 |
[68]
Understanding
the stiffness of macromolecules: From linear chains to
bottle-brushes
Binder K, Hsu HP, Paul W
Eur. Phys. J. Special Topics
144: 154907 Apr 2
Binder K, Hsu HP, Paul W
Eur. Phys. J. Special Topics
144: 154907 Apr 2
The intrinsic local
stiffness of a polymer is characterized by its persistence
length. However, its traditional definition in terms of
the exponential decay of bond orientational correlations
along the chain backbone is accurate only for Gaussian
phantom-chain-like polymers. Also care is needed to
clarify the conditions when the Kratky-Porod wormlike
chain model is applicable. These problems are elucidated
by Monte Carlo simulations of simple lattice models for
polymers in both d = 2 and d = 3
dimensions. While the asymptotic decay of the bond
orientational correlations for real polymers always is of
power law form, the Kratky-Porod model is found to be
applicable for rather stiff (but not too long) thin
polymers in d = 3 (but not in d = 2).
However, it does not describe thick chains, e.g.,
bottle-brush polymers, where stiffness is due to grafted
flexible side-chains, and the persistence length grows
proportional to the effective thickness of the
bottle-brush. A scaling description of bottle-brushes is
validated by simulations using the bond fluctuation
model.
[67]
Static
and dynamic properties of large polymer melts in
equilibrium
Hsu HP, Kremer K
Journal of Chemical Physics
144: 154907 Apr 2
Hsu HP, Kremer K
Journal of Chemical Physics
144: 154907 Apr 2
We present a
detailed study of the static and dynamic behaviors of long
semiflexible polymer chains in a melt. Starting from
previously obtained fully equilibrated high molecular
weight polymer melts [G. Zhang et al., ACS Macro Lett. 3,
198 (2014)], we investigate their static and dynamic
scaling behaviors as predicted by theory. We find that for
semiflexible chains in a melt, results of the mean square
internal distance, the probability distributions of the
end-to-end distance, and the chain structure factor are
well described by theoretical predictions for ideal
chains. We examine the motion of monomers and chains by
molecular dynamics simulations using the ESPResSo++
package. The scaling predictions of the mean squared
displacement of inner monomers, center of mass, and
relations between them based on the Rouse and the
reptation theory are verified, and related characteristic
relaxation times are determined. Finally, we give evidence
that the entanglement length N-e,N- PPA as determined by a
primitive path analysis (PPA) predicts a plateau modulus,
G(N)(0) = 4/5 (rho k(B)T/N-e), consistent with stresses
obtained from the Green-Kubo relation. These
comprehensively characterized equilibrium structures,
which offer a good compromise between flexibility, small
Ne, computational efficiency, and small deviations from
ideality, provide ideal starting states for future
non-equilibrium studies. Published by AIP Publishing.
[66]
Semiflexible
macromolecules in quasi-one-dimensional confinement:
Discrete versus continuous bond angles
Huang, AQ, Hsu HP, Bhattacharya A, Binder K
Journal of Chemical Physics
143: 243102December 28 , 2015
Huang, AQ, Hsu HP, Bhattacharya A, Binder K
Journal of Chemical Physics
143: 243102
The conformations of
semiflexible polymers in two dimensions confined in a
strip of width D are studied by computer simulations,
investigating two different models for the mechanism by
which chain stiffness is realized. One model (studied by
molecular dynamics) is a bead-spring model in the
continuum, where stiffness is controlled by a bond angle
potential allowing for arbitrary bond angles. The other
model (studied by Monte Carlo) is a self-avoiding walk
chain on the square lattice, where only discrete bond
angles (0 degrees and +/- 90 degrees) are possible, and
the bond angle potential then controls the density of
kinks along the chain contour. The first model is a crude
description of DNA-like biopolymers, while the second
model (roughly) describes synthetic polymers like alkane
chains. It is first demonstrated that in the bulk the
crossover from rods to self-avoiding walks for both models
is very similar, when one studies average chain linear
dimensions, transverse fluctuations, etc., despite their
differences in local conformations. However, in
quasi-one-dimensional confinement two significant
differences between both models occur: (i) The persistence
length (extracted from the average cosine of the bond
angle) gets renormalized for the lattice model when D gets
less than the bulk persistence length, while in the
continuum model it stays unchanged. (ii) The monomer
density near the repulsive walls for semiflexible polymers
is compatible with a power law predicted for the
Kratky-Porod model in the case of the bead-spring model,
while for the lattice case it tends to a nonzero constant
across the strip. However, for the density of chain ends,
such a constant behavior seems to occur for both models,
unlike the power law observed for flexible polymers. In
the regime where the bulk persistence length l(p) is
comparable to D, hairpin conformations are detected, and
the chain linear dimensions are discussed in terms of a
crossover from the Daoud/De Gennes "string of
blobs"-picture to the flexible rod picture when D
decreases and/or the chain stiffness increases.
Introducing a suitable further coarse-graining of the
chain contours of the continuum model, direct estimates
for the deflection length and its distribution could be
obtained.
[65]
Semiflexible
polymer brushes and the brush-mushroom crossover
Egorov SA, Hsu HP, Milchev A, Binder K
Soft Matter
11: 2604Feb 04, 2015
Egorov SA, Hsu HP, Milchev A, Binder K
Soft Matter
11: 2604
Semiflexible polymers
end-grafted to a repulsive planar substrate under good
solvent conditions are studied by scaling arguments,
computer simulations, and self-consistent field theory.
Varying the chain length N, persistence length l(p), and
grafting density sigma(g), the chain linear dimensions and
distribution functions of all monomers and of the free chain
ends are studied. Particular attention is paid to the limit
of very small sigma(g), where the grafted chains behave as "
mushrooms" no longer interacting with each other. Unlike a
flexible mushroom, which has a self-similar structure from
the size (a) of an effective monomer up to the mushroom
height (h/a proportional to N-v, v approximate to 3/5), a
semiflexible mushroom (like a free semiflexible chain)
exhibits three different scaling regimes, h/a proportional
to N for contour length L proportional to Na < l(p), a
Gaussian regime, h/a proportional to (Ll(p))(1/2)/a for l(p)
<< L << R* proportional to (l(p)(2)/a), and a
regime controlled by excluded volume, h/a f
(l(p)/a)N-1/5(v). The semiflexible brush is predicted to
scale as h/a f (l(p)a sigma(g))N-1/3 in the excluded volume
regime, and h/a f (l(p)a(3)sigma(2))N-1/4 in the Gaussian
regime. Since in the volume taken by a semiflexible mushroom
excluded-volume interactions are much weaker in comparison
to a flexible mushroom, there occurs an additional regime
where semiflexible mushrooms overlap without significant
chain stretching. Moreover, since the size of a semiflexible
mushroom is much larger than the size of a flexible mushroom
with the same N, the crossover from mushroom to brush
behavior is predicted to take place at much smaller
densities than for fully flexible chains. The numerical
results, however, confirm the scaling predictions only
qualitatively; for chain lengths that are relevant for
experiments, often intermediate effective exponents are
observed due to extended crossovers.
Full Text: Soft
Matter[64]
Lattice
Monte Carlo simulations of polymer melts
Hsu HP
Journal of Chemical Physics
141: 234901December 21, 2014
Hsu HP
Journal of Chemical Physics
141: 234901
We use Monte Carlo simulations to study
polymer melts consisting of fully flexible and moderately
stiff chains in the bond fluctuation model at a volume
fraction 0.5. In order to reduce the local density
fluctuations, we test a pre-packing process for the
preparation of the initial configurations of the polymer
melts, before the excluded volume interaction is switched on
completely. This process leads to a significantly faster
decrease of the number of overlapping monomers on the
lattice. This is useful for simulating very large systems,
where the statistical properties of the model with a
marginally incomplete elimination of excluded volume
violations are the same as those of the model with strictly
excluded volume. We find that the internal mean square
end-to-end distance for moderately stiff chains in a melt
can be very well described by a freely rotating chain model
with a precise estimate of the bond-bond orientational
correlation between two successive bond vectors in
equilibrium. The plot of the probability distributions of
the reduced end-to-end distance of chains of different
stiffness also shows that the data collapse is excellent and
described very well by the Gaussian distribution for ideal
chains. However, while our results confirm the systematic
deviations between Gaussian statistics for the chain
structure factor S-c(q) [minimum in the Kratky-plot] found
by Wittmer et al. [EPL 77, 56003 (2007)] for fully flexible
chains in a melt, we show that for the available chain
length these deviations are no longer visible, when the
chain stiffness is included. The mean square bond length and
the compressibility estimated from collective structure
factors depend slightly on the stiffness of the chains.
Full Text: JCP
or cond-mat[63]
Monte
Carlo simulations of lattice models for single polymer
systems
Hsu HP
Journal of Chemical Physics
141: 164903 October 28, 2014
Hsu HP
Journal of Chemical Physics
141: 164903 October 28, 2014
Single linear polymer chains in dilute
solutions under good solvent conditions are studied by Monte
Carlo simulations with the pruned-enriched Rosenbluth method
up to the chain length N similar to O(10(4)). Based on the
standard simple cubic lattice model (SCLM) with fixed bond
length and the bond fluctuation model (BFM) with bond
lengths in a range between 2 and root 10, we investigate the
conformations of polymer chains described by self-avoiding
walks on the simple cubic lattice, and by random walks and
non-reversible random walks in the absence of excluded
volume interactions. In addition to flexible chains, we also
extend our study to semiflexible chains for different
stiffness controlled by a bending potential. The persistence
lengths of chains extracted from the orientational
correlations are estimated for all cases. We show that
chains based on the BFM are more flexible than those based
on the SCLM for a fixed bending energy. The microscopic
differences between these two lattice models are discussed
and the theoretical predictions of scaling laws given in the
literature are checked and verified. Our simulations clarify
that a different mapping ratio between the coarse-grained
models and the atomistically realistic description of
polymers is required in a coarse-graining approach due to
the different crossovers to the asymptotic behavior.
