Crumpling Transition
Ping-Cheng
December 2024
1
Introduction
When we drink hot soy milk, we might notice there’s a thin membrane on
the surface, sometimes the membrane is so flat that we even can’t see it, but
sometimes it crumples and we can notice it at first glance. This is an example
of the crumpling phase from our daily life experience, it seems like this phase
can easily be observed in nature, but what makes the surface behave differently,
and what makes the phase transition happen?
We know the world is made of many tiny things, such as atoms, molecules,
etc. Usually, these tiny things will bond with each other and form a larger
stuff. If this large stuff is made of many identical elements, which are called
monomers, bond together and repeat the structure many times, then we call it
polymer. For the previous example, the membrane on soy milk is a polymer
made of many proteins and lipid molecules. In the reference I find, they try to
be as general as possible. In that case, the polymer doesn’t need to live in 3
dimension world, and the polymer itself can be a D-dimensional object, we say
it’s a D-dimensional object embedded in d-dimensional space, it’s lattice is label
by a vector r(x), where r represent lattice’s position in external d-dimensional
space, and x is position in internal D-dimensional space which do not change
as deformation happen. Like soy milk membrane and the membrane discussed
in the following is the D = 2 polymer embedded in d = 3 space, r(x) represent
it’s position in 3-D space x tells us it’s place in that surface.
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what causes phase transition
Two main factors cause this phase transition to happen, one is the rigidity of
the surface. If a surface is hard to bend, stretch, or squeeze then it is obvious
that the surface is hard to be found in a crumpling state, since a crumpling
state means there are many bending stretching and other kinds of deformation,
which takes a lot of energy to do so if the surface is very rigid (high elastic
constant, strong bonding ).
Another factor is temperature or the entropy effect. Temperature is mainly
related to equilibrium, two systems that can exchange energy with each other
reaching equilibrium if they both have the same temperature. Entropy is more
1
or less the quantity that describes the number of configurations of system S =
log(Ω), it is a natural thing that the realization of a system is more likely to be
in a macroscopic state that has more configuration.
The crumpling surface has more configuration than the flat one since the
flat one has only one configuration, which is all the monomers stay at the same
plane. Thus from an entropy point of view surface should always tend to be
crumple. Then how do we know whether the system prefers the crumpling
phase or the flat phase? This is when the temperature and free energy comes
in. When system can change energy with heat bath or environment, it’s steady
state would be those minimize free energy
F = E − TS
(1)
If temperature is small then system would be control by internal energy, which
is bending and stretch energy, thus surface stays in flat phase. If temperature
is high entropy term would dominate and system would more likely to stay in
crumpling phase.
3
method to estimate free energy
free energy give us a way to deal with phase transition by finding behavior of
local minimum, but at this level we don’t know how to find it, in order to solve
this problem let’s go back to Ising problem.
Ising model can be use to explain phase transition in ferromagnetic material,
in ferromagnetic materials spin of atoms align at same direction corresponding
to lower energy state, which means spins are more likely to align and form a magnetization on macro scale. However ferromagnetic material doesn’t always have
magnetization, due to reason similar to crumpling transition, random direction
with zero net magnetization has more configuration than align one. System is
depicted by some free energy that determine wether spin of atom align together
or not.
Good news is the free energy of Ising model had been well studied by many
physicists. In this final report we would choose Landau theory since it would
give us simple analogy to crumpling transition.
Z
b
γ
(∇η)2 + atη 2 + η 4
(2)
L = dd r
2
2
L is Landau free energy, η is order parameter which describe the phase of the
system, in Ising case order parameter is magnetization η = m, if η = 0 system
is in paramagnetic state, for η ̸= 0 system is in freeomagnetic state. a, b >
0 are some phenomenological parameter that give the system right behavior.
c
t = T −T
is related to temperature. Now we can see why a, b > 0 give us
T
right behavior, if we want system to have finite η, b must greater than zero
otherwise minimum of L always happen at η → ±∞. a > 0 tells us above
critical temperature local free energy density would be single well potential of
2
η with minimum located at η = 0 and below critical temperature L is double
well potential of η which give us non-zero η. Thus a, b must greater than zero.
Notice that I did not mention γ2 (∇η)2 above since it give us non-local effect.
For γ > 0 difference between η and it’s neighbor would coast energy, thus for
large γ we would expect η changing slowly in space.
Although L seems just a phenomenological stuff, we can relate it to free
energy in thermal dynamics.
X
e−βL ∼ e−βF = Z =
(3)
e−βHΩ
Ω
By changing the way we sum partition function
!
Z=
XX
η(r) Ω
=
X
δ
X
Si − N η(r) e−βHΩ
(4)
i
e−βL[η(r)]
(5)
η(r)
Ω means all the configuration and Si is spin on site i, delta symbol change the
sum from all possible configuration of Si to order parameter η, it is like some
kind of coarse grain and N is how many lattice we want to do coarse grain.
