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MATLAB代写-3MA010
时间：2021-10-04
1Computational/Mathematical Physics 3MA010 2Three Subjects • Partial differential equations and numerical methods (PDE) (Prof. dr. Federico Toschi, Dr. Gianluca Di Staso, MSc. Alessandro Corbetta, MSc Ilian Pihlajamaa) • Brownian dynamics (BD) (Dr. Alexey Lyulin, MSc. I. Pihlajamaa, MSc. Maarten Boomstra, Msc Nikos Sigalas) • Monte-Carlo (MC) (Dr. Henk Huinink, MSc. I. Pihlajamaa, MSc. Maarten Boomstra, Msc Nikos Sigalas) 3● Check Canvas and OnCourse pages for the detailed planning, and all course material ● PDE: 6 lectures/instructions ● BD: 4 lectures/instructions ● MC: 4 lectures/instructions Software: MatLab 4PDE: lectures, instructions, home work. OnCourse testing will be used. BD and MC parts: students will work on theoretical and modelling exercises. - Written reports are expected, with findings from the simulations Deadlines for reports: November 6 retake 5• Evaluation: testing for PDE part, 2 reports for BD and MC parts, NO final exam Final mark: 40% – PDE 30% - BD report 30% - MC report Lecture 1: Random numbers and distributions 6 - Stochastic modelling - Random number generators - Distribution functions and its momenta - Central Limit Theorem - Non-uniform distributions - Markov processes - Random walks and applications 7 John von NeumannNicholas Metropolis 8 Andrey Markov 9 Random walks in polymer physics t0=0 t1=t Idea: the collective activity of many randomly moving participants can be effectively predictable, even if their individual motions are not 10 Polymers as long molecular chains Number of monomer units in a chain N>>1 (for DNA N~109 – 1010) 11 12 Freely-jointed flexibility mechanism The flexibility is located in the freely rotating junction points. This mechanism is normally not characteristic for real chains, but it is used for model theoretical calculations. 13 Portrait of a polymeric coil Typical conformations of few PE-like chains simulated on a computer Chain conformation - trajectory of a Brownian particle?? 14 Let us consider ideal N-segment freely-jointed chain ( each segment of length l ) 15 Thus, the conformation of an ideal chain is far from the rectilinear one. The chain conformation is equivalent to the trajectory of a Brownian particle. 16 17 - Stochastic modelling - Random number generators - Distribution functions and its momenta - Central Limit Theorem - Non-uniform distributions - Markov processes - Random walks and applications Lecture 1 recap 18 Lecture 2: Correlation functions and diffusion equations - Time series analysis - ACF computation in practice - Brownian motion - Diffusion equation - Langevin equation - Ermak-McCammon algorithm 19 20 Autocorrelation functions time correlation function: if A=B – autocorrelation function: ( ) (0) ( )AAC t A A t= ( ) (0) ( )ABC t A B t= (0) ( ) ( ) (0)A B t A t B= − 2(0)AAC A= normalization: 2 22 (0) ( ) ( )AA A A t AC t A A − = − (0) 1AAC = ( ) 0AAC t = ∞ = 21 Time correlation function may be evaluated as a time average, assuming the system is ergodic – the phase space average is equal to a time average 0 1( ) lim ( ) ( ) T t AA T C t d A A t T t τ τ τ − →∞ = + − ∫ simulations: phase space trajectory is determined at discrete time steps, the integral is expressed as a sum − + = ∆ = = − ∑1 1( ) ( ) ( ), 0,1,2..., N k AA j k j c j C k t A t A t k N N k N – total number of time steps ∆t – time step Nc << N why? 22 time AC F 1 0 exp(-t/τ) τ Reality, i.e. simulations Nc << N why? 23 Practical realization tNti+1tit1 t3t2 time a(tN)a(t1) a(ti) ACF=
N
+ + +
=
1 2 2 3 1( ) ( ) ( ) ( ) ... ( ) ( ) .. (0) ( )
1
i ia t a t a t a t a t a t a a t
N
++ + + + = ∆
−
t2
1 3 2 4 2)
3
( ) ( ) ( ) ( ) ... ( ) ( )
( )
2
N Na t a t a t a t a t a t ACF t
N
−+ + + =
−
t3
…..
tN
equidistant time intervals
How many terms exist in order to
calculate this point?
