© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L09 Ch121a Atomic Level Simulations of Materials and Molecules William A. Goddard

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L09

Ch121a Atomic Level Simulations of Ch121a Atomic Level Simulations of Materials and MoleculesMaterials and Molecules

William A. Goddard III, [email protected]

Charles and Mary Ferkel Professor of Chemistry, Materials

Science, and Applied Physics, California Institute of Technology

BI 115Hours: 2:30-3:30 Monday and Wednesday

Lecture or Lab: Friday 2-3pm (+3-4pm)

Teaching Assistants Wei-Guang Liu, Fan Lu, Jose Mendoza, Andrea Kirkpatrick

Lecture 9, April 18, 2011Monte Carlo – Polymer Monte Carlo – Polymer

distributionsdistributions


Monte Carlo Method and applications in Monte Carlo Method and applications in chemistry, biological, polymer related fieldschemistry, biological, polymer related fields

Youyong Li and William A. Goddard III

Continuous self-avoiding walk with application to the description of polymer chains; Youyong Li and WAG; J. Phys. Chem. B, 110: 18134-18137 (2006)

Efficient Monte Carlo Method for Free Energy Evaluation of Polymer Chains; Jiro Sadanobu and WAG; Fluid Phase Equilibria 144, 415 (1998)

The Continuous Configurational Boltzmann Biased Direct Monte Carlo Method for Free Energy Properties of Polymer Chains; Jiro Sadanobu and WAG; J. Chem. Phys. 106, 6722 (1997)

references


The Monte Carlo method provides approximate solutions to a variety of mathematical problems by performing statistical sampling experiments (on a computer).

The method applies to problems with no probabilistic content as well as to those with inherent probabilistic structure.

What is Monte Carlo?What is Monte Carlo?


1. Partition the space into grids for numerical evaluation

2. Tedious, thorough study: enumerate all N2 grid points and evaluate whether satisfies r<R

3. Monte Carlo study: sample part of the grids (e.g. 10%) to get the conclusion.

4. For interesting problems, tedious and thorough study is seldom practical.

R

Simple example: Evaluate from the area of a circle from numerical evaluation (dartboard method)


Named after the city in the Monaco principality of France, famous for its gambling casinos. Roulette is a simple random number generator. The name and the systematic development of Monte Carlo methods dates from LASL ~ 1944.

In 1873: A. Hall performed Needle-board experiment to determine ( “On an experimental determination of Pi”). R

In 1899, Lord Rayleigh showed that a one-dimensional random walk without absorbing barriers could provide an approximate solution to a parabolic differential equation.

In 1931 Kolmogorov showed the relationship between Markov stochastic processes and certain integro-differential equations.

In 1948, Harris and Herman Kahn systematically developed the ideas used in the study of random neutron diffusion in fissile material of the atomic bomb.

~ 1948 Fermi, Metropolis, and Ulam obtained Monte Carlo estimates for the eigenvalues of Schrodinger equation

History of Monte Carlo methodHistory of Monte Carlo method


Major components of Monte Carlo Major components of Monte Carlo AlgorithmAlgorithm

Probability distribution functions (pdf's) - the physical (or mathematical) system must be described by a set of pdf's.

Random number generator - a source of random numbers uniformly distributed on the unit interval must be available.

Sampling rule - a prescription for sampling from the specified pdf's, assuming the availability of random numbers on the unit interval, must be given.

Scoring (or tallying) - the outcomes must be accumulated into overall tallies or scores for the quantities of interest.

Error estimation - an estimate of the statistical error (variance) as a function of the number of trials and other quantities must be determined.


RefinementsRefinements

Variance reduction techniques - methods for reducing the variance in the estimated solution to reduce the computational time for Monte Carlo simulation

Parallelization and vectorization - algorithms to allow Monte Carlo methods to be implemented efficiently on advanced computer architectures.


Two main reasons:

1. Global minimization (avoiding being trapped)

2. Ensemble description: Ergodicity (avoiding being trapped)

MD: straightforward algorithm, but only samples the high frequency region efficiently, may not be ergotic

MC: Complicated algorithm, but much more practical to realize ergodicity in difficult problems

Why we need MC or MD?Why we need MC or MD?


