Patterning Soft Materials at the Nanoscaleerjank/Lectures/557/557-Lecture9-TDGL.pdf ·...

ChE/MSE 557 Lecture 9 Fall 2006 1Computational Nanoscience of Soft [email protected]

Patterning Soft Materials at theNanoscale:

Phase-field methods

Recommended reading:• S.C. Glotzer and W. Paul, Annual Reviews of MaterialsResearch 32 (401-436), 2002 and references therein(e.g. Castellano and Glotzer, Lee, Douglas and Glotzer,Glotzer, DiMarzio and Muthukumar)• Larson book, pp389

Lecture 9


Phase Field Methods

• Phase field methods are continuum methods.• Partial differential equations solved by finite

difference methods.• Key variable is one or more fields that depend on

space and time (like a concentration field ordensity field).

• Typically based on a free energy formalismwhere free energy is a function of the fieldvariable and where free energy is minimized tofind equilibrium values of the field.

• First phase field method: TDGL/Cahn-Hilliard.


Phase separation and nanotechnology

• We have seen that patterning materials on the nanoscalecan often be achieved by exploiting the process of phaseseparation.– Templating surfaces for positioning of

nanodevices.– Creating nanostructured

surfaces for catalysis.– Creating photonic crystals.– Manipulating arrays

of nanoparticles intonanowires, etc.

– Producing layerednanostructures ofvarious shapes andsizes.


Phase Separation (“Unmixing”)

• When “quenched” to the immiscible region of the phasediagram, a mixture will separate into two coexistingphases with equilibrium concentrations given by the valuesof φ1 and φ2 on the binodal.

• The dynamics of the separation process is controlled by– The final quench temperature Tf– the overall composition φ– the rate of the quench dT/dt– the viscous or viscoelastic properties of the phases

formed– and the interfacial tension Γ between the two phases


Phase Separation Mechanisms

• Nucleation and growth– Metastable - stable with respect to small amplitude

variations in composition, but unstable with respect tolarge variations.

– Mixture will phase separate via nucleation and growth.– Phase separation occurs only after a nucleus of one

phase forms that is large enough to grow.

• Spinodal decomposition– Unstable with respect to small amplitude variations in

composition.– Mixture will phase separate via spinodal decomposition.– Phase separation begins immediately if variations are

characterized by a long enough wavelength.



f(φ)

φφ1 φs1 φs2 φ2

T > Tc

T < Tc

single well

double well

In metastable region, ∂2f/ ∂φ2 > 0. Thus, if a small patchof homogeneous fluidat A suffers a weak compositionalfluctuation, therebyproducing two patcheswith compositionscorresponding to Band C, then the avefree energy willincrease, which is unfavorable, so fluidwill remix.

ABC



f(φ)

φφ1 φs1 φs2 φ2

T > Tc

T < Tc

single well

double well

In unstable region, ∂2f/ ∂φ2 < 0. Thus, if a small patchof homogeneous fluidat D suffers a weak compositionalfluctuation, therebyproducing two patcheswith compositionscorresponding to Eand F, then the avefree energy willdecrease, which is favorable. Thus weakfluctuations growspontaneously.

D

EF


The Kinetics of Phase Separation bySpinodal Decomposition

• Fick’s first law relating flux to a concentration gradient:

• Continuity equation:

• Fick’s second law = Fick’s first law + continuity:

!

"#i(x,t)

"t= $% & j

i(x,t)!

ji(x,t) = "D

i#$

i

!

"#i(x,t)

"t= D

i$2#

i

Recall Fick’s Laws


The Cahn-Hilliard Cook Model

• The CHC model is a non-linear “uphill” diffusion or massbalance equation for an immiscible mixture involving thefree energy of the mixture.

• Main steps in derivation:1. Generalized transport equation relating flux (= mass

current density) to gradient in chemical potential foreach species.

2. Continuity equation relating the spatial and timedependence of the concentration of each componentin the mixture to the flux that expresses theconservation of mass.

3. Combine the two.


The Cahn-Hilliard Cook Model

• The mass current is related to the chemical potential µkthrough:

where Mik is the mobility of species i due to k and jT(r,t) isthe mass current arising from thermal noise.

• Conservation of mass requires:

!

