Download Describing many-particle systems

Describing many‐particle systems
Prof. Judit Fidy
2015, October 14
Complicated example
Simple example
V, p, T
particle: ‐ atom, atomic group
‐ molecule
‐ molecular complexes, macromolecules
‐ etc.
many: 6x1023
Air molecules in the room…
What do the particles „do”, when the
macroscopic parameters are constant: V,p,T? Complex system of
large number of atoms and molecules in
interaction with
each other
Energetic states
In the field of interactions?
Electronic states
In the field of interactions?
Boltzmann distribution
Boltzmann distribution ‐ conclusions
N distinguishable independent particles at
thermal equilibrium, T= 0 in a closed system,
exposed to a force field
ε
E = ∑ n jε j
j
εj possible energy status for one particle
nj number of particles having εj energies
k
ε
j
j
ε ,n
k
k
−
ni
=e
n0
k
Boltzmann distribution function
εi, ni
−
nk
=e
nj
εk −ε
kT
j
=e
− Δε
kT
Boltzmann
factor
1
εi, ni
kT
j
=e
− Δε
kT
εi − ε0
kT
=e
−
εi
Same system of N particles
at different temperatures
supposing ε0=0
kT
As the temperature is lowered, more and more
particles are in the energy minimum state
at T=0 all particles are there: ni=0, n0=N
n0 is the same
ε ,n
1
1
0 ,,n0
εk −ε
T1<T2<T3
ε ,n
ε
−
nk
=e
nj
j
Populations of potential energy states – nk: number of particles in εk energy state
„distribution” of particles is supposed
T1<T2<T3
N = ∑ nj
E = ∑ n jε j
ε ,n
Ludwig Eduard Boltzmann
1844-1906, Austrian physicist
N = ∑ nj
ε
Valid for any (j,k) combinations of energy levels
1
0 ,,n0
At a given ε k > ε 0 energy level, the relative
population related to the energy minimum
increases with the temperature
−
ni
=e
n0
εi − ε0
kT
εk > ε0
Boltzmann distribution – a general rule in nature
Boltzmann distribution – a general rule in nature
Considering the classes of materials – structural consequences
Considering the classes of materials – structural consequences
1. Gasous systems (ideal gas)
-> lack of order in the structure
2. Classes of materials with structural order.
‐ solid crystalline materials
Order is based on
‐ ideal crystals -> case of perfect, long range order
primary bonds
‐ real crystals
-> case of imperfect, long range order
‐ liquid crystals
Order is based on
secondary bonds
‐ liquids
-> case of short range order
1. Gasous systems (ideal gas)
-> lack of order in the structure
2. Classes of materials with structural order.
‐ solid crystalline materials
Order is based on
‐ ideal crystals -> case of perfect, long range order
primary bonds
‐ real crystals
-> case of imperfect, long range order
‐ liquid crystals
Order is based on
secondary bonds
‐ liquids
-> case of short range order
FIRST EXAMPLE
Absolute value of velocity
Absolute value of velocity
2. Classes of materials with structural order.
Examples for the functional form of attractive interactions
Order is based on interactions (bonds) between the constituting atoms or molecules
The distance dependence in the energy formulae
of electrostatic interactions
General concept of bond formation (applicable for all kinds of interactions)
repulsion
E
pot
= B
rm
Interaction
m>n
Distance in the
energy function
ion-ion
r −1
200 - 300
ion – static dipole
r −2
10 - 20
static dipole –
- static dipole
r −3
1 -2
Eb
attraction
E
pot
=− A
rn
r= distance of two atoms forming the bond
ro= bond distance
Ebond= bond energy
The concrete expression for A, B and the values of n and m depend on the form of interactions
Magnitude of the
Energy of interaction
(kJ/mól)