Full Text: JCP
or cond-mat[62]
Coil-bridge transition in a single polymer chain as
an unconventional phase transition: Theory and simulation
Klushin LI, Skvortsov AM, Polotsky AA, Hsu
HP, Binder K
Journal of Chemical Physics
140: 204908 May 58, 2014
Journal of Chemical Physics
140: 204908 May 58, 2014
The coil-bridge transition in a self-avoiding
lattice chain with one end fixed at height H above the attractive planar surface is investigated by theory and Monte Carlo
simulation. We focus on the details of the first-order phase transition between the
coil state at large height H
⩾ H tr and a bridge state
at H ⩽ H tr , where H tr corresponds to the
coil-bridge transition
point. The equilibrium properties of the chain were
calculated using the Monte Carlo pruned-enriched Rosenbluth
method in the moderate adsorption
regime at (H/Na) tr ⩽ 0.27 where N is the number of monomer units of linear size
a. An analytical theory of the coil-bridge transition for lattice chains
with excluded volume interactions is presented in this
regime. The theory
provides an excellent quantitative description of numerical
results at all heights, 10 ⩽ H/a ⩽ 320 and all chain lengths 40
< N < 2560 without
free fitting parameters. A simple theory taking into account
the effect of finite extensibility of the lattice chain in
the strong adsorption
regime at (H/Na) tr ⩾ 0.5 is presented.
We discuss some unconventional properties of the coil-bridge
transition: the absence
of phase coexistence, two micro-phases involved in the
bridge state, and abnormal behavior in the microcanonical
ensemble.
[59]
Pulling Single Adsorbed Bottle-Brush Polymers off a Flat Surface: A Monte Carlo Simulation
Force versus extension behavior of
flexible chains and semiflexible bottle-brush polymers
adsorbed from a good solvent on a planar substrate is
studied by Monte Carlo simulation of the bond fluctuation
model. The properties of the polymers (fraction of adsorbed
monomers, height of the free end of the macromolecule above
the surface, gyration radius components parallel and
perpendicular to the surface, etc.) are studied in full
thermal equilibrium as well as out of equilibrium, varying
the pulling speed over 3 orders of magnitude. The
equilibrium extension vs force curve reveals that the
transition force (where force-induced desorption occurs)
increases with increasing side chain length N of the bottle
brushes, while further extension is almost independent of N,
and can be described by a model due to Odijk, in agreement
with a recent experiment.
[58]
Semiflexible macromolecules with discrete bond angles confined in nanoslits: A Monte Carlo test of scaling concepts
Hsu HP, Binder K
Macromolecules
46: 8017 September 23, 2013
Abstract:46: 8017 September 23, 2013
Single semiflexible polymer chains confined in a planar slit geometry between parallel nonadsorbing repulsive walls a distance D apart are studied by Monte Carlo simulations of a lattice model, for the case of good solvent conditions. The polymers are modeled as self-avoiding walks on the simple cubic lattice, where every 90° kink requires a bending energy εb. For small qb = exp(−εb/kBT) the model has a large persistence length (given by ≈ 1/(4qb) in the bulk three-dimensional dilute solution, in units of the lattice spacing). Unlike the popular Kratky–Porod model of worm-like chains, this model takes both excluded volume into account and approximates the fact that bond angles between subsequent carbon–carbon bonds of real chains are (almost) restricted to large nonzero values, and the persistence length is controlled by torsional potentials. So the typical local conformation in the model is a straight sequence of (on average) lp bonds (roughly corresponding e.g. to an all-trans sequence of an alkane chain) followed by a 90° kink. While under weak confinement (D lp) the model (for very long chains) still is compatible with the Daoud–de Gennes scaling theory, for strong confinement (D ≤ lp) strong deviations from the predictions based on the Kratky–Porod model are found.
[57]
Semi-flexible polymer chains in quasi-one-dimensional confinement: a Monte Carlo study on the square lattice
Hsu HP, Binder K
Soft Matter
9: 10512 June 26, 2013
9: 10512 June 26, 2013
Abstract:
Single semi-flexible polymer chains are modeled as self-avoiding walks (SAWs) on the square lattice with every 90° kink requiring an energy εb. While for εb = 0 this is the ordinary SAW, varying the parameter qb = exp(−εb/kBT) allows the variation of the effective persistence length
p over about
two decades. Using the pruned-enriched Rosenbluth method
(PERM), chain lengths up to about N = 105
steps can be studied. In previous work it has already been
shown that for contour lengths L = Nb (the bond
length b is the
lattice spacing) of order p a smooth
crossover from rods to two-dimensional self-avoiding walks
occurs, with radii R ∝ p1/4L3/4,
the Gaussian regime predicted by the Kratky–Porod model for
worm-like chains being completely absent. In the present
study, confinement of such chains in strips of width D
is considered, varying D from 4 to 320 lattice
spacings. It is shown that for narrow strips (D <
p) the
effective persistence length of the chains (in the direction
parallel to the confining boundaries) scales like p2/D,
and R‖ ∝ L (with
a pre-factor of order unity). For very wide strips, D
≫ p, the
two-dimensional SAW behavior prevails for chain lengths up to
Lcross ∝ p(D/p)4/3,
while for L ≫ Lcross
the chain is a string of blobs of diameter D, i.e.
R‖ ∝ L(p/D)1/3.
In the regime D < p, the chain is
a sequence of straight sequences with length of the order p2/D
parallel to the boundary, separated by sequences with length
< D perpendicular to the boundary; thus Odijk's
deflection length plays no role for discrete bond angles.Full Text: Soft Matter
[56]
Estimation of persistence lengths of semiflexible polymers: Insight from simulations
Hsu HP, Paul W, Binder K
Polymer Science
Ser. c
55: 39 September 1, 2013
55: 39 September 1, 2013
Abstract:
The persistence length of macromolecules is one of their basic characteristics, describing their intrinsic local stiffness. However, it is difficult to extract this length from physical properties of the polymers, different recipes may give answers that disagree with each other. Monte Carlo simulations are used to elucidate this problem, giving a comparative discussion of two lattice models, the self-avoiding walk model extended by a bond bending energy, and bottle-brush polymers described by the bond fluctuation model. The conditions are discussed under which a description of such macromolecules by Kratky-Porod worm-like chains holds, and the question to what extent the persistence length depends on external conditions (such as solvent quality) is considered. The scattering function of semiflexible polymers is discussed in detail, a comparison to various analytic treatments is given, and an outlook to experimental work is presented.
[55]
Effect of Chain Stiffness on the Adsorption
Transition of Polymers
Hsu HP, Binder K
Macromolecules
87: 022604 Febuary 27, 2013
87: 022604 Febuary 27, 2013
Abstract:
The coil-bridge transition in a self-avoiding lattice chain with one end fixed at height H above the attractive planar surface is investigated by theory and Monte Carlo simulation. We focus on the details of the first-order phase transition between the coil state at large height H ⩾ H tr and a bridge state at H ⩽ H tr , where H tr corresponds to the coil-bridge transition point. The equilibrium properties of the chain were calculated using the Monte Carlo pruned-enriched Rosenbluth method in the moderate adsorption regime at (H/Na) tr ⩽ 0.27 where N is the number of monomer units of linear size a. An analytical theory of the coil-bridge transition for lattice chains with excluded volume interactions is presented in this regime. The theory provides an excellent quantitative description of numerical results at all heights, 10 ⩽ H/a ⩽ 320 and all chain lengths 40 < N < 2560 without free fitting parameters. A simple theory taking into account the effect of finite extensibility of the lattice chain in the strong adsorption regime at (H/Na) tr ⩾ 0.5 is presented. We discuss some unconventional properties of the coil-bridge transition: the absence of phase coexistence, two micro-phases involved in the bridge state, and abnormal behavior in the microcanonical ensemble.
Polymers grafted with one chain end to an impenetrable flat hard wall which attracts the monomers with a short-range adsorption potential (of strength ε) are studied by large scale Monte Carlo simulations, using the pruned–enriched Rosenbluth method (PERM). Chain lengths up to N = 25600 steps are considered, and the intrinsic flexibility of the chain is varied via an energy penalty for nonzero bond angles, εb. Choosing qb = exp(−εb/kBT) in the range from qb = 1 (fully flexible chains) to qb = 0.005 (rather stiff chains with a persistence length of about
lattice spacings), the adsorption transition is
found to vary from about ε/kBTc
≈ 0.286 to ε/kBTc
≈ 0.011, confirming the theoretical expectation that 
for large 
. The simulation data are compatible with a
continuous adsorption transition for all finite values of 
, while in the rigid rod limit (
) a first order transition seems to emerge. Scaling
predictions and blob concepts on the structure of weakly
adsorbed semiflexible polymers absorbed at interfaces are
briefly discussed.[54]
Adsorption of a single polymer chain on a surface:
Effects of the potential range
Klushin LI, Polotsky
AA, Hsu HP, Markelov, DA, Binder K, Skvortsov AM
Physics Review E
87: 022604 Febuary 27, 2013
87: 022604 Febuary 27, 2013
Abstract:
We investigate the effects of the range of adsorption potential on the equilibrium behavior of a single polymer chain end-attached to a solid surface. The exact analytical theory for ideal lattice chains interacting with a planar surface via a box potential of depth U and width W is presented and compared to continuum model results and to Monte Carlo (MC) simulations using the pruned-enriched Rosenbluth method for self-avoiding chains on a simple cubic lattice. We show that the critical value U-c corresponding to the adsorption transition scales as W-1/nu,where the exponent nu = 1/2 for ideal chains and nu approximate to 3/5 for self-avoiding walks. Lattice corrections for finite W are incorporated in the analytical prediction of the ideal chain theory U-c approximate to (pi(2)/24)(W + 1/2)(-2) and in the best-fit equation for the MC simulation data U-c = 0.585(W + 1/2)(-5/3). Tail, loop, and train distributions at the critical point are evaluated by MC simulations for 1 <= W <= 10 and compared to analytical results for ideal chains and with scaling theory predictions. The behavior of a self-avoiding chain is remarkably close to that of an ideal chain in several aspects. We demonstrate that the bound fraction theta and the related properties of finite ideal and self-avoiding chains can be presented in a universal reduced form: theta(N, U, W) = theta(NUc, U/U-c). By utilizing precise estimations of the critical points we investigate the chain length dependence of the ratio of the normal and lateral components of the gyration radius. Contrary to common expectations this ratio attains a limiting universal value < R-g perpendicular to(2)>/< R-g parallel to(2)> = 0.320 +/- 0.003 only at N similar to 5000. Finite-N corrections for this ratio turn out to be of the opposite sign for W = 1 and for W >= 2. We also study the N dependence of the apparent crossover exponent phi(eff)(N). Strong corrections to scaling of order N-0.5 are observed, and the extrapolated value phi = 0.483 +/- 0.003 is found for all values of W. The strong correction to scaling effects found here explain why for smaller values of N, as used in most previous work, misleadingly large values of phi(eff)(N) were identified as the asymptotic value for the crossover exponent.
