Transient Interfacial Phenomena in Miscible Polymer Systems

(TIPMPS)

John A. Pojman

University of Southern Mississippi

Vitaly Volpert

Université Lyon I

Hermann Wilke

Institute of Crystal Growth,

Berlin

A flight project in the Microgravity Materials Science Program

2002 Microgravity Materials Science Meeting2002 Microgravity Materials Science MeetingJune 25, 2002June 25, 2002

2

What is the question to be answered?

Can gradients of composition and temperature in miscible polymer/monomer systems create

stresses that cause convection?

3

Why?The objective of TIPMPS is to confirm Korteweg’s The objective of TIPMPS is to confirm Korteweg’s model that stresses could be induced in miscible fluids model that stresses could be induced in miscible fluids by concentration gradients, causing phenomena that by concentration gradients, causing phenomena that would appear to be the same as with immiscible fluids. would appear to be the same as with immiscible fluids. The existence of this phenomenon in miscible fluids The existence of this phenomenon in miscible fluids will open up a new area of study for materials science.will open up a new area of study for materials science.

Many polymer processes involving miscible monomer Many polymer processes involving miscible monomer and polymer systems could be affected by fluid flow and polymer systems could be affected by fluid flow and so this work could help understand miscible and so this work could help understand miscible polymer processing, not only in microgravity, but also polymer processing, not only in microgravity, but also on earth.on earth.

4

How

An interface between two miscible fluids will be be An interface between two miscible fluids will be be created by photopolymerization, which allows the created by photopolymerization, which allows the creation of precise and accurate concentration gradients creation of precise and accurate concentration gradients between polymer and monomer. between polymer and monomer.

Optical techniques will be used to measure the Optical techniques will be used to measure the refractive index variation caused by the resultant refractive index variation caused by the resultant temperature and concentration fields.temperature and concentration fields.

5

Impact

Demonstrating the existence of this phenomenon in Demonstrating the existence of this phenomenon in miscible fluids will open up a new area of study for miscible fluids will open up a new area of study for materials sciencematerials science..

““If gradients of composition and temperature in miscible If gradients of composition and temperature in miscible polymer/monomer systems create stresses that cause polymer/monomer systems create stresses that cause convection then it would strongly suggest that stress-induced convection then it would strongly suggest that stress-induced flows could occur in many applications with miscible materials. flows could occur in many applications with miscible materials. The results of this investigation could then have potential The results of this investigation could then have potential implications in implications in polymer blending (phase separation), colloidal polymer blending (phase separation), colloidal formation, fiber spinning, polymerization kinetics, membrane formation, fiber spinning, polymerization kinetics, membrane formation and polymer processing in space.”formation and polymer processing in space.” SCR Panel, December 2000.SCR Panel, December 2000.

6

Science Objectives

• Determine if convection can be induced by variation of the width of a miscible interface

• Determine if convection can be induced by variation of temperature along a miscible interface

• Determine if convection can be induced by variation of conversion along a miscible interface

7

Surface-Tension-Induced Convection

convection caused by gradients in surface (interfacial) tension between immiscible fluids, resulting from gradients in concentration and/or temperature

high temperaturehigh temperature low temperaturelow temperature

fluid 1fluid 1

fluid 2fluid 2

interface

8

Consider a Miscible “Interface”

xx

yy

“interface” = ∂C/∂y

C = volume fraction of AC = volume fraction of A

fluid Afluid A

fluid Bfluid B

9

Can the analogous process occur with miscible fluids?

high temperature low temperaturelow temperature

fluid 1fluid 1

fluid 2

10

Korteweg (1901)• treated liquid/vapor interface as a continuum

• used Navier-Stokes equations plus tensor depending on density derivatives

• for miscible liquid, diffusion is slow enough that “a provisional equilibrium” could exist (everything in equilibrium except diffusion)– i.e., no buoyancy

• fluids would act like immiscible fluids

11

Korteweg Stress• mechanical interpretation (“tension”)

k has units of N

has units of N m–1

yyxx

Change in a concentration gradient along the interface causes a volume force

pN −pT =kdcdx

⎛ ⎝

⎞ ⎠

2

σ = kdcdx

⎛ ⎝

⎞ ⎠

2

dx∫

FV

=k∂∂y

∂c∂x

⎛ ⎝

⎞ ⎠

2

12

Hypothesis: Effective Interfacial Tension

σ =k(ΔC)2

δ

aα

C

δk is the same parameter in the Korteweg model

(Derived by Zeldovich in 1949)

13

Theory of Cahn and Hilliard

= k ∇

2

∫ C ( r ) dr

also called the surface tension (N m–1)

square gradient contribution to surface free energy (J m–2)

Cahn, J. W.; Hilliard, J. E. "Free Energy of a Nonuniform System. I. Interfacial Free Cahn, J. W.; Hilliard, J. E. "Free Energy of a Nonuniform System. I. Interfacial Free

Energy,"Energy,"J. Chem. Phys. J. Chem. Phys. 19581958, , 28, 28, 258-267258-267.

