Equation of state for macromolecules of variable flexibility in good solvents: a comparison of techniques for Monte Carlo simulations of lattice models.

Abstract

The osmotic equation of state for the athermal bond fluctuation model on the simple cubic lattice is obtained from extensive Monte Carlo simulations. For short macromolecules (chain length N=20 ) we study the influence of various choices for the chain stiffness on the equation of state. Three techniques are applied and compared in order to critically assess their efficiency and accuracy: the "repulsive wall" method, the thermodynamic integration method (which rests on the feasibility of simulations in the grand canonical ensemble), and the recently advocated sedimentation equilibrium method, which records the density profile in an external (e.g., gravitationlike) field and infers, via a local density approximation, the equation of state from the hydrostatic equilibrium condition. We confirm the conclusion that the latter technique is far more efficient than the repulsive wall method, but we find that the thermodynamic integration method is similarly efficient as the sedimentation equilibrium method. For very stiff chains the onset of nematic order enforces the formation of an isotropic-nematic interface in the sedimentation equilibrium method leading to strong rounding effects and deviations from the true equation of state in the transition regime.

Full text

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arXiv:0706.4192v2 [cond-mat.soft] 23 Jul 2007
Equation of State for Macromolecules of Variable Flexibility in
Good Solvents: A Comparison of Techniques for Monte Carlo
Simulations of Lattice Models
V. A. Ivanov, E. A. An, L. A. Spirin, M. R. Stukan,∗
Faculty of Physics, Moscow State University, Moscow 119992, Russia
M. M¨ uller,
Institut f¨ ur Theoretische Physik, Georg-August-Universit¨ at,
Friedrich-Hund-Platz 1, 37077 G¨ ottingen, Germany
W. Paul, and K. Binder
Institut f¨ ur Physik, Johannes-Gutenberg-Universit¨ at,
Staudinger Weg 7, 55099 Mainz, Germany
(Dated: February 1, 2008)
Abstract
The osmotic equation of state for the athermal bond fluctuation model on the simple cubic lattice is
obtained from extensive Monte Carlo simulations. For short macromolecules (chain length N=20) we
study the influence of various choices for the chain stiffness on the equation of state. Three techniques
are applied and compared in order to critically assess their efficiency and accuracy: the “repulsive
wall” method, the thermodynamic integration method (which rests on the feasibility of simulations
in the grand canonical ensemble), and the recently advocated sedimentation equilibrium method,
which records the density profile in an external (e.g. gravitation-like) field and infers, via a local
density approximation, the equation of state from the hydrostatic equilibrium condition. We confirm
the conclusion that the latter technique is far more efficient than the repulsive wall method, but we
find that the thermodynamic integration method is similarly efficient as the sedimentation equilibrium
method. For very stiff chains the onset of nematic order enforces the formation of an isotropic–nematic
interface in the sedimentation equilibrium method leading to strong rounding effects and deviations
from the true equation of state in the transition regime.
∗Current address: Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge CB3 OEL,
UK