[53]
Scattering function of semiflexible polymer chains
under good solvent conditions
Hsu HP, Paul W, Binder K
Journal of
Chemical Physics
137: 174902 November 6, 2012
137: 174902 November 6, 2012
Abstract:
Using the pruned-enriched Rosenbluth Monte Carlo algorithm, the scattering functions of semiflexible macromolecules in dilute solution under good solvent conditions are estimated both in d = 2 and d = 3 dimensions, considering also the effect of stretching forces. Using self-avoiding walks of up to N = 25 600 steps on the square and simple cubic lattices, variable chain stiffness is modeled by introducing an energy penalty εb for chain bending; varying qb = exp (−εb/kBT) from qb = 1 (completely flexible chains) to qb = 0.005, the persistence length can be varied over two orders of magnitude. For unstretched semiflexible chains, we test the applicability of the Kratky-Porod worm-like chain model to describe the scattering function and discuss methods for extracting persistence length estimates from scattering. While in d = 2 the direct crossover from rod-like chains to self-avoiding walks invalidates the Kratky-Porod description, it holds in d = 3 for stiff chains if the number of Kuhn segments nK does not exceed a limiting value nK* (which depends on the persistence length). For stretched chains, the Pincus blob size enters as a further characteristic length scale. The anisotropy of the scattering is well described by the modified Debye function, if the actual observed chain extension ⟨X⟩ (end-to-end distance in the direction of the force) as well as the corresponding longitudinal and transverse linear dimensions ⟨X2⟩ − ⟨X⟩2, 〈Rg,⊥2〉 are used.
[52]
Stretching semiflexible polymer chains: Evidence
for the importance of excluded volume effects from Monte
Carlo simulation
Hsu HP, Binder K
Journal of
Chemical Physics
136: 024901 January 14, 2012
136: 024901 January 14, 2012
Abstract:
Semiflexible macromolecules in dilute solution under very good solvent conditions are modeled by self-avoiding walks on the simple cubic lattice (d = 3 dimensions) and square lattice (d = 2 dimensions), varying chain stiffness by an energy penalty εb for chain bending. In the absence of excluded volume interactions, the persistence length ℓp of the polymers would then simply be ℓp = ℓb(2d−2)−1qb−1 with qb = exp (−εb/kBT), the bond length ℓb being the lattice spacing, and kBT is the thermal energy. Using Monte Carlo simulations applying the pruned-enriched Rosenbluth method (PERM), both qb and the chain length N are varied over a wide range (0.005 ⩽ qb ⩽ 1, N ⩽ 50 000), and also a stretching force f is applied to one chain end (fixing the other end at the origin). In the absence of this force, in d = 2 a single crossover from rod-like behavior (for contour lengths less than ℓp) to swollen coils occurs, invalidating the Kratky-Porod model, while in d = 3 a double crossover occurs, from rods to Gaussian coils (as implied by the Kratky-Porod model) and then to coils that are swollen due to the excluded volume interaction. If the stretching force is applied, excluded volume interactions matter for the force versus extension relation irrespective of chain stiffness in d = 2, while theories based on the Kratky-Porod model are found to work in d = 3 for stiff chains in an intermediate regime of chain extensions. While for qb ≪ 1 in this model a persistence length can be estimated from the initial decay of bond-orientational correlations, it is argued that this is not possible for more complex wormlike chains (e.g., bottle-brush polymers). Consequences for the proper interpretation of experiments are briefly discussed.
[51]
Computer simulation of bottle-brush polymers with flexible backbone: Good solvent versus theta solvent conditions
PE, Hsu HP, Paul W, Binder K
Journal of Chemical Physics
135: 164903 October 31, 2011
By molecular dynamics
simulation of a coarse-grained bead-spring-type model for a
cylindrical molecular brush with a backbone chain of Nb effective monomers to
which with grafting density σ side chains with N effective monomers are tethered,
several characteristic length scales are studied for
variable solvent quality. Side chain lengths are in the
range 5 ⩽ N ⩽ 40, backbone
chain lengths are in the range 50 ⩽ Nb ⩽
200, and we perform a comparison to results for the bond
fluctuation model on the simple cubic lattice (for which
much longer chains are accessible, Nb ⩽
1027, and which corresponds to an athermal, very good,
solvent). We obtain linear dimensions of the side chains and
the backbone chain and discuss their N-dependence
in terms of power laws and the associated effective
exponents. We show that even at the theta point the side
chains are considerably stretched, their linear dimension
depending on the solvent quality only weakly. Effective
persistence lengths are extracted both from the
orientational correlations and from the backbone end-to-end
distance; it is shown that different measures of the
persistence length (which would all agree for Gaussian
chains) are not mutually consistent with each other and
depend distinctly both on Nb
and the solvent quality. A brief discussion of pertinent
experiments is given.
[50]
Scaling behaviour of lattice animals at the upper critical dimension
von Ferber C, Foster D, Hsu HP, Kenna R
The European Physical Journal B
83: 245-249 15 September, 2011
We perform numerical
simulations of the lattice-animal problem at the upper
critical dimension d = 8 on hypercubic lattices in
order to investigate logarithmic corrections to scaling
there. Our stochastic sampling method is based on the
pruned-enriched Rosenbluth method (PERM), appropriate to
linear polymers, and yields high statistics with animals
comprised of up to 8000 sites. We estimate both the
partition sums (number of different animals) and the radii
of gyration. We re-verify the Parisi-Sourlas prediction for
the leading exponents and compare the logarithmic-correction
exponents to two partially differing sets of predictions
from the literature. Finally, we propose, and test, a new
Parisi-Sourlas-type scaling relation appropriate for the
logarithmic-correction exponents.
Full Text: EPJB or cond-mat
[49]
Breakdown of the Kratky-Porod Wormlike Chain Model for Semiflexible Polymers in Two Dimensions
Hsu HP, Paul W, Binder K
Europhysics Letters
95: 68004 September 6, 2011
By large-scale Monte Carlo
simulations of semiflexible polymers in d=2
dimensions the applicability of the Kratky-Porod model is
tested. This model is widely used as "standard model" for
describing conformations and force vs. extension
curves of stiff polymers. It is shown that semiflexible
polymers in d=2 show a crossover from hard rods to
self-avoiding walks, the intermediate Gaussian regime
(implied by the Kratky-Porod model) is completely absent.
Hence the latter can also describe force vs.
extension curves only if the contour length is only a few
times larger than the persistence length. Consequences for
experiments on biopolymers at interfaces are briefly
discussed.
[48]
A Review of Monte Carlo Simulations of Polymers with PERM
Hsu HP, Grassberger P
Journal of Statistical Physics
144: 597-637 July 20, 2011
In this review, we describe
applications of the pruned-enriched Rosenbluth method
(PERM), a sequential Monte Carlo algorithm with resampling,
to various problems in polymer physics. PERM produces
samples according to any given prescribed weight
distribution, by growing configurations step by step with
controlled bias, and correcting “bad” configurations by
“population control”. The latter is implemented, in contrast
to other population based algorithms like e.g. genetic
algorithms, by depth-first recursion which avoids storing
all members of the population at the same time in computer
memory. The problems we discuss all concern single polymers
(with one exception), but under various conditions:
Homopolymers in good solvents and at the Θ point,
semi-stiff polymers, polymers in confining geometries,
stretched polymers undergoing a forced globule-linear
transition, star polymers, bottle brushes, lattice animals
as a model for randomly branched polymers, DNA melting, and
finally—as the only system at low temperatures, lattice
heteropolymers as simple models for protein folding. PERM is
for some of these problems the method of choice, but it can
also fail. We discuss how to recognize when a result is
reliable, and we discuss also some types of bias that can be
crucial in guiding the growth into the right directions.
[47]
Structure of Bottle brush Polymers on Surfaces: Weak versus Strong Adsorption
Hsu HP, Paul W, Binder K
Journal of Physical Chemistry B
DOI: 10.1021/jp204006z
Abstract:
Large-scale
Monte Carlo simulations are presented for a
coarse-grained model of cylindrical molecular brushes
adsorbed on a flat structureless substrate, varying both
the chain length N of the side chains and the
backbone chain length Nb. For the case of good
solvent conditions, both the cases of weak adsorption
(only 10 to 15% of the monomers being bound to the
surface) and strong adsorption (
40% of the monomers being
bound to the surface, forcing the bottle brush into an
almost 2D conformation) are studied. We focus on the
scaling of the total linear dimensions of the
cylindrical brush with both chain lengths N and Nb, demonstrating a crossover
from rod-like behavior (for not very large Nb) to the scaling of 2D
self-avoiding walks. Despite the fact that snapshot
pictures suggest a “worm-like” picture as a
coarse-grained description of such cylindrical brushes,
the Kratky–Porod worm-like chain model fails because
there is no regime where Gaussian statistics applies. We
compare the stiffness (orientational correlations of
backbone bonds, persistence length estimates, etc.) of
the adsorbed bottle brush polymers with their
corresponding 3D nonadsorbed counterparts. Consequences
for the discussion of pertinent experiments are briefly
discussed.
