Nucleation rates of ethanol and Nucleation rates of ethanol and methanol using an equation of methanol using an equation of state.state.

NouraNoura AlAl--Zoubi Zoubi

Advisor:Advisor:Dr. Abdalla ObeidatDr. Abdalla Obeidat

CoCo--advisor:advisor:Dr. Maen GharaibehDr. Maen Gharaibeh

Out LinesOut Lines::

•• Introduction.Introduction.•• First order phase transition.First order phase transition.•• Classical nucleation theory (CNT).Classical nucleation theory (CNT).•• Gradient theory (GT).Gradient theory (GT).•• Computational methodology.Computational methodology.•• Results and conclusion.Results and conclusion.

DefinitionDefinition

•• Nucleation is the process which the formation of Nucleation is the process which the formation of new phases begins and is thus a widely spread new phases begins and is thus a widely spread phenomenon in both nature and technology.phenomenon in both nature and technology.

•• Nucleation refers to the kinetic processes involved Nucleation refers to the kinetic processes involved in the initiation of the first order phase transition in in the initiation of the first order phase transition in non equilibrium systems.non equilibrium systems.

•• Condensation and evaporation, crystal growth, Condensation and evaporation, crystal growth, deposition of thin films and overall crystallization are deposition of thin films and overall crystallization are only a few of the processes in which nucleation plays only a few of the processes in which nucleation plays a prominent role. a prominent role.

..

First Order Phase TransitionsFirst Order Phase Transitions

pressure pressure –– volume diagram for pure fluid volume diagram for pure fluid for different isotherms.for different isotherms.

dependence of the reduced pressure dependence of the reduced pressure and Gibbs energy on the reduced volume and Gibbs energy on the reduced volume using VDWusing VDW

Classical Nucleation TheoryClassical Nucleation TheorySurface and volume energies of formation a Surface and volume energies of formation a cluster versus cluster radius. The energy of cluster versus cluster radius. The energy of formation has a maximum at critical radius.formation has a maximum at critical radius.

In (CNT) , the work of formation In (CNT) , the work of formation have different formshave different forms

•• PP--form: form:

•• --form:form:

•• SS--form:form:

( )23 3/16 pWcl Δ= ∞πγ

( )232

316

μγπ

Δ= ∞l

clvW

( ) ve

v

blcl P

PSwhereSTk

W == ∞2

3

ln316

ρπγ

μ

Experimental data for methanol illustratingExperimental data for methanol illustratingthe inadequate temperature dependence the inadequate temperature dependence predicted by Spredicted by S--form.form.

2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3100000

1000000

1E7

1E8

1E9

1E10

1E11

272K

Expt S-form/1010

Methanol

257K

J (c

m3 s-1

)

S

Experimental data for ethanol illustrating the Experimental data for ethanol illustrating the inadequate temperature dependence inadequate temperature dependence predicted by Spredicted by S--form.form.

2.5 3.0100000

1E8

1E11

Ethanol

expt

S-form/107

J (c

m-3s-1

)

S

286K293K

Density functional theory and Density functional theory and the road to gradient theorythe road to gradient theoryDFTDFTAdvantagesAdvantages

•• Powerful technique Powerful technique that has been used to that has been used to explore varies systems.explore varies systems.

DisadvantagesDisadvantages

•• Require exact Require exact intermolecular intermolecular Potential. Potential.

GTGTAdvantagesAdvantages

•• Requires a Cubic Requires a Cubic EOSEOS

DisadvantagesDisadvantages

•• An approximation An approximation to DFTto DFT

•• The Helmholtz free energy density is given as:The Helmholtz free energy density is given as:

•• The density profile can be determined by The density profile can be determined by integrating the following equation:integrating the following equation:

•• The work of formation is given as:The work of formation is given as:

Gradient TheoryGradient Theory

( ) ( ) ( )20 2ρρρ ∇+=

cff

( )

ρμρρρ

ρ

−=−=Δ

⎥⎦⎤

⎢⎣⎡ ∇+Δ= ∫

0

2

)()()(2

fwandwwwwhere

dVcwW

b

( )02

2 12 μμρρ−=+

cdrd

rdrd

Experimental nucleation rates of Experimental nucleation rates of methanol compared to the predictionsmethanol compared to the predictionsof GT with the CPHB EOS.of GT with the CPHB EOS.

2.2 2.4 2.6 2.8 3.0 3.2105

106

107

108

109

JGT/107

Jexp

Methanol

T=272 KT=257 K

J (c

m-3s-1

)

S

Experimental nucleation rates of ethanol Experimental nucleation rates of ethanol compared to the predictions of GT with compared to the predictions of GT with the CPHB EOSthe CPHB EOS..