Side chain torsions- Number of side chain torsions vary for each aa

Main chain torsions (peptide bond is planar)

180o

Backbone and Side chain torsions in Backbone and Side chain torsions in Proteins --- Flexible and complicated systemProteins --- Flexible and complicated system


Ramachandran plotRamachandran plot


Rc

i

i-1

i-2

i-3

i

i-1’

If we sample 10 points for each torsion of 40 monomers or residue, there will be 10400 possible conformations!

Systems with Many Degrees of Freedom

The complexity of The complexity of polymer, protein, and polymer, protein, and other macromoleculesother macromolecules


Markov ProcessesMarkov ProcessesLet us set up a so-called Markov chain of configurations Ct by the introduction of a fictitious dynamics. The ``time'' t is computer time (marking the number of iterations of the procedure), NOT real time -- our statistical system is considered to be in equilibrium, and thus time invariant.

Let P(A,t) be the probability of being in configuration A at time t.

Let W(A->B) be the probability per unit time, or transition probability, of going from A to B. Then:

At large t, once the arbitrary initial configuration is ``forgotten,'' want


Detailed balanceDetailed balance

Clearly a sufficient (but not necessary) condition for an equilibrium (time independent) probability distribution is the so-called detailed balance condition

This method can be used for any probability distribution, but if we choose the Boltzmann distribution

Note: Z does not appear in this expression! It only involves quantities that we know or can easily calculate


The Metropolis AlgorithmThe Metropolis Algorithm

This dynamic method of generating an arbitrary probability distribution was invented by Metropolis, Teller , and Rosenbluth in 1953 (supposedly at a Los Alamos dinner party).

There are many possible choices of the W's which will satisfy detailed balance. They chose a very simple one:

So if

And if


Realize Metropolis AlgorithmRealize Metropolis Algorithm

So we have a valid Monte Carlo algorithm if:

1. We have a means of generating a new configuration B from a previous configuration A such that the transition probability W(A->B) satisfies detailed balance

2. The generation procedure is ergodic, i.e. every configuration can be reached from every other configuration in a finite number of iterations

The Metropolis algorithm satisfies the first criterion for all statistical systems. The second criterion is model dependent, and not always true (e.g. at T=0).


Folding KineticsFolding Kinetics

New View: Energy Bias


Simple sampling

Independent Rotational Sampling



Monte Carlo Sampling for protein structureMonte Carlo Sampling for protein structure

dZ

kTZE

kTXE

XP)(

exp

)(exp

)(

The probability of finding a protein (biomolecule) with a total energy E(X) is:

Partition function

Estimates of any average quantity of the form:

dXXPXAA )()(

using uniform sampling would therefore be extremely inefficient.

Metropolis et al. developed a method for directly sampling according to the actual distribution.

Metropolis et al. Equation of state calculations by fast computing machines. J. Chem. Phys. 21:1087-1092 (1953)


Monte Carlo for the canonical ensembleMonte Carlo for the canonical ensemble

• The canonical ensemble corresponds to constant NVT.• The total energy (Hamiltonian) is the sum of the kinetic

energy and potential energy: E=Ek(p)+Ep(X)

• If the quantity to be measured is velocity independent, it is enough to consider the potential energy:

dXkT

XE

dXkT

XEXA

dXdpkT

XE

kTpE

dXdpkT

XE

kTpE

XA

A

p

p

pk

pk

)(exp

)(exp)(

)(exp

)(exp

)(exp

)(exp)(

The kinetic energydepends on themomentum p; it can be factored and canceled.



dZ

kT

ZEkT

XE

XPp

p

)(exp

)(exp

)(Let:

)( YX let be transition probability from state X to state Y.

Suppose we carry out a large number of Monte Carlo simulations, such that the number of points observed in conformation X is proportional to N(X). The transition probability must satisfy one obvious condition: it should not destroy this equilibrium once it is reached. Metropolis proposed to realize this using the detailed balance condition:

or



)()(1

)()()()(

exp)(

XEYEif

XEYEifkT

XEYEYX

pp

pppp

There are many choices for the transition probability that satisfy the balance condition. The choice of Metropolis is:

The Metropolis Monte Carlo algorithm:1. Select a conformation X at random. Compute its energy E(X)2. Generate a new trial conformation Y. Compute its energy

E(Y)3. Accept the move from X to Y with probability:

kT

XEYEP pp )()(

exp,1min(

Pick a random number RN, uniform in [0,1].If RN < P, accept the move.

4. Repeat 2 and 3.



Notes:1. There are many types of Metropolis Monte Carlo

simulations, characterized by the generation of the trial conformation.

