Transcript Slide 1

Interfacial Tension and Interfacial
profiles:
The essentials of the microscopic approach
Costas Panayiotou
University of Thessaloniki, Greece
Aerogel
Production of porous materials:
Phase inversion with antisolvent (scCO2)
Polymer
2
1
L-L
solution
Antisolvent
Solvent
(α)
Production of porous membranes:
3. Phase inversion using supercritical CO2 as antisolvent
Polymer
V-L
V-L-L
V-L
Pressure
L-L
V-L-L
L-L
V-L
V-L
Solvent
Antisolvent
The concentration profiles through
an interface in a binary system
H
z
INHOMOGENEOUS SYSTEMS:
The Interfacial Tension
The key equations:
t
d   SdT  P e dV  dA    ie dN i
i 1
 e
 t e

   P V     i N i    A     e A
 i 1




INHOMOGENEOUS SYSTEMS:The Interfacial Tension
Alternative Formulation
The key equations:
t
dE  TdS  P e dV  dA   ie dN i
i 1
Apply Euler relation:
t


e
   E  P V  TS    ie N i  A  Gint f  G e A
i 1




where: Gintf : free-energy of the system WITH interface
:
Ge : free-energy of the system WITHOUT interface
Using ρi(z) and ψi(z)
or, density gradient is considered
N i   A i ( z )dz
V   Adz
H
H
t
    A i ( z ) i ( z )dz
i 1
H
   P  P( z )dz
e
H
t


t
e
e

P ( z )    i ( z )  i   i ( z )    i ( z ) i   0 ( z )
i 1
i 1
As a consequence, γ is given by

    0 ( z)dz

No contribution from
density gradients yet
where
t
 0   0 ( z )   e V   0 ( z )   i ( z )ie  P e
i 1
With density gradient contributions:
   0    c1   c2    ...
2
2

 


 cd dz  dz
2
0

or

  2   0 ( z)dz

Density gradient in mixtures
Quadr. truncation
Δψ0 : no dens. gradients

di d j 
1
    ( z )dz    0 ( z )   cij
dz
2
dz
dz
i
j





Applying Euler-Lagrange
minimization (calc. var.)
d 2 j
 0
  cij
0
2
 i
dz
j
Multiplying these equations
by dρi/dz, summing over
all components i, and
integrating, we obtain:
d i d j
1
 0 ( z )   cij
2 i j
dz dz

or
  2   0 ( z)dz

i, j  1,2,...,t
Density gradient in mixtures II
With last equation go from
z-space to ρ-space
This equation when integrated gives the interfacial profile z(ρ) or ρ(z)
Combining with eq. for γ
we obtain:


 0 ( z )
d1

 
dz
 c  2 c d j   c  d j
1j
jj
 11
j 1
d1 j 1  d1



2 
 

 
d j
1  z 
z  z0  
1  z 0 
 d j
c11  2 c1 j
  c jj 
j 1
d1 j 1  d1
 0

d j
 d j
  2  c11  2 c1 j
  c jj 
d1 j 1  d1
j 1
1 

1



2
2


 d
1
1
 2
1
  0 2 d1

Density gradient in mixtures III
From Euler-Lagrange eq.:
d 2 j
 0
'
e
e
  i ,0  z , P   i   cij
 i
dz 2
j

which upon differentiation
gives
i  1,2,...,t
d 2 1
d 2 2
 z , P    c11
 c12
2
dz
dz 2


'
1, 0

'
2, 0
For binaries they are:
Assuming cij = ciicjj we get

e

e
1
d 2 1
d 2 2
 z , P    c 21
 c 22
2
dz
dz 2

 
e

e
2


 

c22 1',0  z , P e  1e  c11  2' ,0  z , P e   2e
d 2

d1
  2' ,0
c11 
 
1


 1' , 0

 c 22 

 
T ,P,2
 1



T ,P,2
 1' , 0
c 22 
 
2


  2' , 0

 c11 

 
2
 T , P , 1




 T , P , 1

Density gradient / calculations
Influence parameters
κii dimensionless adjustable
In hydrogen-bonded:
For multimers we may write
φ is the segment fraction
 
cii   v
*
i
 ii
H H
 *

cii    i  
E  TS H  vi
2



t
t
i 1
i 1
 
53
 ii
  i ( z)   i ( z)  ( z)   ( z)

Local point thermodynamics
Requires in quadr. truncation
β=2/3 universal !
 53
i




    P  PEOS dz  2   0 dz 2  P e  PCP dz
e

Is the above truncation
adequate?
Requirement for internal consistency:
All four alternative ways of calculating γ
should give equivalent results.
In general, then:

  2     dz
or


   P

e






2
 d 
e








 PEOS dz  2    dz  2    P  PCP dz  2    c  dz
dz 


 
And without truncation?
From the general equation:



