Introduction to Soft Matter Physics - Andreas B. Dahlinadahlin.com/onewebmedia/TIF015/introduction.pdf · 2015-09-11 2 2015-09-01 Soft Matter Physics 3 Soft? What is a “soft”

2015-09-11

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1Soft Matter Physics

Introduction to Soft Matter Physics

2015-09-01

Lecture 1Jones: 1.1-1.2, 2.1-2.3, 5.5, A

[email protected]

http://www.adahlin.com/

2015-09-01 Soft Matter Physics 2

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.

http://www.adahlin.com/

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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:

• Polymers (long flexible molecules).

• Colloidal suspensions (small particles in a liquid).

• Liquid crystals (LCD technology).

• Organisms (highly ordered) and food!

AGA

http://www.aga.se/


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


Plastic Fantastic

Plastics are polymers!

• Pipes

• Furniture

• Vehicles

• Tools

Not as tough as other materials, but good enough! Cheap, simple to produce and light.

Tupperware

http://www.tupperware.com/

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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!


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: ReptationWikipedia: Diffusion

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Basic Thermodynamics

The first law (conservation of energy) says the internal energy U of a system changes as:

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:

Entropy (thermodynamics) for reversible process:

The second law (in the most common formulation) says that the total entropy change (a

system and its surroundings) always increases:

wqU ddd

PVUH

T

qS

dd

0ddd sursystot SSS


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:

In differential form we get:

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.

TSUF

TSVP

TSSTVPSTTSSTwqTSSTUTSUF

dd

dddddddddddddd

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Gibbs Free Energy

The definition of Gibbs free energy is:

The differential becomes:

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?)

TSHTSPVUG

TSPVTSSTPVVPVPSTTSSTPVVPwq

TSSTPVVPUTSPVUG

dddddddddddddd

ddddddddd

Free Energy Minimization

12Soft Matter Physics2015-09-01

“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.

For Gibbs: A system at constant T and P with strive towards a minimum in G.

Why???

heatmatter

P, T

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Gibbs Energy Minimization

13Soft Matter Physics2015-09-01

Using the first law we can see that:

Hence the enthalpy change is equal to the heat transferred to the surroundings if the

system is kept at constant pressure:

The second law then gives:

So this is why a process with negative ΔG is thermodynamically favorable!

T

HS

dd sur

PVqVPPVVPqHPVU

PVVPqPVU

VPSTwqU

dddddddd

ddddd

ddddd

0ddd0dd

dd syssystot STHGT

G

T

HSS


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.

1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

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Boltzmann’s Entropy Formula

Most general entropy formula:

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:

n

i

ii ppkS1

B log

WkWW

kWW

kSW

i

W

i

loglog11

log1

B

1

B

1

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)

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)?


Test: Entropy of Gender

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Classical thermodynamics: An ideal gas expands isothermally. The work it does is:

Same number of particles at same temperature means U is unchanged so:

q = w

The entropy change is then:


Isothermal Ideal Gas Expansion

i

fB log

V

VNk

T

w

T

qS

q

T

Vi

Vf

N

i

fBBB loglogd

1d f

i

f

i

f

iV

VTNkVTNkV

VTNkVPw

VV

VV

V

V

V

V

TNkPV B


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:

dV

Vi

Vf

i

fB

i

fB

i

fBifBiBfB

log

d/

d/loglogloglogloglog

V

Vk

VV

VVk

W

WkWWkWkWkS

dV

To get the total entropy change

we must multiply with the

number of particles N.

Sanity check: Agrees with

previous result!

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Boltzmann Statistics

i

ii

ii

i

Tk

Gg

Tk

Gg

N

N

B

B

exp

exp

Tk

G

Tk

G

Tk

G

Tk

G

Tk

G

Tk

G

Tk

G

Tk

G

Tk

G

N

N

i

i

i

i

BB

A

B

B

B

A

B

B

B

B

A

B

B

B

A

B expexp

exp

exp

exp

exp

exp

exp

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:


Chemical Potential

The chemical potential μ is per definition the free energy required to introduce molecules

into the system:

N is some measure of number of molecules. Boltzmann statistics makes it possible to

relate concentration to chemical potential:

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).

PTVT N

G

N

F

,,

TkΦ

B

exp

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The basic metabolism of glucose supports life and obviously releases energy. So why

does sugar not just disintegrate if we have ΔG < 0?

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!


Equilibrium vs Kinetics

? ? ?

C6H12O6 + 6O2 → 6H2O + 6CO2


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”.

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:

reaction progression

free

en

ergy

ΔG*

ΔG

Tk

Gk

B

exp

Most molecules “wobble

around” with frequencies

on the order of GHz…

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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.


Generic Interaction Potential

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.

As T and P changes, a pure substance

assumes different phases:

Solid ↔ Liquid (melting, freezing)

Liquid ↔ Gas (boiling, condensation)

Solid ↔ Gas (sublimation)distance (r)

ener

gy

0 r*

ε

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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.

A liquid with extremely high viscosity is essentially a

hard solid.

Contextual Feed

http://www.contextualfeed.com/

For a Hookean solid this leads to a shear

strain e = Δx/d (dimensionless) and the

proportionality constant is the shear

modulus G.

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:


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.

stressΔx

flow

d

Gesstrain

top plate

(moving)

v

t

e

d

vs

bottom plate

(stationary)

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Non-Newtonian Liquids

Newtonian

shear thinning

shear thickening

strain rate (γ)

shea

r st

ress

(σ)

0

0

shear thickening

strain rate (γ)

vis

cosi

ty (η)

0

0

shear thinning

Newtonian

Bingham plastic

Viscosity may depend on strain rate!


Shear-Thinning and Bingham Plastics

Shear-thinning: Flows easier with higher shear stress!

Bingham plastics: Threshold (yield) stress!

Wikipedia: Mayonnaise

Heinz

http://www.heinz.com/

wiseGEEK

http://www.wisegeek.com/

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Video: Oobleck

Maizena

http://www.maizena.se/

Corn starch in water (50%) is shear-thickening!


Time Dependent Viscosity

In special cases the viscosity changes with the time during which the stress is applied:

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!)

t 0 t

Arla

http://www.arla.se/

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Viscoelasticity

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.

time (t)

stra

in (

e)

σ0 applied0

elastic response

flow

τ time (t)

stre

ss (σ)

0

elastic response

flow

τ

e0 applied


Maxwell Approximation

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:

time (t)

stra

in (

e)

σ0 appliedτ

γ = σ0/η

σ0/G0

0G

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/


Oscillatory Deformations

One interesting and important case is oscillatory strain:

The stress response can be written with a phase delay δ:

The resulting stress can be described by a complex dynamic modulus G* as:

It is related to the stress relaxation modulus by the transform:

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!

0

diexpi* ttGtG

tt sin0

tete sin0

cos*Re0

0

eG

sin*Im

0

0

eG

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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.

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.

Maxwell Voigt


Reflections and Questions

?

Documents

Introduction to Soft Matter Physics - Andreas B. Dahlinadahlin.com/onewebmedia/TIF015/introduction.pdf · 2015-09-11 2 2015-09-01 Soft Matter Physics 3 Soft? What is a “soft”