40% of the monomers being
bound to the surface, forcing the bottle brush into an
almost 2D conformation) are studied. We focus on the
scaling of the total linear dimensions of the
cylindrical brush with both chain lengths N and Nb, demonstrating a crossover
from rod-like behavior (for not very large Nb) to the scaling of 2D
self-avoiding walks. Despite the fact that snapshot
pictures suggest a “worm-like” picture as a
coarse-grained description of such cylindrical brushes,
the Kratky–Porod worm-like chain model fails because
there is no regime where Gaussian statistics applies. We
compare the stiffness (orientational correlations of
backbone bonds, persistence length estimates, etc.) of
the adsorbed bottle brush polymers with their
corresponding 3D nonadsorbed counterparts. Consequences
for the discussion of pertinent experiments are briefly
discussed.Full Text: J. Phys. Chem. B
[46]
A fast
Monte Carlo algorithm for studying bottle-brush polymers
Hsu HP, Paul W
Computer Physics Communications
182: 2115-2121 October 25, 2011
Hsu HP, Paul W
Computer Physics Communications
182: 2115-2121 October 25, 2011
Abstract:
Obtaining reliable estimates of the statistical
properties of complex macromolecules by computer
simulation is a task that requires high computational
effort as well as the development of highly efficient
simulation algorithms. We present here an algorithm
combining local moves, the pivot algorithm, and an
adjustable simulation lattice box for simulating dilute
systems of bottle-brush polymers with a flexible
backbone and flexible side chains under good solvent
conditions. Applying this algorithm to the bond
fluctuation model, very precise estimates of the mean
square end-to-end distances and gyration radii of the
backbone and side chains are obtained, and the
conformational properties of such a complex
macromolecule are studied. Varying the backbone length
(from Nb=67 to Nb=1027), side chain length (from
N=0 to N=24 or 48), the scaling
predictions for the backbone behavior as well as the
side chain behavior are checked. We are also able to
give a direct comparison of the structure factor between
experimental data and the simulation results.
[45]
New Development of Monte Carlo Techniques for
Studying Bottle-brush polymers
Hsu HP
Physics Procedia
15: 44-53 July 22, 2011
Hsu HP
Physics Procedia
15: 44-53 July 22, 2011
Due to the complex
characteristics of bottle-brush polymers, it became a
challenge to develop an e_cient algorithm for studying
such macromolecules under various solvent conditions or
some constraints in the space by using computer
simulations. In the limit of a bottle-brush polymer with a
rather sti_ backbone (straight rigid backbone), we
generalize the variant of the biased chain growth
algorithm, the pruned-enriched Rosenbluth method, for
simulating polymers with complex architecture, from star
polymers to bottle-brush polymers, on the simple cubic
lattice. With the high statistics of our Monte Carlo
results, we check the theoretical predictions of side
chain behavior and radial monomer density profile. For the
comparison of the experimental data for bottle-brush
polymers with a flexible backbone and flexible side
chains, based on the bond fluctuation model we propose
another fast Monte Carlo algorithm combining the local
moves, the pivot move, and an adjustable simulation
lattice box. By monitoring the autocorrelation functions
of gyration radii for the side chains and for the
backbone, we see that for fixed side chain length there is
no change in the behavior of these two functions as the
backbone length increases. Our extensive results cover the
range which is accessible for the comparison to
experimental data and for the checking of the
theoretically predicted scaling laws.
Full Text: Physics Procedia or cond-mat
[44]
Understanding
the Multiple Length Scales Describing the Structure of
Bottle-brush Polymers by Monte Carlo Simulation Methods
Hsu HP, Paul W, Binder K
Macromolecular Theory and Simulations
20: 510-525 August 25, 2011
Hsu HP, Paul W, Binder K
Macromolecular Theory and Simulations
20: 510-525 August 25, 2011
Abstract:
Bottle-brush polymers contain a long flexible macromolecule as a backbone to which flexible side chains are grafted. Through the choice of the grafting density and the length of the side chains the local stiffness of this cylindrical molecular brush can be controlled, but a quantitative understanding of these phenomena is lacking. Monte Carlo simulation results are presented and discussed which address this issue, extracting mesoscopic length scales (such as the cross-sectional radius, persistence length, and contour length of these objects). Large-scale simulations of the bond fluctuation model are combined with simulations of the simple self-avoiding walk (SAW) model with flexibility controlled by a bond-angle potential, using the pruned-enriched Rosenbluth algorithm. It is shown that under good solvent conditions the bottle-brush polymers never display a pre-asymptotic Gaussian regime that would be described by the Kratky–Porod worm-like chain model, unlike the semiflexible SAW model. Implications of these results for the proper interpretation of experiments are discussed.
Bottle-brush polymers contain a long flexible macromolecule as a backbone to which flexible side chains are grafted. Through the choice of the grafting density and the length of the side chains the local stiffness of this cylindrical molecular brush can be controlled, but a quantitative understanding of these phenomena is lacking. Monte Carlo simulation results are presented and discussed which address this issue, extracting mesoscopic length scales (such as the cross-sectional radius, persistence length, and contour length of these objects). Large-scale simulations of the bond fluctuation model are combined with simulations of the simple self-avoiding walk (SAW) model with flexibility controlled by a bond-angle potential, using the pruned-enriched Rosenbluth algorithm. It is shown that under good solvent conditions the bottle-brush polymers never display a pre-asymptotic Gaussian regime that would be described by the Kratky–Porod worm-like chain model, unlike the semiflexible SAW model. Implications of these results for the proper interpretation of experiments are discussed.
[43]
Polymer
Chain Stiffness vs. Excluded Volume: A Monte Carlo Study of
the Crossover Towards the Worm-like Chain Model
Hsu HP, Paul W, Binder K
EUROPHYS LETTER
92: 28003 (6 pages) 15 November 2010
Hsu HP, Paul W, Binder K
EUROPHYS LETTER
92: 28003 (6 pages) 15 November 2010
Abstract:
When the local intrinsic stiffness of a polymer chain varies over a wide range, one can observe both a crossover from rigid-rod–like behavior to (almost) Gaussian random coils and a further crossover towards self-avoiding walks in good solvents. Using the pruned-enriched Rosenbluth method (PERM) to study self-avoiding walks of up to Nb=50000 steps and variable flexibility, the applicability of the Kratky-Porod model is tested. Evidence for non-exponential decay of the bond-orientational correlations ⟨cos θ(s)⟩ for large distances s along the chain contour is presented, irrespective of chain stiffness. For bottle-brush polymers on the other hand, where experimentally stiffness is varied via the length of side-chains, it is shown that these cylindrical brushes (with flexible backbones) are not described by the Kratky-Porod worm-like chain model, since their persistence length is (roughly) proportional to their cross-sectional radius, for all conditions of practical interest.
When the local intrinsic stiffness of a polymer chain varies over a wide range, one can observe both a crossover from rigid-rod–like behavior to (almost) Gaussian random coils and a further crossover towards self-avoiding walks in good solvents. Using the pruned-enriched Rosenbluth method (PERM) to study self-avoiding walks of up to Nb=50000 steps and variable flexibility, the applicability of the Kratky-Porod model is tested. Evidence for non-exponential decay of the bond-orientational correlations ⟨cos θ(s)⟩ for large distances s along the chain contour is presented, irrespective of chain stiffness. For bottle-brush polymers on the other hand, where experimentally stiffness is varied via the length of side-chains, it is shown that these cylindrical brushes (with flexible backbones) are not described by the Kratky-Porod worm-like chain model, since their persistence length is (roughly) proportional to their cross-sectional radius, for all conditions of practical interest.
[42]
Conformational
Studies
of Bottle-brush Polymers Adsorbed on a Flat Solid Surface
Hsu HP, Paul W, Binder K
JOURNAL OF CHEMICAL PHYSICS
133: 134902 (14 pages) 7 October 2010
Hsu HP, Paul W, Binder K
JOURNAL OF CHEMICAL PHYSICS
133: 134902 (14 pages) 7 October 2010
Abstract:
The adsorption of a bottle-brush polymer end-grafted with one chain end of its backbone to a flat substrate surface is studied by Monte Carlo simulation of a coarse-grained model, that previously has been characterized in the bulk, assuming a dilute solution under good solvent conditions. Applying the bond fluctuation model on the simple cubic lattice, we vary the backbone chain length Nb from Nb = 67 to Nb = 259 effective monomeric units, the side chain length N from N = 6 to N = 48, and set the grafting density to σ = 1, i.e., parameters that correspond well to the experimentally accessible range. When the adsorption energy strength ϵ is varied, we find that the adsorption transition (which becomes well-defined in the limit Nb→∞, for arbitrary finite N) roughly occurs at the same value ϵc as for ordinary linear chains (N = 0), at least within our statistical errors. Mean square end-to-end distances and gyration radii of the side chains are obtained, as well as the monomer density profile in the direction perpendicular to the adsorbing surface. We show that for longer side chains the adsorption of bottle-brushes is a two-step process, the decrease of the perpendicular linear dimension of side chains with adsorption energy strength can even be nonmonotonic. Also, the behavior of the static structure factor S(q) is analyzed, evidence for a quasi-two-dimensional scaling is presented, and consequences for the interpretation of experiments are discussed.
The adsorption of a bottle-brush polymer end-grafted with one chain end of its backbone to a flat substrate surface is studied by Monte Carlo simulation of a coarse-grained model, that previously has been characterized in the bulk, assuming a dilute solution under good solvent conditions. Applying the bond fluctuation model on the simple cubic lattice, we vary the backbone chain length Nb from Nb = 67 to Nb = 259 effective monomeric units, the side chain length N from N = 6 to N = 48, and set the grafting density to σ = 1, i.e., parameters that correspond well to the experimentally accessible range. When the adsorption energy strength ϵ is varied, we find that the adsorption transition (which becomes well-defined in the limit Nb→∞, for arbitrary finite N) roughly occurs at the same value ϵc as for ordinary linear chains (N = 0), at least within our statistical errors. Mean square end-to-end distances and gyration radii of the side chains are obtained, as well as the monomer density profile in the direction perpendicular to the adsorbing surface. We show that for longer side chains the adsorption of bottle-brushes is a two-step process, the decrease of the perpendicular linear dimension of side chains with adsorption energy strength can even be nonmonotonic. Also, the behavior of the static structure factor S(q) is analyzed, evidence for a quasi-two-dimensional scaling is presented, and consequences for the interpretation of experiments are discussed.
