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Lecture notes for Computational Nanotechnogoly and Molecular
Engineering Workshop, Pan American Advanced Study
Institutes
1/14/2004
Shiang-Tai Lin, Youyong Li, Seung Soon Jang, Prabal Maiti
Tahir Çağın, Mario Blanco, and William A. Goddard III
Materials and Process Simulation Center, Caltech
Free Energy MD and Nanoscale Polymers
Free Energy Calculation in Nanotechnology
Free Energy is a key parameter in Nanofabrication
• Equilibrium structure is determined by free energy– formation of self-assembled monolayers (SAM)– nanoscale patterns in liquid crystals and block copolymers– secondary structures of DNA, RNA
• However, “Many of the ideas that are crucial to the development of this
area--"molecular shape", the interplay between enthalpy and entropy, the nature of
non-covalent forces that connect the particles in self-assembled molecular
aggregates--are simply not yet under the control of investigators.”
George M. Whitesides (http://www.zyvex.com/nanotech/nano4/whitesidesAbstract.html)
Weak interactions(vdW, Coulomb, Hb)
Free Energy
NanoStructuresFunctions
Outline
• How to obtain Free Energies from MD Simulations
–Test Particle method
–Perturbation method
–Nonequilibrium method
–Normal mode analysis
• A new 2PT approach for efficient Free Energy Estimation
–Basic Ideas (with Blanco and Goddard)
–Test of method with LJ fluids (with Blanco and Goddard)
• Applications of 2PT methods in the study of Dendrimers
–Zimmerman H-bond dendrimer (with Jang, Çağın and Goddard)
–PAMAM dendrimer (with Maiti and Goddard)
–Percec dendrimer (with Li and Goddard)
Free Energy Calculation from Molecular Simulations
• Test Particle Method (insertion or deletion)
•Good for low density systems •Available in the Sorption Module of Cerius2
• Perturbation Method (Thermodynamic integration, Thermodynamic perturbation)
•Applicable to most problems •Require long simulations to maintain “reversibility”
• Nonequilibrium Method (Jarzinski’s equality)•Obtaining differential equilibrium properties from irreversible processes •Require multiple samplings to ensure good statistics
• Normal model Method•Good for gas and solids•Fast•Not applicable for liquids
Reference: Frenkel, D.; Smit, B. Understanding Molecular Simulation from Algorithms to Applications. Academic press: Ed., New York, 2002. McQuarrie, A. A. Statistical Mechanics. Harper & Row: Ed., New York, 1976. Jarzynski, C. Nonequilibrium Equality for Free Energy Differences. Phys. Rev. Lett. 1997, 78, 2690.
From Normal Modes to Free Energy
• Total number of normal modes = 3N, N=number of particles
For an isolated molecule with N atoms
3 translation, 3 rotation (or 2 for linear molecules, 0 for monoatomic molecules), 3N -6 vibration
For a crystal with N particles
3 translation (acoustic modes), 3N-3 vibrational modes
• Each vibrational mode can be treated as a harmonic oscillator
• The partition function is the sum of contributions from HOs
• All the thermodynamic properties are defined
)/2âexp(-1
)/2âexp()âexp()( n υ
υευh
hq
n
QHO −
−=−=∑
)(ln)(ln0
υυυ HOqSdQ ∫∞
=
υε hnn )2
1( +=
VNT
QTVE
,
10
lnâ ⎟
⎠
⎞⎜⎝
⎛∂
∂+= −
VNT
QQkS
,
1 lnâln ⎟
⎠
⎞⎜⎝
⎛∂
∂+= −
)()(âlnâ0
10
10 υυυ AWSdVQVA ∫
∞−− +=−=
Determine Normal Modes from Molecular Simulations
•The density of states S(υ)
•S(υ)dυ number of modes between υ and υ+dυ
•
•Determination of S(υ)
• The eigenvalues of the Hessian matrix (vibrational analysis, phonon spectrum )
• Covariance matrix of atomic position fluctuations
• Fourier transform of velocity autocorrelation function
)()0()( tvvtC ⋅=dtetCkT
S ti∫−−
∞−=
τ
ττ
∂õ2)(lim 2
)õ(
NdS 3)(0
=∫∞
υυ
??
S (? )
Liquid
??
S (? )
Solid
Debey crystalS(v) ~ 2
The Density of States Distribution S(υ)
• Singularity at zero frequency
• Strong anharmonicity at low frequency regime
??
S (? )
Gas
exponentialdecay
??
