Download Introduction to Soft Matter Physics

2015-09-11
Introduction to Soft Matter Physics
Lecture 1
Jones: 1.1-1.2, 2.1-2.3, 5.5, A
[email protected]
http://www.adahlin.com/
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Overview
As an introduction we will look a little bit at:
• What is soft matter?
• Who cares about soft materials?
• Typical assumptions in SMP!
• Basic thermodynamics.
• Intermolecular forces.
• Non-Newtonian liquids and viscoelasticity.
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Soft?
What is a “soft” material? Perhaps it is easier to say what it is not soft:
• Crystalline hard solids.
• Pure “ordinary” (Newtonian) liquids.
• Gases (no intermolecular forces).
Examples of materials relevant for SMP:
AGA
http://www.aga.se/
• Polymers (long flexible molecules).
• Colloidal suspensions (small particles in a liquid).
• Liquid crystals (LCD technology).
• Organisms (highly ordered) and food!
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SMP for Food Science
Mezzenga et al.
Nature Materials 2005
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Dry Water
Water microdroplets encapsulated by hydrophobic colloids.
95% water!
Wikipedia: Dry water
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Plastic Fantastic
Plastics are polymers!
• Pipes
• Furniture
• Vehicles
• Tools
Tupperware
http://www.tupperware.com/
Not as tough as other materials, but good enough! Cheap, simple to produce and light.
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Lengthscales in SMP
The interesting stuff happens on the nanoscale!
Polymers, micelles and colloids are in this size regime.
High surface to volume ratio, interfaces matter!
The coarse grained approximation: Ignore individual atoms!
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Timescales in SMP
SMP incudes very fast and very slow processes! From diffusion of small molecules to
reptation of polymers…
Quite often the understanding of a phenomenon is directly related to the timescale of the
processes involved.
Sometimes dynamics are so slow that equilibrium is never reached for the system. We
must consider the kinetics to understand what is happening.
Wikipedia: Diffusion
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Wikipedia: Reptation
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Basic Thermodynamics
The first law (conservation of energy) says the internal energy U of a system changes as:
dU  dq  dw
Here q is the heat supplied to the system and w the work performed by the system (note
signs). Usually dw = PdV (mechanical work) but it can also be related to other things
like addition or removal of matter (chemical potential)!
Enthalpy definition:
H  U  PV
Entropy (thermodynamics) for reversible process:
dS 
dq
T
The second law (in the most common formulation) says that the total entropy change (a
system and its surroundings) always increases:
dS tot  dSsys  dSsur  0
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Helmholtz Free Energy
We will use “free energy” a lot in this course. By this one means energy that is available
for performing thermodynamic work, i.e. work mediated by thermal energy.
Helmholtz free energy is defined as:
F  U  TS
In differential form we get:
dF  dU  dTS   dU  TdS  SdT  dq  dw  TdS  SdT  TdS  PdV  TdS  SdT 
 PdV  SdT
Here we used the first law and the entropy definition. We see that if V is constant only T
influences F.
Helmholtz free energy is mostly used by physicists and engineers who are interested in
mechanical work and gases.
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Gibbs Free Energy
The definition of Gibbs free energy is:
G  U  PV  TS  H  TS
The differential becomes:
dG  dU  dPV   dTS   dU  PdV  VdP  TdS  SdT 
dq  dw  PdV  VdP  TdS  SdT  TdS  PdV  PdV  VdP  TdS  SdT  VdP  SdT
This is similar to how we treated Helmholz before. We see that if P is constant only
temperature influences G.
Gibbs free energy ignores mechanical work, which chemists enjoy. Gibbs free energy is
especially useful in biological systems!
In practice the choice of free energy is mainly a way to emphasize how experiments
were performed. (Was it constant pressure or volume?)
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Free Energy Minimization
“The first law says something about how things must happen while the second law
explains why things do happen.”
In this course we will work a lot with the principle that the free energy of a closed
system (exchanges heat but not matter with the environment) will strive towards a
minimum.
P, T
heat
matter
For Gibbs: A system at constant T and P with strive towards a minimum in G.
Why???
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Gibbs Energy Minimization
Using the first law we can see that:
dU  dq  dw  TdS  PdV
dU  dPV   dq  PdV  dPV 
dU  PV   dH  dq  PdV  VdP  PdV  dq  VdP
Hence the enthalpy change is equal to the heat transferred to the surroundings if the
system is kept at constant pressure:
dSsur  
dH
T
The second law then gives:
dS tot  dSsys 
dH
dG