k > 0k > 0

Thermodynamic approach

g= g0 +k∇C∫2

Square gradient theorySquare gradient theory

PN −PT =kdcdx

⎛ ⎝

⎞ ⎠

2

Cahn-Hilliard Theory of Cahn-Hilliard Theory of Diffuse Interfaces (1958)Diffuse Interfaces (1958)

Convection: experiment and theoryConvection: experiment and theory

Korteweg StressKorteweg Stress

Effective Interfacial Tension

σ =k ∇C∫2

spinning drop tensiometryspinning drop tensiometry

f = f0 +k∇C 2

19011901

MechanicsMechanics ThermodynamicsThermodynamics

Cahn-Hilliard Theory of Cahn-Hilliard Theory of Phase SeparationPhase Separation

Nonuniform MaterialsNonuniform Materials

PPTTPPNNkk

simulationsimulation

material propertiesmaterial properties 1414

Zeldovich TheoryZeldovich Theory(1949)(1949)

van der Waalsvan der Waals

18931893

15

Why is this important?• answer a 100 year-old question of

Korteweg

• stress-induced flows could occur in many applications with miscible materials

• link mechanical theory of Korteweg to thermodynamic theory of Cahn & Hilliard

• test theories of polymer-solvent interactions

16

How?

• photopolymerization of dodecyl acrylate

• masks to create concentration gradients

• measure fluid flow by PIV or PTV

• use change in fluorescence of pyrene to measure viscosity

17

Photopolymerization can be used to rapidly prepare

polymer/monomer interfaces

We have an excellent model to predict reaction rates and conversion

0

50

100

150

200

0 10 20 30 40 50 60

Temperature - model / CConversion - model / %Temperature - exp / C

Temperature (˚C) or % conversion

Time (s)

+

Experimental conversion after 10 seconds exposure

18

Schematic

a

UV

laser sheet illumination

videocameramaskno reaction

reaction1 cm6 cm

100 W Hg arc lampfilter (330 - 400 nm)

19

20

Variable gradient

21

Why Microgravity?

• buoyancy-driven convection destroys polymer/monomer transition

QuickTime™ and aSorenson Video decompressorare needed to see this picture.

2.25 cm

1 g

(Polymer)UV exposure region

22

Simulations

• add Korteweg stress terms in Navier-Stokes equations

• validate by comparison to steady-state calculations with standard interface model

• spinning drop tensiometry to obtain k

• theoretical estimates of k

• predict expected fluid flows and optimize experimental conditions

23

Model of Korteweg Stresses in Miscible Fluids

∂T∂t

+v∇T =κΔT

∂C∂t

+v∇α =DΔC

∂v∂t

+ v∇( )v =−1ρ

∇p+νΔv+1ρ

∂K11

∂x1

+∂K12

∂x2

∂K 21

∂x1

+∂K 22

∂x2

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

div v=0

K11 =k∂C∂x2

⎛

⎝ ⎜

⎞

⎠ ⎟

2

K12 =K21 =−k∂α∂x1

∂C∂x2

K22 =k∂C∂x1

⎛

⎝ ⎜

⎞

⎠ ⎟

2

where k is system specific, with units of N.

Navier-Stokes equations + Korteweg terms

24

Essential Problem: Estimation of Square Gradient Parameter “k”

• using spinning drop tensiometry

• use thermodynamics (work of B. Nauman)

25

Spinning Drop Technique

• Bernard Vonnegut, brother of novelist Kurt Vonnegut, proposed technique in 1942

Vonnegut, B. "Rotating Bubble Method for the Vonnegut, B. "Rotating Bubble Method for the Determination of Surface and Interfacial Determination of Surface and Interfacial Tensions,"Tensions,"Rev. Sci. Instrum. Rev. Sci. Instrum. 19421942, , 13, 13, 6-9.6-9.

rpm

1 cm

σ =Δρω2r3

4

26

Image of drop

1 mm

27

Evidence for Existence of EIT

0

0.1

0.2

0.3

0.4

0.5

0 100 200 300 400 500

EIT (mN/m) 2000 RPMEIT (mN/m) 3000 RPMEIT (mN/m) 4000 RPMEIT (mN/m) 5000 RPMEIT (mN/m) 6000 RPMEIT (mN/m) 7000 RPMEIT (mN/m) 7900 RPM

Time (sec)

drop relaxes rapidly and reaches a quasi-steady valuedrop relaxes rapidly and reaches a quasi-steady value

28

Estimation of k from SDT

• estimates from SDT: k = 10–8 N – k = EIT * δ– k = 10-4 Nm–1 * 10–4 m

• Temperature dependence– ∂k/∂T = -10–9 N/k– results are suspect because increased T

increases diffusion of drop

29

Balsara & Nauman: Polymer-Solvent Systems

k=Rgyr

2 RT

6Vmolar

Χ +3

1−C⎡ ⎣ ⎢

⎤ ⎦ ⎥

X = the Flory-Huggins interaction parameter for a polymer and a good solvent, 0.45.