[41]
A
Stevedore's Protein Knot
Bolinger D, Sulkowska, JI, Hsu HP, Mirny LA, Kardar M, Onuchic JN, Virnau P
PLOS COMPUTATIONAL BIOLOGY
6(4): e1000731 APRIL 2010
Bolinger D, Sulkowska, JI, Hsu HP, Mirny LA, Kardar M, Onuchic JN, Virnau P
PLOS COMPUTATIONAL BIOLOGY
6(4): e1000731 APRIL 2010
Abstract:
Protein knots, mostly regarded as intriguing oddities, are gradually being recognized as significant structural motifs. Seven distinctly knotted folds have already been identified. It is by and large unclear how these exceptional structures actually fold, and only recently, experiments and simulations have begun to shed some light on this issue. In checking the new protein structures submitted to the Protein Data Bank, we encountered the most complex and the smallest knots to date: A recently uncovered alpha-haloacid dehalogenase structure contains a knot with six crossings, a so-called Stevedore knot, in a projection onto a plane. The smallest protein knot is present in an as yet unclassified protein fragment that consists of only 92 amino acids. The topological complexity of the Stevedore knot presents a puzzle as to how it could possibly fold. To unravel this enigma, we performed folding simulations with a structure-based coarse-grained model and uncovered a possible mechanism by which the knot forms in a single loop flip.
Protein knots, mostly regarded as intriguing oddities, are gradually being recognized as significant structural motifs. Seven distinctly knotted folds have already been identified. It is by and large unclear how these exceptional structures actually fold, and only recently, experiments and simulations have begun to shed some light on this issue. In checking the new protein structures submitted to the Protein Data Bank, we encountered the most complex and the smallest knots to date: A recently uncovered alpha-haloacid dehalogenase structure contains a knot with six crossings, a so-called Stevedore knot, in a projection onto a plane. The smallest protein knot is present in an as yet unclassified protein fragment that consists of only 92 amino acids. The topological complexity of the Stevedore knot presents a puzzle as to how it could possibly fold. To unravel this enigma, we performed folding simulations with a structure-based coarse-grained model and uncovered a possible mechanism by which the knot forms in a single loop flip.
[40]
Standard
Definitions of Persistence Length Do Not Describe the Local
"Intrinsic" Stiffness of Real Polymer Chains
Hsu HP, Paul W, Binder K
MACROMOLECULES
43(6): 3094-3102 MARICH 23 2010
Hsu HP, Paul W, Binder K
MACROMOLECULES
43(6): 3094-3102 MARICH 23 2010
Abstract:
On the basis of extensive Monte Carlo simulations of lattice models for linear chains under good and Theta solvents conditions, and for bottle-brush polymers under good solvent conditions, different methods to estimate the persistence lengths of these polymers are applied and compared to each other. While for chain molecules at the Theta point standard textbook definitions of the persistence length yield consistent results, under good solvent conditions the persistence length (according to its standard definitions) diverges when the chain length of the macromolecules tends to infinity. Accurate simulation results for chain lengths up to N-b = 6400 allow us to verify the theoretically predicted power laws for the decay of the bond orientational correlation function. For the case of bottle-brush polymers, this dependence of "the" persistence length on the backbone chain length obscures the dependence on the side chain length, that is controversially discussed in the literature. Alternative definitions Or a persistence length that do not suffer from this problem, based on the total linear dimension of the chain or on the scattering function via the so-called "Holtzer plateau" are studied as well. We show that the backbone contour length of the bottle-brush needs to be very large (about 100 persistence lengths in typical cases) to reach the asymptotic limit where the bottle-brush satisfies the self-avoiding walk statistics, and where a well-defined persistence length can be extracted. An outlook to pertinent experimental work is given.
On the basis of extensive Monte Carlo simulations of lattice models for linear chains under good and Theta solvents conditions, and for bottle-brush polymers under good solvent conditions, different methods to estimate the persistence lengths of these polymers are applied and compared to each other. While for chain molecules at the Theta point standard textbook definitions of the persistence length yield consistent results, under good solvent conditions the persistence length (according to its standard definitions) diverges when the chain length of the macromolecules tends to infinity. Accurate simulation results for chain lengths up to N-b = 6400 allow us to verify the theoretically predicted power laws for the decay of the bond orientational correlation function. For the case of bottle-brush polymers, this dependence of "the" persistence length on the backbone chain length obscures the dependence on the side chain length, that is controversially discussed in the literature. Alternative definitions Or a persistence length that do not suffer from this problem, based on the total linear dimension of the chain or on the scattering function via the so-called "Holtzer plateau" are studied as well. We show that the backbone contour length of the bottle-brush needs to be very large (about 100 persistence lengths in typical cases) to reach the asymptotic limit where the bottle-brush satisfies the self-avoiding walk statistics, and where a well-defined persistence length can be extracted. An outlook to pertinent experimental work is given.
[39]
Characteristic
Length
Scales and Radial Monomer Density Profiles of Molecular
Bottle-Brushes: Simulation and Experiment
Hsu HP, Paul W, Rathheber S, Binder K
MACROMOLECULES
43(3): 1592-1601 JANUARY 7 2010
Hsu HP, Paul W, Rathheber S, Binder K
MACROMOLECULES
43(3): 1592-1601 JANUARY 7 2010
Abstract:
Extensive Monte Carlo simulations are presented for bottle-brush polymers under good solvent conditions, using the bond fluctuation model on the simple cubic lattice. Varying the backbone length (from Nb = 67 to Nb = 259 effective monomers) as well as the side chain length (from N = 6 to N = 48), for a physically reasonable grafting density of one chain per backbone monomer, we find that the structure factor describing the total scattering from the bottle-brush provides an almost perfect match for some combinations of (Nb, N) to experimental data of Rathgeber et al. [ J. Chem. Phys. 2005, 122, 124904], when we adjust the length scale of the simulation to reproduce the experimental gyration radius of the bottle-brush. While in the experiment other length scales (gyration radius of side chains, backbone persistence length, scale characterizing the radial monomer density profile in the plane normal to the backbone) can be extracted only via fitting to a complicated and approximate theoretical expression derived by Pedersen and Schurtenberger, all these properties can be extracted from the simulation directly. In this way, quantitatively more reliable estimates for the persistence length and side chain gyration radius of the experimental systems can be extracted. In particular, we show that the popular assumption of a Gaussian radial monomer density profile is inaccurate, in the very good solvent regime studied by the simulation, and show that alternative forms based on scaling theory work better. We also show that the persistence length of the bottle brush in the simulation depends systematically on the backbone length and not only on the side chain length. For the cases where an explicit comparison with the experimental results (based on their evaluation within the Pedersen−Schurtenberger model) is possible, simulation and experiment are consistent with each other and some of the (rather minor) differences between simulation and experiment can be attributed to the weaker strength of excluded volume in the latter. Thus, we show that by suitable mapping between simulation and experiment on length scales of the local concentration fluctuations (here <2 nm) the analysis of experimental data can be systematically refined.
Extensive Monte Carlo simulations are presented for bottle-brush polymers under good solvent conditions, using the bond fluctuation model on the simple cubic lattice. Varying the backbone length (from Nb = 67 to Nb = 259 effective monomers) as well as the side chain length (from N = 6 to N = 48), for a physically reasonable grafting density of one chain per backbone monomer, we find that the structure factor describing the total scattering from the bottle-brush provides an almost perfect match for some combinations of (Nb, N) to experimental data of Rathgeber et al. [ J. Chem. Phys. 2005, 122, 124904], when we adjust the length scale of the simulation to reproduce the experimental gyration radius of the bottle-brush. While in the experiment other length scales (gyration radius of side chains, backbone persistence length, scale characterizing the radial monomer density profile in the plane normal to the backbone) can be extracted only via fitting to a complicated and approximate theoretical expression derived by Pedersen and Schurtenberger, all these properties can be extracted from the simulation directly. In this way, quantitatively more reliable estimates for the persistence length and side chain gyration radius of the experimental systems can be extracted. In particular, we show that the popular assumption of a Gaussian radial monomer density profile is inaccurate, in the very good solvent regime studied by the simulation, and show that alternative forms based on scaling theory work better. We also show that the persistence length of the bottle brush in the simulation depends systematically on the backbone length and not only on the side chain length. For the cases where an explicit comparison with the experimental results (based on their evaluation within the Pedersen−Schurtenberger model) is possible, simulation and experiment are consistent with each other and some of the (rather minor) differences between simulation and experiment can be attributed to the weaker strength of excluded volume in the latter. Thus, we show that by suitable mapping between simulation and experiment on length scales of the local concentration fluctuations (here <2 nm) the analysis of experimental data can be systematically refined.
[36]
How to
Define Variation of Physical Properties Normal to an
Undulating One-Dimensional Object
Hsu HP, Binder K, Paul W
PHYSICAL REVIEW LETT
103(4): 198301 (4 pages), NOV 2009
Hsu HP, Binder K, Paul W
PHYSICAL REVIEW LETT
103(4): 198301 (4 pages), NOV 2009
One-dimensional flexible
objects
are
abundant in physics, from polymers to vortex lines to defect lines and
many more. These objects structure their environment and it is
natural to assume that the influence these objects
exert on their environment depends on the distance from the line
object. But how should this be defined? We argue here that
there is an intrinsic length scale along the undulating
line that is a measure of its
stiffness (i.e., orientational persistence), which
yields a natural way of
defining the variation of physical properties normal to
the undulating line. We
exemplify how this normal variation can be determined from a computer
simulation for the case of a so-called bottle-brush
polymer, where side chains are grafted onto a flexible backbone.