2.4 2.5 2.6 2.7 2.8 2.9 3.0

106

107

108

109

1010

JG T/104

Jexp

E thanol

T=286 K

T=293 K

J (c

m-3s-1

)

S

SAFTSAFT--EOSEOS•• Exact EOS for low TExact EOS for low T•• Cubic EOSCubic EOS

Computational MethodologyComputational Methodology•• Equilibrium Densities of liquid and vaporEquilibrium Densities of liquid and vapor

1. Conditions1. Conditions

2. Calculations using Newton2. Calculations using Newton--Raphson methodRaphson methoddodorow(1)=guess1row(1)=guess1row(2)=guess2row(2)=guess2k(1,1)=dp(row(1),t)k(1,1)=dp(row(1),t)k(1,2)=k(1,2)=--dp(row(2),t)dp(row(2),t)k(2,1)=dmew(row(1),t)k(2,1)=dmew(row(1),t)k(2,2)=k(2,2)=--dmew(row(2),t)dmew(row(2),t)f(1)=p(row(2),t)f(1)=p(row(2),t)--p(row(1),t) p(row(1),t) f(2)=mew(row(2),t)f(2)=mew(row(2),t)--mew(row(1),t)mew(row(1),t)z=k(2,1)/k(1,1)z=k(2,1)/k(1,1)k(2,1)=0.0d0k(2,1)=0.0d0f(2)=f(2)f(2)=f(2)--(z*f(1))(z*f(1))k(2,2)=k(2,2)k(2,2)=k(2,2)--(z*k(1,2))(z*k(1,2))u(2)=f(2)/k(2,2)u(2)=f(2)/k(2,2)u(1)=(f(1)u(1)=(f(1)--k(1,2)*u(2))/k(1,1)k(1,2)*u(2))/k(1,1)row=row+urow=row+uend doend do

)()()()(

leveleve

leveleve

orPorPPP

ρμρμμμρρ

====

Influence ParameterInfluence Parameter•• Experimental surface tensionExperimental surface tension

Surface=(23.88Surface=(23.88--0.08807*(T0.08807*(T--273))*10273))*10--77• GT surface tension

•• Equivalence between GT and experimental surface tensionsEquivalence between GT and experimental surface tensionsC=C=(Surface/integral)2 /2(Surface/integral)2 /2 where C: influence parameterwhere C: influence parameterIntegral=

•• Calculations using Composite Simpson method Calculations using Composite Simpson method step=4001step=4001h1=(row(1)h1=(row(1)--row(2))/steprow(2))/stepsum=0.d0sum=0.d0

sum1=0.d0sum1=0.d0do i=1,(step/2.d0)do i=1,(step/2.d0)--11x1=row(2)+2.d0*i*h1x1=row(2)+2.d0*i*h1x2=row(2)+(2.d0*ix2=row(2)+(2.d0*i--1)*h11)*h1sum=sum+(2.d0*h1/3.d0)*deltaW(x1,row(2),t)sum=sum+(2.d0*h1/3.d0)*deltaW(x1,row(2),t)sum1=sum1+(4.d0*h1/3.d0)*deltaW(x2,row(2),t)sum1=sum1+(4.d0*h1/3.d0)*deltaW(x2,row(2),t)end doend dointegral=(h1/3.d0)*(deltaW(row(2),row(2),t)+deltaW(row(1),row(2)integral=(h1/3.d0)*(deltaW(row(2),row(2),t)+deltaW(row(1),row(2),t))+,t))+sum+sum1sum+sum1

( ) ( )[ ] ρρργ dwwcle

ve

p

pve∫ −=∞ 2

ρρρρ

ρ

dwwle

ve

ve∫ − ))()((

Droplet Density Profile Droplet Density Profile •• Second order differential EquationSecond order differential Equation

wherewhere

•• Boundary conditionsBoundary conditions

as and as and asas

( )02

22 12 μμρρμρ −=+→Δ=∇

Cdrd

rdrdC

0→drd ρ 0→r bρρ→ ∞→r

hdrdand

hdrd iiiii

22 11

211

2

2−+++ −

=+−

=ρρρρρρρ

Tridiagonal MatrixTridiagonal Matrix

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−−)(1)(000000

)1()1(1)1(000000000000000)()(1)(00000...000000.)3()3(1)3(0000.0)2()2(1)2(000.00)1()1(1

NbNAaNCcNbNAa

iCcibiAa

CcbAaCcbAa

Ccb

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

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⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

N

N

i

N

N

i

RRRR

RR

RRRRRR

1

3

2

1

1

3

2

1

.

.

.

.