dipole – dipole
with Brown motion
r −6
dispersion
r −6
0.3
2
2. Classes of materials with structural order.
Atomic/molecular bond distances (ro= rA + rB) and binding energies Eb depend
on the type of interactions (functions a and b)
Order is based on interactions (bonds) between the constituting atoms or molecules
General concept of bond formation
Rends
zám
Z
Van
der
van der
Waals
Waals
radius (nm)
sugár (nm)
Kovalens
Covalent
sugár
radius
(nm)
(nm)
Ionic
Ionsugá
radius
r (nm)
(nm)
Ion
H
1
0,120
0,037
−
H+
C
6
0,170
0,077
0,029
C+
N
7
0,155
0,075
0,025
N+
O
8
0,152
0,073
0,140
O2-
F
9
0,147
0,071
0,117
F-
Atom
Elem
Eb
repulsion
E
pot
= B
rm
m>n
Eb
attraction
P
15
0,180
0,106
0,058
P3+
S
16
0,180
0,102
0,184
S2-
rA, rB atomic radii
of atoms A and B
„atomic radius”
E
pot
Is it true that all constituting atoms
are at bond distances from each other
all the time?!
=− A
rn
r= distance of two atoms forming the bond
ro= bond distance
Ebond= bond energy
The concrete expression for A, B and the values of n and m depend on the form of interactions
2. Classes of materials with structural order.
Order is based on interactions (bonds) between the constituting atoms or molecules
General concept of bond formation
repulsion
E
pot
Is it true that all constituting atoms are at bond
distances from each other all the time?
= B
rm
Boltzmann distribution allows for
having broken bonds!
m>n
nbrokenbonds nb
− Δε
=
= e kT
nint actbonds ni
Eb
attraction
E
pot
Δε = Ebond
=− A
rn
The probability of breaking bonds by thermal
fluctuations depends on the relation:
r= distance of two atoms forming the bond
ro= bond distance
Ebond= bond energy
Δε
The concrete expression for A, B and the values of n and m depend on the form of interactions
kT
Bond energies in materials with structural order
Great variety!
electronvolt
1 eV= 23 kcal/mole ~
~ 100 kJ/mól
Eb ~ primary bonds: covalent
ionic
metallic
Eb ~ secondary bonds
H-bond
hydrophobic
dipole – point charge
van der Waals dipole – dipole
dipole – induced dipole
dispersion
(temporary dipoles)
kT ~ 0.027 eV
T=310 K,
k=1.38x10-23JK-1 Boltzmann constant
2.1.Structure of crystalline materials = ideal state of order
Particles: atoms - bonds: primary bonds -> ordered structure
Single crystal
(ideal)
microcrystalline state
(real crystals)
grain
long range order in ideal crystals
large number of atoms in periodic array
Characteristics of „crystalline” state
‐anisotropy: properties depends on the direction
‐mechanical stability
‐defined volume
‐defined shape
‐structure: long range order
grain boundary:
accumulation of
crystal defects
NaCl
unit cells – periodic repetition Æ crystal lattice
14 kinds of units cells in nature: Bravais lattices
Si
long range order: distance of periodic repetition >> 100 times r0(=0.15nm)
2 – 10 eV/bond
The 14 kinds of unit cells
found in nature
Eb(eV)
several x 0.1 (water:0.2 eV)
~ 0.1
~ 0.1-0.2
~ 0.02
~ 0.01
~ 0.02
crystal defects in real crystals ?
Considering two states of bonds – intact
‐ broken
nb
−
=e
ni
Eb
kT
2.2. Liquid state – case of partial (short range) and temporary order
−
≈ n = e 0.023 = e − 270 = 0
N
ni + nb = N
6.31
Particles: molecules
Interaction energy: Eb~ secondary bonds
ordered and disordered regions Æ average Eb is small Æmany broken bonds
nb << ni
e.g. NaCl, Eb=6.31eV, kT(room tempr.)=0.023eV
But: imperfections in crystal growth
n broken
≅ 1 % - 0.05% (T = 300 K)
n0
Primary bonds can not be broken by thermal fluctuations at
room temperature
Elocal <Eb
E
− local
kT
n≈e
Properties of liquid state
‐isotropy
‐deformability
‐it has a volume
‐the shape of its volume is defined by the container
‐short range order in temporary regions
Liquid water
physical/chemical point
defects
interstitium
vacancy
Crystalline water
Frenkel pair
point defects Ædiffusion to grain boundaries Æ
Æline defectsÆsurface defectsÆfracture
diffusion takes time!
