POLYMER PHASE EQUILIBRIUM CALCULATION SEYED ABBAS ... - …

o^^

POLYMER PHASE EQUILIBRIUM CALCULATION

BY A DIFFERENTIAL METHOD

by

SEYED ABBAS KHALAFI, B.S.

A THESIS

IN

CHEMICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

CHEMICAL ENGINEERING

Approved

Accepted

May, 1977

m- \^\ I

ACKNOWLEDGMENTS

The author wishes to express his appreciation for the stimulat

ing discussion and wise counsel given by Dr. D. C. Bonner. I v;ould

like to express my special appreciation to the committee members.

Dr. D. C. Bonner, Dr. R. W. Tock and Dr. L. D. Clements, for reading

the entire manuscript and for their individual assistance.

My grateful thanks are also due to Dr. H. R. Heichelheim for

his valuable suggestions.

Finally, I wish to express my sincere appreciation to my

parents for having provided me with their support throughout my

academic career, and to my wife for her help during these two years.

n

CONTENTS

Page

ACKNOWLEDGMENTS ii

LIST OF TABLES v

LIST OF FIGURES vi

CHAPTER I

GENERAL INTRODUCTION 1

Literature Cited 3

CHAPTER II

DERIVATION OF A DIFFERENTIAL EQUATION FOR POLYMER

PHASE EQUILIBRIUM CALCULATIONS 4

Introduction 4

Solution Theory 5

Binary Mixtures 8

Chemical Potential 10

X Parameters 11

Derivation of the Phase Equilibrium Equation for Polymers 11

Differential Approach for Polymer Phase Equilibrium Relationships 11

Algorithm for Solution of the Differential

Equation 17

Results 23

Conclusions 33

Recommendations 34

Literature Cited 35

• • •

Page

LIST OF REFERENCES 36

APPENDIX

A. DETERMINATION OF PARTIAL MOLAR VOLUME AND PARTIAL MOLAR ENTHALPY 38

B. DETERMINATION OF PRESSURE AND TEMPERATURE VARIATION OF ACTIVITY AND DERIVATION OF GIBBS-DUHEM EQUATION FOR BINARY MIXTURE 47

C. DIFFERENTIAL METHOD TO CALCULATE POLYMER PHASE EQUILIBRIUM FOR A TERNARY SYSTEM 55

D. COMPUTER SIMULATION 63

TV

LIST OF TABLES

Table Page

I Pure-Component Characteristic Parameters for PIB/Benzene 25

II Molecular Weights and Physical Constants for PIB/Benzene 26

III Pure-Component Characteristic Parameters for PIP/MEK 27

IV Molecular Weights and Physical Constants for PIP/MEK 28

LIST OF FIGURES

Figure Page

1 Algorithm for the Solution of the Differential Phase Equilibrium Equation 18

2 Calculated Temperature vs. Volume Fraction of Polymer in a Phase for PIB/Benzene 30

3 Calculated Temperature vs. Volume Fraction of Polymer in a Phase for PIP/MEK 31

VI

CHAPTER I

GENERAL INTRODUCTION

The most widely known theoretical treatment of polymer solutions

is that of Flory and Huggins. Flory (3, 4) and Huggins (6) derived an

expression for the entropy of mixing for athermal solutions containing

monomeric solvent molecules and long-chain polymer solute molecules

which consist of a number of contiguous segments, each equal in size

to a solvent molecule. Their treatment applies only at concentrations

such that the randomly coiled polymer molecules overlap one another

extensively. Extension of the theoretical athermal equation to non-

athermal solutions was achieved semiempirically by the adoption of a

Van Laar term (9) for representation of the heat of mixing.

The Flory-Huggins equation which we are going to discuss in

Chapter II does not always provide a quantitative description of the

thermodynamic properties of polymer solutions. It has been inadequate

because,1) contrary to the theory, experimentally determined values

of X (Flory-Huggins interaction parameter) are often strong functions

of solution concentration [Eichinger and Flory (2)], 2) x often does

not vary as 1/T as originally proposed by the Van Laar model, 3) no

equation of state is given by the lattice treatment, and 4) no lower

critical solution temperature is predicted if x shows temperature

dependence as 1/T [if x has a minimum with respect to T, the Flory-

Huggins theory can predict both upper and lower critical solution

temperature (1)].