2. The random number generator is crucial3. Metropolis Monte Carlo simulations are used for finding

thermodynamics quantities, for optimization, …4. An extension of the Metropolis algorithm is often used

for minimization: the simulated annealing technique, where the temperature is lowered as the simulation evolves, in an attempt to locate the global minimum.


Grand Canonical Monte CarloGrand Canonical Monte Carlo

A classical force-field method where molecules interact and are ...

X

MOVED

INSERTEDDESTROYED

... such that the proper distribution of positions, energies, and numbers of molecules is achieved for a system at fixed chemical potential (or fugacity or pressure) and temperature. GCMC is often used to study the adsorption of small molecules in inorganic materials such as zeolites.

,V,T N,E


Molecular Size via Computational SievingMolecular Size via Computational SievingUse a Force-Field and Grand Canonical Monte Carlo to adsorb a gas molecule into a confined system at a standard T & P. If significant numbers of the molecule fit into the system, the characteristic size of the system is related to the size of the molecule.An unbiased method which uses the information inherent in the FF.

Dslit

Dtube

Nthreshold

D = Dslit -

0

4

8

12

16

20

5.6 5.7 5.8 5.9 6 6.1 6.2

Dslit (A)

Nu

mb

er

Ad

so

rbe

d


Static Monte Carlo: CCBBStatic Monte Carlo: CCBB

SS-DMC (Simple sampling direct Monte Carlo)

IRS (Independent rotational sampling)

fixed bond lengths and angles, random sampling of all torsions

Drastic sampling attritionRotational biased sampling, using the normalized torsion weighting function (TWF), WIRS:

IRS

IRSIRS z

gW

)()(

2

0)( dgz IRSIRS

)](exp[)( tIRS Eg

NonBond bias corrected TWF

Excludes high torsion energiesStill have spatial overlaps

),...,(

),...,;(),...,;(

14*

14*

14*

ii

iiiiii

z

gW

4

114

* )(exp)(),...,;(i

jijLJiIRSiii rEgg

2

0 14*

14* ),...,;(),...,( iiiiii dgz

W* updated every stepToo expensive for large system

CCB (Continuous configurationally biased direct Monte Carlo)

Off-lattice, Rc introduced into TWF

4

114 )()(exp)(),...,;(

i

jijLJijCiIRSiii rErRgg CCB

),...,(

),...,;(),...,;(

14

144

i

CCBi

iiCCB

iii

CCBi

z

gW

iiiCCB

iiCCB

i dgz

2

0 1414 ),...,;(),...,(


CCB (Continuous configurationally biased direct Monte Carlo)

4

114 )()(exp)(),...,;(

i

jijLJijCiIRSiii rErRgg CCB

),...,(

),...,;(),...,;(

14

144

i

CCBi

iiCCB

iii

CCBi

z

gW

iiiCCB

iiCCB

i dgz

2

0 1414 ),...,;(),...,(

United atomistic FFIsolated chain

Sadanobu, J. and Goddard, W. A., III. J. Chem. Phys. 1997, 106, 6722

CCBTX (Continuous configurationally biased TX direct Monte Carlo)

NumTX

kikLJikC

i

jijLJijCiIRSiii rErRrErRgg CCBTX

1

4

114 )()()()(exp)(),...,;(

),...,(

),...,;(),...,;(

14

144

i

CCBTXi

iiCCBTX

iii

CCBTXi

z

gW

iiiCCBTX

iiCCBTX

i dgz

2

0 1414 ),...,;(),...,(

Use CCB works for isolated UA chain, also for full atomistic model, multiple chains, dense polymer/ dendrimer

Description of CCBTX method


We developed the efficient Continuous Configurational Biased TX (CCBTX) method to generate high quality amorphous polymer and dendrimer atomistic structures directly.

The method developed here can also be used for protein fold prediction.

The code is implemented in python environment and CMDF package, which provides friendly interface.