"#i(x,t)

"t= $% & j

i(x,t)

!

ji(x,t) = " M

ik#µ

k

k

n"1

$ + jT(x,t)


The Cahn-Hilliard-Cook Model

• The chemical potential µk is related thermodynamically tothe free energy functional F{φi(x,t), φj(x,t), φk(x,t), …,φn-1(x,t)} by:

where n is the total number of species and the derivativeis taken subject to the constraint φi + φj + φk + … φn = 1.

• For an incompressible, binary A/B mixture where φA + φB =1, there is only one independent concentration φ, oneindependent chemical potential µ, and one independentcontinuity equation.

!

µi="F

"#i



• Thus for an incompressible, binary mixture, wehave:

!

"#(x,t)

"t= $% & j(x,t)

!

j(x,t) = "M#µ + jT(x,t)

!

µ ="F

"#

(1)

(2)

(3)



• Combining the three equations gives:

which is usually written as:!

"#(x,t)

"t= $% & $M%

'F

'#+ j

T(x,t)

(

) *

+

, -

!

"#(x,t)

"t=$ % M$

&F

&#+ ' (x,t)

µ

j

Cook term



• The noise term satisfies the conditions

where the average is performed over the thermalfluctuations in the system.

!

" (x,t) = 0

and

" (x,t)" (x',t') = #2MkBT$

2%(x # x')%(t # t ')



• If the mobility M is independent of concentration φ, then:

• If the free energy functional takes a particular form, thisequation is known as the Cahn-Hilliard-Cook equation.

• Often, the noise term is neglected, since it does not affectscaling behavior of domain growth, in which case theequation is called the Cahn-Hilliard equation.

!

"#(x,t)

"t= M$2 %F

%#+ & (x,t)


Form of CH Free Energy Functional

• The Cahn-Hilliard free energy functional originally proposedfor binary mixtures has the following form:

!

FCH ["(x)]

kBT= dx

f "( )kBT

+#

2$"(x)

2%

& '

(

) * +

Bulk free energy of mixing

“Square gradient term”: describes the energy penalty for interfaces between A-rich and B-rich domains.


Form of CH Free Energy Functional

• For small molecule mixtures, the interfacial energyparameter κ arises due to increases in potential energy atan interface and is related to the interaction energies byκ = χλ2, where λ is the effective interaction distancebetween molecules, and

• For polymer blends, κ has an additional contribution due toentropy penalties at the interface.

!

" =z

kBT#AB$1

2#AA

+ #BB( )

%

& '

(

) *


Landau Theory: The TDGL Model

• Recall, the Helmholtz free energy of mixing of anincompressible binary mixture is f = U - TS:

!

f (")

kBT= " ln" + (1#")ln(1#") + $"(1#")

where

$ =z

kBT%AB #

1

2%AA + %BB( )

&

' (

)

* +


Taylor Expansion of Bulk Free Energy

• Close to φc = 1/2, the free energy expression can beapproximated via a Taylor expansion by the form:

• Note that this form (with only even terms) holds only forsymmetric mixtures.

• With higher order terms neglected, this “ψ2,ψ4” form ofthe free energy is called the Landau free energy.!

f (")

kBT# $

r

2" 2

+g

4" 4

+ ...

where " % &A $&B = & $ (1$&) = 2& -1,

r = 1

2(' $ 'c ) and g =

1

3

ψ is the “orderparameter”


Landau Free Energy

• Note that when χ - χc changes sign, f(ψ) changes form:

• Thus the form of f will be different below and above χc.

!

f (")

kBT# $

r

2" 2

+g

4" 4

+ ...

where " % &A $&B = & $ (1$&) = 2& -1,

r = 1

2(' $ 'c ) and g =

1

3


Free Energy

f(ψ)

ψψ1ψs1 ψs2ψ2

χ < χc

single well

double well

ψ1, ψ2 defined by:

ψs1, ψs2 defined by:∂2f/ ∂ ψ2 = 0

For every χ > χc, there exists uniquevalues of ψ1, ψ2,

ψs1, and ψs2.

χ > χc!

"f

"#

$

% &

'

( ) #1

="f

"#

$

% &

'

( ) #2

This schematicdepicts the free energyfor a symmetricmixture.


Free Energy

f(ψ)

ψψ1 ψs1 ψs2 ψ2

χ < χc

single well

double well

ψ1, ψ2 defined by:

ψs1, ψs2 defined by:∂2f/ ∂ ψ2 = 0

For every χ > χc, there exists uniquevalues of ψ1, ψ2,

ψs1, and ψs2.

χ > χc!