„fatique”
Short range order: clusters of 5‐10 molecules in
continuous rearrangement
Practical aspects
2.1.‐2.2. Mesomorphous materials – liquid crystals
1.
Average interaction energy between molecules is small like in liquids
Long range order (but not so strict) ~ like in crystals
Deformability like in liquids
Constituting molecules are of special shape ~ string‐like, disc‐like
polarizable
Cholesteric order Æ sensitivity to temperature of the distance of ordered layersÆ diffraction change in reflectionÆ
Æcolor change
Contact thermography
Color ~ T thermometers
Forms of order of string‐like molecules
2. Electro‐optical properties: electric fieldÆstructural order changeÆ
Ælight transmission change Æ liq.crys. displays based on reflection
smectic
Weak bonding energy Æ order can be easily
perturbed by
Æ pixels of LCD monitors in transmission mode
nematic
Unpolarized
Illumination
cholesteric
‐temperature (by Boltzmann distr.)
‐concentration and polarity of solvents
‐electric field
Polarizer1
Orienting layer
LC molecules
Cholesteric crystal
voltage
Orienting layer
Polarizer2
trans
mission
No trans
mission
p. 467
3. Lyotropic liq.crys.: membranes formed by amphiphilic molecules (like phopholipids)
in proper solvents– lipid membranes: bi‐layers or multi‐layers
2.3. Macromolecular systems (e.g. cells): order is stabilized by a wide range of interactions
mitochondrion
Cell‐membrane: lipid bilayer
nucleus
cytoskeleton
Hierarchy in
Liposomes
‐bond strengths
‐bond distances
‐distance dependence of interaction energies
‐binding water molecules and ions
‐structural constraint of prosthetic groups
Targeting of medications
Real structure of „parallel” lipids (cell membrane)
Boltzmann distribution!
rough
endoplasmic
reticulum
Golgi
apparatus
smooth
endoplasmic
reticulum
Schematics of Figures
Blue: nucleus
Wide range of interaction energies Æ „structural dynamics”
How is this observed in the fuctioning of a dsDNA? Are the H-bonds always intact?
Red: actin filament
nbroken
− Δε
= e kT
nbound
Δε = Ebond
if nbroken << nbound ⇒
nbroken nbroken
≅
nbound
N
Example: The T7 DNA with about 40000 base pairs Æ about 100 000 H-bonds Æ N= 100 000
nbroken
n
− Δε
− 0.2
= e kT = e 0.026 = 0.00046 ≅ broken
nbound
N
Δε = Ebond = 0.2 eV
nbroken ~ 46/DNA
time-average!
Green: microtubular system
Significant number of secondary bonds are broken at body
temperature Æ flexibility of macromolecular complexes Æ
Possibility for ligand binding and chemical reactions
Get back to gases
Boltzmann distribution - more examples (see textbook)
1. Gasous systems (ideal gas) -> lack of order in the structure
3. Equilibrium rate of chemical reactions
Rules based on Thermodynamics
and Boltzmann distribution
1. average of kinetic energy is
determined by the temperature
Etotal = N 12 mv 2
1 mv 2 = f kT
2
2
f =3
pV = NkT
ε i = 12 mvi2
Characteristics:
- composed of uniform, point-like particles
- no interaction energy – no bond formation – no „structure”
- isotropy
- deformability
- fills the volume of the container
- only kinetic energy
degree of freedom of
motion for point-like
particles
Reaction :
εbarrier
A
B
The kAB and kBA rates are proportional to the number of reactants which are
higher in energy, reaching the top of the barrier
Reaction coordinate
k AB = const × e
k BA = const × e
Frequency of occurance =n/N
K=
2. deviations from the average velocity are
determined by the
−
k BA
=e
k AB
−
−
ε barrier −ε A
Arrhenius plot
kT
ε barrier −ε B
log K = − log e
εA −εB 1
kT
ε A −ε B
kT
Experimental determination of the energy of activation
Maxwell – Boltzmann distribution
function
4. Barometric formula
Density of air in the atmosphere decreases with the altitude (h) by the formula:
ρ (h) − mgh
= e kT
ρ ( 0)
m average mass of particles in the air
g gravitational acceleration
Absolute value of velocity
Thank you for your attention!
k
T