S (? )
The 2PT idea: Liquid Solid+Gas
• Decompose liquid S(v) to a gas and a solid contribution• S(0) attributed to gas phase diffusion• Gas component contains anharmonic effects• Solid component contains quantum effects• Two-Phase Thermodynamics Model (2PT)
solid-likegas-like
The 2PT Method
• The basic idea
•The DoS
•Thermodynamic properties
• The gas component
• VAC for hard sphere gas
• DoS for hard sphere gas
• Two unknowns ( and Ngas) or (so and f)
) exp(3
)( tm
kTtcHS −=
2
0
0222
61
4
12)(
⎥⎦
⎤⎢⎣
⎡+
=+
=
fNs
sNS
gasHS
υπυπυ
tcoefficienfriction :),,( HSHST σρ
gasHS
gas
NSs
fNN
12)0(0 ==
=
)()()( υυυ solidgas SSS +=
)()()()(00
υυυυυυ gP
gHOP
s WSdWSdP ∫∫∞∞
+=
??
S (? )
Gas
f
S0
Lin, S. T.; Blanco, M.; Goddard, W. A. The Two-Phase Model for Calculating Thermodynamic Properties of Liquids from Molecular Dynamics: Validation for the Phase Diagram of Lennard-Jones Fluids. J. Chem. Phys. 2003, 119, 11792.
Determining so and f from MD Simulation
• so (DoS of the gas component at υ=0)
• completely remove S(0) of the fluid
•
• f (gas component fraction)
• T or ρ0 : f1 (all gas)
ρ : f0 (all solid)
•
• one unknown σHS
0)0( ),0(0 == solidSSs
); ,(
),(
0HSHS TD
TDf
σρρ
= mN
kTSTD
12
)0(),( =ρ
Enskog)-(Chapman )(1
8
3);,( 2/1
20 m
kTTD
HS
HSHS
πρσσρ =
Determining σHS
• σHS (hard sphere radius for describing the gas molecules)
• gas component diffusivity should agree with statistical
mechanical predictions at the same T and ρ
• gas component diffusivity from MD simulation
• HS diffusivity from the Enskog theory
mfN
kTsfTDHS
12),( 0=ρ
1)(
4);,(),( 0 −
=fyzfy
fTDfTD HSHSHS σρρ
3
6HSy ρσπ
=
3
32
)1(
1)(
y
yyyyz
−−++
=
At Last…
• A universal equation for f
• Graphical representation
022662 2/52/32/72/3532/152/9 =−+Δ+Δ−Δ−Δ −−−− ffyff
3/23/12/100 )
6()(
9
2),,,( :ydiffusivit normalized
πρπρ
mkT
Ns
smT =Δ
0.0
0.2
0.4
0.6
0.8
1.0
1.E-05 1.E-03 1.E-01 1.E+01 1.E+03? ( )normalized diffusivity
f or f y
ffy
Comparison of the 1PT and 2PT methods
Run a MD simulation(trajectory information saved)
Calculate VAC
Calculate DoS (FFT of VAC)
Apply HO approximation
To S(υ)
1PT thermodynamic
predictions
Calculate S(0) and Δ
Solve for f
Determine Sgas(υ), Ssolid(υ)
Apply HO statisticsTo Ssolid(υ)
Apply HS statistics to Sgas(υ)
2PT thermodynamic
predictions
An Overview of the 2PT Method
To Phase Behaviors
MD simulations
0
10
20
30
40
50
60
70
80
90
100
0 20 40 60 80 100
frequency v(cm-1)
DoS S(cm)
???????????????????????????????????
DoS
VAC
-800
-400
0
400
800
1200
1600
0 0.5 1 1.5 2 2.5 3time (ps)
C (t)
???????????????????????????????????
)()0()( tvvtC ⋅=
dtetCkT
S ti∫−−
∞−=
τ
ττ
∂õ2)(lim 2
)õ(
)()(â0
1 υυυ AWSdG ∫∞
−=
-30
-25
-20
-15
-10
-5
0
5
0 0.2 0.4 0.6 0.8 1 1.2? *
G *
*=1.8T*=1.4T*=1.1T*=0.9T
2 ( )PT Q2 ( )PT C
From MD Simulations
Test the 2PT Method Using the LJ System
●stable
●metastable
●unstable
T - ? diagram for Lennard Jones Fluid
0.6
1.0
1.4
1.8
0.0 0.4 0.8 1.2?*
*T
Solid
LiquidGas
Supercritical Fluid
• Intermolecular potential
• Phase diagram
⎥⎦
⎤⎢⎣
⎡ −= 612 )()(4)(rr
rVσσ
ε
Lennard-Jones Potential
r = V(r)
r
0-ε
(T*=kT/ *= )
– critical point
– triple point
006.0304.0
006.0316.1*
*
±=
±=
c
cT
ρ
69.0* ≈tpT
VAC and DoS of LJ Fluids
0
10
20
30
40
50
60
70
80
90
100
0 20 40 60 80 100
frequency v(cm-1)
DoS S(cm)
???????????????????????????????????