In thermal dynamic limit (we know L ∝ N ) only minimum of L in summation
would be left, thus βF ∼ βL[η(r)]
3.1
analogy to crumpling transition
Let’s go back to crumpling transition. Note that in crumpling transition energy
mostly comes from bending and stretching, we can describe it by tangent vector
tα = ∂r(x)/∂xα , information of deformation is contain in this surface tangent
vector. The magnitude of this vector tell us the extent to which the surface
is stretched. And the derivative of this vector tell us bending intensity. Most
crucial thing is this tangent vector has property similar to spin in Ising model. In
Ising model spin like align to it’s neighbor, so do the tangent vector in crumpling
transition since align of tangent vector means little bending energy coast.
Thus the order parameter in here is the coarse-grained tangent vector, let’s
keep it’s original form tα = ∂r(x)/∂xα , then we can find it’s Landau free energy
by analogy
Z
1
1
(6)
βF {tα } = dD x κ(∂α tα )2 + t(tα )2 + u(tα · tβ )2 + ṽ(tα · tα )2
2
2
Z
1
+ b dD xdD x′ δ d (r(x) − r(x′ ))
(7)
2
first integral is analogy of Ising model to crumpling transition, and the coefficient
like κ, t, u, ṽ is related to bending rigidity and stretching energy, notice that
t ≡ a(T − Tc ) also relate to temperature . Second integral is due to self-avoiding
interaction. If surface overlap to itself, system need to pay extra energy, thus it
can prevent self overlap.
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4
Crumpling transition
Now let’s study crumpling transition first from flat state where T < Tc , t < 0.
In flat state effect of self-avoiding can be neglect since surface is almost flat and
cannot touch itself. Thus we assume position vector r have the form r(xα ) =
ζxα eα , where {eα } are unit vectors specifying the orientation of the surface,
and xα ranging from 0 to L (size of surface). We may notice that in this case
ζ is the α component of order parameter tα = ∂(ζxα eα )/∂xα which measure
the shrinkage of the surface. minimization of free energy give us ζ = 21 (−t/Dv),
this also tell us the size of surface, represented by radius of gyration1 scales as
RG = ζL ∼ |t|1/2 L
Next let’s move to crumpled state which require the understanding of selfavoiding interactions. Although interaction of self-avoiding has already included
in free energy, however it’s not related to order parameter, at least not explicitly.
In the article they use a method called Flory approximation to deal with selfavoiding interaction.
4.1
Flory approximation
To see what’s Flory approximation, we turn to polymer chain. An ideal chain
composed by many small monomer can be describe by random walk, and we
know when number of step random walk takes N is large enough, the number
r2
of configuration of end-to-end distant to be r is Ω(r) ∼ exp( −d
2N a2 ), and entropy
2
r
is S(r) = ln(Ω(r)) ∼ −d
2N a2 .
Locally self-avoiding effect is due to monomer-monomer interaction, which is
proportional to number of pair of monomer we can find in a small volume. Thus
we can write repulsive energy as Frep = 21 T vc2 , where c is local concentration of
monomer, v is excluded parameter and it proportional to monomer volume ad .
First approximation Flory made is saying ⟨c2 ⟩ → ⟨c⟩2 by ignoring correlation
between monomer, also it’s like kind of mean field. And ⟨c⟩ is just proportional
to number of monomer over size of chain ⟨c⟩ ∼ N/Rd , in this approximation
total repulsive energy is just local repulsive energy time size of chain Frep,tot ∼
=
1
N2
T
v
.
Then
we
add
repulsive
energy
to
free
energy
of
ideal
chain,
we
would
2
Rd
get approximated free energy and minimum of this free energy would give us
approximated radius of gyration
F ∼ N2
R2
=v d +
T
R
N a2
d+2 ∼
2 3
RG = va N
(8)
(9)
There’s an interesting fact that above 4 dimension exponent relation between
radius of gyration and number of particle will come back to ideal case. With
help of Flory approximation we can turn back to estimate free energy
−d 2D
2 D−4
2 D−2
4 D−4
2βF/D ≈ κRG
L
+ tRG
L
+ DvRG
L
+ bRG
L /D
1 definition of R
G is mean square displacement from center of mass
4
(10)
Figure 1:
In order to get this approximation we need to assume every derivative will give
us 1/L where L is size of polymer without deformation. and the last term is
replaced by repulsive energy in Flory approximation (L ∝ N ). Then we get
the exponent relation between radius of gyration and t, which is temperature
dependent parameter, and we can plot it out
5
conclusion
We studied the crumpling transition by making analogy the coarse-grained tangent vector tα to order parameter in Ising model m. Notice that in Ising model
we have Z2 symmetric, and in crumpling the symmetry is all continuous rotation. Then we use Flory approximation to help us deal with self-avoiding effect.
The result of radius of gyration have same exponent as Ising one when T < Tc .
For T > Tc due to effect of self-avoiding, radius of gyration would not drop to
zero immediately, instead it would slowly approaching to zero.
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reference
• Paczuski, M., Kardar, M., & Nelson, D. R. (1988). Landau theory of the
crumpling transition. Physical Review Letters, 60(25), 2638–2640
• de Gennes, P.-G. (1979). Scaling concepts in polymer physics. Cornell
University Press.
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