ρρ

ρ

ρρρ

Tridiagonal MatrixTridiagonal Matrix

•• Construction of tridiagonal matrixConstruction of tridiagonal matrixdo do do i=2,stepsdo i=2,stepsAa(iAa(i)=i)=i--22B1(i)=B1(i)=--1*(i1*(i--1)*(2.d0+hv2*1)*(2.d0+hv2*dmew(U(i),t)/CValuedmew(U(i),t)/CValue))Cc(i)=iCc(i)=iRR(i)=(iRR(i)=(i--1)*hv2*1)*hv2*Gg(U(i),nb,t)/CValueGg(U(i),nb,t)/CValueend doend doAa(1)=0Aa(1)=0B1(1)=B1(1)=--(6.d0+hv2*dmew(U(1),t)/CValue)(6.d0+hv2*dmew(U(1),t)/CValue)Cc(1)=6.d0Cc(1)=6.d0Cc(steps)=0Cc(steps)=0RR(1)=hv2*Gg(U(1),nb,t)/CValueRR(1)=hv2*Gg(U(1),nb,t)/CValueRR(steps)=(stepsRR(steps)=(steps--1)*hv2*1)*hv2*Gg(U(steps),nb,t)/CValueGg(U(steps),nb,t)/CValue--(steps)*(steps)*nbnb

Flat Density ProfileFlat Density Profile•• Second order differential equationSecond order differential equation

•• First order First order differentioaldifferentioal EquationEquation

•• Calculations using Calculations using RungaRunga--KuttaKutta methodmethodx11=(row1+row2)/2.d0x11=(row1+row2)/2.d0o i=1.d0,500.d0o i=1.d0,500.d0dx1=((mew(x11,t)dx1=((mew(x11,t)--mew(row2,t))/dmew(x11,t))mew(row2,t))/dmew(x11,t))x11=x11x11=x11--dx1dx1if((dabs(dx1))<upsilon) exitif((dabs(dx1))<upsilon) exitend doend dointerval=35.dinterval=35.d--8/(step8/(step--1), mm=22.d1), mm=22.d--8/interval, 8/interval, ub(mmub(mm)= x11)= x11do i=(22.ddo i=(22.d--8/interval),1.d0,8/interval),1.d0,--1.d01.d0f11=interval*diff(Ub(i),row2,t), f12=interval*diff(Ub(i)+0.5d0*ff11=interval*diff(Ub(i),row2,t), f12=interval*diff(Ub(i)+0.5d0*f11,row2,t)11,row2,t)f13=interval*diff(ub(i)+0.5d0*f12,row2,t), f14=interval*diff(ub(f13=interval*diff(ub(i)+0.5d0*f12,row2,t), f14=interval*diff(ub(i)+f13,row2,t)i)+f13,row2,t)Ub(iUb(i) =Ub(i)+(f11+2.0d0*f12+2.0d0*f13+f14)/6.0d0 , ub(i) =Ub(i)+(f11+2.0d0*f12+2.0d0*f13+f14)/6.0d0 , ub(i--1)=1)=ub(iub(i))end doend doUb(mmUb(mm)=x11)=x11do i=(22.ddo i=(22.d--8/interval),step8/interval),stepf11=interval*diff(Ub(i),row2,t), f12=interval*diff(Ub(i)f11=interval*diff(Ub(i),row2,t), f12=interval*diff(Ub(i)--0.5d0*f11,row2,t) 0.5d0*f11,row2,t) f13=interval*diff(Ub(i)f13=interval*diff(Ub(i)--0.5d0*f12,row2,t), f14=interval*diff(Ub(i)0.5d0*f12,row2,t), f14=interval*diff(Ub(i)--f13,row2,t)f13,row2,t)Ub(iUb(i)=Ub(i))=Ub(i)--(f11+2.0d0*f12+2.0d0*f13+f14)/6.0d0 (f11+2.0d0*f12+2.0d0*f13+f14)/6.0d0 ub(i+1)=ub(i+1)=ub(iub(i))end doend do

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−=

Cw

dxd 2ρ

( ) ( )02

2

02

2 112 μμρμμρρ−=→−=+

Cdxd

cdrd

rdrd

Flat density profileFlat density profile

0 1 2 3 4

0.000

0.005

0.010

0.015

ρ(m

ol/c

m3 )

r(nm)

Tridiagonal MatrixTridiagonal Matrix

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−−)(1)(000000

)1()1(1)1(000000000000000)()(1)(00000...000000.)3()3(1)3(0000.0)2()2(1)2(000.00)1()1(1

NbNAaNCcNbNAa

iCcibiAa

CcbAaCcbAa

Ccb

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

N

N

i

N

N

i

RRRR

RR

RRRRRR

1

3

2

1

1

3

2

1

.

.

.

.

ρρ

ρ

ρρρ

Solution of Tridiagonal MatrixSolution of Tridiagonal Matrix

•• Lower Upper (LU) Decomposition methodLower Upper (LU) Decomposition methodn=stepsn=stepsbet=b1(1)bet=b1(1)u(1)=rr(1)/betu(1)=rr(1)/betdo j=2,ndo j=2,ngam(jgam(j)=cc(j)=cc(j--1)/bet1)/betbet=b1(j)bet=b1(j)--aa(j)*aa(j)*gam(jgam(j))u(j)=(u(j)=(rr(j)rr(j)--aa(jaa(j)*u(j)*u(j--1))/bet1))/betend doend dodo j=ndo j=n--1,1,1,1,--11u(j)=u(j)u(j)=u(j)--gam(j+1)*u(j+1)gam(j+1)*u(j+1)end do end do

The S value to start from!The S value to start from!

