For up to date version of this document, see http://imechanica.org/node/288
Z. Suo
FREELY-JOINTED CHAIN
A polymer is a chain of a large number of monomers. The monomers are
joined by covalent bonds, and two bonded monomers may rotate relative to each
other. Consequently, the polymer may be modeled as a chain of many links, each
link representing a monomer. At a finite temperature, the monomers restlessly
rotate relative to one another, and the polymer rapidly changes from one
configuration to another. When the two ends of the polymer is pulled by a force,
the distance between the two ends change. Thus, the polymer chain is known as
an entropic spring.
Boltzmann distribution. For a reminder see the notes on the
Boltzmann distribution (http://imechanica.org/node/293). Here we list the key
results.
A small system is in thermal contact with a large system. To keep track of
names, we will call the small system just the system, and the large system the
reservoir of energy, or just reservoir for brevity. The system and the reservoir
exchange energy by heat. All other modes of interactions between the system and
the reservoir are blocked: no exchange of matter, and no exchange energy by
work.
The system flips rapidly and ceaselessly among a set of states. Because the
system is not isolated, these states may have different amounts of energy, and are
not equally probable. What is the probability for the system to be in each of the
states?
The reservoir is much larger than the system, so that the temperature of
the reservoir is held at a fixed value, being kT in the unit of energy. We label the
states of the system as 1, 2, …, s..., and denote the energy of the states by
U 1 ,U 2 ,...,U s ,... The probability for the molecule to be in state s is proportional to
" U %
exp $$ − s '' .
# kT &
This probability distribution is known as the Boltzmann distribution.
Mean value of a property. Let Y be a property of the system. When
the system is in the states 1, 2,…, s,…, this property takes values Y1 ,Y2 ,...,Ys ,...
When the system is in thermal equilibrium with the reservoir, the mean value of
the property is
February 18, 2013
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For up to date version of this document, see http://imechanica.org/node/288
Z. Suo
" U %
s
∑Y exp $$− kT ''
s
Y =
#
" U %
∑exp $$− kTs ''
#
&
&
.
The sums are taken over all the states of the system.
A rigid rod subject to a fixed force. A rigid rod, length b, is subject to
a force of constant magnitude f and fixed direction. The rod can rotate freely. As
the rod rotates, the potential energy of the rod
changes. We regard the rod and the force
together as a system. The system is in thermal
contact with a reservoir of energy held at a fixed y
b θ
temperature kT.
The state of the system is specified by
two angles: the angle between the direction of
f
the rod and that of the force, θ , as well as the
angle around the direction of the force, φ . The
two angles vary in the intervals 0 ≤ θ ≤ π and
0 ≤ ϕ ≤ 2π .
Reservoir, kT
The energy of the system is the potential energy of the constant force:
U = − fbcosθ .
The potential energy is minimal when θ = 0 , increases with the angle θ , and is
maximal when θ = π . The minimal-energy state is a single state. By contrast,
states with θ ≠ 0 are numerous. Consequently, the expectation value of the angle
will not be zero.
The mean value of the displacement.
projected onto the axis of the force is
y = bcosθ .
The length of the rod
As the orientation θ of the rod fluctuates, so does y.
The number of states is proportional to the area of the element on the unit
sphere, 2π sin θdθ . Consequently, the mean value of y is given by
π
y =
∫ (bcosθ ) 2π exp (β cosθ ) sinθ dθ
0
π
,
∫ 2π exp (β cosθ ) sinθ dθ
0
where β = fb / kT is the normalized force. Integrating, we obtain that
February 18, 2013
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y
b
=
Z. Suo
1
1
− .
tanh β β
The expression on the right-hand side is known as the Langevin function.
In the absence of the force, β = 0 ,
the orientation of the rod is equally
probable in any direction, and the mean
value of the projection is y = 0 .
force
The
mean value y can be interpreted as the
displacement. The above expression gives
the displacement as a function of the
applied
force.
Because
thermal
fluctuation, the rigid rod acts as a spring—
displacement
the entropic spring.
When the force is small, fb / kT << 1 , we obtain that
y
b
=
b
fb
.
3kT
The force f is linear in the displacement y , and the stiffness is 3kT / b2 .
When the force is large, fb / kT >> 1 , we obtain that
y
b
=1−
1
.
fb / kT
The large force prevails over the temperature, and the rod will be nearly vertical,
y →b.
Exercise. Carry out the integral. Confirm the two limits.
(
)
Free energy. The Gibbs free energy is a function g kT , f , and the
following relation holds:
y =−
(
∂g kT , f
∂f
).
Integrating the force-displacement relation in the previous paragraph, we obtain
that
February 18, 2013
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Z. Suo
g
β
= log
.
kT
sinh β
We have dropped an additive term independent of the force.
The Helmholtz free energy w relates to the Gibbs free energy g as
w = g+ y f .
giving
w
β
β
=
+ log
.
kT tanh β
sinh β
Freely jointed chain. This model represents a polymer by a chain of n
links, each link being of length b. Two neighboring links can rotate freely relative
to each other. The chain is pulled at the two ends by a constant force f in a fixed
direction. Every link is subject to the same force f, and behaves in the same way
as a rod. Let y be the length of the entire chain projected onto the axis of the
force. The mean value of the displacement is given by
y
nb
=
1
1
− ,
tanh β β
where β = fb / kT is the normalized force. The Helmholtz free energy is
w
β
β
=
+ log
.
nkT tanh β
sinh β
•
•
•
•
REFERENCES
W. Kuhn and F. Grun, Kolloidzschr. 101, 248 (1942).
H.M. James and E. Guth, Theory of the elastic properties of rubber. The
Journal of Chemical Physics 11, 455 (1943).
L.R.G. Treloar, The Physics of Rubber Elasticity, Third Edition, Oxford
University Press, 1975.
M. Rubinstein and R. Colby, Polymer Physics. Oxford University Press,
2003.
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