Surfactant in a Polyol–CO2 Mixture: Insights from a Classical Density Functional Theory Study

Surfactant

in

a Polyol

−

CO

2

Mixture:

Insights

from

a Classical

Density

Functional

Theory

Study

Sriteja

Mantha,

Huikuan

Chao,

Andrew

S. Ylitalo,

Thomas

C. Fitzgibbons,

Weijun

Zhou,

Valeriy

V. Ginzburg,

and Zhen-Gang

Wang

*

Cite

This:

Langmuir

2022,

38, 16172−16182

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ABSTRACT:

Silicone

−

polyether

(SPE)

surfactants,

made

of

a polydimethyl-siloxane

(PDMS)

backbone

and

polyether

branches,

are

commonly

used

as

additives

in

the

production

of

polymeric

foams

with

improved

properties.

A

key

step

in

the

production

of

polymeric

foams

is

the

nucleation

of

gas

bubbles

in

the

polymer

matrix

upon

supersaturation

of

dissolved

gas.

However,

the

role

of

SPE

surfactants

in

the

nucleation

of

gas

bubbles

is

not

well

understood.

In

this

study,

we

use

classical

density

functional

theory

to

investigate

the

effect

of

an

SPE

surfactant

on

the

nucleation

of

CO

2

bubbles

in

a polyol

foam

formulation.

We

find

that

the

addition

of

an

SPE

surfactant

leads

to

a

∼

3-fold

decrease

in

the

polyol

−

CO

2

interfacial

tension

at

the

surfactant’s

critical

micelle

concentration.

Additionally,

the

surfactant

is

found

to

reduce

the

free

energy

barrier

and

affect

the

minimum

free

energy

pathway

(MFEP)

associated

with

CO

2

bubble

nucleation.

In

the

absence

of

a

surfactant,

a

CO

2

-rich

bubble

nucleates

from

a

homogeneous

CO

2

-supersaturated

polyol

solution

by

following

an

MFEP

characterized

by

a

single

nucleation

barrier.

Adding

a

surfactant

results

in

a

two-step

nucleation

process

with

reduced

free

energy

barriers.

The

first

barrier

corresponds

to

the

formation

of

a spherical

aggregate

with

a liquid-like

CO

2

core.

This

spherical

aggregate

then

grows

into

a CO

2

-rich

bubble

(spherical

aggregate

with

a vapor-like

CO

2

core)

of

a critical

size

representing

the

second

barrier.

We

hypothesize

that

the

stronger

affinity

of

CO

2

for

PDMS

(than

polyether)

stabilizes

the

spherical

aggregate

with

the

liquid-like

CO

2

core,

leading

to

a lower

free

energy

barrier

for

CO

2

bubble

nucleation.

Stabilization

of

such

an

aggregate

during

the

early

stages

of

the

nucleation

may

lead

to

foams

with

more,

smaller

bubbles,

which

can

improve

their

microstrustural

features

and

insulating

abilities.

1.

INTRODUCTION

Polymer

foams

are

lightweight

materials

with

gaseous

voids

trapped

in

a

polymer

matrix.

1

−

7

Their

properties

depend

strongly

on

microscopic

features

such

as

the

size,

density,

and

connectivity

of

the

gaseous

voids

in

the

material.

8,9

The

gaseous

pores

can

be

entirely

separated

from

each

other

by

the

polymer

matrix,

leading

to

a

foam

with

a

closed-cell

structure.

2,4,5,10

Alternatively,

the

gaseous

pores

can

be

interconnected

within

the

polymer

matrix,

forming

an

open-

cell

foam.

2,10

The

closed-cell

foams

are

rigid

and

are

good

thermal

insulators.

Consequently,

they

have

found

applications

as

materials

in

the

construction,

refrigeration,

and

automotive

industries.

Open-cell

foams,

on

the

contrary,

are

soft

and

flexible.

These

foams

are

used

as

materials

for

sound

insulation

and

cushions

for

furniture,

among

many

other

applications.

Microscopic

features

of

a foam

are

significantly

influenced

by

the

foam

production

process.