Density of StatesVelocity Autocorrelation
-800
-400
0
400
800
1200
1600
0 0.5 1 1.5 2 2.5 3time (ps)
C (t)
???????????????????????????????????
gasliquidsolid
gasliquidsolid
)()0()( tvvtC ⋅= dtetCkT
S ti∫−−
∞−=
τ
ττ
∂õ2)(lim 2
)õ(
0
200
400
600
800
1000
1200
0 5 10? [cm-1]
S(υ) [ ]cm
0
5
10
15
20
25
30
0 50 100 150? [cm-1]
S(υ) [ ]cm
0
5
10
15
20
25
30
35
0 50 100 150? [cm-1]
S(υ) [ ]cm
2PT DoS Decomposition
• Examples
gas liquid solid
solid-likegas-likegas-like
solid-like
solid-likegas-like
Pressure and Energy
Pressure
-2
0
2
4
6
8
10
12
14
16
18
0 0.2 0.4 0.6 0.8 1 1.2? *
P *
*=1.8T
*=1.4T
*=1.1T
*=0.9T
MD
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
0 0.2 0.4 0.6 0.8 1 1.2? *
E *
*=1.8T*=1.4T*=1.1T*=0.9T
MD2 ( )PT Q
Pressures and MD Energies agree with EOS valuesQuantum Effect (ZPE) most significant for crystals (~2%)
Total Energy
Entropy
4
6
8
10
12
14
16
18
20
0 0.2 0.4 0.6 0.8 1 1.2? *
S *
*=1.8T
*=1.4T
*=1.1T
*=0.9T
1 ( )PT Q
4
6
8
10
12
14
16
18
20
0 0.2 0.4 0.6 0.8 1 1.2? *
S *
*=1.8T*=1.4T*=1.1T*=0.9T
2 ( )PT Q2 ( )PT C
1PT 2PT model
• Overestimate entropy for low density gases
• Underestimate entropy for liquids• Accurate for crystals
• Accurate for gas, liquid, and crystal• Accurate in metastable regime• Quantum Effects most important for crystals (~1.5%)
gas
liquid
crystal
Gibbs Free Energy
-30
-25
-20
-15
-10
-5
0
5
0 0.2 0.4 0.6 0.8 1 1.2? *
G *
*=1.8T
*=1.4T*=1.1T
*=0.9T1 ( )PT Q
1PT 2PT model
• Underestimate free energy for low density gases
• overestimate entropy for liquids• Accurate for crystals
• Accurate for gas, liquid, and crystal• Accurate in metastable regime
gas
liquid
crystal
-30
-25
-20
-15
-10
-5
0
5
0 0.2 0.4 0.6 0.8 1 1.2? *
G *
*=1.8T*=1.4T*=1.1T*=0.9T
2 ( )PT Q2 ( )PT C
0
1
2
3
4
5
0 50 100 150? ?[cm-1]
WS(?)
0
5
10
15
20
25
30
0 50 100 150? [cm-1]
(Sυ) [ ]cm
0
200
400
600
800
1000
1200
0 2 4 6 8 10? [cm-1]
(Sυ) [ ]cm
Why does 2PT work?
HS fy = 0.036
QHO
HS fy = 0.309
CHO
)()()()(00
υυυυυυ gP
gHOP
s WSdWSdP ∫∫∞∞
+=
gas
liquid • 1PT overestimates Wsgas for gas for
modes < 5 cm-1
• 1PT underestimates Wsgas for liquid for
modes between 5 and 100 cm-1
• 2PT properly corrects these errors
Convergence of 2PT
6.5
7.5
8.5
9.5
10.5
11.5
12.5
13.5
14.5
15.5
100 1000 10000 100000 1000000
MD steps
S*
2PT(Q)
2PT(C)
MBWR EOS
gas (ρ*=0.05 T*=1.8)
liquid (ρ*=0.85 T*=0.9)
• For gas, the entropy
converges to within 0.2% with
2500 MD steps (20 ps)
• For liquid, the entropy
converges to within 1.5% with
2500 MD steps (20 ps).
2PT for Melting and Solidification
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.00 0.40 0.80 1.20
??