"f

"#

$

% &

'

( ) #1

="f

"#

$

% &

'

( ) #2

This schematicdepicts the free energyfor an asymmetricmixture.


Example of Asymmetric Free Energy

!

f (")

kBT="

NA

ln" +(1#")

NB

ln(1#") + $"(1#")

!

"c

=N

B

NA

+ NB( )

#c

=N

A+ N

B( )2

2NAN

B


Phase Diagram

T~1/χone phase(miscible)

For each χ > χc, plot ψ1, ψ2,

ψs1, and ψs2.

This defines boundaries between (i) a miscible and immiscible regionand (ii) a metastable and unstable immiscible region.

two-phase(immiscible)

binodal or coexistence curve

spinodal

unstable metastable

UCST

ψψ1ψs1 ψs2ψ2


Time-dependent Ginzburg-Landau Equation

• Recall, the CHC equation is:

• The TDGL equation is the CH (or CHC) equation with theGinzburg-Landau form of the free energy functional:

!

"#(x,t)

"t= M$2 %FCH

%#+ & (x,t)

!

FCH ["(x)]

kBT= dx

f "( )kBT

+#

2$"(x)

2%

& '

(

) * +

!

FGL["(x)]

kBT= dx #

r

2" 2

+g

4" 4

+$

2%"

2&

' ( )

* + ,


TDGL

• The functional derivative δF/δψ is given by:

• Then, the TDGL equation becomes

!

"#(x,t)

"t= M $r%2# + g%2# 3 $&%4#( ) + ' (x,t)

!

"FGL[#(x)]

"#= $r# + g# 3 $%&2#

Usually setto zero


Dimensionless TDGL

• Through the following change of variables:

it’s possible to rewrite the TDGL equation in adimensionless form:

!

"#(x,t)

"t= $%2# +%2# 3 $%4# + &'(x,t)

Depth of quench controlled entirely by noise term.

!

x" r #( )12 x

t" Mr2 #( )t

$ " g r( )12$


Dimensionless TDGL

• In the dimensionless equation, the noise term satisfies thefollowing conditions:

• Then the only parameters left to specify the dynamics are theconserved average concentration ψ0 = <φ> - φc and thedimensionless noise strength parameter ε:

!

"(x,t) = 0

and

"(x,t)"(x',t') = #$2%(x # x')%(t # t ')

!

" = 2g # d 2r4$d( ) 2( )

χ and χc arecontained in κ and r!

ε usually setto zero.


Three Dynamical Regimes ofPhase Separation

• Early-time (linear) regime– Immediately following a quench to the unstable, two-phase

region– Concentration fluctuations of wavelength λ grow or decay

exponentially in time at constant λ.

• Intermediate-time (“late”-time) regime– When average domain size is larger than the interfacial width– Domains coarsen and average size grows as a power law in

time

• Very late time (hydrodynamic) regime (fluids only)– At very late times hydrodynamic interactions dominate the

coarsening process– Domain size grows linearly with time


Dimensionless TDGL

• From the dimensionless TDGL, studies have shown there is onlyone “fixed point” for this model at T=0.

• That is, all scaling behavior describing how the domains grow intime is T-independent (χ-independent).

• The parameters that were scaled out of this equation describethe speed at which the process will occur at a given T (or χ), butthis affects only amplitudes in front of scaling laws.

!

R" t1/ 3

True both in d=2 and d=3.

!

R = A(",M,#)t1/ 3


Solving the TDGL Equation

• Often, the noise term is neglected in studies of SD, sothat the TDGL is:

• This equation follows the time evolution of the orderparameter everywhere in space following a quench fromthe one-phase to the two-phase region.

• To solve it numerically, we must turn the differentialequation into a finite difference equation, and numericallyintegrate the resulting equation of motion for ψ.

!

"#(x,t)

"t= $%2# +%2# 3 $%4#


Finite Difference Equation for TDGL

• First, space is discretized by replacing the continuousspace of position vectors x by a simple square (2d) orcubic (3d) grid with L2 or L3 sites, respectively, and meshsize δx. We’ll consider 2-d for simplicity; generalization to3-d is straightforward.

• The order parameter field (or “composition difference”field) ψ(x,t) then becomes ψij

m at the mth time step at gridpoint i,j.

• Time is discretized by replacing the continuous timevariable t by a series of m discrete time steps of durationδt.



• The TDGL may be numerically integrated using a first-order Euler finite difference scheme:

!

