1 CE 530 Molecular Simulation Lecture 20 Phase Equilibria David A. Kofke Department of Chemical Engineering SUNY Buffalo [email protected]

1

CE 530 Molecular Simulation

Lecture 20

Phase Equilibria

David A. Kofke

Department of Chemical Engineering

SUNY Buffalo

[email protected]

2

Thermodynamic Phase Equilibria

Certain thermodynamic states find that two (or more) phases may be equally stable• thermodynamic phase coexistence

Phases are distinguished by an order parameter; e.g.• density

• structural order parameter

• magnetic or electrostatic moment

Intensive-variable requirements of phase coexistence• T = T

• P = P

• i = i

, i = 1,…,C

3

Phase Diagrams Phase behavior is described via phase diagrams

Gibbs phase rule• F = 2 + C – P

F = number of degrees of freedom– independently specified thermodynamic state variables

C = number of components (chemical species) in the system

P = number of phases

• Single component, F = 3 – P

Fiel

d va

riab

lee.

g.,

tem

pera

ture

Density variable

L+V

tie line

V LS

S+V

L+S

triple point

critical point

Fiel

d va

riab

le(p

ress

ure)

Field variable(temperature)

V

L

S

critical point

isobar

two-phase regions two-phase lines

4

Approximate Methods for Phase Equilibria by Molecular Simulation 1.

Adjust field variables until crossing of phase transition is observed• e.g., lower temperature until condensation/freezing occurs

Problem: metastability may obsure location of true transition

Try it out with the LJ NPT-MC applet

Pres

sure

Temperature

V

L

S

Metastable region

Tem

pera

ture

Density

LS

S+V

L+S

isobar

L+VV Hysteresis

5

Approximate Methods for Phase Equilibria by Molecular Simulation 2.

Select density variable inside of two-phase region• choose density between liquid and vapor values

Problem: finite system size leads to large surface-to-volume ratios for each phase• bulk behavior not modeled

Metastability may still be a problem

Try it out with the SW NVT MD appletT

empe

ratu

reDensity

LS

S+V

L+SL+VV

6

Rigorous Methods for Phase Equilibria by Molecular Simulation

Search for conditions that satisfy equilibrium criteria• equality of temperature, pressure, chemical potentials

Difficulties• evaluation of chemical potential often is not easy

• tedious search-and-evaluation process

Che

mic

al p

oten

tial

Simulation data

Simulation data orequation of state

T constant

Pressure

7





Che

mic

al p

oten

tial


T constant

Pressure

8





Che

mic

al p

oten

tial


T constant

Pressure

9





Che

mic

al p

oten

tial


T constant

Pressure

10





Che

mic

al p

oten

tial


Coexistence T constant

Pressure

11





Che

mic

al p

oten

tial


Coexistence T constant

Pressure

Pres

sure

Temperature

VL

S

+

Many simulations to get one point on phase diagram

12

Gibbs Ensemble 1.

Panagiotopoulos in 1987 introduced a clever way to simulate coexisting phases without an interface

Thermodynamic contact without physical contact• How?

Two simulation volumes

13

Gibbs Ensemble 2.

MC simulation includes moves that couple the two simulation volumes

Try it out with the LJ GEMC applet

Particle exchange equilibrates chemical potential

Volume exchange equilibrates pressure

Incidentally, the coupled moves enforce mass and volume balance

14

Gibbs Ensemble Formalism

Partition function

Limiting distribution

General algorithm. For each trial:• with some pre-specified probability, select

molecules displacement/rotation trial

volume exchange trial

molecule exchange trial

• conduct trial move and decide acceptance

1

1 1 1 1 10 0

( , , ) ( , , ) ( , , )VN

GE

N

Q N V T dV Q N V T Q N N V V T

2 1

2 1

N N N

V V V

1 21 21 2 1 1 2 2

3( , ) ( , )

1 1 1 1 2( , , )N NN

N NN N N U V N U VNGE

N V dV d d V d e V d eQ

s sr s s s s

15Molecule-Exchange Trial Move Analysis of Trial Probabilities

Detailed specification of trial moves and probabilities

Event[reverse event]

Probability[reverse probability]

Select box A[select box B]

1/21/2

Select molecule k[select molecule k]

1/NA

[1/(NB+1)]

Move to rnew

[move back to rold]ds/1

[ds/1]

Accept move[accept move]

min(1,)[min(1,1/)]

Forward-step trial probability

Scaled volume

1min(1, )

2 A

d

N

s

Reverse-step trial probability

11min(1, )