*T
Simulation conditions
solid
supercritical fluid
3
4
5
6
7
8
0.80 1.20 1.60 2.00T*
S*liquid (EOS)
solid (EOS)
heating
cooling
classical
En
tro
py
solidmetatstable
unstablesupercritical
fluid
starting withfcc crystal
starting with amorphous liquid
• Initial amorphous structure is used in the cooling process
• The fluid remains amorphous in simulation even down to T*=0.8 (supercooled)
• The predicted entropy for the fluid and supercooled fluid agree well with EOS for LJ fluids
• Initial fcc crystal is used in the heating process
• The crystal appears stable in simulation even up to T*=1.8 (superheated)
• The predicted entropies for the crystal and superheated crystal agree well with EOS for LJ solids
Extension to Mixtures
LJ mixture
-300
-200
-100
0
100
200
300
0 0.2 0.4 0.6 0.8 1
x(Ar)
Gmix (kJ/mol)
LJ Mixtures at T*=0.85
Combination Rules:
σij= lij ( σii+ σjj)/2εij= bij ( εii+ εjj)1/2
22/ 11=0.8022/11=0.85l12=0.95b12=0.70
2PT Literaturex1(I) ~0.02 0.04x1(II) ~0.79 0.84
Efficiency of 2PT for Mixtures
T*=0.95 P*=0.125
-2.5E-04
-2.0E-04
-1.5E-04
-1.0E-04
-5.0E-05
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0 0.2 0.4 0.6 0.8 1
x(Ar)
Gmix/RT
400ps
T*=0.95 P*=0.125
-2.5E-04
-2.0E-04
-1.5E-04
-1.0E-04
-5.0E-05
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0 0.2 0.4 0.6 0.8 1
x(Ar)
Gmix/RT
100ps
T*=0.95 P*=0.125
-2.5E-04
-2.0E-04
-1.5E-04
-1.0E-04
-5.0E-05
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0 0.2 0.4 0.6 0.8 1
x(Ar)
Gmix/RT
25ps
T*=0.95 P*=0.125
-2.5E-04
-2.0E-04
-1.5E-04
-1.0E-04
-5.0E-05
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0 0.2 0.4 0.6 0.8 1
x(Ar)
Gmix/RT
12.5ps
400 ps 100 ps
25 ps 12.5 ps
An Overview of the 2PT Method
To Phase Behaviors
MD simulations
0
10
20
30
40
50
60
70
80
90
100
0 20 40 60 80 100
frequency v(cm-1)
DoS S(cm)
???????????????????????????????????
DoS
VAC
-800
-400
0
400
800
1200
1600
0 0.5 1 1.5 2 2.5 3time (ps)
C (t)
???????????????????????????????????
)()0()( tvvtC ⋅=
dtetCkT
S ti∫−−
∞−=
τ
ττ
∂õ2)(lim 2
)õ(
)()(â0
1 υυυ AWSdG ∫∞
−=
-30
-25
-20
-15
-10
-5
0
5
0 0.2 0.4 0.6 0.8 1 1.2? *
G *
*=1.8T*=1.4T*=1.1T*=0.9T
2 ( )PT Q2 ( )PT C
From MD Simulations
Summary of 2PT
1. Thermodynamic and transport properties are determined simultaneously.
2. Only short simulation times (20 ps) are needed to obtain high accuracy. For a
system with N particles, we expect 2PT to be N times faster than methods
such as particle insertion and thermodynamic integration.
3. The efficiency of 2PT does not deteriorate with increasing density (a severe
limitation in most other techniques).
4. The properties are obtained under real equilibrium conditions (no
perturbation in the simulation itself).
5. Zero point energy and corrections for quantum effects are included.
6. 2PT can be used to determine the properties in metastable and unstable
regimes.
7. 2PT could also be used for nonequilibrium systems to estimate effects of
transient effects, reaction, and phase transitions, since it is only necessary to
have stabilities over time scales of ~20 ps.