CCBTX Monte Carlo method


Application of CCBBApplication of CCBBThermodynamic Functions and Critical Exponents for Polymer Chains from CCBB

1. Polymer chain dimensions22 NRg

32 / grep RNkTF

)/( 22 NkTRF gel

5

62

Flory theta temperature: 2=1

Rci

i-1i-2

i-3

i

i-1’

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L095.4 5.6 5.8 6.0

6.2

6.4

6.6

6.8

7.0

5.4 5.6 5.8 6.0

6.2

6.4

6.6

6.8

7.0

5.4 5.6 5.8 6.0

6.2

6.4

6.6

6.8

7.0

5.4 5.6 5.8 6.0

6.2

6.4

6.6

6.8

7.0

5.4 5.6 5.8 6.0

6.2

6.4

6.6

6.8

7.0

ln(N-1)

1800K

ln<

Rg

2>

ln(N-1)

2160K

ln<

Rg

2>

ln(N-1)

2520K

ln<

Rg

2>

ln(N-1)

3600K

ln<

Rg

2>

ln(N-1)

7200K

The radius of gyration, ln<Rg2>, as a function of polymer length,

ln(N-1) for a range of temperatures (CCBB calculations using SKSFF).

The uncertainties are shown, but are less than the size of the symbols.

22 NRg

Where = 1/2

This confirms that ln<Rg2> is

proportional to ln(N-1), as in following equation.

Here the slope, 2, is a function of temperature


NN NZ 1

ANB

N CNNZTk

A ln)1(lnln

EB

n CNeTk

E 0

SB

N CNNsk

S ln)1(0

Free energy theta temperature: =1

i=0

i=1

i=2

i=4

i=3

(a)

Rci

i-1i-2

i-3

i

i-1’

Application of CCBB: Application of CCBB: Thermodynamic functions of polymers Thermodynamic functions of polymers

Old methods of MC assumed atoms on a lattice. This gives bad estimates of entropy

CCBB allows a continuous description (no lattice)


3 4 5

-4

-3

-2

3 4 5

-4

-3

-2

3 4 5

-4

-3

-2

3 4 5

-4

-3

-2

3 4 5

-4

-3

-2

log

(/x

)

log(NumClusters)

log(/<R2>)

log

(/x

)

log(NumClusters)

log(-/A)

log

(/x

)

log(NumClusters)

log(/E)

log

(/x

)

log(NumClusters)

log(/S)

log

(/x

)

log(NumClusters)

log(/<Rg

2>)

Standard deviation among 10 independent samples of CCBB calculations as a function of the number of clusters sampled.


0 100 200 300 400-700

-600

-500

-400

-300

-200

-100

0

0 100 200 300 400-700

-600

-500

-400

-300

-200

-100

0

0 100 200 300 400-700

-600

-500

-400

-300

-200

-100

0

0 100 200 300 400-700

-600

-500

-400

-300

-200

-100

0

0 100 200 300 400-700

-600

-500

-400

-300

-200

-100

0

A/k

T

N

1800K 2160K 2520K 3600K 7200K

Free energy A/kT for polymer chains of length N at different temperatures from CCBB calculations with SKS-FF.


ANB

N CNNZTk

A ln)1(lnln

Linearity is expected from mean field description.

We see that A/kT is proportional to N, where the slope gives -ln,

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L090 100 200 300 400

0

20

40

60

80

100

120

140

160

0 100 200 300 400

0

20

40

60

80

100

120

140

160

0 100 200 300 400

0

20

40

60

80

100

120

140

160

0 100 200 300 400

0

20

40

60

80

100

120

140

160

0 100 200 300 400

0

20

40

60

80

100

120

140

160

E/k

T

1800K

N

2160K 2520K

3600K 7200K

Internal energy E/kT for polymer chains of length N at different temperatures from CCBB calculations with SKS-FF.


We find that E/kT is proportional to N, where the slope represents e0,

EB

n CNeTk

E 0


0 100 200 300 400

0

100

200

300

400

500

600

700

0 100 200 300 400

0

100

200

300

400

500

600

700

0 100 200 300 400

0

100

200

300

400

500

600

700

0 100 200 300 400

0

100

200

300

400

500

600

700

0 100 200 300 400

0

100

200

300

400

500

600

700

N

1800K

S/k

2160K 2520K 3600K 7200K

Entropy S/k for polymer chains of length N at different temperatures from CCBB calculations with SKSFF.

The uncertainties are shown, but are less than the size of the symbols. We find that S/k is proportional to N, where the slope is s0.