3

A

standard

procedure

for

producing

foams

2,4

involves

generating

bubbles

and

stabilizing

them

within

a

polymer

matrix.

Reactive

foaming

takes

advantage

of

chemical

reactions

between

the

blending

reactants

for

gas

evolution

and

their

nucleation

within

a

dense

polymer

medium.

On

the

contrary,

in

a physical

foaming

process,

the

polymer

is

first

saturated

with

gas

at

a

desired

pressure.

Then

the

system

conditions

are

instantly

changed

to

initiate

nucleation

of

gas

bubbles

in

the

system.

This

process

yields

a

metastable

condition

in

which

the

system

is

supersaturated

with

gas

in

the

polymer.

Such

a supersaturated

system

evolves

through

nucleation

of

gas

bubbles

and

subsequent

growth

of

thus

nucleated

bubbles.

The

presence

of

gas

bubbles

in

a liquid

makes

a foam

an

inherently

unstable

system.

11,12

As

a consequence,

a foaming

material

ages

over

time.

Drainage

of

liquid

from

the

film

between

the

bubbles,

13,14

bubble

coarsening,

15

and

bubble

coalescence

16

are

three

main

processes

that

contribute

to

foam

instability.

Surfactants

are

commonly

used

to

stabilize

foams

against

aging.

12,17,18

These

molecules

adsorb

at

the

gas

−

liquid

Received:

October

26,

2022

Revised:

November

26,

2022

Published:

December

16,

2022

Article

pubs.acs.org/Langmuir

The Authors.

Published

by

American

Chemical

Society

16172

https://doi.org/10.1021/acs.langmuir.2c02913

Langmuir

2022,

38, 16172

−

16182

interface,

improve

interfacial

properties,

and

constrain

bubble

coalescence

and

coarsening.

19

−

21

Many

studies

in

the

literature

have

investigated

the

effect

of

a

surfactant

on

the

insulating

properties

and

the

mechanical

strength

of

surfactant-stabilized

foams.

22

−

30

The

most

striking

observation

from

these

studies

is

that

the

addition

of

a

surfactant

yields

a foam

with

a reduced

cell

size,

an

increased

cell

density,

and

an

improved

uniformity

in

cell

size.

Though

the

microstructural

features

of

a

foam

are

affected

by

both

bubble

nucleation

and

growth,

Zhang

et

al.

highlighted

that

they

are

more

sensitive

to

the

parameters

governing

bubble

nucleation

than

those

governing

bubble

growth.

31

While

the

role

of

a

surfactant

in

stabilizing

bubbles

to

achieve

these

properties

has

been

reported,

32

its

role

in

the

nucleation

of

bubbles

has

not

been

reported,

despite

the

high

sensitivity

of

the

nucleation

barrier

to

interfacial

tension.

33,34

We

attempt

to

bridge

that

gap

through

this

study.

Classical

nucleaton

theory

(CNT),

35

classical

density

functional

theory

(cDFT),

36

and

molecular

simulation

techniques

37

are

commonly

employed

to

investigate

nucleation

phenomena.

CNT

describes

nucleation

as

formation

of

a new

phase

within

a

bulk

phase.

The

formalism

includes

essential

physics

that

governs

the

thermodynamic

driving

force

for

the

formation

of

a new

phase

and

the

penalty

for

the

formation

of

an

interface

between

the

new

phase

and

the

bulk

phase.

However,

CNT

uses

equilibrium

interfacial

tension

(IFT)

and

assumes

a sharp

interface.

Such

a theory

is

valid

only

near

the

coexistence.

A

semiempirical

correction,

called

the

Tolman

correction,

38

is

often

employed

to

model

the

radius-dependent

interfacial

tension.

When

nanoscale

bubbles

are

being

studied,

such

a modification

is

less

accurate.

39

There

are

also

reports

that

invalidate

the

high

free

energy

barriers

predicted

by

CNT.

40

Additionally,

the

theory

does

not

include

any

molecule

specific

aspects

of

the

system.

Molecular

simulations

can

alleviate

some

of

the

problems

posed

by

CNT.