[35]
Structure of
bottle-brush polymers in solution: A Monte Carlo test of
models for the scattering function
Hsu HP, Paul W, Binder
K
JOURNAL OF CHEMICAL PHYSICS
129: 204904-1-204904-11
Nov 28 2008
Abstract:
Extensive Monte Carlo
results are presented for the structure of a bottle-brush
polymer under good solvent or theta solvent conditions.
Varying the side chain length, backbone length, and the
grafting density for a rigid straight backbone, both
radial density profiles of monomers and side chain ends
are obtained as well as structure factors describing the
scattering from a single side chain and from the total
bottle-brush polymer. To describe the structure in the
interior of a very long bottle brush, a periodic boundary
condition in the direction along the backbone is used, and
to describe effects due to the finiteness of the backbone
length, a second set of simulations with free ends of the
backbone is performed. In the latter case, the
inhomogeneity of the structure in the direction along the
backbone is carefully investigated. We use these results
to test various phenomenological models that have been
proposed to interpret experimental scattering data for
bottle-brush macromolecules. These models aim to extract
information on the radial density profile of a bottle
brush from the total scattering via suitable convolution
approximations. Limitations of this approach and the
optimal way to perform the analysis of the scattering data
within this approach are discussed.
Full Text: JCP or cond-mat
Full Text: JCP or cond-mat
[34]
Kenna, R.; Hsu, HP, von Ferber, C
JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT
Art. No. L10002 OCT 2008
Abstract:
For continuous phase transitions characterized by power-law divergences, Fisher renormalization prescribes how to obtain the critical exponents for a system under constraint from their ideal counterparts. In statistical mechanics, such ideal behaviour at phase transitions is frequently modified by multiplicative logarithmic corrections. Here, Fisher renormalization for the exponents of these logarithms is developed in a general manner. As for the leading exponents, Fisher renormalization at the logarithmic level is seen to be involutory and the renormalized exponents obey the same scaling relations as their ideal analogues. The scheme is tested in lattice animals and the Yang Lee problem at their upper critical dimensions, where predictions for logarithmic corrections are made.
Full Text: JStatM or cond-mat
[33]
Escape transition of a polymer
chain from a nanotube: How to avoid spurious results by
use of the force-biased pruned-enriched Rosenbluth
algorithm
Hsu HP, Binder K, Klushin LI, Skvortsov AM
PHYSICAL REVIEW E
78 (4): 041803-1 - 041803-11, OCT 2008
Abstract:
A polymer chain containing N monomers confined in a finite cylindrical tube of diameter D grafted at a distance L from the open end of the tube may undergo a rather abrupt transition, where part of the chain escapes from the tube to form a "crownlike" coil outside of the tube. When this problem is studied by Monte Carlo simulation of self-avoiding walks on the simple cubic lattice applying a cylindrical confinement and using the standard pruned-enriched Rosenbluth method (PERM), one obtains spurious results, however, with increasing chain length the transition gets weaker and weaker, due to insufficient sampling of the "escaped" states, as a detailed analysis shows. In order to solve this problem, a new variant of a biased sequential sampling algorithm with resampling is proposed, force-biased PERM: the difficulty of sampling both phases in the region of the first order transition with the correct weights is treated by applying a force at the free end pulling it out of the tube. Different strengths of this force need to be used and reweighting techniques are applied. Using rather long chains (up to N=18 000) and wide tubes (up to D=29 lattice spacings), the free energy of the chain, its end-to-end distance, the number of "imprisoned" monomers can be estimated, as well as the order parameter and its distribution. It is suggested that this algorithm should be useful for other problems involving state changes of polymers, where the different states belong to rather disjunct "valleys" in the phase space of the system.
Full Text: PRE or cond-mat
A polymer chain containing N monomers confined in a finite cylindrical tube of diameter D grafted at a distance L from the open end of the tube may undergo a rather abrupt transition, where part of the chain escapes from the tube to form a "crownlike" coil outside of the tube. When this problem is studied by Monte Carlo simulation of self-avoiding walks on the simple cubic lattice applying a cylindrical confinement and using the standard pruned-enriched Rosenbluth method (PERM), one obtains spurious results, however, with increasing chain length the transition gets weaker and weaker, due to insufficient sampling of the "escaped" states, as a detailed analysis shows. In order to solve this problem, a new variant of a biased sequential sampling algorithm with resampling is proposed, force-biased PERM: the difficulty of sampling both phases in the region of the first order transition with the correct weights is treated by applying a force at the free end pulling it out of the tube. Different strengths of this force need to be used and reweighting techniques are applied. Using rather long chains (up to N=18 000) and wide tubes (up to D=29 lattice spacings), the free energy of the chain, its end-to-end distance, the number of "imprisoned" monomers can be estimated, as well as the order parameter and its distribution. It is suggested that this algorithm should be useful for other problems involving state changes of polymers, where the different states belong to rather disjunct "valleys" in the phase space of the system.
Full Text: PRE or cond-mat
[32]
Dragging a
polymer chain into a nanotube and subsequent release
Klushin
LI, Skvortsov AM, Hsu HP,
Binder K
MACROMOLECULES
41(15): 5890-5898 JULY 2
2008
Abstract:
We present a scaling theory and Monte Carlo (MC) simulation results for a flexible polymer chain slowly dragged by one end into a nanotube. We also describe the situation when the completely confined chain is released and gradually leaves the tube. MC simulations were performed for a self-avoiding lattice model with a biased chain growth algorithm, the pruned-enriched Rosenbluth method (PERM). The nanotube is a long channel opened at one end and its diameter D is much smaller than the size of the polymer coil in solution. We analyze the following characteristics as functions of the chain end position x inside the tube: the free energy of confinement, the average end-to-end distance, the average number of segments imprisoned in the tube, and the average stretching of the confined part of the chain for various values of D and for the number of repeat units in the chain, N. We show that when the chain end is dragged by a certain critical distance x* into the tube, the polymer undergoes a first-order phase transition whereby the remaining free tail is abruptly sucked into the tube. This is accompanied by jumps in the average size, the number of imprisoned segments, and the average stretching parameter. The critical distance scales as x* similar to ND1-1/v. The transition takes place when approximately 3/4 of the chain units are dragged into the tube. The theory presented is based on constructing the Landau free energy as a function of an order parameter that provides a complete description of equilibrium and metastable states. We argue that if the trapped chain is released with all monomers allowed to fluctuate, the reverse process in which the chain leaves the confinement occurs smoothly without any jumps. Finally, we apply the theory to estimate the lifetime of confined DNA in metastable states in nanotubes.
We present a scaling theory and Monte Carlo (MC) simulation results for a flexible polymer chain slowly dragged by one end into a nanotube. We also describe the situation when the completely confined chain is released and gradually leaves the tube. MC simulations were performed for a self-avoiding lattice model with a biased chain growth algorithm, the pruned-enriched Rosenbluth method (PERM). The nanotube is a long channel opened at one end and its diameter D is much smaller than the size of the polymer coil in solution. We analyze the following characteristics as functions of the chain end position x inside the tube: the free energy of confinement, the average end-to-end distance, the average number of segments imprisoned in the tube, and the average stretching of the confined part of the chain for various values of D and for the number of repeat units in the chain, N. We show that when the chain end is dragged by a certain critical distance x* into the tube, the polymer undergoes a first-order phase transition whereby the remaining free tail is abruptly sucked into the tube. This is accompanied by jumps in the average size, the number of imprisoned segments, and the average stretching parameter. The critical distance scales as x* similar to ND1-1/v. The transition takes place when approximately 3/4 of the chain units are dragged into the tube. The theory presented is based on constructing the Landau free energy as a function of an order parameter that provides a complete description of equilibrium and metastable states. We argue that if the trapped chain is released with all monomers allowed to fluctuate, the reverse process in which the chain leaves the confinement occurs smoothly without any jumps. Finally, we apply the theory to estimate the lifetime of confined DNA in metastable states in nanotubes.
[31]
Adsorption
of
Multi-block
and Random Copolymer on a Solid Surface: Critical
Behavior and Phase Diagram
Bhattacharya S, Hsu HP, Milchev,
Rostiashvili VG, Vilgis TA
MACROMOLECULES
41(8): 2920-2930 MARCH 29
2008
Abstract: The adsorption
of a single multi-block $AB$-copolymer on a solid planar
substrate is investigated by means of computer simulations
and scaling analysis. It is shown that the problem can be
mapped onto an effective homopolymer adsorption problem. In
particular we discuss how the critical adsorption energy and
the fraction of adsorbed monomers depend on the block length
$M$ of sticking monomers $A$, and on the total length $N$ of
the polymer chains. Also the adsorption of the random
copolymers is considered and found to be well described
within the framework of the annealed approximation. For a
better test of our theoretical prediction, two different
Monte Carlo (MC) simulation methods were employed: a)
off-lattice dynamic bead-spring model, based on the standard
Metropolis algorithm (MA), and b) coarse-grained lattice
model using the Pruned-enriched Rosenbluth method (PERM)
which enables tests for very long chains. The findings of
both methods are fully consistent and in good agreement with
theoretical predictions.
Full Text: MACROMOLECULES
[30]
One- and two-component
bottle-brush polymers: simulations compared to
theoretical predictions
Hsu HP, Paul W, Binder
K
MACROMOLECULAR THEORY AND SIMULATIONS
16 (7): 660-689 SEP 25
2007
Abstract:
Scaling predictions for
bottle-brush polymers with a rigid backbone and flexible
side chains under good solvent conditions are discussed
and their validity is assessed by a comparison with Monte
Carlo simulations of a simple lattice model. It is shown
that typically only a rather weak stretching of the side
chains is realized, and then the scaling predictions are
not applicable. Also two-component bottle brush polymers
are considered, where two types (A,B) of side chains are
grafted, assuming that monomers of different kind repel
each other. In this case, variable solvent quality is
allowed. Theories predict "Janus cylinder"-type phase
separation along the backbone in this case. The Monte
Carlo simulations, using the pruned-enriched Rosenbluth
method (PERM) give evidence that the phase separation
between an A-rich part of the cylindrical molecule and a
B-rich part can only occur locally. The corelation length
of this microphase separation can be controlled by the
solvent quality. This lack of a phase transitions
interpreted by an analogy with models for ferromagnets in
one space dimension.
