible equilibrium polymers
FIG. 8. !Color online" Chain growth vs concentration from MC simulations.
The dashed curves are from Eq. !10". Open circles: ( = 0; filled circles, (
= 0.432kBT; squares, ( = 0.451kBT; diamonds, ( = 0.470kBT; upright triangles,
( = 0.495kBT; left-facing triangles, ( = 0.513kBT.
J. Chem. Phys. 126, 134906 !2007"
FIG. 9. !Color online" Fraction of monomers in rings and mean ring size in
ring+ chain containing system at Kassoc = 500 000&3. The symbols indicate
square-well depth ( as in Fig. 8. The curves are from equilibrium distributions based on free energy of Eq. !8".
B. Chains and rings at equilibrium
likelihood, in the size range covered by the simulations, Kcyc
is influenced by the bending stiffness in the model. In ideal
three-dimensional wormlike chains, the limiting scaling behavior for cyclization is not seen until contour lengths over
ten times the persistence length.18,28 Although the present
model bears little resemblance to a wormlike chain, as a
persistence length of &4& can be obtained from the bond
orientation correlation function, it is not unreasonable that
the small bending stiffness present in the model will influence the cyclization equilibrium for structures containing up
to tens of monomers. The observation from Sec. II and the
Appendix that the critical point coincides with the onset of
the ring-chain transition should not be sensitive to the details
of the ring size distribution; numerical calculations indicate
Simulations of equilibrium polymers including both
chain and ring structures were performed only at the highest
bond strength investigated, Kassoc = 500 000&3, where the
greatest cyclization constants and therefore the greatest influence of rings are expected. As has been well
documented,15–17 at high association constant the fraction of
monomers belonging to rings is near unity when the concentrations are low !but not so dilute as to promote dissociation
into monomers", but drops rapidly above some threshold as
shown in Fig. 9. The mean number of monomers per ring, in
contrast to the mean chain length, reaches a plateau near this
crossover. The crossover concentration and the average ring
size in simulations are both somewhat higher than the theoretical predictions of Eq. !9". Although the theoretical model
has been adjusted to give the same pentamer ring/chain ratio
Kcyc!5" as for the tangent hard-sphere simulation model, it
assumes that the equilibrium constant scales as i−5/2 thereafter. Looking at this ratio, plotted in Fig. 10!a" for a simulated
ensemble of rings and chains in dilute solution, we see that
Kcyc drops off approximately as i−5/2 in the absence of attractions, but nearly as i−2 with strong attractions. It is very
probable that neither of these represents the limiting scaling
law; even the apparently “correct” exponent of −5 / 2 is not
appropriate for purely self-avoiding chains in dilute solution,
where we expect Kcyc / i−35/12 in the limit of large i.18,27 In all
FIG. 10. Ratio of ring to chain concentration vs aggregate size i !a" and
mean square radius of gyration !b" of rings with no attractions !open circles"
and ( = 0.470kBT !filled diamonds" at # = 0.01&−3. The straight lines in !a"
indicate power law scaling for reference: i−5/2 !solid" and i−2 !dashed". The
curves in !b" show mean square radius of gyration of chains, scaled by 0.55.
without attractions, the end of a chain is more exposed to
interaction with other chains: when interactions are purely
repulsive, an additional free energy penalty is effectively imposed on the breaking of a bond to create a free end, while
attractive interactions reduce the effective bond strength by
stabilizing the free end. The extreme limit of this type of
effect, where polymerization reduces the ability of monomers to interact via nonbonded interactions, may lead to an
actual maximum in mean aggregate size with increasing #,26
in the present simulations, it has been difficult to reach converged length distributions in the limit of high # and strong
associations where such a maximum might be found.
Downloaded 04 Apr 2007 to 170.140.190.50. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
134906-8
J. Chem. Phys. 126, 134906 !2007"
Pam, Spell, and Kindt
FIG. 11. Mean chain-only and all-aggregate contour lengths in chain+ ring
containing system at Kassoc = 500 000&3. The symbols indicate square-well
depth ( as in Fig. 8. The solid curves are from equilibrium distributions
based on free energy of Eq. !8", while the dashed curve is growth law for
ideal chain-only system from Eq. !10", shown for comparison.
!data not shown" qualitatively similar limiting behavior when
the ideal exponent of −2.5 is replaced by −2.1 or when the
minimum ring size is changed from 5 to 50.