However,

they

suffer

from

finite

size

effects

that

arise

while

simulating

nucleation

in

a small

system

with

a fixed

number

of

particles.

41

Though

one

can

simulate

a very

large

system

41

or

simulate

by

imposing

a constant

chemical

potential,

42

such

simulations

are

computationally

intensive.

cDFT

43,44

coupled

with

the

string

method

45

provides

an

appropriate

framework

for

capturing

the

essential

structure

and

thermodynamics

associated

with

bubble

nucleation.

cDFT

is

a

mean-field

approach

in

which

the

free

energy

of

the

system

is

expressed

as

a function

of

spatially

varying

molecular

densities.

The

density

profile

representing

the

equilibrium

state

of

a

system

is

determined

by

variational

extremization

of

the

free

energy

functional.

If

we

have

two

such

states,

for

example,

state

A

being

the

homogeneous

bulk

and

state

B

being

a

fully

formed

bubble

of

the

prescribed

size,

a transition-state

path-

finding

algorithm

like

the

string

method

45

finds

a

minimum

free

energy

path

(MFEP)

that

connects

states

A

and

B.

Though

such

an

approach

does

not

yield

any

dynamic

information

to

go

from

state

A

to

state

B,

it

is

still

a

very

powerful

technique

for

characterizing

the

MFEP

and

associated

free

energy

barriers

for

the

formation

of

a critical

nucleus.

Xu

et

al.

46

−

49

have

successfully

employed

cDFT

with

the

string

method

to

characterize

the

barriers

for

the

nucleation

of

CO

2

bubbles

in

homopolymers

like

poly-methyl

methacrylate

(PMMA)

and

polystyrene

(PS).

In

all

of

these

studies,

the

authors

have

modeled

their

free

energy

functional

for

the

cDFT

based

on

the

perturbed

chain

statistical

associating

fluid

theory

(PC-SAFT)

equation

of

state

(EoS).

50,51

Because

the

PC-SAFT

EoS

and

cDFT

have

been

demonstrated

to

quantitatively

describe

the

gas

solubility

and

interfacial

properties

in

CO

2

−

PS

and

CO

2

−

PMMA

systems,

we

employ

the

same

approach

to

model

surfactants

and

study

their

effect

on

bubble

nucleation

in

polymer

foams.

We

specifically

investigate

the

effect

of

a silicone

−

polyether

(SPE)

surfactant

on

CO

2

bubble

nucleation

in

polyol.

SPE

surfactants

are

made

of

a

polydimethylsiloxane

(PDMS)

backbone

and

polyether

branches.

18,52,53

While

alkylethoxylate

surfactants

19

are

ineffective

in

reducing

the

air

−

polyol

interfacial

tension,

SPE

surfactants

are

reported

to

significantly

reduce

the

corresponding

interfacial

tension.

19

This

makes

SPE

surfactants

indispensable

as

stabilizers

in

the

production

of

ubiquitous

polyol-based

foams

like

polyurethanes.

The

addition

of

co-solvents

is

known

to

promote

or

inhibit

the

surface-active

abilities

and

aggregation

behavior

of

surfactants

in

solution.

54

−

57

However,

there

have

been

no

studies

on

how

the

presence

of

gas

molecules

would

influence

a surfactant’s

activity.

Understanding

these

properties

and

how

they

manifest

in

the

nucleation

of

gas

bubbles

is

particularly

relevant

to

characterizing

the

role

of

surfactants

in

foam

production.

Using

the

cDFT

based

on

the

PC-SAFT

EoS

approach

described

above,

we

characterize

the

interfacial

properties

and

aggregation

behavior

of

SPE

surfactants

in

a mixture

of

polyol

and

CO

2

. Then,

using

the

string

method,

we

compute

the

MFEP

associated

with

the

nucleation

of

a CO

2

bubble

from

a

homogeneous

mixture

of

CO

2

, polyol,

and

an

SPE

surfacant.

Our

main

finding

is

that

the

SPE

surfactant

opens

a low-energy

barrier

nucleation

pathway.

