JOURNAL OF CHEMICAL PHYSICS
VOLUME 113, NUMBER 10
8 SEPTEMBER 2000
First-order transition of a homopolymer chain with Lennard-Jones potential
Haojun Lianga) and Hanning Chen
The Open Laboratory for Bond-Selective Chemistry, Department of Polymer Science & Engineering,
University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China
͑
Received 5 April 2000; accepted 13 June 2000͒
The thermodynamics of a homopolymer chain with the Lennard-Jones ͑LJ͒ potential was studied by
the multicanonical Monte Carlo method. The results confirm there indeed exists a liquid–solid-like
first-order transition at lower temperatures for a free-joint chain, revealing that the transition is a
characteristic of a homopolymer chain, independent of the algorithms and potential used in
simulation. © 2000 American Institute of Physics. ͓S0021-9606͑00͒50534-0͔
NϪ1
NϪ2
N
I. INTRODUCTION
Hϭ
U͑l ͒ϩ
E .
͚
i
͚ ͚ ij
In addition to the temperature and solubility dependent
of second order coil-to-globule transition,1,2 both theoretical
and experimental studies suggested more complex behaviors
iϭ1
iϭ1 jϭiϩ2
In this study, we simply set ϭ1 and ϭ1.
In the Metropolis algorithm, the evolution of conforma-
tion of a polymer chain is allowed with a probability
for a homopolymer in solution.3 Recently, Zhou et al.
–9
10,11
used the discontinuous molecular dynamics ͑DMD͒ simula-
tion to show that there exists a first-order liquid–solid-like
transition for a square-well free-joint homopolymer chain.
exp(Ϫ⌬E/k T), where ⌬E is positive, reflecting an attractive
B
energy between two segments. At low temperatures, the
probability is so small that it is difficult, if not impossible, to
sample sufficient configurations for an accurate statistical
1
2
Noguchi et al. observed the similar behavior for a freely
jointed square-well homopolymer chain with a bending po-
tential by multicanonical Monte Carlo simulation. L o´ pez13
calculation.10 To overcome this difficulty, we used the mul-
ticanonical Monte Carlo algorithms,1
5,16
in which all ener-
found that the application of J-walking Monte Carlo method
on a cluster of 55 LJ atoms also yielded a liquid-to-solid-like
transition. However, two questions remained, namely,
whether the transition is induced by the discontinuous fea-
ture of the square-well potential, and how it will change if
different MC algorithms are used. We present our recent
Monte Carlo simulation on a free-joint Lennard-Jones ͑LJ͒
homopolymer chain and address these two questions.
gies have an equal weight in the multicanonical ensemble, so
the energy is forced into a one-dimentional random walk, so
the system can overcome any energy barrier.
In multicanonical algorithms, the P ϰexp(␣(E)
B
ϩ(E)E) is used, unlike exp(ϪE) which is used in the Me-
tropolis algorithm, where E is the configuration energy.
The values of ␣(E) and (E) are determined as follows.
Perform a canonical Monte Carlo simulation with N
Ϫ1
energy bins at sufficiently high temperature, e.g., 0
ϭk T ϭ1000, to approximate P ( ,E) with a histogram
II. MODEL AND SIMULATION
B
0
B
0
P ( ,E ) (iϭ1,2, . . . ,N), and determine the histogram
B
0
i
The conformation of a homopolymer chain made up of n
made of Emax at which the histogram reaches its maximum.
The simulation is limited in the energy range EрEmax , out-
side of which (E)ϭ0 and ␣(E)ϭ0. Therefore, (E )E
segments is defined by n coordinates r , r , . . . ,r of beads
1
2
n
in a three-dimensional space. In this study, the off-lattice
Monte Carlo model developed by Binder et al.14 was used, in
which each randomly selected bead on the chain was allowed
to move around its position with a restriction of the bond
fluctuation between 1.001 (lmax) and 0.999 (lmin). The inter-
i
i
Ϫ␣(E )ϭln(P( , E ))ϩconstϵy . The parameters (E )
i
0
i
i
i
and ␣(E ) are obtained from a straight-line connection be-
i
tween two adjacent points (E ,y ) and (E ,yiϩ1). The
i
i
iϩ1
iMs ed t er ofi pn oe dl isa sc riterion of the transition probability w(E→EЈ)
action potential between two successive bonded segments
2
was treated as U(l)ϭk(lϪl ) , where k is a constant, set as
0
W͑E→EЈ͒ϭ1 if ⌬ϵB͑EЈ͒ϪB͑E͒р0,
a unit, and l is the equilibrium distance between the seg-
0
ments, which is set as 1 Å. The interaction potential between
Ϫ⌬
W͑E→EЈ͒ϭe
if ⌬ϵB͑EЈ͒ϪB͑E͒Ͼ0,
two nonbonded segments i and j is defined as E
ij
1
2
6
ϭ͓(/r ) Ϫ(/r ) ͔, where r (ϭ͉riϪrj͉) is the inter-
ij
ij
ij
where B(E)ϭ͓ ϩ(E)͔Eϩ␣(E). In this way, the prob-
0
segments distance, and and are two adjustable param-
eters controlling, respectively, the energy scale and the inter-
action length between two segments. The system has a
Hamiltonian of
ability distribution is flatter and the low energy region is
explored as the iteration number increases. The iteration
stops when the obtained probability distribution is reason-
ably flat in a chosen energy range. It is expected that near the
ground state, such a flat distribution would abruptly go to
zero like a step function, a criterion for an optimal choice of
(E) and ␣(E). With a pair of optimal (E) and ␣(E), we
a͒Author to whom correspondence should be addressed; electronic mail:
0021-9606/2000/113(10)/4469/3/$17.00
4469
© 2000 American Institute of Physics
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