do i=1,stepsdo i=1,stepsU(i)=(U(i)=(U(i)U(i)--rhogrhog)*()*(nrefnref--nb)/(rholnb)/(rhol--rhog)+nbrhog)+nbend doend dodo do do i=2,stepsdo i=2,stepsAa(iAa(i)=i)=i--22B1(i)=B1(i)=--1*(i1*(i--1)*(2.d0+hv2*1)*(2.d0+hv2*dmew(U(i),t)/CValuedmew(U(i),t)/CValue))Cc(i)=iCc(i)=iRR(i)=(iRR(i)=(i--1)*hv2*1)*hv2*Gg(U(i),nb,t)/CValueGg(U(i),nb,t)/CValueend doend doAa(1)=0Aa(1)=0B1(1)=B1(1)=--(6.d0+hv2*dmew(U(1),t)/CValue)(6.d0+hv2*dmew(U(1),t)/CValue)Cc(1)=6.d0Cc(1)=6.d0Cc(steps)=0Cc(steps)=0RR(1)=hv2*Gg(U(1),nb,t)/CValueRR(1)=hv2*Gg(U(1),nb,t)/CValueRR(steps)=(stepsRR(steps)=(steps--1)*hv2*1)*hv2*Gg(U(steps),nb,t)/CValueGg(U(steps),nb,t)/CValue--(steps)*(steps)*nbnb

Comparison between flat density profile Comparison between flat density profile and density profile at S=1.65and density profile at S=1.65

0 1 2 3 4

0.000

0.005

0.010

0.015

Flat Density profile density profile at S=1.65

ρ(m

ol/c

m3 )

r(nm)

11

Density profiles from S=1.65 to S=6 at Density profiles from S=1.65 to S=6 at T=300KT=300K

filename1='Density_at_S_1_6_5_0.txt'filename1='Density_at_S_1_6_5_0.txt'do l1=1,6do l1=1,6do mm=0,9do mm=0,9do qq=0,8,9do qq=0,8,9do vv=0,8,9do vv=0,8,9sasa=l1+mm/10.0d0+qq/100.0d0 +vv/1000.0d0 , =l1+mm/10.0d0+qq/100.0d0 +vv/1000.0d0 , if(saif(sa<=1.650d0) cycle<=1.650d0) cyclenbnb==sasa**rhogrhogdo i=1,MAXdo i=1,MAXdnbdnb=(=(p(nb,t)p(nb,t)--sasa**p(rhog,t))/dp(nb,tp(rhog,t))/dp(nb,t), ), nbnb==nbnb--dnbdnb, , if((dabs(dnbif((dabs(dnb))<upsilon) exit))<upsilon) exitend doend doopen(13,file=filename1,status="old",action="open(13,file=filename1,status="old",action="read",iostatread",iostat==fstatfstat))do k=1,stepsdo k=1,stepsread(13,'(f18.16)') U(k)read(13,'(f18.16)') U(k)end doend dodo k=1,stepsdo k=1,stepsU(k)=U(k)+nbU(k)=U(k)+nb--nb1nb1end doend do

22

dododo i=1,stepsdo i=1,stepsAa(iAa(i)=i)=i--22B1(i)=B1(i)=--1.d0*(i1.d0*(i--1)*(2.d0+hv2*1)*(2.d0+hv2*dmew(U(i),t)/CValuedmew(U(i),t)/CValue))Cc(i)=iCc(i)=iRR(i)=(iRR(i)=(i--1)*hv2*1)*hv2*Gg(U(i),nb,t)/CValueGg(U(i),nb,t)/CValueend doend doAa(1)=0Aa(1)=0B1(1)=B1(1)=--(6.d0+hv2*dmew(U(1),t)/CValue)(6.d0+hv2*dmew(U(1),t)/CValue)Cc(1)=6.d0Cc(1)=6.d0Cc(steps)=0Cc(steps)=0RR(1)=hv2*Gg(U(1),nb,t)/CValueRR(1)=hv2*Gg(U(1),nb,t)/CValueRR(steps)=(stepsRR(steps)=(steps--1)*hv2*1)*hv2*Gg(U(steps),nb,t)/CValueGg(U(steps),nb,t)/CValue--(steps)*(steps)*nbnbend doend doend do end do end doend doend doend docall tridag_ser(aa,b1,cc,RR,X1,steps)call tridag_ser(aa,b1,cc,RR,X1,steps)error=0.d0 error=0.d0 do i=1,stepsdo i=1,stepserror=error+(X1(i)error=error+(X1(i)--U(i))**2U(i))**2end doend doerror=error=sqrt(dabs(errorsqrt(dabs(error))))U=X1U=X1if (error<Upsilon) exitif (error<Upsilon) exitend doend do

Results for density profiles with Results for density profiles with different supersaturation values.different supersaturation values.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.000

0.005

0.010

0.015

0.020

S=1.65 S=2 S=2.65 S=3 S=3.65

ρ(m

ol/c

m3 )

r(nm)