Balsara, N.P. and Nauman, E.B., "The Entropy of Inhomogeneous Polymer-Solvent Balsara, N.P. and Nauman, E.B., "The Entropy of Inhomogeneous Polymer-Solvent Systems", Systems", J. Poly. Sci.: Part B, Poly. Phys., 26, J. Poly. Sci.: Part B, Poly. Phys., 26, 1077-1086 (1988)1077-1086 (1988)

k = 5 x 10–8 N at 373 K

Rgyr = radius of gyration of polymer = 9 x 10–16 m2

Vmolar = molar volume of solvent

30

Simulations

• Effect of variable transition zone, δ• Effect of temperature gradient along

transition zone

• Effect of conversion gradient along transition zone

31

We validated the model by comparing to “true interface

model”

• Interfacial tension calculated using Cahn-Hilliard formula:

• same flow pattern

• Korteweg stress model exceeds the flow velocity by 20% for a true interface

σ =k(ΔC)2

δ

32

Variation in δ

Korteweg Interface

33

Simulation of Time-Dependent Flow

• we used pressure-velocity formulation

• adaptive grid simulations

• temperature- and concentration-dependent viscosity

• temperature-dependent mass diffusivity

• range of values for k

34

Concentration-Dependent Viscosity: = 0ec

1

10

100

1000

0 0.2 0.4 0.6 0.8 1

=1=2==4=5

Relative Viscosity

Conversion

= 5 gives a polymer that is 120 X more viscous than the monomer

35

Variable transition zone, δ0.2 to 5 mm

k = 2 x 10–9 N

polymer

monomer

36

streamlines

37

Maximum Displacement Dependence on k

δ1=0.2mm; δ2=5mm; k1 *106, N =0; μ0, Pa*s =0.006; σC =3; σT =0; d0*106, kg/(m*s) =0.1; d 1*106, kg/(m*s) =0.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300 400 500 600 700 800 900 1000

seconds

k0 = 1e-7, Nk0 = 1e-8, Nk0 = 2e-9, N

38

Dependence on variation in δIsothermal case with variable transition zone:

k0 *106, N =0.01; k 1 *106, N =0; ,Pa* =.;sC=; T =;d*1 , /(kg m* )=.1;s d1*1 , /( * )=.kg m s

.1

.2

.

.4

.5

.

1 2 4 5 7 8 9 1

seconds

1=.2 , 2=5epsilon mm epsilon mm

1=.2 , 2=2epsilon mm epsilon mm

1=.2 , 2=1epsilon mm epsilon mm

39

Effect of Temperature Gradient

Temperature is scaled so T = 1 corresponds to T = 50 K..

Simulations with different k(T) -does it increase or decrease with T?

Simulations with temperature-dependent diffusion coefficient

40

viscosity 10X monomer (0.01 Pa s)

k0 = 1.3 x 10–7 N

k1 = 0.7 x 10–7 N

41

side heating, δ= 0.9 mm

polymer = 10 cm2 s–1

monomer = 10 cm2 s–1

T0T0 + 50K

Two vortices are observed

42

side heating, δ= 0.9 mm, (c) =0 .01 Pa s * e5 c

polymer = 12 cm2 s–1

monomer = 0.1 cm2 s–1

asymmetric vortices are observed with variable viscosity model

43

effect of temperature-dependent D

• k independent of T

• D increases 4 times across cell

• higher T means larger δ• effect is larger than k depending on T even

with viscosity 100 times larger

44

significant flow

TT00TT11 TT00 T1

45

ε, =1;mm k*1 , =1.;N k1*1 , =;N ,Pa* =.1;sC=5; T =;d*1 , /(kg m* )=1;s d1*1 , /( * )=4.kg m s

.2

.4

.

1 2 4 5 7 8 9 1

seconds

46

Effect of Conversion Gradient

• T also varies because polymer conversion and temperature are coupled through heat release of polymerization

• T affects k

• T affects D

47

9.4

C =1 C =0

monomermonomer

k = 10–8 N

T = 150 ˚C

T = 25 ˚C

polymer viscosity independent of T

48

with D(T)

C =1 C =0

monomer

k = 10–8 N

T = 150 ˚C

T = 25 ˚C

D=1.2 x 10-5 cm2 s-1 D=1 x 10-6

cm2 s-1

49

Limitations of Model

• two-dimensional

• does not include chemical reaction

• neglects temperature difference between polymer and monomer– polymer will cool by conduction to monomer

and heat loss from reactor

50

Conclusions

• Modeling predicts that the three TIPMPS experiments will produce observable fluid flows and measurable distortions in the concentration fields.

• TIPMPS will be a first of its kind investigation that should establish a new field of microgravity materials science.

51

Quo Vademus?

• refine estimations of “k” using spinning drop tensiometry

• completely characterize the dependence of all parameters on T, C and molecular weight

• prepare 3D model that includes chemistry

• include volume changes

52

Acknowledgments

• Support for this project was provided by NASA’s Microgravity Materials Science Program (NAG8-973).

• Yuri Chekanov for work on photopolymerization

• Nick Bessonov for numerical simulations