2 1B

d

N s

is formulated to satisfy detailed balance

16

Molecule-Exchange Trial MoveAnalysis of Detailed Balance

Detailed balance

i ij j ji=



1min(1, )

2 A

d

N

s Reverse-step trial probability

11min(1, )

2 1B

d

N s

3( , ) ( , )( , , )

N NA BA BA B A A B B

NN NN N N U V N U VN

A A A BAGEN V dV d d V d e V d e

Q

s sr s s s s

17


Detailed balance

i ij j ji=


1min(1, )

2 A

d

N


11min(1, )

2 1B

d

N s

111 11 1

3 3

min(1, )min(1, )

2 2( 1)

old newA B A BB A BN N N NN N N NU U

A AB BA AN GE N GE

A B

de V d V d dV e V d V d dVd

N NQ Q

ss s s ss


3( , ) ( , )( , , )



A A A BAGEN V dV d d V d e V d e

Q

s sr s s s s

18


Detailed balance

111 11 1

3 3

min(1, )min(1, )

2 2( 1)

old newA B A BB A BN N N NN N N NU U

A AB BA AN GE N GE

A B

de V d V d dV e V d V d dVd

N NQ Q

ss s s ss

i ij j ji=


1min(1, )

2 A

d

N


11min(1, )

2 1B

d

N s

1 1

1

old newU U B

A A B

Ve e

N V N

1totUB A

A B

V Ne

V N

Acceptance probability

19

Volume-Exchange Trial Move Analysis of Trial Probabilities

Take steps in = ln(VA/VB)

Detailed specification of trial moves and probabilities

Event[reverse event]

Probability[reverse probability]

Step to new

[step to old]d/d/

Accept move[accept move]

min(1,)[min(1,1/)]


min(1, )d


1min(1, )d

1 1

1

A B A B

VA AV V V V

A A BV

d dV dV

dV V V d

20

Volume-Exchange Trial MoveAnalysis of Detailed Balance

Detailed balance

i ij j ji=




31 1( , ) ( , )( , , )



A A BAGE

dN V d d d V d e V d e

VQ

s sr s s s s

min(1, )d

1min(1, )d

21


Detailed balance

i ij j ji=

1 11 11 1,,,,

3 3

min(1, )min(1, )newold

A BA BA B B N NN NN N UN NUB jA jB iA i

N GE N GE

de V d V de V d V d d

Q Q

s ss s




31 1( , ) ( , )( , , )



A A BAGE

dN V d d d V d e V d e

VQ

s sr s s s s

min(1, )d

1min(1, )d

22


Detailed balance

i ij j ji=



1 1 1 1, ,, ,

old newA B A BN N N NU U

B i B iA i A ie V V e V V

1 1A B

tot

N Nnew newUA B

old oldA B

V Ve

V V

Acceptance probability

1 11 11 1,,,,

3 3

min(1, )min(1, )newold

A BA BA B B N NN NN N UN NUB jA jB iA i

N GE N GE

de V d V de V d V d d

Q Q

s ss s

min(1, )d

1min(1, )d

23

Gibbs Ensemble in the Molecular Simulation API

U ser 's P erspe ctiv e o n the M o lecu la r S im u la tion A P I

S pa ce

In teg ra to rG E M C

C on tro lle r

M ete r B ou nda ry C on fig ura tion

P ha se S p ec ies P o te n tia l D isp lay D ev ice

S im ula tion

24

Gibbs Ensemble MCExamination of Java Code. General

//Performs basic GEMC trial//At present, suitable only for single-component systemspublic void doStep() { int i = (int)(rand.nextDouble()*iTotal); //select type of trial to perform if(i < iDisplace) { //displace a randomly selected atom if(rand.nextDouble() < 0.5) {atomDisplace.doTrial(firstPhase);} else {atomDisplace.doTrial(secondPhase);} } else if(i < iVolume) { //volume exchange trialVolume(); } else { //molecule exchange if(rand.nextDouble() < 0.5) { trialExchange(firstPhase,secondPhase,firstPhase.parentSimulation.firstSpecies); } else { trialExchange(secondPhase,firstPhase,firstPhase.parentSimulation.firstSpecies); } }}