T=300K

11

Density profiles from T=300K to T=230K Density profiles from T=300K to T=230K at S=6at S=6

filename1='Density_at_S_6_0_0_0.txtfilename1='Density_at_S_6_0_0_0.txt''

do T=300.d0,230.d0,do T=300.d0,230.d0,--1.d01.d0if(T==300.d0)cycleif(T==300.d0)cycledodorow(1)=guess1row(1)=guess1row(2)=guess2row(2)=guess2k(1,1)=dp(row(1),t)k(1,1)=dp(row(1),t)k(1,2)=k(1,2)=--dp(row(2),t)dp(row(2),t)k(2,1)=dmew(row(1),t)k(2,1)=dmew(row(1),t)k(2,2)=k(2,2)=--dmew(row(2),t)dmew(row(2),t)f(1)=p(row(2),t)f(1)=p(row(2),t)--p(row(1),t) p(row(1),t) f(2)=mew(row(2),t)f(2)=mew(row(2),t)--mew(row(1),t)mew(row(1),t)z=k(2,1)/k(1,1)z=k(2,1)/k(1,1)k(2,1)=0.0d0k(2,1)=0.0d0f(2)=f(2)f(2)=f(2)--(z*f(1))(z*f(1))k(2,2)=k(2,2)k(2,2)=k(2,2)--(z*k(1,2))(z*k(1,2))

u1(2)=f(2)/k(2,2)u1(2)=f(2)/k(2,2)u1(1)=(f(1)u1(1)=(f(1)--k(1,2)*u1(2))/k(1,1)k(1,2)*u1(2))/k(1,1)row=row+u1row=row+u1error1=0.0d0error1=0.0d0do i=1,2do i=1,2error1=error1+f(i)**2error1=error1+f(i)**2end doend doerror1=dsqrt(error1)error1=dsqrt(error1)if (error1<Upsilon) exitif (error1<Upsilon) exitguess1=row(1)guess1=row(1)guess2=row(2)guess2=row(2)end doend do

Ts=TTs=T--273.15d0273.15d0gamagama = (23.88d0= (23.88d0--.08807d0*Ts)*1.d.08807d0*Ts)*1.d--77row1=row(1)row1=row(1)row2=row(2)row2=row(2)steps=4001steps=4001h1=(row1h1=(row1--row2)/stepsrow2)/steps

22

do k1=1,stepsdo k1=1,stepsread(13,'(f18.16)') U(k1)read(13,'(f18.16)') U(k1)end doend doclose(13)close(13)do k1=1,stepsdo k1=1,stepsU(k1)=U(k1)+nbU(k1)=U(k1)+nb--nb1nb1end doend docounter=0 counter=0 do sum=0.d0do sum=0.d0sum1=0.d0sum1=0.d0do i=1,(steps/2.d0)do i=1,(steps/2.d0)--11x1=row2+2.d0*i*h1x1=row2+2.d0*i*h1x2=row2+(2.d0*ix2=row2+(2.d0*i--1)*h11)*h1sum=sum+(2.d0*h1/3.d0)*deltaW(x1,rowsum=sum+(2.d0*h1/3.d0)*deltaW(x1,row

2,t)2,t)sum1=sum1+(4.d0*h1/3.d0)*deltaW(x2,rsum1=sum1+(4.d0*h1/3.d0)*deltaW(x2,r

ow2,t)ow2,t)end doend dointegral=(h1/3.d0)*(deltaW(row2,row2,t)integral=(h1/3.d0)*(deltaW(row2,row2,t)

+deltaW(row1,row2,t))+sum+sum1+deltaW(row1,row2,t))+sum+sum1cvaluecvalue=(=(gamagama/integral)**2/2.d0/integral)**2/2.d0rhogrhog=row(2)=row(2)rholrhol=row(1)=row(1)sasa=6.d0=6.d0

nbnb==sasa**rhogrhogdo i=1,MAXdo i=1,MAXdnbdnb=(=(p(nb,t)p(nb,t)--sasa**p(rhog,t))/dp(nb,tp(rhog,t))/dp(nb,t))nbnb==nbnb--dnbdnbif((dabs(dnbif((dabs(dnb))<upsilon) exit))<upsilon) exitprint*,print*,i,dnbi,dnbend doend dohvhv=(35.d=(35.d--8/(steps8/(steps--1))1))hv2=hv2=hvhv**2**2nb1=nb1=nbnbopen(13,file=filename1,status="old",action="open(13,file=filename1,status="old",action="rere

ad",iostatad",iostat==fstatfstatread(13,'(I7)') stepsread(13,'(I7)') steps

counter=counter+1counter=counter+1do i=2,stepsdo i=2,stepsAa(iAa(i)=i)=i--22

B1(i)=B1(i)=--1.d0*(i1.d0*(i--1)*(2.d0+hv2*1)*(2.d0+hv2*dmew(U(i),t)/CValuedmew(U(i),t)/CValue))

Cc(i)=iCc(i)=iRR(i)=(iRR(i)=(i--1)*hv2*1)*hv2*Gg(U(i),nb,t)/CValueGg(U(i),nb,t)/CValue