Full Text: MTS
Full Text: MTS
[28]
What is the order of the
two-dimensional polymer escape transition?
Hsu HP, Binder K, Klushin LI, Skvortsov AM
PHYSICAL REVIEW E
76 (2): 021108-1 - 021108-14, AUG 2007
Abstract:
An end-grafted flexible polymer chain in three-dimensional space between two pistons undergoes an abrupt transition from a confined coil to a flowerlike conformation when the number of monomers in the chain, N, reaches a critical value. In two-dimensional (2D) geometry, excluded-volume interactions between monomers of a chain confined inside a strip of finite length 2L transform the coil conformation into a linear string of blobs. However, the blob picture raises questions about the nature of this escape transition. To check theoretical predictions based on the blob picture we study 2D single-polymer chains with excluded-volume interactions and with one end grafted in the middle of a strip of length 2L and width H by simulating self-avoiding walks on a square lattice with the pruned-enriched Rosenbluth method. We estimate the free energy, the end-to-end distance, the number of imprisoned monomers, the order parameter, and its distribution. It is shown that in the thermodynamic limit of large N and L but finite L/N, there is a small but finite jump in several average characteristics, including the order parameter. We also present a theoretical description based on the Landau free energy approach, which is in good agreement with the simulation results. Both simulation results and the analytical theory indicate that the 2D escape transition is a weak first-order phase transition.
Full Text: PRE or cond-mat
An end-grafted flexible polymer chain in three-dimensional space between two pistons undergoes an abrupt transition from a confined coil to a flowerlike conformation when the number of monomers in the chain, N, reaches a critical value. In two-dimensional (2D) geometry, excluded-volume interactions between monomers of a chain confined inside a strip of finite length 2L transform the coil conformation into a linear string of blobs. However, the blob picture raises questions about the nature of this escape transition. To check theoretical predictions based on the blob picture we study 2D single-polymer chains with excluded-volume interactions and with one end grafted in the middle of a strip of length 2L and width H by simulating self-avoiding walks on a square lattice with the pruned-enriched Rosenbluth method. We estimate the free energy, the end-to-end distance, the number of imprisoned monomers, the order parameter, and its distribution. It is shown that in the thermodynamic limit of large N and L but finite L/N, there is a small but finite jump in several average characteristics, including the order parameter. We also present a theoretical description based on the Landau free energy approach, which is in good agreement with the simulation results. Both simulation results and the analytical theory indicate that the 2D escape transition is a weak first-order phase transition.
Full Text: PRE or cond-mat
[27]
A single polymer grafted to a porous membrane
Hsu HP, Grassberger P
Hsu HP, Grassberger P
EUROPHYSICS LETTERS
77 (1): 18003-P1-18003-P4 JAN 2007
77 (1): 18003-P1-18003-P4 JAN 2007
Abstract:
We study a single flexible chain molecule grafted to a membrane which has pores of size slightly larger than the monomer size. On both sides of the membrane there is the same solvent. When this solvent is good, i.e. when the polymer is described by a self-avoiding walk, it can fairly easily penetrate the membrane, so that the average number of membrane crossings tends, for chain length N→∞, to a positive constant. The average numbers of monomers on either side of the membrane diverges in this limit, although their ratio becomes infinite. For a poor solvent, in contrast, the entire polymer is located, for large N, on one side of the membrane. For good and for theta solvents (ideal polymers) we find scaling laws, whose exponents can in the latter case be easily understood from the behaviour of random walks.
We study a single flexible chain molecule grafted to a membrane which has pores of size slightly larger than the monomer size. On both sides of the membrane there is the same solvent. When this solvent is good, i.e. when the polymer is described by a self-avoiding walk, it can fairly easily penetrate the membrane, so that the average number of membrane crossings tends, for chain length N→∞, to a positive constant. The average numbers of monomers on either side of the membrane diverges in this limit, although their ratio becomes infinite. For a poor solvent, in contrast, the entire polymer is located, for large N, on one side of the membrane. For good and for theta solvents (ideal polymers) we find scaling laws, whose exponents can in the latter case be easily understood from the behaviour of random walks.
Full Text: EPL or cond-mat
[26]
Intramolecular phase
separation of copolymer "bottle brushes": No sharp
phase transition but a tunable length scale
Hsu HP, Paul W , Binder K
EUROPHYSICS LETTERS
76 (3): 526-532 NOV 2006
76 (3): 526-532 NOV 2006
Abstract:
A lattice model for a symmetrical copolymer "bottle brush" molecule, where two types (A, B) of flexible side chains are grafted with one chain end to a rigid backbone, is studied by a variant of the pruned-enriched Rosenbluth method (PERM), allowing for simultaneous growth of all side chains in the Monte Carlo sampling. Choosing repulsive binary interactions between unlike monomers and varying the solvent quality, it is found that phase separation into an A-rich part of the cylindrical molecule and a B-rich part can occur only locally. Lon-grange order (in the direction of the backbone) does not occur, and hence the transition from the randomly mixed state of the bottle brush to the phase separated structure is strongly rounded, in contrast to corresponding mean-field predictions. This lack of a phase transition can be understood from an analogy with spin models in one space dimension. We predict that the range of microphase separation along the bottle brush backbone can be controlled on the nanoscale by varying the solvent quality.
A lattice model for a symmetrical copolymer "bottle brush" molecule, where two types (A, B) of flexible side chains are grafted with one chain end to a rigid backbone, is studied by a variant of the pruned-enriched Rosenbluth method (PERM), allowing for simultaneous growth of all side chains in the Monte Carlo sampling. Choosing repulsive binary interactions between unlike monomers and varying the solvent quality, it is found that phase separation into an A-rich part of the cylindrical molecule and a B-rich part can occur only locally. Lon-grange order (in the direction of the backbone) does not occur, and hence the transition from the randomly mixed state of the bottle brush to the phase separated structure is strongly rounded, in contrast to corresponding mean-field predictions. This lack of a phase transition can be understood from an analogy with spin models in one space dimension. We predict that the range of microphase separation along the bottle brush backbone can be controlled on the nanoscale by varying the solvent quality.
Full Text: EPL or cond-mat
[25]
wedges, cones, and branch points of Riemann surfaces
Hsu HP, Nadler W, Grassberger P
PHYSICAL REVIEW E
71 (6): Art. No. 065104 Part 2, JUN 2005
Abstract:
Lattice animals are one
of the few critical models in statistical mechanics
violating conformal invariance. We present here
simulations of 2-d site animals on square and triangular
lattices in non-trivial geometries. The simulations are
done with the newly developed PERM algorithm which gives
very precise estimates of the partition sum, yielding
precise values for the entropic exponent $\theta$ ($Z_N
\sim \mu^N N^{-\theta}$). In particular, we studied
animals grafted to the tips of wedges with a wide range
of angles $\alpha$, to the tips of cones (wedges with
the sides glued together), and to branching points of
Riemann surfaces. The latter can either have $k$ sheets
and no boundary, generalizing in this way cones to
angles $\alpha > 360$
degrees, or can have boundaries, generalizing wedges. We find conformal invariance behavior, $\theta \sim 1/\alpha$, only for small angles ($\alpha \ll 2\pi$), while $\theta \approx const -\alpha/2\pi$ for $\alpha \gg 2\pi$. These scalings hold both for wedges and cones. A heuristic (non-conformal) argument for the behavior at large $\alpha$ is given, and comparison is made with critical percolation.
degrees, or can have boundaries, generalizing wedges. We find conformal invariance behavior, $\theta \sim 1/\alpha$, only for small angles ($\alpha \ll 2\pi$), while $\theta \approx const -\alpha/2\pi$ for $\alpha \gg 2\pi$. These scalings hold both for wedges and cones. A heuristic (non-conformal) argument for the behavior at large $\alpha$ is given, and comparison is made with critical percolation.
Hsu HP, Grassberger P
JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT
Art. No. P06003 JUN 2005
Abstract:
We present high statistics
simulations of weighted lattice bond animals and lattice
trees on the square lattice, with fugacities for each
non-bonded contact and for each bond between two
neighbouring monomers. The simulations are performed using
a newly developed sequential sampling method with
resampling, very similar to the pruned-enriched Rosenbluth
method (PERM) used for linear chain polymers. We determine
with high precision the line of second order transitions
from an extended to a collapsed phase in the resulting
2-dimensional phase diagram. This line includes critical
bond percolation as a multicritical point, and we verify
that this point divides the line into two different
universality classes. One of them corresponds to the
collapse driven by contacts and includes the collapse of
(weakly embeddable) trees, but the other is {\it not yet}
bond driven and does not contain the Derrida-Herrmann
model as special point. Instead it ends at a multicritical
point $P^*$ where a transition line between two collapsed
phases (one bond-driven and the other contact-driven)
sparks off. The Derrida-Herrmann model is representative
for the bond driven collapse, which then forms the fourth
universality class on the transition line (collapsing
trees, critical percolation, intermediate regime, and
Derrida-Herrmann). We obtain very precise estimates for
all critical exponents for collapsing trees. It is already
harder to estimate the critical exponents for the
intermediate regime. Finally, it is very difficult to
obtain with our method good estimates of the critical
parameters of the Derrida-Herrmann universality class. As
regards the bond-driven to contact-driven transition in
the collapsed phase, we have some evidence for its
existence and rough location, but no precise estimates of
critical exponents.
[23]
Hsu HP, Grassberger P
JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT
Art. No. P01007 JAN 2005
Abstract:
We study long polymer chains
in a poor solvent, confined to the space between two parallel
hard walls. The walls are energetically neutral and impose
only a geometric constraint which changes the properties of
the coil - globule (or 'theta') transition. We. nd that the
theta temperature increases monotonically with the width D
between the walls, in contrast to recent claims in the
literature. Put in a wider context, the problem can be seen as
a dimensional crossover at a tricritical point of a phi(4)
model. We roughly verify the main scaling properties expected
for such a phenomenon, but we. nd also somewhat unexpected
very long transients before the asymptotic scaling regions are
reached. In particular, instead of the expected scaling R
similar to N-4/7 exactly at the (D-dependent) theta point, we
found that R increases less fast than N-1/2, even for
extremely long chains.