public class IntegratorGEMC extends Integrator

25

Gibbs Ensemble MCExamination of Java Code. Molecule Exchange/** * Performs a GEMC molecule-exchange trial */private void trialExchange(Phase iPhase, Phase dPhase, Species species) { Species.Agent iSpecies = species.getAgent(iPhase); //insertion-phase speciesAgent Species.Agent dSpecies = species.getAgent(dPhase); //deletion-phase species Agent double uNew, uOld; if(dSpecies.nMolecules == 0) {return;} //no molecules to delete; trial is over Molecule m = dSpecies.randomMolecule(); //select random molecule to delete uOld = dPhase.potentialEnergy.currentValue(m); //get its contribution to energy m.displaceTo(iPhase.randomPosition()); //place at random in insertion phase uNew = iPhase.potentialEnergy.insertionValue(m); //get its new energy if(uNew == Double.MAX_VALUE) { //overlap; reject m.replace(); //put it back return; } double bFactor = dSpecies.nMolecules/dPhase.volume()//acceptance probability * iPhase.volume()/(iSpecies.nMolecules+1) * Math.exp(-(uNew-uOld)/temperature); if(bFactor > 1.0 || bFactor > rand.nextDouble()) { //accept iPhase.addMolecule(m,iSpecies); //place molecule in phase } else { //reject m.replace(); //put it back }}


26

Gibbs Ensemble MCExamination of Java Code. Volume Exchange

private void trialVolume() { double v1Old = firstPhase.volume(); double v2Old = secondPhase.volume(); double hOld = firstPhase.potentialEnergy.currentValue() + secondPhase.potentialEnergy.currentValue(); double vStep = (2.*rand.nextDouble()-1.)*maxVStep; //uniform step in volume double v1New = v1Old + vStep; double v2New = v2Old - vStep; double v1Scale = v1New/v1Old; double v2Scale = v2New/v2Old; double r1Scale = Math.pow(v1Scale,1.0/(double)Simulation.D); double r2Scale = Math.pow(v2Scale,1.0/(double)Simulation.D); firstPhase.inflate(r1Scale); //perturb volumes secondPhase.inflate(r2Scale); double hNew = firstPhase.potentialEnergy.currentValue() //acceptance probability + secondPhase.potentialEnergy.currentValue(); if(hNew >= Double.MAX_VALUE || Math.exp(-(hNew-hOld)/temperature + firstPhase.moleculeCount*Math.log(v1Scale) + secondPhase.moleculeCount*Math.log(v2Scale)) < rand.nextDouble()) { //reject; restore volumes firstPhase.inflate(1.0/r1Scale); secondPhase.inflate(1.0/r2Scale); } //accept; nothing more to do}


27

Gibbs Ensemble Example Applications

Seen in first lecture

28

Water and Aqueous Solutions GEMC simulation of water/methanol mixtures

• Strauch and Cummings, Fluid Phase Equilibria, 86 (1993) 147-172; Chialvo and Cummings, Molecular Simulation, 11 (1993) 163-175.

29

Phase Behavior of Alkanes

Panagiotopoulos group• Histogram reweighting with finite-size scaling

30

Critical Properties of Alkanes Siepmann et al. (1993)

31

Alkane Mixtures

Siepmann group• Ethane +

n-heptane

32

Gibbs Ensemble Limitations Limitations arise from particle-exchange requirement

• Molecular dynamics (polarizable models)

• Dense phases, or complex molecular models

• Solid phases N = 4n3 (fcc)

and sometimes from the volume-exchange requirement too

?

33

Approaching the Critical Point Critical phenomena are difficult to capture by molecular simulation Density fluctuations are related to compressibility Correlations between fluctuations important to behavior

• in thermodynamic limit correlations have infinite range

• in simulation correlations limited by system size

• surface tension vanishes

Pres

sure

Density

L+VV LS

S+V

L+S

Cisotherm

0P 1 1

T

V

V P P

2

34

Critical Phenomena and the Gibbs Ensemble

Phase identities break down as critical point is approached• Boxes swap “identities” (liquid vs. vapor)

• Both phases begin to appear in each boxsurface free energy goes to zero

35

Gibbs-Duhem Integration

GD equation can be used to derive Clapeyron equation

• equation for coexistence line

Treat as a nonlinear first-order ODE• use simulation to evaluate

right-hand side

Trace an integration path the coincides with the coexistence line

ln p

–h

Z

Fiel

d va

riab

le(p

ress

ure)

Field variable(temperature)

V

L

S

critical point

36

lnp

lnp

lnp

GDI Predictor-Corrector Implementation Given initial condition and slope

(= –h/Z), predict new (p,T) pair.

Evaluate slope at new state condition…

…and use to correct estimate of new (p,T) pair

37

GDI Simulation Algorithm

Estimate pressure and temperature from predictor Begin simultaneous NpT simulation of both phases at the

estimated conditions As simulation progresses, estimate new slope from ongoing

simulation data and use with corrector to adjust p and T Continue simulation until the iterations and the simulation

averages converge Repeat for the next state point

Each simulation yields a coexistence point.Particle insertions are never attempted.