1

This inadequacy of the Flory-Huggins equation has been recognized

for many years and various improved theories, especially for polar

solutions, have been proposed, notably by Huggins (7) and Yamakawa

et_ al_. (10). These improved theories are not only very complicated

but also require extensive data in order to determine numerous para

meters. As a result, these theories are of very little use for

practical applications. We have therefore derived and tested a

differential equation which with a minimum number of parameters can

calculate polymer phase equilibria. The derivation of such differential

equations will be discussed in detail in Chapter II.

Literature Cited

1. Bonner, D. C , J. Macromol. Sci.-Revs. Macromol. Chem., C13(2), 263-319 (19751:

2. Eichinger, B. E., and P. J. Flory, Trans. Faraday Soc., 64,

2053 (1968).

3. Flory, P. J., J. Chem. Phys., Kl, 51 (1942).

4. Flory, P. J., J. Chem. Phys., 9, 660 (1941).

5. Huggins, M. L., J. Phys. Chem., 46., 151 (1942).

6. Huggins, M. L., J. Chem. Phys., i, 440 (1941).

7. Huggins, M. L., J. Amer. Chem. Soc, 8£, 3535 (1964).

8. Meyer, K. H., "Natural and Synthetic High Polymers," Interscience Publishers, Inc., New York, pp. 582-595 (1942).

9. Van Laar, J. J., Z. Phys. Chem., 72 , 723 (1910).

10. Yamakawa, H., S. A. Rice, R. Cornelinsen, and L. Kotin, J. Chem. Phys., 38, 1759 (1963).

CHAPTER II '

DERIVATION OF A DIFFERENTIAL EQUATION FOR

POLYMER PHASE EQUILIBRIUM CALCULATIONS

Introduction

The thermodynamic properties of pure fluids and mixtures ob

tained from statistical mechanics can be divided into two categories:

combinatorial and non-combinatorial (11). The entropy of athermal

mixing is a combinatorial property, while pressure, volume, temper

ature (PVT) properties due to intermolecular forces are non-

combinatorial .

Prigogine (11) developed a corresponding-states theory for

polymer solutions. The major feature of this theory is that the

energy modes that give rise to internal and external degrees of

freedom are implicitly treated. This concept has been used by

several workers for polymer solutions, notably Flory (7) and Bonner

and Prausnitz (1).

In 1965 Flory (8) developed another version of the correspond

ing-states theory of polymer solutions.

Our goal in this chapter is to discuss the solution theory

developed by Flory for binary mixtures and use this theory to

derive a differential equation to calculate polymer phase equilibrium.

Solution Theory

The first qualitatively correct theory of polymer solutions was

proposed independently in 1941 by Huggins (9) and Flory (5, 6); they

extended lattice method to athermal mixtures of monomers and chain

polymers.

Flory suggested (5) that the Gibbs energy of mixing of a non-

athermal polymer solution could be calculated empirically by adding

a Van Laar type heat of mixing term to the entropy of athermal mixing:

AG^ = AH^ (empirical) - TAs'^ (athermal) (1)

In this case the athermal entropy of mixing, also called the com

binatory entropy of mixing, is derived (5, 7).

AS^ (athermal) = -k(N^ In ^^ + N^ In $2) (2)

In this equation

AS (athermal) = athermal entropy of mixing

k = Boltzman's Constant

N- = number of molecules of component i

$. = volume fraction of component i

The volume fraction of component i is related to mole fraction

of component i by the following expression:

X.v-$. = 1 ^ ^ (3) 1 X,v-j + XpVp

In this and all following expressions the subscript "1" refers to

solvent and the subscript "2) refers to polymer. Subsequent differ

entiation of equation 1 gives the chemical potential of solvent:

o

^r^i 1 ? — R T ^ = In $1 + (1 - ) $2 X 4 ( )

where

y^ = chemical potential of the solvent in the mixture o

y,- = chemical potential of pure solvent at system temperature and pressure

T = absolute temperature

R = gas constant

X = Flory-Huggins interaction parameter

In the development of the theory, x was assumed to be independent of

concentration and proportional to 1/T.

This theory gives only a rough representation of the activities

of polymer solution, assuming constant x- As noted earlier, this

theory is based on a rigid lattice model and gives no equation of

state.