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I. INTRODUCTION
Understanding the equation of state of macromolecules in solution has been a longstanding
problem of polymer science, which is both important as a fundamental problem in the statis-
tical mechanics of soft matter1,2,3,4,5,6,7and relevant for various applications of polymers. The
interplay of excluded volume effects2,3,4,6, solvent quality and variable chain stiffness4,8already
is very difficult to describe for macromolecules in very dilute solution9, withstanding an analytic
solution and making the application of computer simulation methods5,10necessary. The exper-
imental situation on measurements of the equation of state of polymer solutions is addressed
in chapter 5 of Ref. 3 and chapter 3 of Ref. 11. Considering semidilute and concentrated so-
lutions, the enthalpic and/or entropic interactions among the polymer chains create nontrivial
correlations in the structure of the solutions involving many chains, and phase transitions such
as phase separation under bad solvent conditions into a dilute polymer solution and a concen-
trated one may occur1,2,12,13,14. Alternatively, under good solvent conditions one may observe
for semiflexible or for stiff chains a transition from an isotropic to a nematic solution when
the polymer concentration increases15,16,17,18,19. Of particular interest is the situation when ne-
matic ordering and the tendency to phase separation under bad solvent conditions compete20.
Preliminary computer simulation studies of a corresponding model21,22gave only rather rough
information on the phase behavior, and it was concluded that the accuracy with which the
osmotic equation of state could be determined needs to be improved.
Thus, there are many reasons why establishing methods that allow the reliable estimation of
the equation of state for various coarse-grained lattice models for polymers is of interest. While
in an off-lattice model the estimation of the pressure tensor from the virial theorem in principle
is straightforward23, it often is not possible to study very large systems with the desirable
accuracy5, and hence simulations of lattice models still have their place5,10. However, for lattice
models different techniques to calculate the osmotic pressure of polymers in solution must be
sought in order to derive their equation of state. The standard method, particularly valuable
for dilute solutions and/or not too large chain lengths, calculates the chemical potential from
the insertion probability of a test chain24,25and the osmotic pressure then follows from standard
thermodynamic integration5,23,26,27. As is well known, insertion of a polymer chain into a volume
containing already other chains is very difficult due to very small acceptance probabilities, and
hence requires advanced Monte Carlo methods26,28,29to be practically feasible.
As an alternative method Dickman30,31proposed the repulsive wall thermodynamic integra-
tion (RWTI) method, which remains applicable also for very dense systems since it works in the
canonic (NV T) ensemble, where the number of chains N in the box remains fixed. However,
this method requires substantial simulation effort, since for each state point in the bulk several
simulation runs with increasing wall–monomer repulsion and subsequent thermodynamic inte-
gration need to be performed. Moreover the RWTI method may suffer from important finite
size effects32.
More recently, it was proposed33to extract the osmotic equation of state from Monte Carlo
simulations where the equilibrium monomer and center-of-mass concentration profiles of lattice
polymers in a gravitation-like potential are computed. From these concentration profiles, the
equation of state can be inferred if a local density approximation like in hydrostatic equilibrium
is invoked. This method has been broadly applied to study the sedimentation equilibrium of
colloidal dispersions containing spherical34, rod-like35or disk-like36particles. More recently,
successful applications to colloid-polymer mixtures37and solutions of block copolymers38or
binary polymer solutions39have been made as well.
Despite these successes, this “sedimentation equilibrium” (SE) method also may have draw-
backs, when other large length scales appear in the system, that compete with the characteris-
tic “sedimentation length”. This length that scales inversely with the “gravitational constant”
characterizing the gravitation-like potential and hence can be made arbitrarily large33, but the
linear dimension of the simulated system in the z-direction in which this gravitation-like force

