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