33

end doend doAa(1)=0Aa(1)=0B1(1)=B1(1)=--(6.d0+hv2*dmew(U(1),t)/CValue)(6.d0+hv2*dmew(U(1),t)/CValue)Cc(1)=6.d0Cc(1)=6.d0Cc(steps)=0.d0Cc(steps)=0.d0RR(1)=hv2*Gg(U(1),nb,t)/CValueRR(1)=hv2*Gg(U(1),nb,t)/CValueRR(steps)=(stepsRR(steps)=(steps--1)*hv2*1)*hv2*Gg(U(steps),nb,t)/CValueGg(U(steps),nb,t)/CValue--(steps)*(steps)*nbnbcall tridag_ser(aa,b1,cc,RR,X11,steps)call tridag_ser(aa,b1,cc,RR,X11,steps)errors=0.d0 errors=0.d0 do i=1,stepsdo i=1,stepserror=error+(X11(i)error=error+(X11(i)--U(i))**2U(i))**2end doend doerror=error=sqrt(errorsqrt(error)/steps)/stepsU=X1U=X1if (errors<upsilon) exitif (errors<upsilon) exitend doend docount1=int((T+.1d0)/100.d0)count1=int((T+.1d0)/100.d0)count2=int((T+.1d0count2=int((T+.1d0--count1*100)/10)count1*100)/10)count3=int(T+.1d0count3=int(T+.1d0--count1*100count1*100--count2*10)count2*10)count4=int((Tcount4=int((T--int(T+.1dint(T+.1d--4))*10+.1d4))*10+.1d--2)2)count5=int((Tcount5=int((T--int(T+.1dint(T+.1d--4))*1004))*100--count4*10+.1dcount4*10+.1d--2)2)

filename2='Density_at_T_'//achar(count1+48)//achar(count2+48)//afilename2='Density_at_T_'//achar(count1+48)//achar(count2+48)//achar(count3+48)//achar(count4+48)//acchar(count3+48)//achar(count4+48)//achar(count5+48)//'.txt' har(count5+48)//'.txt'

open(12,file=filename2,status="replace",action="write",position=open(12,file=filename2,status="replace",action="write",position=""rewind",iostatrewind",iostat==gstatgstat))do k1=1,stepsdo k1=1,stepswrite(12,'(f18.16)') U(k1)write(12,'(f18.16)') U(k1)end doend do

close(12)close(12)end doend do

Results for density profiles with Results for density profiles with different temperatures.different temperatures.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.000

0.005

0.010

0.015

0.020S=6

T=300 T=290 T=280 T=270

ρ(m

ol/c

m3 )

r(nm)

11

Density profiles from S=6 to S=2 at T=293KDensity profiles from S=6 to S=2 at T=293K

T=293.d0T=293.d0Ts=TTs=T--273.15d0273.15d0gamagama = (23.88d0= (23.88d0--.08807d0*Ts)*1.d.08807d0*Ts)*1.d--77cvaluecvalue=5.974465658636744d=5.974465658636744d--1010sasa=6.d0 =6.d0 rhogrhog=3.656930116927259d=3.656930116927259d--66rholrhol=1.711839175934329d=1.711839175934329d--22nbnb==sasa**rhogrhogdo i=1,MAXdo i=1,MAXdnbdnb=(=(p(nb,t)p(nb,t)--sasa**p(rhog,t))/dp(nb,tp(rhog,t))/dp(nb,t))nbnb==nbnb--dnbdnbif((dabs(dnbif((dabs(dnb))<upsilon) exit))<upsilon) exitend doend dohvhv=(35.d=(35.d--8/(steps8/(steps--1))1))hv2=hv2=hvhv**2**2nb1=nb1=nbnbfilename1='Density_at_T_2_9_3.txt'filename1='Density_at_T_2_9_3.txt'do do sasa=6.0d0,2.0d0,=6.0d0,2.0d0,--.01d0.01d0llll=int(sa/10.d0)=int(sa/10.d0)mm=int(sa+.001d0mm=int(sa+.001d0--ll*10.d0)ll*10.d0)qqqq=int((sa+.001d0=int((sa+.001d0--ll*10.d0ll*10.d0--mm)*10.d0)mm)*10.d0)vv=int((sa+.001d0vv=int((sa+.001d0--ll*10.d0ll*10.d0--mmmm--qq/10.d0)*100.d0)qq/10.d0)*100.d0)nbnb==sasa**rhogrhogdo i=1,MAXdo i=1,MAXdnbdnb=(=(p(nb,t)p(nb,t)--sasa**p(rhog,t))/dp(nb,tp(rhog,t))/dp(nb,t))nbnb==nbnb--dnbdnb

22

B1(1)=B1(1)=--(6.d0+hv2*dmew(U(1),t)/CValue)(6.d0+hv2*dmew(U(1),t)/CValue)Cc(1)=6.d0Cc(1)=6.d0Cc(steps)=0Cc(steps)=0RR(1)=hv2*Gg(U(1),nb,t)/CValueRR(1)=hv2*Gg(U(1),nb,t)/CValueRR(steps)=(stepsRR(steps)=(steps--