38

Gibbs-Duhem Integration in the Molecular Simulation API

U ser 's P erspe ctiv e o n the M o lecu la r S im u la tion A P I

S pa ce

In te gra to r

C o ntro lle rG D I

M ete r B ou nda ry C on fig u ra tion

P ha se S p ec ies P o te n tia l D isp lay D ev ice

S im u la tion

39

GDI Sources of Error

Three primary sources of error in GDI method

1

Stochastic error

lnp

2

Uncertainty in slopes due to uncertainty in simulation averages

Quantify via propagation of error using confidence limits of simulation averages

Finite step of integrator

• Smaller step• Integrate series with

every-other datum• Compare predictor and

corrector values

Stability

• Stability can be quantified

• Error initial condition can be corrected even after series is complete

1

2

1

2True curve

?

Incorrect initial condition

40

GDI Applications and Extensions

Applied to a pressure-temperature phase equilibria in a wide variety of model systems• emphasis on applications to solid-fluid coexistence

Can be extended to describe coexistence in other planes• variation of potential parameters

inverse-power softness

stiffness of polymers

range of attraction in square-well and triangle-well

variation of electrostatic polarizability

• as with free energy methods, need to identify and measure variation of free energy with potential parameter

• mixturesA U

41

Semigrand Ensemble Thermodynamic formalism for mixtures

Rearrange

Legendre transform

• Independent variables include N and 2

• Dependent variables include N2

must determine this by ensemble average

ensemble includes elements differing in composition at same total N

Ensemble distribution

1 1 2 2Ud Pd d dVdA N N

1 1 2 2 1 2

1 2 2

( ) ( )dA d N N dN

d

Ud Pd

N dNUd Pd

V

V

1 2 1 2dN dN

2 2

1 2 2

( )dY d A N

dN N dUd PdV

2 2 23

( , , )1 12 !

, ,N N

NE p r N NN N

h NN e e

r p

42

GDI/GE with the Semigrand Ensemble

Governing differential equation (pressure-composition plane)

2

2 ,

ln

T

xp

Z

0.10

0.09

0.08

0.07

0.06

Pre

ssur

e

1.00.80.60.40.20.0

Mole Fraction of Species 2

Gibbs Ensemble Gibbs-Duhem Integration

11 1 11 1

12 1 12 0.75

22 1 22 1

Simple LJ Binary

43

GDI with the Semigrand Ensemble Freezing of polydisperse hard spheres

• appropriate model for colloids

Integrate in pressure-polydispersity plane

Findings• upper polydisersity bound to freezing

• re-entrant melting1.4

1.3

1.2

1.1

1.0

0.9

Den

sity

0.70.60.50.40.30.20.10.0

Polydispersity

c1 = 1

c2 = 1

solid

fluid

22 / 2

( / )sat

dp m

d V N

Composition(Fugacity)

Diameter

m2

44

Other Views of Coexistence Surface Dew and bubble lines Residue curves Azeotropes

• semigrand ensemble

• formulate appropriate differential equation

• integrate as in standard GDI methodright-hand side involves partial molar properties

,

1

,

1

,

1

,

1

,

1

,

1

)(

PP

azeo

sat

PP

azeo XY

d

dP

P

Y

P

XYX

d

d

)( VL

VL

azeo

sat

vv

hh

d

dP

45

Tracing of Azeotropes Lennard-Jones binary, variation with intermolecular

potential• 11 = 1.0; 12 = 1.05; 22 = variable

• 11 = 1.0; 12 = 0.90; 22 = 1.0

• T = 1.10

0.0425

0.0445

0.0465

0.0485

0.0505

0.0525

0.0545

0 0.1 0.2 0.3 0.4 0.5

X2, Y2

Coe

xist

ence

pre

ssur

e, P

*

46

Summary Thermodynamic phase behavior is interesting and useful Obvious methods for evaluating phase behavior by simulation are too

approximate Rigorous thermodynamic method is tedious Gibbs ensemble revolutionized methodology

• two coexisting phases without an interface

Gibbs-Duhem integration provides an alternative to use in situations where GE fails• dense and complex fluids

• solids

Semigrand ensemble is useful formalism for mixtures GDI can be extended in many ways

Documents

1 CE 530 Molecular Simulation Lecture 20 Phase Equilibria David A. Kofke Department of Chemical Engineering SUNY Buffalo [email protected]