This

has

significant

implications

on

the

propensity

for

bubble

nucleation

and

the

resultant

microstructural

features

of

a foam.

The

rest

of

the

manuscript

is

organized

as

follows.

We

describe

our

models

and

cDFT

approach

in

section

2.

We

report

the

results

from

our

calculations

and

discuss

them

in

section

3.

We

then

conclude

the

article

with

an

outlook

on

the

path

forward

for

developing

foams

with

better

physical

properties.

2.

MODELS

AND

METHODS

2.1.

Molecular

Model

and

the

Helmholtz

Free

Energy

Functional.

We

employ

cDFT

to

model

the

CO

2

−

polyol

−

surfactant

ternary

system.

In

our

study,

polyol

is

a

homopolymer

of

poly(propylene

oxide)

(PPO)

and

the

surfactant

is

a linear

diblock

copolymer

with

one

block

being

PDMS

and

the

other

being

PPO.

Each

of

these

molecule

types

is

modeled

as

a tangentially

connected

chain

of

spherical

beads.

The

Helmholtz

free

energy

functional

of

such

a

system

is

then

expressed

as

a sum

of

different

perturbation

contributions

to

the

reference-state

free

energy

functional.

To

construct

the

cDFT,

Xu

et

al.

used

weight-density

functionals

58

and

extended

the

PC-SAFT

EoS

to

model

the

free

energy

functional

for

the

inhomogeneous

system.

For

the

systems

of

interest

to

this

work,

we

follow

the

same

procedure

and

write

the

Helmholtz

free

energy

functional

[

F

({

ρ

})]

as

{ }

=

{ }

+

{ }

+

{ }

+

{ }

F

(

)

(

)

(

)

(

)

(

)

id

hs

assoc

disp

(1)

where

F

id

is

the

ideal

gas

contribution,

F

hs

is

the

excluded

volume

contribution

due

to

hard

sphere

repulsion,

F

assoc

is

the

association

free

energy

due

to

the

formation

of

a

chain

type

molecule,

and

F

disp

represents

dispersion

interactions

between

segments

of

different

molecule

types

in

the

system.

Each

of

these

perturbation

contributions

to

the

free

energy

density

is

expressed

as

a

function

of

the

spatially

varying

segmental

density

({

ρ

})

of

the

different

components

in

the

system.

Langmuir

pubs.acs.org/Langmuir

Article

https://doi.org/10.1021/acs.langmuir.2c02913

Langmuir

2022,

38, 16172

−

16182

16173

In

the

PC-SAFT

EoS,

the

properties

of

a given

molecule

of

type

i

are

specified

by

the

pure

component

parameters

N

i

,

σ

i

, and

ε

i

, which

represent

the

number

of

segments

per

molecule,

the

size

of

the

segment,

and

the

strength

of

the

interaction

between

segments

of

the

same

type,

respectively.

The

interaction

potential

between

any

two

segments

of

types

i

and

j

is

described

by

=

<

l

m

o

n

o

u

r

(

)

,

(

/

)

,

ij

6

(2)

where

σ

ij

=

(

σ

i

+

σ

j

)/2

and

=

k

(1

)

ij

i

j

.

k

ij

is

the

binary

interaction

correction

term

that

is

used

to

account

for

any

missing

interactions

between

segments

of

types

i

and

j

.

If

ρ

i

(

r

N

i

)

is

the

multidimensional

density

profile

of

molecule

i

with

N

i

segments,

where

r

N

i

=

(

r

1

,

r

2

, ...,

r

ζ

...,

r

N

d

i

),

the

corresponding

segmental

density

[

ρ

i

(

r

)]

can

be

expressed

as

=

r

(

)

d

(

)

(

)

i

N

i

N

1

i

(3)

The

different

contributions

to

the

Helmholtz

free

energy,

listed

in

eq

1,

are

expressed

as

a function

of

these

densities

defined

in

eq

3.

The

ideal

term

(

F

id

) of

the

Helmholtz

free

energy

is

known

exactly:

=

× {[

] +

}

F

v

V

r

d

(

)

ln

(

)

1

(

)

i

N

i

N

i

N

id

B

i

where

v

is

a volume

scale

that

has

no

thermodynamic

consequence

as

it

just

shifts

the

chemical

potential

by

a constant.