1)*hv2*1)*hv2*Gg(U(steps),nb,t)/CValueGg(U(steps),nb,t)/CValue--(steps)*(steps)*nbnb

call tridag_ser(aa,b1,cc,RR,X1,steps)call tridag_ser(aa,b1,cc,RR,X1,steps)error=0.d0 error=0.d0 do i=1,stepsdo i=1,stepserror=error+(X1(i)error=error+(X1(i)--U(i))**2U(i))**2end doend doerror=error=sqrt(errorsqrt(error))U=X1U=X1if (error<epsilon) exitif (error<epsilon) exitend doend doopen(13,file=filename2,status="replace",actiopen(13,file=filename2,status="replace",acti

on="write",position="on="write",position="rewind",iostatrewind",iostat==gstagstatt))

do k=1,stepsdo k=1,stepswrite(13,'(f18.16)') U(k)write(13,'(f18.16)') U(k)end doend doclose(13)close(13)filename1=filename2filename1=filename2nb1=nb1=nbnbend doend doif((dabs(dnbif((dabs(dnb))<upsilon) exit))<upsilon) exitend doend do

if(saif(sa<=1.99d0) cycle<=1.99d0) cycleif(saif(sa>=6.0d0) cycle>=6.0d0) cycleopen(12,file=filename1,status="old",actionopen(12,file=filename1,status="old",action

="="read",iostatread",iostat==fstatfstat))if(fstatif(fstat==0) print*,trim(filename1),' ==0) print*,trim(filename1),'

opened'opened'read(12,'(I7)') stepsread(12,'(I7)') stepsdo k=1,stepsdo k=1,stepsread(12,'(f18.16)') U(k)read(12,'(f18.16)') U(k)end doend doclose(12)close(12)

do k=1,stepsdo k=1,stepsU(k)=U(k)+nbU(k)=U(k)+nb--nb1nb1end doend docounter=0 counter=0 do do counter=counter+1counter=counter+1do i=1,stepsdo i=1,stepsAa(iAa(i)=i)=i--22

B1(i)=B1(i)=--1.d0*(i1.d0*(i--1)*(2.d0+hv2*1)*(2.d0+hv2*dmew(U(i),t)/CValuedmew(U(i),t)/CValue))

Cc(i)=iCc(i)=iRR(i)=(iRR(i)=(i--1)*hv2*1)*hv2*Gg(U(i),nb,t)/CValueGg(U(i),nb,t)/CValueend doend doAa(1)=0Aa(1)=0

Results for density profiles with Results for density profiles with different supersaturation values at different supersaturation values at T=293KT=293K

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.000

0.005

0.010

0.015

0.020T=293K

S=3 S=4 S=5 S=6

ρ(m

ol/c

m3 )

r(nm)

RESULTSRESULTS

Work of formationWork of formation•• Integral EquationIntegral Equation

•• CalculationsCalculationsresults1=0.0results1=0.0do i=4,stepsdo i=4,steps--33results1=results1+(w0(U(i),nb,t)results1=results1+(w0(U(i),nb,t)--w0(nb,nb,t))*hv2*((i1)**2)+w0(nb,nb,t))*hv2*((i1)**2)+((((cValuecValue*dabs((U(i+1)*dabs((U(i+1)--U(iU(i--1)))**2)/8.0d0)*((i1)))**2)/8.0d0)*((i--1)**2)1)**2)end doend doresults1=results1+bb*((w0(U(2),nb,t)results1=results1+bb*((w0(U(2),nb,t)--w0(nb,nb,t))*hv2*((2w0(nb,nb,t))*hv2*((2--1)**2)+1)**2)+((((cValuecValue*dabs((U(3)*dabs((U(3)--U(1)))**2)/8.0d0)*((2U(1)))**2)/8.0d0)*((2--1)**2))1)**2))results1=results1+cc*((w0(U(3),nb,t)results1=results1+cc*((w0(U(3),nb,t)--w0(nb,nb,t))*hv2*((3w0(nb,nb,t))*hv2*((3--1)**2)+1)**2)+((((cValuecValue*dabs((U(4)*dabs((U(4)--U(2)))**2)/8.0d0)*((3U(2)))**2)/8.0d0)*((3--1)**2))1)**2))results1=results1+aa*((w0(U(steps),nb,t)results1=results1+aa*((w0(U(steps),nb,t)--w0(nb,nb,t))*hv2*((stepsw0(nb,nb,t))*hv2*((steps--1)**2)+1)**2)+((((cValuecValue*dabs((nb*dabs((nb--U(stepsU(steps--1)))**2)/8.0d0)*((steps1)))**2)/8.0d0)*((steps--1)**2))1)**2))results1=results1+bb*((w0(U(stepsresults1=results1+bb*((w0(U(steps--1),nb,t)1),nb,t)--w0(nb,nb,t))*hv2*((stepsw0(nb,nb,t))*hv2*((steps--2)**2)+2)**2)+((((cValuecValue*dabs((U(steps)*dabs((U(steps)--U(stepsU(steps--2)))**2)/8.0d0)*((steps2)))**2)/8.0d0)*((steps--2)**2))2)**2))results1=results1+cc*((w0(U(stepsresults1=results1+cc*((w0(U(steps--2),nb,t)2),nb,t)--w0(nb,nb,t))*hv2*((stepsw0(nb,nb,t))*hv2*((steps--3)**2))+3)**2))+((((cValuecValue*dabs((U(steps*dabs((U(steps--1)1)--U(stepsU(steps--3)))**2)/8.0d0)*((steps3)))**2)/8.0d0)*((steps--3)**2))3)**2))GW1=4.d0*Pi*GW1=4.d0*Pi*hvhv*results1/(kb*T)*results1/(kb*T)