 0  dG  dH  TdSsys  0
T
T
So this is why a process with negative ΔG is thermodynamically favorable!
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Entropy in Statistical Mechanics
Entropy is about probabilities and the number of microstates associated with a certain
macrostate. The microstates are not observable! Entropy is lack of information.
Two dice have 36 microstates with equal probability. Entropy can be observed in the
macrostate represented by the sum. Example: Probability of getting macrostate 7 with
two dice is 1/6 (6 out of 36 microstates). The probabilities for getting 2 or 12 are only
1/36 each.
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Boltzmann’s Entropy Formula
Most general entropy formula:
n
S  k B  pi log pi 
i 1
The probability of microstate i is pi. Boltzmann’s
constant kB = 1.3806×10-23 JK-1 relates entropy to
free energy via temperature.
If all W microstates are equally probable p = 1/W for
all i and n = W. We can get the simpler formula:
W
1
1
1
log   k B  logW   k B logW 
W 
i 1 W
i 1 W
W
S  k B 
Wikipedia: Ludvig Boltzmann
The logarithmic dependence essentially comes from combinatorics: If there are WA states
in system A and WB states in system B the total number of states is WAWB, but entropy
becomes additive: SA + SB = kBlog(WA) + kBlog(WB) = kBlog(WAWB)
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Test: Entropy of Gender
Alice and Bob have two kids…
I: One is a boy.
p(the other is also a boy)?
I: The older is a boy.
p(the younger is also a boy)?
I: (nothing more)
p(both are boys)?
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Isothermal Ideal Gas Expansion
Classical thermodynamics: An ideal gas expands isothermally. The work it does is:
PV  Nk BT
V
1
V V
dV  Nk BT logV V Vif  Nk BT log f
V
 Vi
Vi
Vf
Vf
w   PdV  Nk BT 
Vi



Same number of particles at same temperature means U is unchanged so:
q=w
The entropy change is then:
Vf
T
S 
V
q w
  Nk B log f
T T
 Vi



Vi
N
q
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Volumes and Entropy
Consider volume expansion again from the viewpoint of statistical mechanics.
We can discretize the space available into a certain number of positions, each with
volume dV, where a gas molecule can be located. The entropy change is then:
W
S  k B logWf   k B logWi   k B logWf   logWi   k B log f
 Wi
 Vf 
 k B log 
 Vi 

 V / dV
  k B log f

 Vi / dV
To get the total entropy change
we must multiply with the
number of particles N.

 

Vf
Vi
Sanity check: Agrees with
previous result!
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dV
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dV
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Boltzmann Statistics
Ni is number of entities occupying state i with energy Gi and
Ni/N the probability that a given entity is at energy level i at
any point in time. (The parameter g is degeneracy and we
can set g = 1 here.) The ratio of the probabilities of
occupying one state (B) compared to another (A) is then:
NB

NA
 G 
exp  B 
 k BT 
 G 
i exp  k Ti 
 B 
 G 
exp  A 
 k BT 
 G 
i exp  k Ti 
 B 
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Ni

N
 G 
g i exp  i 
 k BT 
 G 
i gi exp  k Ti 
 B 
 G 
exp  B 
 k BT   exp  GB  GA   exp  G 

 k T k T
 k T
 G 
B 
 B
 B 
exp  A 
 k BT 
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Chemical Potential
The chemical potential μ is per definition the free energy required to introduce molecules
into the system:

F
N

T ,V
G
N T , P
N is some measure of number of molecules. Boltzmann statistics makes it possible to
relate concentration to chemical potential:
    

Φ  exp
 k BT 
Here we have introduced a standard chemical potential μ° which is the chemical potential
at a standard state. (Typically T = 25°C and P = 1 bar.) Note that we must work with a
dimensionless concentration Φ (volume fraction, mole fraction).
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Equilibrium vs Kinetics
The basic metabolism of glucose supports life and obviously releases energy. So why
does sugar not just disintegrate if we have ΔG < 0?
C6H12O6 + 6O2 → 6H2O + 6CO2
???
In physical chemistry and thermodynamics we often say that a system will assume a
certain state because it minimizes the energy.
However, sometimes the kinetics for reaching that state are so slow that it cannot happen
in practice. Instead, systems get stuck in local energy minima or find other pathways.
The global minimum is never reached!
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Reaction Kinetics
The basic model of reaction kinetics is that thermal fluctuations can make a system (or a
part thereof) reach the “activated state”, corresponding to the activation energy, after
which the energy change is just “downhill”.
 G  