"ij

m+1 ="ij

m + #t$" ij

m

$t

"ij

m+1 ="ij

m + #t %&2" ij

m +&2 " ij

m( )3

%&4" ij

m( )

Now must discretize these spatial derivatives



• There are several ways of differencing a derivative. Weuse central differences:

!

"2# x,t( )$#i+1, j

m % 2# i, j

m +#i%1, j

m

&x( )2

+# i, j+1

m % 2#i, j

m +# i, j%1

m

&x( )2

i,j i+1,ji-1,j

i,j+1

i,j-1



• For notational convenience, define the following symbolfor any function fij:

• Then the finite difference equation is finally:

!

fij "nn

# f i+1, j + f i$1, j + f i, j+1 + fi, j$1 $ 4 f i, j

!

"ij

m+1 ="ij

m +#t

#x( )2

$" ij

m + " ij

m( )3

$1

#x( )2

" ij

m

nn

%&

' ( (

)

* + +

nn

%


Choosing the time step and mesh size

• When we discretize any differential equation, wechange the model.

• The solution to the discretized finite differenceequation converges to the solution of thedifferential equation only in the limit of δt and δxgoing to zero.

• As with any method, the solution must beindependent of the choice of δt and δx.


Choosing the mesh size and time step

Morphology Evolution

t = 2000dt t = 8000dt t = 40,000dt


Choosing the time step and mesh size

• Choice of time step δt– Too large a time step can cause instabilities and spurious

solutions.– The time step is limited by the mesh size.

• Choice of mesh size δx– For the solution to any finite difference equation to converge

to the true solution of the differential equation, δx must besmaller than the smallest length being resolved in thesystem. For this model, the smallest length is the interfacialwidth, which decreases in time until the latest stages ofunmixing.


Choosing the mesh size

N = 32, dx = 1, t = 100 N = 32, dx = 1, t = 200

Too large a mesh size causes unphysical “pinning”

Here, interface width is equal to the mesh size (pixel size).



Reducing the mesh size below the interfacial width allows demixing

N = 32, dx = 1, t = 200 N = 32, dx = 0.5, t = 200Here, mesh size was decreased by half to be smaller than

the interface width.



Reducing the mesh size below the interfacial width allows demixing

Mesh size of system at right was decreased to half its valuein the system on left, to be smaller than the interface width.

Δx = 1.0 , t = 15,000 Δt Δx = 2.0, t = 15,000 Δt


Why does pinning occur from a too-largemesh size?

Consider the interfacial profile in 1-d.ψij goes from one bulk value, ψ1, at xij - δx to the other bulk value ψ2 at xij + δx across an interface.

For R to increase, the interface mustmove, which means ψ ij must change fromψ1 to ψ2. The driving force for this is the interfacial (square gradient) termin F, which must overcome the barrierbetween the wells in the double-wellbulk free energy for φij to change.

!

1

"x( )2# i+1, j

m +#i$1, j

m $ 2# i, j

m( )

ψ1 ψ2


Why does pinning occur from a too-largemesh size?

But, numerator in Laplacian term is bounded above by ψ1 + ψ2 - 2 ψij = constant, and thus increasing δx reduces the Laplacian while the localterm remains unchanged.

Thus, for δx sufficiently large, the interfacial term cannot overcome thebulk term, and pinning or unphysicalslowing of the growth occurs.

!

1

"x( )2# i+1, j

m +#i$1, j

m $ 2# i, j

m( )

ψ1 ψ2


Analyzing Microstructure Development

• If left to its own devices, an unstable mixture that isdemixing via spinodal decomposition will eventually phaseseparate to two bulk coexisting phases with uninterestingmorphologies.

• Along the way, extremely interesting morphologies(microstructures) can arise. These morphologies can betrapped in by quenching and other schemes and used fordifferent purposes important for nanotechnology.

• Two methods are typically used to monitor morphology:– Light or neutron scattering– Laser scanning confocal microscopy


Experimental Microstructures from SD

Laser scanning confocal microscopy of PS-PB/PB blend

Hashimoto’s group, Kyoto University


Monitoring Growth Kinetics: The StructureFactor

• The static structure factor S(q) is given by the Fouriertransform of the pair correlation function g(r).

• S(q) is obtained from light, x-ray and neutron scatteringexperiments and provides a convenient way of monitoringthe growth kinetics following a quench to the unstabletwo-phase region.– Early-time growth of concentration fluctuations– Intermediate and late-stage scaling behavior of the domain

size

• S(q) is easily calculated in simulations.