Full Text: JStatM or cond-mat
[22]
Hsu HP, Nadler W,Grassberger P
JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL
38 (4): 775-806 JAN 28 2005
Abstract:
The scaling behaviour of
randomly branched polymers in a good solvent is studied in two
to nine dimensions, using as microscopic models lattice
animals and lattice trees on simple hypercubic lattices. As a
stochastic sampling method we use a biased sequential sampling
algorithm with re-sampling, similar to the pruned-enriched
Rosenbluth method (PERM) used extensively for linear polymers.
Essentially we start simulating percolation clusters (either
site or bond), re-weigh them according to the animal (tree)
ensemble, and prune or branch the further growth according to
a heuristic fitness function. In contrast to previous
applications of PERM, this fitness function is not the weight
with which the actual configuration would contribute to the
partition sum, but is closely related to it. We obtain high
statistics of animals with up to several thousand sites in all
dimension 2 less than or equal to d less than or equal to 9.
In addition to the partition sum (number of different
animals.) we estimate gyration radii and numbers of perimeter
sites. In all dimensions we verify the Parisi-Sourlas
prediction, and we verify all exactly known critical exponents
in dimensions 2, 3, 4 and greater than or equal to 8. In
addition, we present the hitherto most precise estimates for
growth constants in d greater than or equal to, 3. For
clusters with one site attached to an attractive surface, we
verify for d greater than or equal to 3 the superuniversality
of the cross-over exponent phi at the adsorption transition
predicted by Janssen and Lyssy, but not for d = 2. There, we
find phi = 0.480(4) instead of the conjectured phi = 1/2.
Finally, we discuss the collapse of animals and trees, arguing
that our present version of the algorithm is also efficient
for some of the models studied in this context, but showing
that it is not very efficient for the 'classical' model for
collapsing animals.
Full Text: JPA or cond-mat
[20]
Hsu HP, Grassberger P
EUROPHYSICS LETTERS
66 (6): 874-880 JUN 2004
Abstract:
We study numerically the
effective pair potential between two star polymers with equal
arm lengths and equal number f of arms. The simulations were
done for the soft-core Domb-Joyce model on the simple cubic
lattice, to minimize corrections to scaling and to allow for
an unlimited number of arms. For the sampling, we used the
pruned-enriched Rosenbluth method (PERM). We find that the
potential is much less soft than claimed in previous papers,
in particular for f much greater than 1. While we verify the
logarithmic divergence of V(r), with r being the distance
between the two cores, predicted by Witten and Pincus, we find
that the Mayer function for f > 20 is hardly
distinguishable from that for a Gaussian potential.
Full Text: EPL or cond-mat
[19]
Hsu HP, Nadler W, Grassberger P
MACROMOLECULES
37 (12): 4658-4663 JUN 15 2004
Abstract:
We present large statistics
simulations of 3-dimensional star polymers with up to f = 80
arms and with up to 4000 monomers per arm for small values of
f. They were done for the Domb-Joyce model on the simple cubic
lattice. This is a model with soft core exclusion which allows
multiple occupancy of sites but punishes each same-site pair
of monomers with a Boltzmann factor v < 1. We use this to
allow all arms to be attached at the central site, and we use
the "magic" value v = 0.6 to minimize corrections to scaling.
The simulations are made with a very efficient chain growth
algorithm with resampling, PERM, modified to allow
simultaneous growth of all arms. This allows us to measure not
only the swelling (as observed from the center-to-end
distances) but also the partition sum. The latter gives very
precise estimates of the critical exponents gamma(f). For
completeness we made also extensive simulations of linear
(unbranched) polymers which give the best estimates for the
exponent gamma.
Full Text: MACROMOLECULES
[17]
Berg BA, Hsu HP
PHYSICAL REVIEW E
Abstract:
We investigate the
solvent-accessible area method by means of Metropolis
simulations of the brain peptide Met-Enkephalin at 300 K. For
the energy function ECEPP/2 nine atomic solvation parameter
(ASP) sets are studied. The simulations are compared with one
another, with simulations with a distance dependent
electrostatic permittivity epsilon(r), and with vacuum
simulations (epsilon=2). Parallel tempering and the biased
Metropolis techniques RM1 are employed and their performance
is evaluated. The measured observables include energy and
dihedral probability densities, integrated autocorrelation
times, and acceptance rates. Two of the ASP sets turn out to
be unsuitable for these simulations. For all other systems
selected configurations are minimized in the search for global
energy minima, which are found for vacuum and the epsilon(r)
system, but for none of the ASP models. Other observables show
a remarkable dependence on the ASPs. In particular, we find
three ASP sets for which the autocorrelations at 300 K are
considerably smaller than those for vacuum simulations.
Full Text: PRE or cond-mat
Hsu HP, Grassberger P
JOURNAL OF CHEMICAL PHYSICS
Single three-dimensional
polymers confined to a slab, i.e., to the region between two
parallel plane walls, are studied by Monte Carlo
simulations. They are described by N-step walks on a simple
cubic lattice confined to the region 1less than or equal
tozless than or equal toD. The simulations cover both
regions D<R-F and D>R-F (where R(F)similar toN(nu) is
the Flory radius, with nuapproximate to0.587), as well as
the cross-over region in between. Chain lengths are up to
N=80 000, slab widths up to D=120. In order to test the
analysis program and to check for finite size corrections,
we actually studied three different models: (a) ordinary
random walks (mimicking Theta polymers); (b) self-avoiding
walks; and (c) Domb-Joyce walks with the self-repulsion
tuned to the point where finite size corrections for free
(unrestricted) chains are minimal. For the simulations we
employ the pruned-enriched-Rosenbluth method with Markovian
anticipation. In addition to the partition sum (which gives
us a direct estimate of the forces exerted onto the walls),
we measure the density profiles of monomers and of end
points transverse to the slab, and the radial extent of the
chain parallel to the walls. All scaling laws and some of
the universal amplitude ratios are compared to theoretical
predictions. (C) 2004 American Institute of Physics.
Full Text: JCP or cond-mat
Hsu HP, Grassberger P
EUROPEAN PHYSICAL JOURNAL B
Abstract:
Single two dimensional
polymers confined to a strip are studied by Monte Carlo
simulations. They are described by N-step self-avoiding
random walks on a square lattice between two parallel hard
walls with distance 1 << D << N-nu (nu = 3/4 is
the Flory exponent). For the simulations we employ the
pruned-enriched-Rosenbluth method (PERM) with Markovian
anticipation. We measure the densities of monomers and of
end points as functions of the distance from the walls, the
longitudinal extent of the chain, and the forces exerted on
the walls. Their scaling with D and the universal ratio
between force and monomer density at the wall are compared
to theoretical predictions.
Full Text: EPJB or cond-mat
[13]
Structure optimization in
an off-lattice protein model
Hsu HP, Mehra V, Grassberger P
PHYSICAL REVIEW E 68
(2): Art. No. 037703 2003
Hsu HP, Mehra V, Grassberger P
PHYSICAL REVIEW E 68
(2): Art. No. 037703 2003
Abstract:
We study an off-lattice
protein toy model with two species of monomers interacting
through modified Lennard-Jones interactions. Low energy
configurations are optimized using the
pruned-enriched-Rosenbluth method (PERM), hitherto employed
to native state searches only for off-lattice models. For
two dimensions we found states with lower energy than
previously proposed putative ground states for all chain
lengths >=13. This indicates that PERM has the potential
to produce native states also for more realistic protein
models. For d = 3, where no published ground states exist,
we present some putative lowest energy states for future
comparison with other methods.
Full Text: PRE or cond-mat
[12]
Hsu HP, Mehra V, Nadler W, Grassberger P
PHYSICAL REVIEW E
Abstract:
An improved version of the
pruned-enriched-Rosenbluth method (PERM) is proposed and
tested on finding lowest energy states in simple models of
lattice heteropolymers. It is found to outperform not only
the previous version of PERM, but also all other fully blind
general purpose stochastic algorithms which have been
employed on this problem. In many cases, it found new lowest
energy states missed in previous papers. Limitations are
discussed.
[11]
Hsu HP, Mehra V, Nadler W, Grassberger P
JOURNAL OF CHEMICAL PHYSICS
Abstract:
Two improved versions of
the pruned-enriched-Rosenbluth method (PERM) are
proposed and tested on simple models of lattice
heteropolymers. Both are found to outperform not only
the previous version of PERM, but also all other
stochastic algorithms which have been employed on this
problem, except for the core directed chain growth
method (CG) of Beutler and Dill. In nearly all test
cases they are faster in finding low-energy states, and
in many cases they found new lowest energy states missed
in previous papers. The CG method is superior to our
method in some cases, but less efficient in others. On
the other hand, the CG method uses heavily heuristics
based on presumptions about the hydrophobic core and
does not give thermodynamic properties, while the
present method is a fully blind general purpose
algorithm giving correct Boltzmann-Gibbs weights, and
can be applied in principle to any stochastic sampling
problem.
Hsu HP, Grassberger P
JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL
Abstract:
Single partially confined
collapsed polymers are studied in two dimensions. They are
described by self-avoiding random walks with
nearest-neighbour attractions below the Theta-point, on the
surface of an infinitely long cylinder. For the simulations
we employ the pruned-enriched-Rosenbluth method (PERM). The
same model had previously been studied for free polymers
(infinite lattice, no boundaries) and for polymers on finite
lattices with periodic boundary conditions: We verify the
previous estimates of bulk densities, bulk free energies and
surface tensions. We find that the free energy of a polymer
with fixed length N has, for N --> infinity, a minimum at
a finite cylinder radius R* which diverges as T --> TB.
Furthermore, the surface tension vanishes roughly as
(T-theta - T)(alpha) for T --> T-theta with alpha
approximate to 1.7. The density in the interior of a globule
scales as (T-theta - T)beta with beta approximate to 0.32.