V

B

is

the

bonding

potential

that

is

used

to

enforce

the

chain

connectivity

between

nearest-neighbor

segments

along

the

chain,

and

β

=

1/

k

B

T

,

where

k

B

is

the

Boltzmann

constant

and

T

the

temperature.

V

B

is

defined

as

[

] =

|

=

+

V

r

exp

(

)

(

)

4

N

j

N

j

i

B

1

2

i

(4)

The

excluded

volume

contribution

(

F

hs

),

due

to

hard

sphere

repulsions,

is

modeled

using

the

fundamental

measure

theory.

59,60

[{ }] =

[

]

F

n

r

d

(

)

hs

(5)

with

[

] =

+

×

Ä

Ç

Å

É

Ö

Ñ

n

n n

n

r

n

(

)

ln(1

)

.

1

ln(1

)

12

1

12

(1

)

(

/3

.

)

V

hs

0

3

1

2

3

2

3

2

3

2

1

2

(6)

where

n

j

[{

ρ

}]

=

∑

i

n

ji

(

j

=

0,

1,

2,

3,

V

1

, or

V

2

) are

the

Rosenfeld

weighted

density

functionals.

58

These

scalar

and

vector

weighted

density

functionals

are

defined

as

=

|

=

|

=

|

=

i

k

j

y

{

z

i

k

j

y

{

z

i

k

j

y

{

z

n

r

n

r

n

r

n

r

n

r

(

)

d

(

)

2

(

)

d

(

)

2

(

)

d

(

)

2

(

)

(

)

,

(

)

(

)

2

(

)

(

)

2

i

V

i

V

i

V

i

2,

3,

,

0,

2,

2

1,

2,

,

2

1

2

(7)

In

these

equations,

δ

(

r

)

is

the

Dirac

delta

function

and

Θ

(

r

)

is

the

Heaviside

step

function.

The

same

weighted

density

functionals

are

used

to

describe

other

short-range

interactions

such

as

association

and

the

local

part

of

the

dispersion

interactions.

The

thermodynamic

perturbation

theory

of

order

1 (TPT-1)

61

−

64

is

used

to

model

the

contributions

to

the

free

energy

due

to

the

association

type

interactions

(

F

assoc

).

These

interactions

represent

correlations

within

a molecule

that

arise

due

to

the

chain

connectivity

between

the

segments.

If

i

=

α

,

β

,

and

γ

index

CO

2

, polyol,

and

surfactant

segments,

respectively,

then

the

TPT-1

expression

for

F

assoc

is

given

by

=

+

[

+

]

F

N

n

g

N

n

g

n

N

g

N

g

r

1

d

(

) ln

(

)

1

d

(

)

ln

(

)

d

(

)

(1

) ln

(

)

(1

) ln

(

)

ln

(

)

assoc

0,

A

AA

B

BB

AB

(8)

where

N

A

and

N

B

are

the

number

of

segments

in

the

A

and

B

type

blocks

of

the

surfactant

chain,

respectively,

and

N

γ

=

N

A

+

N

B

is

the

total

number

of

segments

per

surfactant

chain.

Similarly,

n

0,

γ

(

r

)

=

n

0,A

(

r

)

+

n

0,B

(

r

),

where

n

0,

i

(

r

)

is

the

weighted

density

functional

defined

in

eq

7.

g

ij

(

r

)

is

the

contact

value

of

the

correlation

function

between

the

segments

of

type

i

and

j

and

is

given

by

the

expression

in

eq

9.

=

+

[

]

+

[

]

[

]

+

i

k

j

y

{

z

i

k

j

y

{

z

g

r

n

r

(

)

1

(

)

3

2

(

)

1

(

)

2

1

2

(

)

1

(

)

2

ij

i

j

i

j

i

j

i

j

3

2

3

2

3

2

(9)

The

contribution

of

dispersion

(

F

disp

) to

the

free

energy

has

local

and

nonlocal

components.