Monitoring Growth Kinetics:The Structure Factor

• On a discrete grid, the structure factor at a given time t, S(q;t),is given by:

where the sum is performed over all Ld lattice sites.

• The wavevectors q = (2π/Lδx)n, and |n| = 1, 2, …, L/2.• <…> denotes an average over all possible origins, and over an

ensemble of many statistically independent configurations.!

S(q;t) = "(q;t)"(q';t) =1

L2

e# iq $x

x'

% G(x;t)x

%

=1

L2

e# iq $x " x + x';t)" x';t( )( ) # "

2[ ]x'

%x

%



• When the morphology of the mixture or blend is isotropic,S(q;t) can be further smoothed by spherical averaging --averaging over all wavevectors between q and q + δq.This amounts to averaging over an entire spherical shell inq-space to obtain S(q;t):

• The denominator is the number of grid points in aspherical shell of width δq centered around q. Ideally, δqshould be taken as small as possible, e.g. δq = 2π/Lδx.

!

S(q;t) =S(q;t)

q"#q / 2( )< q < q+#q / 2( )$

1q"#q / 2( )< q < q+#q / 2( )

$

Before averaging


Monitoring Growth Kinetics: The StructureFactor

Evolution of lightscattering patternfollowing a quenchfrom 75C to 25C.

System: Nearlysymmetriccritical mixture(φc = 0.486) ofdeuterated andprotonated1,4-polybutadiene.UCST Tc = 62 C

Bates and Wiltzius, JCP 91, 3258 (1989)



S(q) at various times t followinga quench from 75C to 25C.

System: Nearly symmetric critical mixture (φc = 0.486) of deuterated and protonated 1,4-polybutadiene.UCST Tc = 62 C




• A key quantity used to characterize the evolvingmorphology of a phase separating mixture is thecharacteristic size of the domains, R(t).

• In experiments, the peak wavevector qm is often used as ameasure of the characteristic inverse domain size R-1(t).

• In simulation, the discreteness of the grid used tointegrate the CH equation make the determination of qmdifficult. Instead, one can calculate the first moment q1 ofS(q;t) given by

!

q1t( ) =

q2S(q;t)

q

"

qS(q;t)q

"



• Because this measure of the characteristic wavevectoruses data acquired over the entire range of q, it is acleaner, more accurate measure than the peak position.

• Of course, the two lengths are not the same, but if alllengths scale the same with time, then we can use anydefinition.– Early time: exponential growth of S(q) at constant q. Thus all

lengths equivalent.– Intermediate/late time: power law growth of R(t); self-similar

patterns, all lengths scale the same.

!

q1t( ) =

q2S(q;t)

q

"

qS(q;t)q

"


Monitoring Growth Kinetics:Pair Correlation Function

• Another approach to calculating the time dependence ofthe characteristic domain size is to calculate the first zeroof the real-space, composition-composition correlationfunction G(x), which is the Fourier transform of S(q):

• G(x) is then spherically averaged to obtain G(x), and thennormalized to unity at x=0:

!

G(x) = eiqx

q

" S(q)

!

g(x;t) =G(x;t)

G(0;t)


Monitoring Growth Kinetics:Total Area Method

• A cheap way to estimate a characteristic domain size isfrom

• Here “area of interface” = sum of all transitions frompositive to negative ψ in all spatial directions.

!

R =Ld

area of interface

+ ++ ++ -

+ ++ ++ +

+ -+ -+ +

- -- -+ +

+ ++ ++ ++ +- ++ +

Here R = 36/14


Linear Theory of Spinodal Decomposition

• Early stage kinetics: To study the behavior ofconcentration fluctuations immediately following a quenchfrom the one-phase to the unstable region, we canlinearize the CH equation about the initial averageconcentration difference ψ0 = ψ - δψ, where δψ is a smallfluctuation around ψ0.Note that all of this can be done in terms of φ rather than ψthrough a trivial change of variables since ψ = 2 φ − 1.

• The CH equation before linearizing is:

!

"#(x,t)

"t= M$2 %F

%#= M$2 "f

"#&'$2#

(

) *

+

, -



• Cahn Hilliard before linearizing:

• Cahn-Hilliard after linearizing about ψ0 = ψ - δψ

!