Although the data are noisy—a single large ring persisting through much of a simulation run can influence the average ring size—the trend in Fig. 9 appears to be that attractions lead to slightly larger and more numerous rings, at least
in dilute solution. Figure 10!b" shows that ring dimensions in
dilute solution are influenced by interactions in apparently
identical manner as chain dimensions: the square of the radius of gyration of rings as a function of size scales as 0.55r2g
of the same size chain. The same scaling, with the same
coefficient, appears to hold under the full range of conditions
of concentration and ( investigated. As with chains, the effects of nonbonded interactions on size distribution are diminished at higher # as interchain interactions become more
important. The rings in the present simulations are rarely
observed to undergo topological transitions, although such
transitions should be possible through crankshaft moves that
change the position of a significant segment of a ring; the
range of ring sizes, in any event, is not large enough
to distinguish the known topological effect on ring
dimensions.29
In the ideal model at a given equilibrium concentration
of free monomers, the distribution of chain lengths is the
same whether or not rings are included in the system. Ring
formation only suppresses chain polymerization by absorbing free monomers. The growth law for chains in the ringchain mixture, shown for simulation results and theoretical
calculations in the upper curve of Fig. 11, looks much like
the growth law for the chain-only system displaced by a
constant value of #. When the total mean aggregate size is
calculated, the growth of both chain and ring fractions and
the diminishing ratio of rings to chains combine to give a
growth law that is close to linear, similar to the recently
described behavior of a dipolar system involving a similar
ring-chain equilibrium8 as well as to the theoretically predicted growth law for activated equilibrium polymerization.5
FIG. 12. !Color online" Comparison of *-# isotherms in systems with only
chains !filled circles, solid curves" and chains+ rings !open circles, dashed/
dotted curves", at Kassoc = 500 000&3. The horizontal lines indicate phase coexistence boundaries calculated from fits with parameters given in Table I.
The equilibrium between free monomers and rings can be
regarded as analogous to an equilibrium between activated
and unactivated !nonpolymerizing" forms of the monomer,
suppressing the pool of monomers available to form chains.
Both types of system exhibit a crossover from primarily inactive to primarily active monomer forms at a characteristic
concentration, above which the inactive form stays at a
nearly fixed total concentration. The details of the equilibrium are rather different in the two cases: while in simple
activation, the concentration of inactive monomers is simply
proportional to the concentration of free monomers, in the
case of ring formation it is a nonlinear function that drops to
zero at very low concentrations and rises to an upper bound
that is independent of the polymerization free energy.
The *-# isotherms of systems with and without rings are
shown in Fig. 12. From the simulation data it is clear that the
concentration of the dilute phase at coexistence is shifted
significantly by the presence of rings. The chemical potential
at high # is essentially unchanged by the presence of rings; a
relatively small downward shift in the concentrations of the
higher-density phase can only be inferred from the model
fits. The qualitative similarity to the predictions of Sec. II
provides support for the validity of the trends predicted by
the simple model, which in the limit of large polymerization
energies yields a nonzero limiting critical concentration !see
Appendix". Continuing the analogy to activated equilibrium
polymerization, a nonzero limiting critical concentration is
also predicted for activated polymerization models where the
activation enthalpy is assumed equal to the polymerization
enthalpy !Ahigh model in Ref. 5". In the present case, the
activation step is the conversion of a ring to a chain, which
involves breaking a bond at a free energy penalty that increases with Kassoc.
V. CONCLUSIONS
Monte Carlo simulations confirm the trends predicted by
theory for isotropic phase separation in flexible equilibrium
polymer mixtures: critical monomer concentrations decrease
and critical temperatures rise as the equilibrium constant for
the polymerization increases. The most pronounced effect of
ring formation on phase separation, based on a simple theory
and confirmed through simulation, is to increase the density
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134906-9
J. Chem. Phys. 126, 134906 !2007"
Flexible equilibrium polymers
of the dilute phase at coexistence; a slight decrease in the
density of the concentrated phase is also predicted. The
simple theory predicts that in the limit of high association
constant, the critical density does not vanish when rings are
present, but rather approaches the onset of the ring-chain
transition. This and other features of the association are
analogous to behaviors predicted for activated equilibrium
polymerization events. In dilute solution, nonspecific attractions between unconnected segments of polymer chains raise
the mean chain length above the predictions for an ideal
solution; contour length distributions are distorted away from
a simple exponential. Over the range of systems covered in
the present simulations, the average radius of gyration rg of
rings and chains of a given contour length maintain a constant ratio even as the scaling of rg with length is influenced
by the strength of attractive interactions and the total system
concentration.
$
$
df!c1,c2,c3, . . . "
# f #!ici"
# f #!ici" # f
=%
=%
=
.
d.
#c1
i=1 #!ici" #.
i=1 #c1 #.
!A2"
In solving for the critical condition df 3 / d.3 = 0 of Eq. !9", in
applying the substitution of Eq. !A2" we find
)
( )
d3 f
−2 #.
3 = − c1
d.
#c1
−2
− c−1
1
( )
#.
#c1
−3 2
1
#.
= 0.