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acts must then be huge, also. A well-known case, where even the real gravitation potential on
earth substantially disturbs the equation of state is a fluid very close to the gas-liquid critical
point40, since there the correlation length of density fluctuations diverges, and critical fluctua-
tions undisturbed by gravity can occur in the x,y-directions perpendicular to the gravitational
force only40. Other cases, apart from critical points of second-order transitions, where large
length scales arise that are potentially disturbed by the gravitation-like potential are associated
with the formation of thick wetting layers at walls, for instance.
While Addison at al.33did test the SE method against the RWTI method for a few cases
varying the solvent quality from good solvents to theta solvents, finding good agreement, they
considered only fully flexible chains. Thus, we take up this problem but rather consider semi-
flexible chains as well: the possible occurrence of nematically ordered wetting layers at the wall
where the density is largest41or even the occurrence of the isotropic to nematic transition in
the bulk solution of the semiflexible or stiff polymers may complicate matters. The aim of
the present work is to carefully test the SE method against both the RWTI method and the
standard thermodynamic integration method in the grand canonical µV T ensemble (TIµV T
method, µ denoting the chemical potential), and to assess by detailed comparisons both the
efficiency and the accuracy of these methods. As has been explained above, there is need for
accurate simulation methods for the computation of the equation of state of polymer solutions
in the context of various interesting problems.
The plan for the remainder of this paper is as follows. In Sec. II we briefly review the
theoretical basis for the RWTI, TIµV T and SE methods, while Sec. III describes the simulated
model and mentions the types of Monte Carlo moves used. Sec. IV presents comparisons of
the three methods for fully flexible chains over a wide range of densities, while Sec. V presents
our results for variable chain stiffness. Sec. VI contains our conclusions and gives an outlook
to future work.
II.
ELS OF POLYMER SOLUTIONS
METHODS TO CALCULATE THE OSMOTIC PRESSURE FOR LATTICE MOD-
A. Thermodynamic integration in the grand canonical ensemble (TIµV T method)
We consider a system of chains of length N in a simulation box of volume V with periodic
boundary conditions (typically the box has a cubic shape, V = L3, L being the linear dimension)
at given temperature T and chemical potential µ of the chains. The density of polymer chains
in the system, ρ = N/V , with N the average number of chains contained in the simulation box
then is an output of the simulation.
Utilizing the standard relations in the canonical ensemble, µ = (∂(F/V )/∂ρ)T,V and p =
−(∂F/∂V )T,N, where F is the free energy of the system and p the pressure, one easily derives
that
ρ
?
0
where the integration constant in Eq. (1) can be fixed by reference to the low density limit,
where the system behaves like an ideal gas of chains,
p = ρµ −
µ(ρ′)dρ′+ const,(1)
pV = NkBT,π ≡ p/kBT = ρ. (2)
Denoting then the chemical potential of the ideal gas of chains as µid, and defining µex=
µ − µidthe excess chemical potential per chain, Eqs. (1),(2) imply24
π = ρ(1 + µex) −
ρ
?
0
µex(ρ′)dρ′
(3)

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In practice the integral in Eq. (3) is discretized, so that the reduced osmotic pressure πi at
chain density ρiis obtained from the recursion relation
πi≈ πi−1+ (1 + µex
i)ρi− (1 + µex
i−1)ρi−1− (µex
i + µex
i−1)(ρi− ρi−1)/2. (4)
The disadvantages of the method are clearly obvious from Eq. (4): a large number of state points
{µi,T,V } needs to be studied with small differences between µiand µi−1and hence ρiand ρi−1,
so that the discretization error going from Eq. (3) to Eq. (4) is negligible; and an efficient grand
canonical simulation method is needed, so ρiis sampled with sufficient accuracy. For not too
long chains and not too high densities, however, the method is indeed practically useful, and
since periodic boundary conditions are used, the system for large V always is homogeneous,
and the analysis is not hampered by any interfacial effects, which are present inevitably in the
other techniques described below due to their explicit use of walls.
B. The repulsive wall thermodynamic integration (RWTI) method
This method can be implemented both in the grand canonical µV T ensemble and in the
canonical NV T ensemble. While the latter choice has the advantage that no chain insertions
are necessary and hence the method also works for long chains and at high densities, it suffers
from rather large finite size effects32. We have used here both the canonical and the grand
canonical version of the method.
One considers now a box of linear dimensions L×L×H, with periodic boundary conditions
in x and y directions only, while a hard wall is placed both at z = 0 and z = H. Moreover, one
introduces a repulsive potential of strength εwallwhich acts in the first layer adjacent to z = 0
and in the last layer available to the monomers, z = H. As discussed in30,31,32, the osmotic
pressure is then obtained from the fraction of sites, φz(λ), occupied by monomers in the layers
adjacent to both walls, z = 0 and z = H,
π =
1
?
0
dλ
λ
?φz=0(λ) + φz=H(λ)
2
?
,λ ≡ exp(−εwall/kBT). (5)
Again the integration in Eq. (5) is discretized, and 20 different values of λ turned out to be
sufficient to obtain reliable results.
C.The sedimentation equilibrium (SE) method
This method utilizes the canonical ensemble NV T, and one also considers a box of linear
dimensions L×L×H with periodic boundary conditions in x and y directions only, while again
hard walls are used in z-direction at z = 0 and at z = H. An external potential is applied not
at the walls but everywhere in the system33
Uexternal(z) = −mgz = −kBT
a
λgz. (6)
Here m is the mass of a monomer, g is the acceleration due to the gravity-like potential,
a is the lattice spacing, and λg is a dimensionless constant that characterizes the strength
of this gravitational potential, λg = amg/kBT. It is also useful to introduce characteristic
gravitational lengths ξm,ξcm
ξm= a/λg,ξcm= ξm/N. (7)
As shown by Addison et al.33, for large z the density profile of an ideal gas of monomers at the
lattice would follow the standard barometric formula, ρm(z) ∝ exp(−mgz/kBT) = exp(−z/ξm),