"#$(x,t)

"t= M%2 " 2 f

"$ 2

&

' (

)

* + $0

#$ ,-%2#$&

'

( (

)

*

+ +

< 0 in unstable two-phase region

Write ψ = ψ0 + δψ. Then:

!

"#(x,t)

"t= M$2 "f

"#%&$2#

'

( )

*

+ ,



• Now Fourier transform the linearized equation.– Define δψ(q,t) = ∫ δψ(x,t) e-iqx dx

!

"#$(q,t)

"t= %Mq2 qc

2 & q2( )[ ]#$(q,t)

where

qc2 '

1

%

" 2 f

"$ 2

(

) *

+

, - $0

Solution is an exponential!



• Initial concentration fluctuations thus develop in timeaccording to the following exponential:

• Then the structure factor S(q) evolves with time as

!

"#(q,t) = "#(q,0)e$(q )t

with the wavevector - dependent growth rate

$(q) =%Mq2(qc2 & q2)

!

S(q,t) = S(q,0)e2"(q )t Grows or decays

exponentially atconstant q


!

"#(q,t) = "#(q,0)e$(q )t

$(q) =%Mq2(qc2 & q2)

ω(q)

qqcConcentration fluctuations at allq < qc grow since ω(q) > 0.

Concentration fluctuations at allq > qc decay since ω(q) < 0.

q = 2π/λ , so long wavelength fluctuations grow, and short wavelength fluctuations decay. Spinodal decomposition arises from a long-wavelengthinstability.


Linear theory holds at earliesttimes only.


Evolution of Structure Factor

Early time

S(q) growsexponentiallyat fixed q.Fastest growthis at qmax= qc/√ 2

!

qmax

at "#(q)

"q=" $Mq2

(qc2% q

2)( )

"q= 0

Later time

Beyond linearregime: domains growwith time: S(q) peak moves to smaller q

How to describe growth in later stages of decomposition?


Late Stage Kinetics

• As system unmixes, A-rich and B-rich regions grow largerand the interface between them becomes thinner and lessdiffuse.

• In this stage, dynamical scaling is observed.– The scaling hypothesis: in the asymptotic scaling regime,

there is only one dominant length scale.– When the characteristic domain size R(t) >> interfacial width,

the dynamical scaling hypothesis states that by scaling alllengths by R(t), one can find time-independent functionsG[x/R(t)] and H[qR(t)] such that

!

g(x;t) =G[x /R(t)]

and

S(q;t) = Rd(t)H[qR(t)]


Late Stage Kinetics

• The scaling function for the structure factor is suggestedto have the form

where y = qR(t) and γ = d+1 for an off-critical mixture(dispersed clusters) and γ = 2d for a critical mixture(percolating clusters), where d = dimension.

• At large q, the scaling function at large q follows Porod’slaw for large values of y:

!

H(y) "y2

# /2 + y2+#

!

H(y) " y#4


Late Stage Kinetics

• In the late stages of phase separation, R(t) is the onlyrelevant length scale in the system.

• In the scaling regime, R(t) and the height of the structurefactor peak, S(qmax;t), will scale with time as

• If no hydrodynamics, α = 1/3 (∀ d) and β = 1 (in d = 3).– Lifshitz-Slyozov theory of evaporation/condensation

• If hydrodynamics, α = 1 (in d = 3) and β = 3.

!

R(t) " t#

and

S(qmax;t) " t $

where $ = d#.


How to create and tailor patterns?

• Thermodynamics drives phase separation towards bulk,coexisting phases.

• Kinetics determines how morphology evolves.• We must add constraints or additional physics to affect

kinetics and/or evolving morphologies if we want togenerate specific patterns or stabilize patterns at specificlength scales.

vs.


TDGL/CH for Block Copolymers

• One way of introducing a constraint on pattern size andshape is by covalently bonding two immiscible polymerstogether into a single block copolymer.

• Finite block length prevents macroscopic phaseseparation, and stabilizes patterns. The physics of this iscaptured by adding a term to the GL free energyfunctional:

!

F "(x,t)[ ] = FGL"(x,t)[ ] +

N#2

2MG(x,x')"(x,t)"(x',t)dxdx'$$

block length

A

B



• G(x,x’) is the Green’s function for the equation:

• In d = 3, G(x,x’) = 1/|x - x’|+ Y(x), where ∇2Y = 0

!

"2G(x,x') = #4$%(x # x')

Long-range repulsionShort-rangeattraction

!