SB

N CNNsk

S ln)1(0


5.4 5.6 5.8 6.0

6.2

6.4

6.6

6.8

7.0

5.4 5.6 5.8 6.0

6.2

6.4

6.6

6.8

7.0

5.4 5.6 5.8 6.0

6.2

6.4

6.6

6.8

7.0

5.4 5.6 5.8 6.0

6.2

6.4

6.6

6.8

7.0

5.4 5.6 5.8 6.0

6.2

6.4

6.6

6.8

7.0

ln(N-1)

1800K

ln<

Rg

2>

ln(N-1)

2160K

ln<

Rg

2>

ln(N-1)

2520K

ln<

Rg

2>

ln(N-1)

3600K

ln<

Rg

2>

ln(N-1)

7200K

The radius of gyration, ln<Rg2>, as a function of polymer length,

ln(N-1) for a range of temperatures (CCBB calculations using SKSFF).


22 NRg

This confirms that ln<Rg

2> is proportional

to ln(N-1), as in following equation.

Here the slope, 2, is a function of temperature,


0.0000 0.0002 0.0004 0.0006 0.0008 0.00102.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

0.0000 0.0002 0.0004 0.0006 0.0008 0.00102.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

0.0000 0.0002 0.0004 0.0006 0.0008 0.00102.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

0.0000 0.0002 0.0004 0.0006 0.0008 0.00102.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

0.0000 0.0002 0.0004 0.0006 0.0008 0.00102.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

0.0000 0.0002 0.0004 0.0006 0.0008 0.00102.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

Cri

tical

attr

ition

1/T (1/K)

NoTorNoAttSKSFF

RBFF

SKSFF SKSFF, f=0.5 SKSFF, f=0.25

NoTorSKSFF

Critical attrition as a function of 1/T from CCBB with various force fields.

The uncertainties are shown but are less than the size of the symbols.

We find that increases as the temperature (except for NoTorSKSFF) reflecting the increasing number of available conformations and partition function.


0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1/T (1/K)

NoTorNoAttSKSFF

e0

RBFF SKSFF

SKSFF, f=0.5 SKSFF, f=0.25

NoTorSKSFF

e0 as a function of 1/T from CCBB with different force fields. The

uncertainties are shown, but are less than the size of the symbols.

We find that e0 decreases as temperature increases, except for

NoTorSKSFF.

As the temperature increases, average energy <E> increases slower than kT, except for NoTorSKSFF.


0 10000 20000 30000 40000 50000

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

<R

g

2>

(N=

30

0)

T/K

RBFF SKSFF SKSFF, f=0.5 SKSFF, f=0.25 NoTorSKSFF NoTorNoAttSKSFF

Radius of gyration, <Rg2> as a function of temperature from CCBB MC

with various force fields, all for a chain length of N=300.

When only the repulsive van der Waals energy is present (NoTorNoAttSKSFF), the polymer chains shrink with increasing temperature, which reflects the increasing number of available conformations.

When the attractive van der Waals energy is present (the other five force fields), the polymer chains collapse at low temperature due to the favorable enthalpy.


CR, Cn, CA derived from e0 at different temperatures from CCBB

with NoTorNoAttSKSFF and NoTorSKSFF.

We find that CR, Cn, CA is linear with 1/T, indicating that ER, En, EA

are independent of temperature.

ARBARBng

Bn EETkCCTkCR

NTkE )(

3


0.0000 0.0002 0.0004 0.0006 0.0008 0.00100.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

0.0000 0.0002 0.0004 0.0006 0.0008 0.00100.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

0.0000 0.0002 0.0004 0.0006 0.0008 0.00100.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

0.0000 0.0002 0.0004 0.0006 0.0008 0.00100.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

0.0000 0.0002 0.0004 0.0006 0.0008 0.00100.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

0.0000 0.0002 0.0004 0.0006 0.0008 0.00100.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

2

1/T (1/K)

NoTorNoAttSKSFF

2

1/T (1/K)

RBFF

2

1/T (1/K)

SKSFF

2

1/T (1/K)

SKSFF, f=0.5

2

1/T (1/K)

SKSFF, f=0.25

2

1/T (1/K)

NoTorSKSFF

The critical exponent 2 as a function of 1/T from CCBB with various force fields.

As temperature decreases, the polymer chains collapse and 2 decreases except for NoTorNoAttSKSFF.

The result of NoTorNoAttSKSFF shows that including only self-avoidance leads to 21.2, as deduced by Flory.