The

local

term

(

F

disp

−

local

) is

expressed

as

a

perturbation

to

a

chain-like

reference

fluid.

50,51,61,62,65,66

This

is

obtained

by

directly

extending

the

corresponding

PC-SAFT

EoS

expression

to

the

inhomogeneous

system

using

the

weight

density

functionals

defined

in

eq

7.

The

resulting

expression

is

shown

in

eq

10.

[{ }] =

× [

+

]

=

F

n

J

NM

J

r

d

(

)

(

)

2

(

)

(

)

(

)(

)

i j

i

j

ij

disp

local

,

, A, B

0,

1

2

3

with

=

+

[

]

+

[

]

+

[

]

[

]

[

] [

]

=

[

]

=

M

N

n

N

n

J

a

N

n

r

m

r

(

)

1

8

(

)

2

(

)

(1

(

))

(1

)

20

(

)

27

(

)

1

2

(

)

2

(

)

1

(

)

2

(

)

(

)

(

)

(

)

,

1,

2

m

l

m

l

3

2

3

2

3

2

3

4

3

2

3

2

0

6

(

)

3

and

the

coefficients

=

+

a

N

a

N

a

1

2

l

m

l

m

l

m

l

m

(

)

0

(

)

1

(

)

2

(

)

(10)

where

N

̅

=

∑

i

N

i

x

i

and

x

i

is

the

mole

fraction

of

molecule

i

.

The

constant

coefficients,

{

a

ln

(

m

)

|

m

=

1,

2;

l

=

0,

1,

2,

...,

6;

n

=

0,

1,

2},

are

obtained

by

fitting

the

calculated

binodal

of

the

EoS

with

the

experimental

data

for

a great

number

of

species.

The

values

of

these

coefficients

can

be

found

in

ref

51.

The

F

disp

−

local

term

in

eq

10

alone

is

not

sufficient

to

describe

the

contributions

due

to

dispersion

interactions.

67

A

mean-field

expression

is

included

to

account

for

any

additional

contributions

Langmuir

pubs.acs.org/Langmuir

Article

https://doi.org/10.1021/acs.langmuir.2c02913

Langmuir

2022,

38, 16172

−

16182

16174

due

to

spatial

inhomogeneity.

68

The

corresponding

nonlocal

dispersion

free

energy

term

(

F

disp

−

nonlocal

) is

given

by

[{ }] =

|

× [

]

|

| [

]

=

F

u

r

1

4

d

(

)

(

)

(

)

(

)

(

)

(

)

i j

ij

i

ij

j

disp

nonlo

cal

,

, A, B

(11)

2.2.

Euler

−

Lagrange

Equations

and

the

cDFT

Numerical

Procedure.

In

a cDFT

approach,

the

equilibrium

state

of

the

system

is

determined

by

minimizing

its

grand

potential

(

Ω

[{

ρ

}]).

The

Helmholtz

free

energy

functional

(

F

[{

ρ

}])

defined

in

eq

1 is

the

grand

potential

of

the

system

by

[{ }] =

[{ }]

F

r

d

(

)

i

N

i

N

i

(12)

where

μ

i

is

the

chemical

potential

of

the

i

th

molecule

and

ρ

̂

i

(

r

N

i

) and

ρ

i

(

r

)

are

the

corresponding

molecular

and

segmental

density

profiles,

respectively,

as

defined

in

eq

3.

When

F

ex

[{

ρ

}]

=

F

[{

ρ

}]

−

F

id

[{

ρ

}],

extremizing

the

grand

potential

with

respect

to

the

molecular

densities

[

ρ

̂

i

(

r

N

i

)]

results

in

the

following

Euler

−

Lagrange

equations.