F "(x,t)[ ] = FGL"(x,t)[ ] +

N#2

2MG(x,x')"(x,t)"(x',t)dxdx'$$



• The TDGL/CH equation for block copolymers is

where W = 1/N2, since

!

"#(x,t)

"t= M$2 %F

%#= M$2 %FGL

%#&W#

!

"2G(x,x')#(x,t)#(x',t)dxdx'$$

= %2#



• To see how this extra term, arising from the long-rangeconstraint on the maximum distance two blocks A and Bon the same chain can be, stabilizes microphase-separatedpatterns, re-examine the linear stability analysis for themodified equation:

!

"#(x,t)

"t= M$2 %FGL

%#&W#


• Modified Cahn Hilliard before linearizing:

• Modified Cahn-Hilliard after linearizing about ψ0 = ψ - δψ

!

"#$(x,t)

"t= M%2 " 2 f

"$ 2

&

' (

)

* + $0

#$ ,-%2#$&

'

( (

)

*

+ + ,W#$

< 0 in unstable two-phase region

Write ψ = ψ0 + δψ. Then:

!

"#(x,t)

"t= M$2 "f

"#%&$2#

'

( )

*

+ , %W#

Linear Theory of BCP SD


• Now Fourier transform the linearized equation.– Define δψ(q,t) = ∫ δψ(x,t) e-iqx dx

!

"#$(q,t)

"t= %Mq2 qc

2 & q2( ) &W[ ]#$(q,t)

where

qc2 '

1

%

" 2 f

"$ 2

(

) *

+

, - $0

Solution still an exponential…




!

"#(q,t) = "#(q,0)e$(q )t

$(q) =%Mq2(qc2 & q2) &W

ω(q)

qqc

Constraint arising from finite block length suppresses long-wavelengthinstability, and selects a particular pattern size scale.

Linear theory holds at earliesttimes only.

W

qc1 qc2

Ordinary spinodal decomposition

Microphase orderingof block copolymers

Only concentration fluctuations atqc1 < q < qc2 grow since ω(q) > 0 there.

but with a modified dispersion relation!


TDGL/CH for Reactive Systems

• Another way of stabilizing patterns is to induce chemicalreactions between the two components of the mixture.

• Example of a simple isomerization reaction:

• Modified Cahn-Hilliard equation becomes:

!

"1

A# B

"2

!

"#(x,t)

"t= M$2 %FCH

%#&'

1# + '

21&#( )

A → B B → A



• Modified Cahn-Hilliard equation for isomerization reaction:

• Can rewrite in terms of ψ = 2φ - 1. For Γ1= Γ2:

!

"#(x,t)

"t= M$2 %FCH

%#&'

1# + '

21&#( )

A → B B → A

!

"#(x,t)

"t= M$2 %F

%#&'#

Same equation as for block copolymers!Γ = 1/N2



• Why do reactive systems behave like block copolymers?What constitutes the “constraint” on bulk phase separation?

When A → B inside a domain with equilibrium composition, B will be driven to move out of domain into B-rich domain, and vice versa. Thesize scale is set by the reaction rate: the slower the rate, the fartherB can move; the faster the rate, the closer B can move.



Small ΓLarge block length

Large ΓSmall block length

Γ = 1/N2


MC Simulation of Phase-SeparatingMixtures with Reactions

• This problem may be modeled with other methods. E.g.,with Monte Carlo, add “reaction” terms whereby A canbecome B and vice versa with rates Γ1, Γ2.

Old New


Controlling patterns through competinginteractions

• Block copolymers and chemical reactions demonstratehow pattern selection can be achieved through constraintson phase separation behavior.– Competing interactions: short-range attraction driving phase

separation + long-range repulsion driving demixing• Zeroth order: Symmetric diblock copolymers or

isomerization reaction with equal forward and backwardreaction rates. (Asymmetric diblocks = unequal rxn rates:off-critical ⇒ different patterns)

• Higher order constraints:– Triblock copolymers; multiblock copolymers; non-linear block

copolymers.– A + B ⇔C, etc. Couple reactions to thermodynamics.


Polymer Blend Phase Separation onNanopatterned Surfaces

A. Karim, et al, PRE 57, R6273 (1998).

Modulated surface interactions can influence patterning of immiscible blends, copolymers, and other mixtures in thin films.

AFM image at right shows PS/PB blend that was spuncast onto SAM substrate patterned using micro-contact printing.