k  exp 
 k BT 
Most molecules “wobble
around” with frequencies
on the order of GHz…
free energy
The probability that a reaction occurs is an exponential function of the activation energy.
If the rate constant is k we have according to Arrhenius kinetics:
ΔG*
ΔG
reaction progression
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Intermolecular Forces
An overview of forces and bonds:
• Ionic bonds. Coulomb interaction, scales with ~r-1, 200-300 kBT at room temperature.
• Covalent bonds. Ångström range and directional, ~100 kBT at room temperature.
• Metallic bonds. Delocalized electrons, ~100 kBT at room temperature.
• Hydrogen bonds. H interacting with N or O, 5-15 kBT at room temperature.
• Van der Waals interactions, always present, scales with r-6, ~kBT at room temperature.
• Hydrophobic interactions, due to ordering of water, ~kBT.
In SMP we do not break covalent bonds, that is for chemists... The bonds that are
interesting are those comparable to kBT in strength.
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Molecules generally attract each other,
but there must eventually be repulsion
when electron orbitals start to overlap.
The lowest energy is at a separation r*
which gives an interaction energy ε.
energy
Generic Interaction Potential
As T and P changes, a pure substance
assumes different phases:
Solid ↔ Liquid (melting, freezing)
ε
Liquid ↔ Gas (boiling, condensation)
Solid ↔ Gas (sublimation)
0
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r*
distance (r)
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Rheology
Rheology: Science of deformation and flow of matter.
A solid withstands stress without yielding.
Hard solids are tough to deform!
Soft solids are easy to deform and elastic: They
return to their original shape upon stress release.
Brittle solids break upon stress without deforming.
(Opposite of ductile materials.) They can be hard!
Liquids flow under stress, but more or less easily as
defined by the viscosity.
Contextual Feed
http://www.contextualfeed.com/
A liquid with extremely high viscosity is essentially a
hard solid.
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Hookean Solids and Newtonian Liquids
Consider a material of thickness d sandwiched between two infinite plates. We apply a
shear stress σs, which is the force that the upper plate is “pulled” with divided by its area
(like a pressure). The plates move with a velocity v relative to each other with the
material perfectly “attached” to the plates.
For a Hookean solid this leads to a shear
strain e = Δx/d (dimensionless) and the
proportionality constant is the shear
modulus G.
strain
 s  Ge
d
flow
bottom plate
(stationary)
v
e

 
d
t
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stress
Δx
v
In a Newtonian liquid, the velocity gradient
(strain rate) γ = v/d is linearly related to the
stress. The proportionality constant is the
dynamic viscosity η:
s 
top plate
(moving)
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Non-Newtonian Liquids
Viscosity may depend on strain rate!
Bingham plastic
Newtonian
viscosity (η)
shear stress (σ)
     
shear thickening
shear thinning
shear thinning
shear thickening
Newtonian
0
strain rate (γ)
0
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0
strain rate (γ)
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Shear-Thinning and Bingham Plastics
Shear-thinning: Flows easier with higher shear stress!
Bingham plastics: Threshold (yield) stress!
Wikipedia: Mayonnaise
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Heinz
http://www.heinz.com/
wiseGEEK
http://www.wisegeek.com/
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Video: Oobleck
Corn starch in water (50%) is shear-thickening!
Maizena
http://www.maizena.se/
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Time Dependent Viscosity
In special cases the viscosity changes with the time during which the stress is applied:
   t 
 t    0
This is very complicated and not the same thing as shear thinning and shear thickening!
Then we had a given viscosity for a given stress, but now it also varies with time.
Rheopectic: Viscosity increases with duration of stress. Very uncommon: The shearing
itself must induce structural changes leading to solidification. (Whipping cream!)
Thixotropic: Viscosity decreases with duration of stress. More common: The shearing
disrupts an initial structure which inhibits flow in the material. (Stirring yoghurt!)
Arla
http://www.arla.se/
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Viscoelasticity
strain (e)
stress (σ)
Some materials are viscoelastic: They can behave either as solids or liquids depending on
the timescale of the applied stress in comparison with the relaxation time τ.
elastic response
flow
flow
elastic response
0
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σ0 applied
τ
time (t)
0
τ
time (t)
e0 applied
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Maxwell Approximation
γ = σ0/η
strain (e)
Looking at the strain-time graph we can get a
rough approximation for the viscosity of a
viscoelastic material exposed to long term stress.
Assume G0 is the elastic response modulus to
deformation on very short timescale. The slope
in the liquid response region is:
σ0/G0
  G0
σ0 applied
τ
time (t)
As Jones puts it, this is “at least dimensionally correct”. (The extrapolated line must not
cross origin.)
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Demonstration: Silly Putty
Bromma Kortförlag
http://www.brommakortforlag.se/
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Oscillatory Deformations
One interesting and important case is oscillatory strain: et   e0 sin t 
The stress response can be written with a phase delay δ:  t    0 sin t   
The resulting stress can be described by a complex dynamic modulus G* as:
0

ImG *  0 sin  
cos 
e0
e0
It is related to the stress relaxation modulus by the transform:
ReG * 

G *    i  exp it G t dt
0
Re(G*) represents the “elastic response” and energy storage, while Im(G*) represents
the “liquid response” and energy dissipation.
As ω → ∞ one expects Im(G*) = δ = 0 for elastic behavior while for a liquid Re(G*) →
0 and δ = π/2. The dominating behavior is determined by τ-1 in comparison with ω!
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Real Stress Relaxation
Linear viscoelasticity: Material has no “memory”. Small separate deformations can be
treated as independent. (Easily broken model but we do not go beyond it.)
Exact expressions for G(t) is an entire research field: The Maxwell model uses a
“spring” (elastic element) in series with a “dashpot” (viscous element) while the Voigt
model places these elements in parallel.
Maxwell
Voigt
The Maxwell model is good at predicting stress relaxation, but fairly poor at predicting
deformation (creep). On the other hand, the Voigt model is good at predicting creep but
rather poor at predicting stress relaxation.
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Reflections and Questions
?
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