=

Ä

Ç

Å

É

Ö

Ñ

V

F

r

(

)

exp

(

)

(

)

i

N

i

N

i

N

ex

i

(13)

Using

eq

3

and

the

corresponding

relation

for

the

functional

derivative

[i.e.,

=

F

N

F

r

(

)

1

(

)

i

N

i

ex

],

eq

13

can

be

re-expressed

in

segmental

densities

[

ρ

i

(

r

)]

as

=

×

=

Ä

Ç

Å

É

Ö

Ñ

V

F

r

(

)

exp(

)

d

(

)

exp

(

)

(

)

i

N

i

N

1

ex

i

(14)

Equation

14

can

be

further

simplified

by

introducing

a recursive

function

=

+

Ä

Ç

Å

É

Ö

Ñ

F

I

r

(

)

exp

(

)

(

)

(

)

i

N

ex

1

i

(15)

where

the

recursive

function

I

ζ

(

r

)

is

given

by

=

>

|

|=

l

m

o

n

o

Ä

Ç

Å

É

Ö

Ñ

I

F

r

(

)

1,

1

4

d

(

),

exp

(

)

,

1

i

r

2

1

ex

(16)

Equations

15

and

16

are

the

key

equations

for

numerically

computing

the

equilibrium

density

profiles

of

the

different

components

in

the

system.

In

this

work,

we

solve

these

cDFT

equations

in

one-dimensional

Cartesian

coordinates

and

in

spherical

coordinates.

The

latter

is

used

to

study

the

properties

of

micellar

aggregates

and

in

the

context

of

the

string

method

to

compute

the

MFEP

for

the

nucleation

of

a spherical

bubble.

The

equations

are

solved

in

one-dimensional

Cartesian

coordinates

to

study

the

properties

of

the

planar

interface

between

the

CO

2

-rich

vapor

and

the

polyol-rich

liquid.

To

determine

the

spatial

density

profiles

across

the

planar

interface,

we

first

compute

the

densities

({

ρ

i

v

,

ρ

i

l

})

of

the

different

components

(

i

=

CO

2

, polyol,

or

surfactant)

in

the

coexisting

vapor

(

v

)

and

liquid

(

l

)

bulk

phases.

If

f

({

ρ

})

is

the

Helmholtz

free

energy

density

of

a bulk

phase,

then

the

corresponding

pressure

(

p

)

is

given

by

=

p

f

i

f

i

. At

a

given

temperature,

pressure,

and

surfactant

concentration

in

the

liquid

phase,

the

densities

in

the

coexisting

bulk

phases

are

obtained

by

searching

for

the

condition

of

equality

of

chemical

potential,

i.e.,

μ

i

v

({

ρ

v

})

=

μ

i

l

({

ρ

l

}),

and

the

equality

of

pressure,

i.e.,

p

v

=

p

l

=

p

.

These

densities

({

ρ

i

v

,

ρ

i

l

})

are

then

used

as

Dirichlet

boundary

conditions

[

ρ

i

(0)

=

ρ

i

v

;

ρ

i

(l)

=

ρ

i

l

] to

numerically

solve

eqs

15

and

16

within

the

range

of

0

≤

z

≤

L

.

In

our

calculations,

we

choose

L

= 50

σ

1

and

Δ

z

= 0.02

σ

1

. Here

σ

1

is

the

diameter

of

a polyol

segment.

Our

choice

for

L

is

8

−

10

times

larger

than

the

width

of

the

interface.

This

allows

us

to

accurately

resolve

the

spatial

density

profiles

at

the

interface

and

the

bulk.

The

equations

are

solved

using

Picard

iteration

with

a

convergence

criterion

that

the

deviation

of

the

Euclidean

distance

between

two

consecutive

density

profiles

is

<10

−

6

.

Because

we

solve

cDFT

equations

in

an

open

system,

the

location

of

the

vapor

−

liquid

interface

is

translationally

invariant.

Hence,

we

fix

the

spatial

position

of

the

interface.

At

the

beginning

of

the

iteration

process,

the

position

of

the

interface

is

at

z

=

L

/2

with

the

following

spatial

density

profile:

=

<

l

m

o

n

o

z

L

z

L

(

)

,

0

/2

,

/2

i

v

l

After

every

10th

iteration

step,

the

entire

profiles

are

shifted

left

or

right,

so

that

the

Gibbs

dividing

surface

69

relative

to

the

polyol

is

located

at

L

/2.