Important example - white noise
Brownian motion
t0=0 t1=t
25
( ) ( ) ( ) ( )Ri i i it V t m tζ− − ∇ + =r r F r
Langevin equation
27
if /L m mτ τ ζ<< =
0
( ) ( ) 2 ( ) 2 ( )
R
i
R R
i j ij B i ijt t Z t t k T t tδ δ ζ δ δ
=
′ ′ ′= − = −
F
F F
Random force FR
28
Drag force Fd
mζ >>v a
d
i i iζ= −F v
2~ , ~ , ~t t te e eω ω ωω ωr v a
/m mζ ω ω ζ>> → <<
/BD v mτ τ ζ>> =
( ) ( ) ( ) 0Ri i it V tζ− − ∇ + =r r FLangevin equation in non-inertial approximation
/BD v mτ τ ζ>> =
29
30
Ermak-McCammon algorithm
1 1pot R pot R
i i i i i i i
i i
v F F v F Fζ
ζ ζ
= + ⇒ = +
t
trttr
tv iii ∆
−∆+
=
)()(
)(
1 1( ) ( ) ( ) ( )
i i
pot R
i i i ir t t r t F t t F tζ ζ+ ∆ = + ∆ + ∆
B( ) ( ) 2
R R
i j i ijF t F t k T tζ δ∆ ∆ = ∆
?????t∆ =
31
Time scales
• σ length
• ζ friction
• T temperature
• m mass
3
2 3 10 20 3
23
6 6 3.14 10 10 4 102~ ~ ~ ~ 20
1.38 10 300BD B B
Pa s m ps
k T k T J
σπηζστ
− − −
−
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅
/cor r BDmτ τ ζ τ<< = <<
2, ~ 10BD BDt tτ τ
−∆ << ∆
Lecture 3: Random force, Brownian
dynamics of a polymer chain
32
- Random force, spectral density, magnitude
- Langevin equation (continued)
- Fokker-Planck equation
- Einstein-Smolukhovski equation
- Equilibration of the initial configuration
- Applications for polymer modelling, hydrodynamic interactions
Lecture 4: Brownian dynamics and
diffusion
Brownian dynamics of polymers
35
Ηψ = Εψ
F = ma
exp(-∆E/kT)
domain
quantum
chemistry
Brownian
dynamics,
MD
Monte Carlo
mesoscale continuum
What and Where: Scales in Simulations
Length scale
10-10 M 10-8 M 10-6 M 10-4 M
10-12 S
10-8 S
10-6 S
36
Historical Background
Brownian motion
Robert Brown (1773-1858)
1827:
1905-1966: Einstein, Smolukhovski, Langevin, Fokker, Planck
1978: Ermak, McCammon, BD simulations
of dimers and trimers in solution
37
Brownian Dynamics
rij
v(rij)
i
j
0tot pot d Ri i i i m= + + =F F F F a Langevin equation
38
Fpot, the Force Field
intermolecular
interactions
intramolecular
nonbonded
torsional
bond stretch
valence angle
bend
39
Vstretch =
1
2 kij(rij – r
0
ij)2
Vbend =
1
2 kijk(Θijk – Θ
0
ijk)2
Vtor=
1
2 kijkl(1- cosnφijkl)
VLJ = Aijrij-12 – Cijrij-6
VCoulomb =
qiqj
rij
V(r) = Vstretch + Vbend + Vtor + VLJ + VCoulomb
( )pot V= −∇F r
HYBRANE from DSM
g=1
g=2 g=3
g=4
g=0
NN
O O
O
OO
O
OOC
HO
O
O
N
OOC
OOC
O
O
N
O
O
O
O
N
O
O
O
O
N
O
O
O O
N
OH
O
O
O
N
O
C11H23
O
O
O
O
C11H23
C11H23
C11H23
C11H23
C11H23
C11H23
C11H23
fatty acid ester
41
v
v
l
Hyperbranched Polymers
synthesized in a much
more economical one-pot
reaction
42
( ) ( )ttvFkTtrr 0i0i0j
j
0
ij
0
ii ∆Φ+∆⋅+⋅∆+= ∑
D/
- velocity (the shear force, )
- Lennard-Jones + SHAKE for rigid
bonds (+Coulomb)
- random force