    0 ( z )   i nm,i   n   dz
m

i


n,m
we obtain:


di d j 
e
t
  2      0 ( z )   cij
dz   P  P ( z ) dz
dz dz 
i
j





by setting:

 ( z )   0 ( z )   i nm ,i    n 
i
n,m
  2   
m

0 ( z )   cij
i
β has a universal value of 2 !
j
di d j 

dz dz 
32
28
24
γ / mN/m
20
16
12
8
4
0
-4
100
3
150
200
250
300
350
5 6 7
400
450
500
550
10
600
T/K
Figure 1: Experimental28 (symbols) and calculated (lines) surface
tensions of normal alkanes as a function of temperature. Numbers near
the curves indicate the number of carbons of the n-alkane.
30
25
20
γ / mN/m
Benzene
15
10
5
Acetone
CO2
0
250
300
350
400
450
500
550
600
T/K
Experimental28 (symbols) and calculated (lines) surface tensions
of pure fluids as a function of temperature.
24
-10
Thickness, D / 10 m
20
16
12
8
4
300
350
400
450
500
T/K
The calculated interfacial layer thickness as a function
of temperature for n-Hexane.
25
20
γ / mN/m
15
10
5
0
250
300
350
400
450
500
550
T/K
Experimental28 (symbols) and calculated (lines) surface tensions
of Methanol as a function of temperature.
25
20
γ / mN/m
15
10
5
0
250
300
350
400
450
500
550
T/K
Experimental28 (symbols) and calculated (lines) surface
tensions of Ethanol as a function of temperature.
34
32
PS
30
PIB
28
γ / mN/m
26
linear PE
24
22
20
PDMS
18
16
14
20
40
60
80
100
120
140
160
180
200
0
T/ C
Experimental1 (symbols) and calculated (solid lines)
surface tensions of pure polymers.
25
24
γ / mN/m
23
22
21
20
19
18
17
0,0
0,2
0,4
0,6
0,8
1,0
X1
Experimental29,30 (symbols) and predicted (lines) surface tensions of
Cyclohexane(1) + n-Hexane(2) mixture at 293.15 K as a function of composition
6,0
5,8
-10
Thickness / 10 m
5,9
5,7
5,6
5,5
5,4
0,0
0,2
0,4
0,6
0,8
1,0
X1
The calculated interfacial layer thickness as a function of composition for
Cyclohexane(1) + n-Hexane(2) mixture at 293.15 K.
0,035
303.15 K
0,030
P / MPa
0,025
0,020
293.15 K
0,015
0,010
0,005
0,0
0,2
0,4
0,6
X1
0,8
1,0
y1
Experimental (symbols)[PWN] and predicted VLE data for Ethanol + Hexane.
22
γ / mN/m
21
20
19
18
17
0,0
0,2
0,4
0,6
0,8
1,0
x1
Experimental (symbols)30-32 and predicted (lines) surface tensions of
Ethanol(1) + n-Hexane(2) mixture at 298.15 K as a function of composition.
323.15 K
0,06
P / MPa
0,05
318.15 K
0,04
0,03
0,02
0,01
0,0
0,2
0,4
0,6
X1
0,8
1,0
y1
VLE predictions of the model for the system:
1-Propanol+Hexane
0,7
0,8
Methanol(1) - Toluene(2) at 308.15 K
X1,liq=0.0275
0,6
0,7
X1,liq=0.0589
0,6
X1,liq=0.2988
0,7
0,5
0,5
0,6
X1
0,4
X1
X1
0,4
0,3
0,5
0,3
0,2
0,2
0,4
0,1
0,1
0,0
0,3
-10
-5
0
5
10
15
0,0
20
-15
-10
-5
0
5
-10
z / 10 m
10
15
20
-15
-10
-5
0
5
-10
z / 10 m
X1,liq=0.4188
20
0,98
X1,liq=0.8045
0,75
15
z / 10 m
0,90
0,80
10
-10
X1,liq=0.9485
0,96
(Near Azeotrope)
0,85
0,70
0,94
0,65
0,80
X1
0,60
X1
X1
0,92
0,90
0,75
0,55
0,88
0,50
0,70
0,86
0,45
0,65
0,40
-15
-10
-5
0
5
-10
z / 10 m
10
15
0,84
-15
-10
-5
0
5
-10
z / 10 m
10
15
20
-10
-5
0
5
10
15
20
-10
z / 10 m
Methanol(1) – Toluene(2) at 308.15 K: The evolution of composition
profiles across the interface, as predicted by the present model.
0,055
X1,liq=0.0525
(Near Azeotrope)
1-Propanol(1) - n-Hexane(2) at 298.15 K
0,045
0,050
0,040
0,045
X1,liq=0.0245
0,040
X1
0,035
X1
0,030
0,035
0,030
0,025
0,025
0,020
0,020
0,015
0,015
-10
0,010
-15
-10
-5
0
5
10
15
0
10
20
-10
z / 10 m
20
-10
z / 10 m
0,12
0,8
0,7
0,10
X1,liq=0.7850
0,6
X1,liq=0.1050
0,08
X1
X1
0,5
0,4
0,06
0,3
0,04
0,2
0,1
0,02
0,0
-15
-10
-5
0
5
-10
z / 10 m
10
15
20
-15
-10
-5
0
5
10
15
-10
z / 10 m
1-Propanol(1) – n-Hexane(2) at 298.15 K: The evolution with liquid
phase composition of the interfacial composition profiles as
predicted by the present model.
Inhomogeneous Systems / Interfaces
(Langmuir, 2002, 18, 8841, IEC Res. 2004, 43, 6592) )
30
25
20
γ / mN/m
Benzene
15
10
5
Acetone
CO2
0
250
300
350
400
450
500
550
600
T/K
Experimental (symbols) and calculated (lines) surface tensions
of pure fluids as a function of temperature.
34
32
PS
30
PIB
28
γ / mN/m
26
linear PE
24
22
20
PDMS
18
16
14
20
40
60
80
100
120
140
160
180
200
0
T/ C
Experimental (symbols) and calculated (solid lines)
surface tensions of pure polymers.
25
24
γ / mN/m
23
22
21
20
19
18
17
0,0
0,2
0,4
0,6
0,8
1,0
X1
Experimental (symbols) and predicted (lines) surface tensions of
Cyclohexane(1) + n-Hexane(2) mixture at 293.15 K as a function of composition
Process Design and Development:
Processing Polymeric Materials
with Supercritical Fluids
Pressure Quench:
A Process for Porous-Structure Formation
Tg / K
420
Rubbery
360
Glassy
300
0
4
8
12
Pressure / MPa
16
20
Figure : Porous structures of polystyrene,
80 οC, (α) 180 bar, (β) 230 bar, (γ) 280 bar, (δ) 330 bar, (ε) 380 bar
The Nucleation Theory
• According to the nucleation theory, in a closed isothermal system in
chemical equilibrium the difference of the free energy per unit volume
related to the formation of new phase
3 cluster is given by the following
4 r
equation :
2
G  