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

slope=-0.1523, R2=0.9278

slope=-0.0090, R2=0.9560

slope=0.0850, R2=0.9954[A

(N-m

)-AN]/k

T-y

0

ln[N/(N-m)]

50400K 3600K 1440K 1080K

slope=0.1113, R2=0.9979

The Free energy increment from CCBB calculations (SKSFF) for adding m=100 monomers [A(N-m)-AN] as a function of length N

(plotted as ln[N/(N-m)]). [A(N-m)-AN]/kT is proportional to ln[N/(N-m)], where the slope is (-1).

mN

Nm

Tk

AAA

B

mNNmN ln)1(ln, AN

B

N CNNZTk

A ln)1(lnln

Free energy theta temperature


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

slope=-0.8648, R2=0.8741

slope=-0.2635, R2=0.9721

slope=-0.0322, R2=0.9665

slope=-0.0041, R2=0.1413

[E(N

-m)-E

N]/k

T-y

0

ln[N/(N-m)]

50400K 3600K 1440K 1080K

The internal energy increment from CCBB calculations (SKSFF) for adding m=100 monomers [E(N-m)-EN] as a function of length N

(plotted as ln[N/(N-m)]).

We see that [E(N-m)-EN]/kT is independent to ln[N/(N-m)] at high

temperature limit, while dependent at low temperature.

EB

n CNeTk

E 0

At low temperature, the chain collapses and the average energy in the long chain is less than in the short chain.


400 800 1200 1600 2000 2400

-2.0

-1.5

-1.0

-0.5

0.0

400 800 1200 1600 2000 2400

-2.0

-1.5

-1.0

-0.5

0.0

400 800 1200 1600 2000 2400

-2.0

-1.5

-1.0

-0.5

0.0

400 800 1200 1600 2000 2400

-2.0

-1.5

-1.0

-0.5

0.0

400 800 1200 1600 2000 2400

-2.0

-1.5

-1.0

-0.5

0.0

400 800 1200 1600 2000 2400

-2.0

-1.5

-1.0

-0.5

0.0

-1

T/K

NoTorNoAttSKSFF

-1=0

T/K

RBFF

-1=0

T/K

SKSFF

-1=0

T/K

SKSFF, f=0.5

-1=0

T/K

SKSFF, f=0.25

T/K

NoTorSKSFF

-1=0

The free energy critical exponent ( - 1) as a function of 1/T from CCBB MC using various force fields.

increases with the temperature, reflecting the increased number of available conformations and partition function.


cNfbNaTk

A

B

N )(**

cNbaNCNNZTk

AAN

B

N lnln)1(lnln

cbNaNTk

A

B

N 3/2

T/K Chi^2/DoF a error b error c error

50400 LnN 4.66E-06 1.7087 8.55E-07 -0.15212 1.97E-03 5.20621 0.00859

50400 N^(2/3) 4.11E-06 1.70728 3.00E-05 -0.01938 0.00024 4.77789 0.00286

3600 LnN 3.64E-06 1.44618 8.31E-03 -0.12018 0.00174 4.47254 0.0076

3600 N^(2/3) 2.56E-06 1.44504 2.00E-05 -0.0154 0.00019 4.13526 0.00226 theta/K

1440 LnN 5.63E-06 1.13825 7.38E-07 -0.01961 0.00217 3.5418 0.00945 1399

1440 N^(2/3) 5.26E-06 1.13804 3.00E-05 -0.00277 0.00027 3.48984 0.00323 1394

1368 LnN 6.08E-06 1.11457 0.00001 0.01474 0.00225 3.37896 0.00981

1368 N^(2/3) 6.35E-06 1.11467 8.44E-07 0.00157 0.00029 3.42419 0.00355

1080 LnN 9.50E-06 0.99709 0.00001 0.40513 0.00282 1.74419 0.01227

1080 N^(2/3) 2.59E-06 1.00091 0.00002 0.05173 0.00019 2.88357 0.00227

NN NZ 1

The correction to the f(N) term in the free energy

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L094.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0

