Linear Response Theory
Reinhard Sigel
German University in Cairo (GUC)
Egypt
Reinhard Sigel
SOMATAI T4, 2014
What is Soft Matter ?
Polymers
Colloids
Liquid
Crystal
Reinhard Sigel
SOMATAI T4, 2014
Soft Matter
Why is it called “ Soft Matter “ ?
Softness → Fluctuations are important!
What determines the amplitude of fluctuations?
k BT
The thermal energy
What is kB?
The Boltzmann constant.
O.k., this is the name of kB, but what is kB?
J
Hint: What is the unit of kB? k B  
K
F  U  TS
→ kB is the “unit” of the entropy,
it relates the entropy with the
internal energy entropy T-scale
→ entropic contributions are essential for soft matter
example: polymer chain as an entropic spring
Free energy F
Reinhard Sigel
SOMATAI T4, 2014
Most Simple Example: Harmonic Oscillator
Force?
Hooke’s law
Fx   kx
Energy?
1 2
U    Fdx  kx
2
Reinhard Sigel
SOMATAI T4, 2014
Why do We Find Often Harmonic Oscillations?
Example: atoms in a crystal lattice
xm in
For small deformations, the potential energy
U  U min  k x  xmin 
2
dU by a parabola
is
well
approximated
Force
→ Hooke’s law
1
F
dx
  k  x  x0 
2
→ mass – spring system
→ harmonic oscillations cost   
k: susceptibility
Other examples: houses, cars, engineering constructions
Reinhard Sigel
SOMATAI T4, 2014
Harmonic Oscillator as Most Simple Example
Oscillator with an external force FE
Fx   kx  FE
Equilibrium Position with FE
Fx  0  x 
FE
k
“Linear Response Theory”
Relaxation process: Introduce a speed dependent friction force b dx
dx
kx  b  0
dt
 t
 x  t   x0 exp   
 
dt
b

k
(overdamped system)
Frequency dependent measurements: external force FE cos t 
dx
 FE cos t   0
dt
b
 tan    
 
k
 x t   x cos t  
kx  b
x
 x cos    
x  x s in   
Reinhard Sigel
FE
k
1  
 FE k

1   2 2
2 2
Debye process
SOMATAI T4, 2014
Harmonic Oscillator as Most Simple Example
1 2
kx
Energy
2
U1   FE x  U  U 0  U1
Additional interaction
“Linear Response Theory”
New equilibrium position? dU  0  x  FE
dx
k
kT
Fluctuations: Equipartition theorem U 0  1 kBT
x2  B
2
k
U0 
Fluctuation dynamics: Langevin Equation in a harmonic potential:
x  t  x  t   t 
x2
t
 t
 exp   
 
„Fluctuation Dissipation Theorem“
Fluctuation measurements and Dissipation (Relaxation)
measurements have the same information content.
Parameters of Interest:
Static properties: Susceptibility
Dynamic Properties: Relaxation Time
Reinhard Sigel
SOMATAI T4, 2014
Scattering Measurements
Thermodynamic system: Use the free energy instead of the energy
Free energy density for a fluctuation A q  of the thermodynamic
variable A with wave vector q
2
1
f  A  q    f  A  q   0 
2
 f
A  q 
2
1

f
Susceptibility: k  q     q  
2 A  q 2
Restoring „force“: K  q  
f
2
A q  0
2
3
A  q   O  A  q  


A q  0
2f
A  q     q  A  q 
A  q  A q 0
b q

q

Friction: b  q 
Relaxation time:  
 q

t 
A  q, t   A  q, t  0  exp  
Relaxation Measurements:

 q 

Frequency dependent measurements: Debye process
A  q 
Fluctuation Measurements (DLS):
Reinhard Sigel

2
A  q, t  A  q, t   t 
A  q 
2

t 
 exp  


q
  
SOMATAI T4, 2014
Analogy for fluctuation modes
4 n

q
sin  

2
Reinhard Sigel
SOMATAI T4, 2014

Konzert von HARMONIC BRASS - Kath. Pfarrgemeinde St. Peter u

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The Third Law Entropy of Strontium Molybdate M. Morishita abd H