2 +
#c1 !1 − ."2
!A3"
The total monomer concentration . is the sum of the concentrations .ch and .r of monomers belonging to chains !including free monomers" and of those belonging to rings. The
relationship between c1 and .ch is given by
$
.ch = % ici1Ki−1 = c1!1 + 2c1K + 3c21K2 ¯ " =
i=1
c1
,
!1 − c1K"2
!A4"
ACKNOWLEDGMENTS
This work was supported by NSF Grant No. CHE0316076. Two of the authors !L.L.S. and L.S.P." acknowledge support from the Summer Undergraduate Research Experience program at Emory University, Howard Hughes
Medical Institute Grant No. 52003727. Acknowledgment is
made to the Cherry L. Emerson Center for Scientific Computation of Emory University for the use of its resources.
with K = Kassoc. The derivatives follow
d.ch
= !1 − Kc1"−2 + 2c1K!1 − c1K"−3 ,
dc1
!A5"
d2.ch 2K!c1K + 2"
=
.
!1 − c1K"4
dc21
!A6"
The corresponding series relating c1 and .r,
APPENDIX: EQUIVALENCE OF CRITICAL POINT AND
RING-CHAIN TRANSITION ONSET
We seek to show that in an equilibrium polymer system
of monomers self-assembling into chains and rings in poor
solvent, whose free energy is described by Eq. !9", the critical volume fraction for phase separation .c coincides with
the onset of the ring-chain transition in the limit of high
association constant Kassoc. In Eq. !9", we have defined the
free energy in terms of the concentrations of aggregates of
arbitrary size. The free energy of the system as a function of
total monomer concentration !or volume fraction for a lattice
model" is obtained through the equilibrium condition that the
distribution of aggregate concentrations minimizes the free
energy under the constraint of constant total monomer concentration. Therefore, partial derivatives of the free energy
with respect to concentrations of aggregates of all sizes must
be related,30
$
f!." = f!c1,c2,c3, . . . " ! % ici = .,
i=1
$
.r = b
%
j=jmin
j−3/2ci1Ki ,
!A7"
does not have a closed-form representation. In the limit
c1K → 1, we will approximate this series as an integral, introducing a constant b0 to compensate for extending the
lower limit of the summation from jmin to zero,
.r , b
/
$
jmin
j−3/2c1j K jdj − b0 , b
/
$
0
j−3/2c1j K jdj − b0 .
!A8"
With this approximation, it can be shown that
#.r
, bc−1
1
#c1
.
'
1/2
−1/2
, bc−1
,
1 ' !1 − c1K"
− ln!c1K"
!A9"
# 2. r
1/2
−3/2
, bc−2
− !− ln!c1K""−1/2$
1 ' #!− ln!c1K""
#c21
#f
#f
=
.
#!ici" #c1
!A1"
Derivatives of the free energy with respect to the volume
fraction . can therefore be conveniently related to derivatives of the free energy with respect to the free monomer
concentration c1,
1/2
−3/2
, bc−2
− !1 − c1K"−1/2$.
1 ' #!1 − c1K"
!A10"
Summing the derivatives with respect to ring and chain
monomer populations, making the definition a 0 1 − c1K, and
assuming .c 0 1, the critical condition #Eq. !A3"$ corresponds to
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134906-10
J. Chem. Phys. 126, 134906 !2007"
Pam, Spell, and Kindt
1
K2!6a−4 + b'1/2K!a−3/2 − a−1/2""
!2a−3 + b'1/2Ka−1/2"3
+
K2
, 1.
!2a−3 + b'1/2Ka−1/2"2
!A11"
At high K, the first term dominates over the second, and the
equation will hold for some critical value of a such that ac
= C!b"K−2/5 for large K, where C!b" is a function independent of K.
Having found the scaling of the critical concentration
with K, we now turn to its relationship to the onset of the
ring-chain transition. At high K, the system will be dominated by rings at low . but will undergo a transition from
ring dominated to chain dominated as total monomer concentration increases. We define the onset of this transition as
the value of . at which the growth rate of the chain population equals the growth rate of the ring population: d.ch / d.
= d.r / d.. As . increases monotonically with c1, this onset
concentration also satisfies
d.ch d.r
=
.
dc1
dc1
!A12"
Equating Eqs. !A5" and !A9" and keeping only leading terms
in K yields
1 − c1K = aonset ,
(
b'1/2
K
2
)
−2/5
.
!A13"
At high K, aonset approaches zero while .r approaches a constant limiting value. We see that aonset and ac show the same
power law scaling with respect to K, with different prefactors
that depend on b. As the absolute difference between aonset
and ac therefore vanishes as K–2/5, and as d.ch / da and
d.r / da are both of order −K1/5, the difference between total
monomer concentrations at the critical point and at the onset
of the ring-chain transition must vanish as K−1/5. Therefore,
in the limit of high association constant, the critical volume
fraction for phase separation .c coincides with the onset of
the ring-chain transition. Finally, we note that the concentration of monomers in chains .ch , K−1a−2 at the critical point
in the mixed ring-chain system scales as K−1/5, consistent
with the scaling of total monomer concentration at the critical point in the chain-only system.
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