An improved representation of the properties of polymer

solutions is given by the corresponding-states theory of Prigogine

(11) and Flory (7, 8) which takes into account volumetric changes.

In this approach, a partition function is formulated for the

mixture based on the Flory-Huggins combinatory factor and a reasonable

representation for the intermolecular potential (7, 8). The function

proposed by Flory for the polymer solution (7, 8) is

where

Z . = combinatory factor

X - geometric packing factor

V* = hard-core volume per segment

v=v/v* = >^5p/^* = reduced volume of mixture

T=T/T* = reduced temperature of mixture

v=V/Nr = the volume per segment

V = total volume

N = number of moles

r = number of segments per molecule

V = specific volume

V* = v*N^ r/M = specific hard-core volume

NA = Avogadro's number

M = molecular v/eight

T* = characteristic temperature of mixture

The equation of state, expressed in reduced form, that follows from

equation (5) is

J/3

^ = ^^^—-h (6)

where

T V -1 vT

p = -^ = reduced pressure of mixture

The characteristic parameters p*, v*, T* satisfy the equation

8

p*v* = CkT* (7)

Equations (5) and (6) can be used for mixtures as well as for the

pure components, but, for the mixture, the variables p*, v*, T*, c,

N, and r are mixture properties and they can be calculated from the

pure-component properties (1) with one binary interaction parameter.

Binary Mixtures

Flory (8) uses a one-fluid corresponding states theory for

mixtures. On the assumption that mixing is random and neglibibly

perturbed by differences in the interaction between neighboring

species, the partition function may be written

where v is the reduced volume of the mixture. All parameters with

out subscript refer to the mixture. We can see that the equation of

state will remain unchanged, but in this case the parameters are

those of the mixture, not pure-component properties. The same

relation between the characteristic parameters is valid. The fol

lowing definitions for N, c, r are valid in equation (8) for

binary mixtures

N = N^ + N^ (9)

r = ^ + ^ (10) ^1 ^2

c = c,y, + c.f. (11) ri "2'2

where 4' (segment fraction) is

- " r^N^ + r^N^ ^ ^

As shown by Flory (8), we can obtain an approximate expression for

Eo by accounting for all binary contacts in solution and by assuming

random mixing of segments. This leads to (1)

= ^(4^) (13) r^N^ + r2N2 v r^N^

where the mixture characteristic parameters are defined by

P* = îîPf + ^2®2P2 " 2(f^^2®l®2^^^^Pf2^* ^ ^

P* T* - ^ p^ ^ p^ (15)

^ri ^ ^2^2 T* T* 'l '2

V* ' vt = V* (16)

Pl2 ^ (p-^Pp^^î^^^) (17)

where 6. = s.r.N./(s^r^N^ + S2N2r2) is the site fraction

s. - number of intermolecular contact sites per segment 1

A - a binary parameter indicating deviation from the geometric-mean assumption for the binary interaction energy density pt^-

10

We discuss calculation of p* in a later section. Note that the

assumption v* = v^ is non-restrictive, since the size of each segment

may be arbitrarily chosen.

The reduced volume of the mixture \3 must be obtained from

equation (6). It can either be obtained graphically from a plot of

T versus v or numerically. In our case we chose Newton's method

to calculate v using as a first approximation the estimate

V = ¥^v^ + ^2^2 (1^)

Since the parameter v* is a measure of molecular size, the

segment ratio is given by

_1 = 1 Isp (19) ^2 V 2 s p

As Flory suggested (8), one must arbitrarily fix either r, or r2

to determine the other. We have set r, equal to unity.

Chemical Potential

From equation (8) the solvent chemical potential is given at

low pressures by

Vir^i° 'l Pl^l^lsD

>< [31n(4j73-) + (Ji-V -1 T-j

+ 2 1 Isp /v 1 (20)

11

where X^2 = Pf + (51^2)^2 ' 2(5^^2^^^^ P*2

A similar expression can be derived for the polymer chemical potential.

^ Parameters

The Flory-Huggins interaction parameter can be obtained in

several ways, but we obtain x ^^om the corresponding states theory

by equating equations (4) and (20) and solving for x to get

X = -±J_^ [3 M-l^yj-) . i (f - )] . - i ^ (x ) (21) RT*$2 ^ -1 T] 1

where

^1 ^ ^lsp/^*sp' 2 = ^2sp/^2sp ( ^ solution temp.)