-2

-1

0

1

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0

-2

-1

0

1

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0

-2

-1

0

1

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0

-2

-1

0

1

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0

-2

-1

0

1

lnN

1080K, -0.40475*lnN-0.00165, R2=0.9996

-A/k

T-a

N+

c

50400K, 0.1529*lnN-0.00339, R2=0.9986

3600K, 0.11995*lnN+0.00101, R2=0.9982 1440K, 0.02017*lnN-0.00241, R2=0.9121

1368K, -0.01523*lnN+0.0021, R2=0.8458

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0

-2

-1

0

1

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0

-2

-1

0

1

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0

-2

-1

0

1

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0

-2

-1

0

1

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0

-2

-1

0

1

lnN

1080K, -0.40475*lnN-0.00165, R2=0.9996

-A/k

T-a

N+

c

50400K, 0.1529*lnN-0.00339, R2=0.9986

3600K, 0.11995*lnN+0.00101, R2=0.9982 1440K, 0.02017*lnN-0.00241, R2=0.9121

1368K, -0.01523*lnN+0.0021, R2=0.8458

cNbaNCNNZTk

AAN

B

N lnln)1(lnln

NN NZ 1

cbNaNTk

A

B

N 3/2


cNfbNaTk

A

B

N )(**

cNfbNaTk

A

B

N )2(*2*2

Coil I with length N

Coil II with length N

cNfbNaccTk

Ac

Tk

A

Tk

A

B

N

B

N

B

N

)(*2*2)()(2

)(*2)2(*22 NfbNfbTk

A

Tk

A

Tk

A

B

N

B

N

B

The physics of the lnN term and gamma


4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.00

2

4

6

8

10

12

14

16

18

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.00

2

4

6

8

10

12

14

16

18

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.00

2

4

6

8

10

12

14

16

18

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.00

2

4

6

8

10

12

14

16

18

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.00

2

4

6

8

10

12

14

16

18E

/kT

+a

N-c

lnN

1080K, 2.9385*lnN+0.00331, R2=0.9996

E/k

T+

aN

-c

lnN

50400K, 0.00228*lnN+0.00084, R2=0.0519

E/k

T+

aN

-c

lnN

3600K, 0.08672*lnN-0.0019, R2=0.9267

E/k

T+

aN

-c

lnN

1440K, 0.56512*lnN+0.00176, R2=0.9972

E/k

T+

aN

-c

lnN

1368K, 0.7851*lnN+0.00408, R2=0.9974

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.00

2

4

6

8

10

12

14

16

18

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.00

2

4

6

8

10

12

14

16

18

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.00

2

4

6

8

10

12

14

16

18

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.00

2

4

6

8

10

12

14

16

18

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.00

2

4

6

8

10

12

14

16

18

E/k

T+

aN

-c

lnN

1080K, 2.9385*lnN+0.00331, R2=0.9996E

/kT

+a

N-c

lnN

50400K, 0.00228*lnN+0.00084, R2=0.0519

E/k

T+

aN

-c

lnN

3600K, 0.08672*lnN-0.0019, R2=0.9267

E/k

T+

aN

-c

lnN

1440K, 0.56512*lnN+0.00176, R2=0.9972

E/k

T+

aN

-c

lnN

1368K, 0.7851*lnN+0.00408, R2=0.9974

Critical exponent ’s components: E and S


2 3 4 5 6

0.0

0.5

1.0

2 3 4 5 6

0.0

0.5

1.0

2 3 4 5 6

0.0

0.5

1.0

2 3 4 5 6

0.0

0.5

1.0

2 3 4 5 6

0.0

0.5

1.0

2 3 4 5 6

0.0

0.5

1.0

(-A

/kT

-Nln

)+

CA

lnN

1368K

(-A

/kT

-Nln

)+

CA

lnN

50400K

=1.1567, R2=0.9997

(-A

/kT

-Nln

)+

CA

lnN

28800K

(-A

/kT

-Nln

)+

CA

lnN

3600K

=1.1172, R2=0.9989

(-A

/kT

-Nln

)+

CA

lnN

1800K

=1.0494, R2=0.9889

(-A

/kT

-Nln

)+

CA

lnN

1440K

=0.9850, R2=0.9885

Derive of for various T (CCBB using SKSFF)

Plot (-AN/kBT-Nln) vs. lnN

(-1) from slope of each line (eliminate CA).

increases with T, because increased number available conformations and partition function.

AB

N CNNTk

A ln)1(ln Critical exponent and Free energy theta temperature

change in g between 28800K and 50400K is negligible, indicating that g has reached its asymptotic value by 50400K.

(-AN/kBT-Nlnm) increases monotonically with increasing N for T 1800K while it decreases monotonically with increasing N for T 1440K.

Thus the free energy theta temperature (g = 1) must be in the range of 1440K<T<1800K. Our fit leads to TqFE(=1.0) = 1524K.