Application: Zimmerman H-bond Dendrimers
Dendrimer study on Zimmerman systemInitiative: Science, 271, 1095 (1996)
dendrons
R= g 1 g 2 g 3 g 4
OO O
OO
O
OO
O
O
OO
O
O
OO
OO O
OO
OO
OO O
OO
O
O
O
O
O
O
OO
OO
O
O
OO
OO
O
O
O
O
O
O
O
OO
O
OO
core
R
H
OO
HO
H
O
O
O
HOO
N
O
linear
OOH
O HOH O
O
O HO
H O
OO
OO H
O
H
OH
O
HOH O
O HO O
O HO
O
O
H OO
HOH O
HOO
O
linear
OOH
O HOH O
O
O HO
H O
OO
OO H
O
H
OH
O
HOH O
O HO O
O HO
O
O
H OO
HOH O
HOO
O
OOH
O HOH O
O
O HO
H O
OO
OO H
O
H
OH
O
HOH O
O HO O
O HO
O
O
H OO
HOH O
HOO
O
circular
H OOOHOOH O O
H O
OH H
O
O
O
OH OO
HOOH O OOH HO OH
O OH
O
O
O
OH HO O HO OH O
O OH O
OO
OH HO
HO
O HO OH OOOHO
circular
H OOOHOOH O O
H O
OH H
O
O
O
OH OO
HOOH O OOH HO OH
O OH
O
O
O
OH HO O HO OH O
O OH O
OO
OH HO
HO
O HO OH OOOHO
H OOOHOOH O O
H O
OH H
O
O
O
OH OO
HOOH O OOH HO OH
O OH
O
O
O
OH HO O HO OH O
O OH O
OO
OH HO
HO
O HO OH OOOHO
Experimental Observations
• Aggregates found in non-polar solvents such as CH2Cl2 and CHCl3 but not in polar solvents such as THF and DMSO
• Circular and ladder coexist in lower generations (gen 1)
• Circular type is dominant for generations 2, 3 and 4
• The generation dependent stability behavior is attributed to the subtle interplay between H-bond, vdW and steric repulsions
Application: Zimmerman H-bond Dendrimers
• Relative stability determined by the free energy differences is in consistent with experimental observations
• Linear and circular forms are energetically similar
• Stability of Zimmerman H-bond mediated dendrimers are dominated by entropic effects
• Opens up possibility to design thermodynamically stable suprastructures
• This method is potentially useful for study other supramolecules such as Protein, DNA, etc
Energy Entropy Free EnergyDifference
0
200
400
600
800
1000
g_0 g_1 g_2 g_3 g_4
E (kJ/mol)
ladder-circular
Difference
-9000-8000-7000-6000-5000-4000-3000-2000-1000
01000
g_0 g_1 g_2 g_3 g_4
S (J/mol K)
ladder-circular
Difference
0500
100015002000250030003500
g_0 g_1 g_2 g_3 g_4
A (kJ/mol)
ladder-circular
Application: PAMAM dendrimers
EDA core
Generation 3
NH2
NH2
NH2
NH
O
repeating (monomer) unit
Applications of PAMAM dendrimer
• Catalysts
• Environment applications (metal encapsulation)
• Medical applications (drug delivery, gene therapy)
• Surface active agents
• Viscosity modifier
• New electrical materials
Application: PAMAM dendrimers
Free Water Domain Surface Water DomainInner Water Domain
Schematic Atomistic
solution overall inner surface free SESA (Å2)
high pH 9406 376 1014 1440 10642±185
neutral pH 8948 547 1328 1806 13216±198low pH 12977 747 1729 2472 18526±723
• Number of water molecules
Water molecules in PAMAM
Application: PAMAM dendrimers
-12
-11
-10
-9
-8
-7
-6
inner surface free
water location
E (kcal/mol)
high pH
neutral
low pH
0
1
2
3
4
5
6
inner surface free
water location
TS (kcal/mol)
high pH
neutral
low pH
-17
-16
-15
-14
-13
-12
-11
inner surface free
water location
A (kcal/mol)
high pH
neutral
low pH
• Engetically slightly less favored for water at the PAMAM surface
• Entropically less favored for water inside PAMAM
• Entropically slightly less favored for water at PAMAM surface
• Surface and Inner Waters are in a higher free energy state compared to the bulk
Application: Percec Dendrimer Crystals
Free energy profile over volume at 277K
1000
1050
1100
1150
1200
1250
1300
1350
1400
1450
1500
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
Volume (A^3)
A(kJ/mol/dendron)
AA15
Critical pressure: 0.033GPa
a. Condense phase
b. Isolated Micelle phase