( ) dVcwW ∫ ⎥⎦⎤

⎢⎣⎡ ∇+Δ= 2

2ρ

Results for work of formationResults for work of formation

2.4 2.6 2.8

25

30

35

40

45

GT-form S-form P-form

W* /K

T

S

Ethanol

T=293

Nucleation RateNucleation Rate

•• EquationEquation

WhereWhere

•• CalculationCalculation

GJ1=dsqrt(2.d4*GJ1=dsqrt(2.d4*gamagama/(PI*ma))*(/(PI*ma))*(sasa**P(rhol,tP(rhol,t)/(kb*T))**2/rhom*)/(kb*T))**2/rhom*1.d0*dexp(1.d0*dexp(--GW1)GW1)

( )[ ]TKrWJJ B/*exp0 −=

( )20 //2 TkpvmJ Bvlπγ ∞=

Comparison of the experimental ratesComparison of the experimental rates(open circles) for methanol with two versions of (open circles) for methanol with two versions of CNT and GT based on the SAFT EOS.CNT and GT based on the SAFT EOS.

2.5 3.0 3.5100000

1000000

1E7

1E8

1E9

1E10

1E11

GT/106

P-form/107

S-form/1010

o Expt

J (c

m-3s-1

)

S

257K272K

Methanol

Comparison of the experimental rates Comparison of the experimental rates (open circles) for ethanol with two versions (open circles) for ethanol with two versions

of CNT, and GT based on the SAFT EOS.of CNT, and GT based on the SAFT EOS.

2.5 3.0100000

1E8

1E11

Ethanol

GT/104

P-form/106

S-form/107

expt

J (c

m-3s-1

)

S

286K293K

Number of molecules in critical nucleusNumber of molecules in critical nucleus

•• Integral EquationIntegral Equation

•• CalculationsCalculationshw=35.d0/(stepshw=35.d0/(steps--1)1)results3=0.0results3=0.0do i=4,stepsdo i=4,steps--33results3=results3+(U(i)results3=results3+(U(i)--nb)*(inb)*(i--1)**2*hw**21)**2*hw**2end doend doresults3=results3+bb*(U(2)results3=results3+bb*(U(2)--nb)*(2nb)*(2--1)**2*hw**21)**2*hw**2results3=results3+cc*(U(3)results3=results3+cc*(U(3)--nb)*(3nb)*(3--1)**2*hw**21)**2*hw**2results3=results3+aa*(results3=results3+aa*(U(steps)U(steps)--nbnb)*(steps)*(steps--1)**2*hw**21)**2*hw**2results3=results3+cc*(U(stepsresults3=results3+cc*(U(steps--1)1)--nb)*(stepsnb)*(steps--2)**2*hw**22)**2*hw**2results3=results3+bb*(U(stepsresults3=results3+bb*(U(steps--2)2)--nb)*(stepsnb)*(steps--3)**2*hw**23)**2*hw**2

ngtngt=4*pi*results3*hw=4*pi*results3*hw

( ) ( )[ ] drrrrn b2

0

* 4 ∫∞

−= ρρπ

The number of methanol molecules in the The number of methanol molecules in the critical nucleus.critical nucleus.

20 30 40 50 60 7020

30

40

50

60

70

Expt P GT Gibbs

n*

n* Gibbs-Thomson

The number of ethanol molecules in The number of ethanol molecules in the critical nucleusthe critical nucleus

30 40 50 60 7030

35

40

45

50

55

60

65

70

Ethanol

Gibbs Expt P GT

n*

n* Gibbs-Thomson

ConclusionsConclusions::

•• GT improved temperature dependence of GT improved temperature dependence of nucleation rates for both ethanol and nucleation rates for both ethanol and methanol.methanol.

•• GT improved supersaturation dependence GT improved supersaturation dependence of nucleation rates exactly for methanol, of nucleation rates exactly for methanol, but for ethanol, GT couldnbut for ethanol, GT couldn’’t improve thet improve thesupersaturation dependence of nucleation supersaturation dependence of nucleation rates. rates.

Thank youThank you