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while for an ideal gas of polymer chains the density profile of monomer units ρm(z) = Nρ(z) ∝
exp(−z/ξcm).
While the variation of the density profile for large z, where the system is very dilute and
the ideal gas behavior holds, hence is trivially known, and this knowledge is an important
consistency check of the method, for smaller z the density profile is nontrivial. But from this
profile the osmotic equation of state can be estimated, when one invokes the local density
approximation, such that the equation of hydrostatic equilibrium holds33,34
dp(z)
dz
= −Nmgρ(z), (8)
p(z) being the local osmotic pressure at altitude z. Integration of Eq.(8) yields
π(z) = p(z)/kBT = ξ−1
cm
∞
?
z
ρ(z′)dz′. (9)
Thus one can record both ρ(z) (and the associated monomer profile ρm(z)) and π(z) simul-
taneously, and eliminating z from these relations one obtains the desired equation of state π(ρ)
(or π(ρm), respectively).
Noting that in the local density approximation we expect that the density distribution of
the monomer units satisfies ρm(z) = Nρ(z), we find from Eqs. (7), (9) that π(z) can also be
written as
?
z
However, the validity of the local density approximation needs to be considered carefully.
E.g., near the hard wall at z = 0, ρm(z) may exhibit strong oscillations (“layering”), and also
ρ(z) exhibits a nontrivial structure. Thus it is clear that the local density approximation breaks
down near the hard wall at z = 0, and in fact one should use Eqs. (9),(10) only for z ≥ Rg, the
gyration radius of the chains. Similarly, rapid density variations invalidating the local density
approximation are also expected near an interface between coexisting phases (as it may occur
for bad solvent conditions, for instance). In addition, one must require that on the scale of
∆z = Rgthe change of the potential, Eq. (6), is negligibly small. This implies
π(z) = ξ−1
m
∞
ρm(z′)dz′. (10)
|∆Uexternal(∆z)/kBT| = λgRg/a = Rg/ξm≪ 1.
√N in concentrated solutions, while for stiff chains we have
Rg∝ N. This implies that for stiff chains considerably larger values of ξm(and hence smaller
values of λg) need to be chosen than for flexible ones. Of course, since one must accommodate
in the simulation box the full density profile from a value appropriate for concentrated solutions
near z = 0 down to the dilute regime near z = H, a linear dimension H ≫ ξmis mandatory;
otherwise, the distortion of the density profile due to this upper boundary at z = H leads
to systematic errors as well. Thus, the SE method requires a careful choice of simulation
parameters to avoid such errors and ensure the desired very good accuracy, and the large size
H in z-direction to some extent will reduce the advantage, that the whole equation of state can
be estimated from a single run.
Finally we note that the approach can be generalized to other forms of the external potential
Uexternal(z), different from a gravitation-like potential. E.g., if one uses
(11)
For flexible chains, Rgscales like
U′
external(z) = −kBT
a
λ′
gzκ, (12)