3
P  4 r 
dG
2
 0  rc 
dr
P
ΔG *hom
16 3

3ΔP 2
Modeling the foaming of polymers with scCO2:
Nucleation Theory
Activation energy for homogeneous nucleation:
ΔG *hom
16 3

3ΔP 2
Nucleation rate:
  ΔG *hom 
N O  CO f O exp
,
kT


f O  ZRimp (4 r )
2
C
Total number of nuclei:
N total  
t ,vitr
0
N O dt  
P ,vitr
P , sat
NO
dP
dP / dt
2
rc 
P
0.14
80
o
420
C
390
100 C
o
120 C
0.12
0.10
360
0.08
330
Tg / K
CO2 weight fraction
o
0.06
0.04
Rubbery
300
270
Glassy
0.02
240
0.00
0
10
20
30
40
Pressure / MPa
Figure: Sorption of CO2 in polystyrene.
Experimental data:
(o) 100 οC, () 120 οC, (—) NRHB
50
210
0
5
10
15
20
25
30
Pressure / MPa
Figure: Glass transition temperature for the
system polystyrene-CO2, () experimental
data, (----) CO2 vapor pressure, (—) NRHB
System CO2 – polystyrene:
γmixr = (1-wCO2) γpolr
Surface tension / mN m-1
40
30
20
10
340
360
380
400
420
440
460
480
Temperature / K
Surface tension of polystyrene versus temperature,
() experimental data, (—) NRHB
Critical radius for nucleus formation in PS-CO2
20
o
80 C
o
100 C
o
120 C
Rc / nm
15
10
5
0
0
10
20
Pressure / MPa
30
40
ΝRΗΒ combined with nucleation theory
500
o
80 C
o
100 C
o
120 C
ΔG*hom / kT
400
300
200
100
0
0
10
20
30
40
Pressure / MPa
Activation energy for homogeneous nucleation (polystyrene-CO2 )
11
10
10
10
9
10
8
10
11
10
10
10
9
10
Nuclei density / nuclei cm-3
Cell density / cells cm-3
ΝRΗΒ combined with nucleation theory
8
10
10
20
30
40
50
60
Pressure / MPa
Figure: Observed cell density, (), and calculated nuclei density, (—), versus
pressure for the system polystyrene-CO2 at 80 oC
ΝRΗΒ combined with nucleation theory
10
11
10
10
9
10
8
10
Cell density / cells cm-3
10
10
9
10
8
10
80
90
100
110
Nuclei density / nuclei cm-3
11
10
120
Temperature / oC
Observed cell density, (), and calculated nuclei density, (—), versus
temperature for the system polystyrene-CO2 at 330 bar