Elastomeric stamps of PDMS are used to stamp ink of hydrophobic andhydrophilic end-group alkanethiolsonto SAM substrate to modulatesurface interaction. This results in 1 micron wide stripes that attract

the PB polymers.


Polymer Blend Phase Separationon Nanopatterned Surfaces

A. Karim, et al, PRE 57, R6273 (1998).


TDGL/CH on Nanopatterned Surfaces

• Surface nanopatterning of immiscible blends can bemodeled using the TDGL/CH method.

• A surface energy term can be added to the GL free energyof mixing:

• h0 is the magnitude of the chemical potential favoring aparticular component at the surface. The interaction ismodulated along the surface in the x-direction with aperiod lp determining the stripe patterning length scale.

• g accounts for changes in the polymer-polymerinteractions near the boundary.


TDGL/CH on Nanopatterned Surfaces

• Cross-section of films showing surface-modulated spinodaldecomposition.

• Here surface pattern wavelength lp is twice the maximallyunstable wavelength λmax.

A. Karim, et al, PRE 57, R6273 (1998).

Thin film

Thicker film

A A A A AB BB B B


TDGL/CH Models of Nanofilled Polymers

• Presence of nanoparticles in immiscible mixtures and blendsaffects the phase separation process since one component ofthe mixture will always preferentially attract the nanoparticle,even by just a small amount.

Questions:• How might nanoparticles be used to manipulate phase separation

morphologies?• How might phase separation be used to manipulate ordering of

nanoparticles?


TDGL/CH for Nanofilled Polymers

• Preferential attraction for polymer A by the nanoparticlemodeled by adding a local surface interaction energy

Fs[φ]= ∫dd-1x[hφ+gφ2+…]

subject to boundary conditions of zero flux and local equilibriumat surface.

h is a surface field that breaks the symmetry between thetwo phases; related to εmf in MD simulations.

Attractive interaction between nanoparticle and polymer Acauses wetting of the particle by the polymer.

*B.P. Lee, J.F. Douglas and SCG, Phys. Rev. E 60, 5812 (1999).



Deep quench

Shallow quench

B.P. Lee, J.F. Douglas and S.C. Glotzer, Phys. Rev. E 60, 5812 (1999).

Spinodal decomposition of a thin film with immobilized nanoparticles.

AFM image of PS/PVMEthin film blend with diluteconcentration of 100 nm silica beads. Karim, et al., Macromolecules 32, 5917 (1999).



Interference patterns can arise from interplay between nanoparticle spacing and spinodal wavelength

Simulation image: B.P. Lee, J.F. Douglas and S.C. Glotzer, PRE 60, 5812 (1999).

Phase contrast microscopy image of photocrosslinked blend. Q. Tran-Cong, 1998.


TDGL/CH for Nanofilled Polymer Blends

• If nanoparticles are allowed to move during the phaseseparation process, different patterns are observed.

Here, MC is used to move the particles around on the grid.

Ginzburg, et al, PRL 82, 4026 (1999) Peng, et al, Science 288, 1802 (2000)


Pattern Selection on the Nanoscale

• Again, all these examples demonstrate the richpossibilities of tailoring nanoscale patterns using softmaterials.

• Method-wise: the TDGL/CH approach is useful, but canbecome cumbersome when complex geometries arise, orwhen hydrodynamic flow or shear or viscoelasticity isincluded. Then, particle-based mesoscopic methods likeDPD may work better, provided you can find a good wayto include the extra physics in the simulation withoutlosing the benefits of the method.


MesodynTM from Accelrys

• Example applications of Mesodyn from www.accelrys.com

Polymer blend with compatibilizers

Latex seed formation Reverse micelles


Other Field Theoretic Methods

• Field Theoretic Polymer Simulation - Fredrickson– Goes beyond mean field - includes fluctuations!– Promising for predicting equilibrium mesophases of

multicomponent soft materials.

Examples of mesophases in ABCA tetrablock melts obtained with fieldtheoretic simulation techniques (Drolet & Fredrickson, PRL 83, 4317 (1999))


Bridging length and time scales

Use output from one level of simulation as input in another

Mesoscale

Atomistic

Macroscale

Molecular/mesoscale

“Designed” polymernanocomposite

First principles

Future goal: “seamless zooming”

Integrated “in silico” soft materials design

Predictmorpho-logies

Specify system:quantum dot, colloidalsilica, etc. in polymer

Predict melt structure anddynamics near nanoparticle

Predict bulk properties

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