Critical exponent and Free energy theta temperature

AB

N CNNTk

A ln)1(ln

RBFF SKSFF SKSFF, f=0.5

SKSFF, f=0.25

1.1582 1.1567 1.1668 1.1649

Deviation 0.0003 0.0005 0.0002 0.0004

RGT21 RGT22 Enum- lattice16

Enum- lattice17

MC- lattice28

Exptl23 Exptl29

1.1596 1.187 1.1595 1.16193 1.1575 1.25 1.233/1.25

Deviation 0.002 N/A 0.0012 0.0001 0.0006 0.01 0.01

Force fields RBFF SKSFF SKSFF, f=0.5 SKSFF, f=0.25

TFE(CCBB) 2411K 1524K 773K 413K


0.2 0.4 0.6 0.8 1.0

300

600

900

1200

1500

0.2 0.4 0.6 0.8 1.0

300

600

900

1200

1500

0.2 0.4 0.6 0.8 1.0

300

600

900

1200

1500

Th

eta

tem

pe

ratu

re (

K)

Scaling factor (f)

CCBB, =1 definition

Th

eta

tem

pe

ratu

re (

K)

Scaling factor (f)

CCBB, 2definition

Th

eta

tem

pe

ratu

re (

K)

Scaling factor (f)

MD, 2definition

Dependence of the theta temperature on the solvent scale factor f from CCBB calculations with SKSFF. For the Flory definition, the CCBB result is the same as for MD. CCBB MC does not allow fluctuations in bond stretch and bond angle, which underestimates the partition function and , leading to an overestimate of the free energy theta temperature.

612

4)(ijij

ijLJ rrfrE


0.2 0.4 0.6 0.8 1.0

300

600

900

1200

1500

0.2 0.4 0.6 0.8 1.0

300

600

900

1200

1500

0.2 0.4 0.6 0.8 1.0

300

600

900

1200

1500

Th

eta

tem

pe

ratu

re (

K)

Scaling factor (f)

TFE, =1 definition

Based on Free Energy from CCBB

Th

eta

tem

pe

ratu

re (

K)

Scaling factor (f)

TF, 2definition

Based on <Rg

2> from CCBB

Th

eta

tem

pe

ratu

re (

K)

Scaling factor (f)

TF, 2definition

Based on <Rg

2> from MD

Dependence of the theta temperature on the solvent scale factor f from CCBB calculations using SKS-FF. We find that TθF is the same

for CCBB and for MD.

But TθFE is ~20% higher than TθF for all solvent scale factors.

By including a van der Waals scaling factor in the Lennard-Jones potential to represent solvation, we find a linear relationship between this factor and the theta temperature

The shift of the theta point


We derive a general mean field model that may be useful in deriving general scaling laws. The mean field description of the non-bond interactions is found to be temperature independent. It also explains that increases and e0 decreases regularly with increasing temperature.

EB

n CNeTk

E 0

We also developed a mean field model and evaluated the critical energy increment e0 and


1. We use the CCBB MC to derive thermodynamic properties [A, S, E, <Rg2>)] for an ensemble of isolated polymer chains

2. Throughout the range of high temperature, we find that

a. Rg2 ~ N , with Flory critical exponent of 2=1.168.

b. ZN ~ N(-1) N, with =1.157, =5.649.

c. the critical energy increment e0 = 0.0187 4 (confirm Des Cloizeau’s prediction that the internal energy increment is uniform at high temperatures)

3. Defining theta point at which the increment in A with N is a constant (=1). This is at ~ 20% higher temperature than the temperature at which the chain describes a Gaussian coil (2=1, that is, <Rg2> scales linearly with N). Three-body term causes this shift of the theta point and is confirmed by our results.

4. We construct a model to explain the physics of gamma and its components.

Summary


•S. Koonin, Computational Physics •H. Gould and J. Tobochnik, An Introduction to Computer Simulation Methods, Vol. 2 •O. Mouritsen, Computer Studies of Phase Transitions and Critical Phenomena •M.H. Kalos and P.A. Whitlock, Monte Carlo Methods, Vol. I. Basics

These are a bit more advanced: •K. Binder ed., Monte Carlo Methods in Statistical Physics •K. Binder ed., Applications of the Monte Carlo Method in Statistical Physics •D.P. Landau et. al, eds. Computer Simulation Studies in Condensed Matter Physics: Recent Developments •K. Kremer and K. Binder, Comput. Phys. Reports 7, 259 (1988).

References to books with introductions to Monte Carlo methods, especially for statistical physics:

Documents

© copyright 2011 William A. Goddard III, all rights reservedCh121a-Goddard-L09 Ch121a Atomic Level Simulations of Materials and Molecules William A. Goddard