2.3.

Incipient

Phase

Calculation

to

Initiate

Bubble

Nucleation.

The

primary

objective

of

this

work

is

to

investigate

the

effect

of

the

SPE

surfactant

on

the

nucleation

of

a spherical

CO

2

bubble

in

polyol.

To

initiate

bubble

nucleation,

we

first

saturate

the

polyol

−

surfactant

mixture

with

CO

2

at

the

desired

high

pressure

and

temperature

(303.8

K).

We

label

this

as

the

saturated

state.

The

initial

high

pressure

dictates

CO

2

solubility

in

the

saturated

state.

The

desired

CO

2

weight

fraction

in

a foam

formulation

is

∼

0.2

−

0.3

(w/

w),

and

this

is

realized

at

pressures

of

6

−

7

MPa.

For

reference,

the

critical

point

in

the

CO

2

phase

diagram

is

at

7.38

MPa

and

303.8

K.

Then,

we

instantly

decrease

the

pressure

to

ambient

conditions

keeping

the

temperature

and

the

CO

2

weight

fraction

in

the

system

fixed.

This

leads

to

a metastable

state

in

which

CO

2

is

supersaturated

in

the

system.

We

refer

to

this

state

as

the

metastable

parent

phase.

We

solve

the

PC-SAFT

EoS

at

1

atm

pressure

and

303.8

K

to

determine

the

densities

of

different

components

in

the

metastable

parent

phase,

while

keeping

the

CO

2

content

unchanged

from

that

in

the

saturated

state.

The

metastable

parent

phase

serves

as

a starting

point

for

the

nucleation

of

a CO

2

-rich

bubble.

Because

nucleation

is

a

rare

event,

the

system

undergoes

local

density

fluctuations

representative

of

the

microstates

that

are

visited

during

the

formation

and

breaking

of

subcritical

nuclei.

In

our

quasi-

thermodynamic

approach

to

nucleation,

we

seek

to

identify

the

CO

2

-

rich

bubble

that

is

in

chemical

potential

equilibrium

with

the

metastable

parent

phase.

We

hypothesize

that

such

a CO

2

-rich

bubble

is

the

incipient

phase

that

the

metastable

parent

phase

tends

to

form;

the

composition

of

the

incipient

phase

represents

that

of

a large,

well-

formed

bubble.

We

note

that

the

pressure

inside

the

incipient

CO

2

-

rich

bubble

is

greater

than

the

ambient

pressure

(i.e.,

the

pressure

of

the

metastable

parent

phase).

As

a

result,

the

nucleated

bubble

eventually

expands.

In

this

work,

we

focus

on

the

MFEP

to

the

incipient

CO

2

-rich

bubble

from

a

metastable

parent

phase

and

the

surfactant

effect

on

the

associated

free

energy

barriers.

We

solve

cDFT

equations

with

the

string

method

45,70

in

spherical

coordinates.

At

each

point

along

the

string,

the

system

is

at

a

constrained

equilibrium,

which

allows

the

free

energy

of

the

bubble

to

be

calculated.

Connecting

the

points

along

the

string

results

in

the

MFEP

for

the

nucleation

of

an

incipient

CO

2

-rich

spherical

bubble.

2.4.

String

Method

for

the

MFEP

of

Bubble

Nucleation.

Within

a mean-field

framework,

the

MFEP

for

the

nucleation

is

the

most

likely

path

that

connects

the

initial

and

final

metastable

states

via

a transition

state.

45,70

On

a hypersurface

characterized

by

the

grand

potential

(

Ω

[{

ρ

}]),

let

{

ρ

i

(

r

,

s

)

|

0

≤

s

≤

1}

be

a smooth

curve

(called

a string)

connecting

the

initial

state

{

ρ

i

(

r

,

s

=

0)}

(metastable

parent

phase,

homogeneous

polyol

−

surfactant

mixture

supersaturated

with

Langmuir

pubs.acs.org/Langmuir

Article

https://doi.org/10.1021/acs.langmuir.2c02913

Langmuir

2022,

38, 16172

−

16182

16175