Author: Tim Verbovšek
Mentor: doc. dr. Primož Ziherl
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
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Entropy
Polymers
Depletion potential


Experiment
Liquid crystals

Simulation
Entropy in soft matter physics

2nd Law of thermodynamics

In equilibrium, the system has maximal entropy
Written in mathematical form by Rudolf Clausius

𝑑𝑆 ≥ 𝑑𝑄
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Free energy


𝑇
ℱ = 𝑈 − 𝑇𝑆
Hard-core interactions

∞
𝑈=
0
Entropy in soft matter physics
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
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Macrostate: property of the system
Microstate: state of a subunit of the system
Ω statistical weight



Different sets of microstates for a given macrostate
𝑆 = 𝑘𝐵 ln Ω if all sets of microstates are equally
probable
𝑆 = − 𝑘𝐵 𝑝𝑖 ln 𝑝𝑖
Entropy in soft matter physics
Entropy in soft matter physics
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
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Long chains
Random walk
Real polymer
chains
Entropic spring
Entropy in soft matter physics
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Random walk
Persistence length


Gaussian probability distribution of the end-to-end
vector size |𝒓|


𝑃 𝒓, 𝑁 =
3
2𝜋𝑁𝑎2
3
2
3𝒓2
exp(−
)
2𝑁𝑎2
Configurational entropy:


Approximate length at which the polymer loses rigidity
𝑆 𝒓 =
3𝑘𝐵 𝒓2
−
2𝑁𝑎2
+ 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Free energy:

ℱ 𝒓 =
3𝑘𝐵 𝑇𝒓2
+
2𝑁𝑎2
+ 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Entropy in soft matter physics
Entropy in soft matter physics
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Correlation of neighbouring bonds
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Finite bond angle
Excluded volume
Self-avoiding walk; the polymer
cannot intersect itself
 The coil takes up more space

Entropy in soft matter physics
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Macrospheres and
microspheres
Exclusion zone
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
Asakura-Oosawa
model (1954)
The result of
overlapping exclusion
zones is an attractive
force between
macrospheres
Microscopic image of milk.
Droplets of fat can be seen.
Entropy in soft matter physics
An excluded zone appears around the plate submerged in a
solution of microspheres
𝑉 = 2𝐴𝜎
Entropy in soft matter physics
Exclusion zones overlap, leading to a larger available volume for
the microspheres
𝑉 = 𝐴(𝜎 + 𝑑)
Entropy in soft matter physics
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Ideal gas of microspheres

Free energy is ℱ = −𝑁𝑘𝐵 𝑇 ln 𝑉′
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Entropic force: 𝐹 =

Two spheres:


1
4
𝐹 = −ρ𝑘𝐵 𝑇π((𝑟 + 𝑅)2 − 𝑑 2 )
Wall-sphere:


𝜕ℱ
−
𝜕𝑑
𝐹 = −ρ𝑘𝐵 𝑇π(3𝑅 + 𝑑)(2r − R − d)
Short ranged interactions
Entropy in soft matter physics
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Silica beads (𝑑 ≈
1.25 𝜇𝑚) were
suspended in a
solution of λ-DNA
polymers (𝑅𝑔 ≈
500 𝑛𝑚)
Measurement of the
positions of the beads
gives the probability
distribution P(r)

𝑃(𝑟) ∝ 𝑒
−𝑈(𝑟)
𝑘𝐵 𝑇
Entropy in soft matter physics
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Optical tweezers hold
the beads in place
The potential as a
result of optical
tweezers was found to
be parabolic
Entropy in soft matter physics
Entropy in soft matter physics
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Experiment gives a good fit to the AsakuraOosawa model
The range of the depletion potential was found
to be 𝑑 = 2(𝑅 + 𝑟)
Depth of the potential increases linearly with
polymer concentration

1
4
𝐹 = −ρ𝑘𝐵 𝑇π((𝑟 + 𝑅)2 + 𝑑 2 )
Entropy in soft matter physics
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Isotropic phase
Nematic phase
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Director
Positions of the centers of mass
are isotropic
Smectic phase
Layers
 Smectic A
 Smectic C
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Columnar

Disk-shaped molecules
Entropy in soft matter physics
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Onsager theory (1949)
Solid rod model
𝑆 = 𝑆𝑜 + 𝑆𝑝

𝑆𝑜 - orientational entropy
 Has a maximum in the isotropic phase

𝑆𝑝 - packing entropy
 It is maximised when the molecules are parallel
 The same role as the depletion potential in colloidal
dispersions
 It is a linear function of the concentration of rods
Entropy in soft matter physics
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Lyotropic liquid crystals:
Phase changes occur by
changing the molecule
concentration (T = const.)
Computer simulations for
hard spherocylinders

Shape anisotropy parameter
𝐿/𝐷

Length-to-width ration
𝐿+𝐷
𝐷
Entropy in soft matter physics
Entropy in soft matter physics

Entropy


Polymers


Entropic spring
Depletion potential



With hard spheres and constant temperature, the
free energy depends only on entropy
Short-range attraction between colloids
Experiment
Liquid crystals


Phase transitions
Simulation
Entropy in soft matter physics