- diffusion tensor,
hydrodynamic interactions via
Rotne-Prager-Yamakawa tensor
γ
0
iv
0
jF
0
iΦ
0
ijD
l - bond length
a - hydrodynamic
bead radius
Brownian dynamics
Langevin equation; Ermak-McCammon algorithm
• Random force
• Potential force
• Diffusion tensor
000 2)()( ijji ttt D∆=∆Φ∆Φ
0
0
0
1
( )
N
k
j k
jk j
UF rr
σλ
=
∂ ∂= − −
∂∂∑ r
2
1/ 2
2 2 2
32( / 3) (3 / 4 )( / ) ( ) ( )
3
ij ij ij ij
ij ij
ij ij ij
R R R RaD h kT l R
R R R
α β α β
αβ
αβ αβπ ζ δ δ
= + + −
+−=
ij
ijijij
ij R
RR
aa
R
kTD
βα
αβ
αβ δζ )
32
3()
32
9
1()/(
2ijR a≥
2ijR a<
44
Problems
• (neutral) dendrimers
in shear flow
• structure of charged
dendrimers
• complexes with linear
polyelectrolytes
2
max
3 3ln ( 1) ln(1 ) ( 1)
2
Q
B B
FF g g s
k T k T
φ
α α φε
φ
= − − + − + + +
Viscosity, shear thinning
• shear viscosity
• dimensionless intrinsic viscosity
• low shear data were fit using
xyτη
γ
= −
* ( )[ ] s
Bk T
η η
η
λ
−
=
[ ] * 20 1[ ] aη η γ
∗
= −
Mark-Houwink: [ ]0
aKMη =
47
Explanation of [η] Maximum
0 1 2 3 4 5 6
1
10
[η
] 0*
/N
generation
dendrimers with () and without (•) spacers
[η] ~ [molecular density]-1 ~ V/M
; M ~ N
Linear Polymer
Dendrimer V ~ (g+1)3 M ~ 2(g+1)
[ ] 2~ ~R Nη
[ ] ( )( )
3
1
1
2 g
g
~η
+
+
INITIALLY
GROWS
FASTER
EVENTUALLY
TAKES OVER
Mark-Houwink behaviour 3 ~ V R
0.5~R N
dendrimer
48
Charged Dendrimers
• Interplay between
excluded-volume and
Coulomb interactions
• influence of
perturbations of the
regular dendrimer
structure
v
v
+
+
+
+
+
+
+ +
+
+
+
+
49
,4)(
612
−
=
rr
rU LJLJ
σσε εLJ = 0.3kbT σ = 0.8l
Lennard-Jones:
/ 2
2( ) ,
4
Dr r
C b B B
b
e eU r k Tz
r k T
λ λ
πε
−
= =
Debye-Hückel:
0 < z2 < 5 λB= l Debye radius rD=100l
Force Field
50
2
max
3 3ln ( 1) ln(1 ) ( 1)
2
Q
LJ
B B
FF g g s
k T k T
φ
α α φε
φ
= − − + − + + +
1. stretching and confinement entropy
2. excluded volume and van der Waals attraction
3. Coulomb
Mean-Field Theory
1 2
single linear branch in a g=6 dendrimer
- linear expansion factor;
( )
g
g
R
R ideal
α =
3
( )
2
2 /
2~ 1 D
Q R rB D
B
F Q r e
k T R
λ −−
- mean segment densityφ
51
Neutral dendrimer
• Linear-expansion factor
vs. solvent quality
• BD and the mean-field
theory are in agreement
52
• Layering of monomers for charged dendrimers
Distribution of Monomers
0 1 2 3 4 5 6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
ρ(r)
r / l
gint = 0
gint = 1
gint = 2
gint = 3
gint = 4
gint = 5
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
ρ(r)
r / l
gint = 0
gint = 1
gint = 2
gint = 3
gint = 4
gint = 5
g=5 dendrimer
uncharged charged
53
Perspectives
• Drug delivery with charged HBPs
low salt, good solvent high salt, poor solvent
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