V = V /v* sp' sp

Derivation of the Phase Equilibrium Equation for Polymers

In this section we are concerned only with binary solutions con

taining one relatively low-molecular weight substance (e.g., benzene)

and one relatively high-molecular weight polymer (e.g., polyisobutylene)

Therefore, the low-molecular weight substance is called solvent and is

denoted by subscript "1", and the polymer is denoted by subscript "2".

Differential Approach for Polymer Phase Equilibrium Relationships

In this section we develop a differential method to calculate

polymer phase equilibrium by treating a binary system of solvent and

a polymer with two coexisting phases, a, 6 in which each component is

present in each phase. As the criteria of equilibrium between co-

12

existing phases, we use the Flory-Huggins chemical potential equations

for solvent and polymer.

V^l . ... .,. .,... ...2 RT ln<D + (l-l/r)$2 + X^2 ^ ^

V^2 . ... ,. . M . ^ . .„ . .2 RT = ln$2 - (r-l)(l-<D2) " ^(1-^2' ^ ^

We also know that the chemical potential differences are related to

activity by

y-j-y^ y2-vi2

—wr = """r -RT = "2

Thus, the criteria of equilibrium for a binary mixture coexisting as

a and 3 phases using the same reference states for all phases are (10):

T ' = T^ (24)

p^ = p^ (25)

Ina^" = Ina/ (26)

lna2°' = lna2^ (27)

In order to obtain differential equations involving p, T, and phase

composition, we are going to take the total derivatives in terms of T,

p, and the pertinent phase composition. Suppose we choose as the

(n+1) variables in each phase, T, p, and $^. Thus, for solvent this

will yield

13

3lna, alna, ' din a - = (-^)p ^a dT . (-^)j^ ,a dp (28)

alna. ' + ( !-).. d$?

8$^

^ alna,^ alna,^ ain a / = (-3J-)p,,e dT . { - ^ ) , ^ ,S dp (29)

3lna/ ^

^ ( r^TD^^l

If we use the same set of intensive variables, the same set of equations

can be written for the polymer activity in the a and 3 phases. We do

not use any phase designation for T and p.

The temperature and pressure variation of activity can be evaluated

to give

3lna. h.-h? (_T.) = - - J i (30)

3lna. v.-v^ (31)

^"8p~~H,$^ " ~RT~

where

h". = partial molar enthalpy of component i

h? = molar enthalpy of pure i as an ideal gas at ^ temperature T

V. = partial molar volume of component i

v° = molar volume of pure i as an ideal gas at ^ temperature T

14

The derivation of equations (30, 31) can be found in Appendix B.

Substituting equations (30) and (31) into equations (28) and (29)

yields

F,-h° v?-v? 31 na.^" dlna ^ = - (-4I) dT + (-V-)dp + ( l-U X (32)

I j j Kl ^ a l,p 1

and

h.-h? v?-v? 31na.^

^' - ' (4^)''' (-V) p ^-Tzrh.p''' ^''^ dlna^ RT'" '' 3$ 1

The same set of equations can be written for polymer activity in

both phases. Now equating the activity expressions

dlna^^' = dlna^^

dlna2°^ = dlna2

and simplifying, we have the following set of equations for solvent

and polymer:

h.-h; Vn-v; 31na, 3Ina, ^ ,( ll)dT + {-\^)dp . ( i-)^ d ^ - ( rH,p^^l = ° ( '

^jd Kl 3^^a i,p I g^^P i,p I

h^-h!: ^9-^0 31na« « 3lna^ ^ , , -(^)dT + (-^)dp . ( ^ ) ^ d - ( )T,pd v - 0 (35)

Equations (34) and (35) are differential equations that must be

15

satisfied simultaneously. We can solve these equations by eliminating

any one of the differentials in T, p, <l>°' and $^. We can thus obtain

a differential equation in three variables. Integration of that

eqaution yields the desired result.

Therefore, to obtain the partial derivatives in T, $°^, $^, we

eliminate dp from equations (34) and (35) simultaneously to get

RT' dT -

(-

31 na a 1

-)

3$ 1 a M,p (-

3lna, a

3^ ) T n _ a M,p

1 /-a ~3\ /~a ~3\ 1

'2 "2

d<|) a

31na^^

( B-)T,P

31na. 3

3$ • r H,l 1

3$ 1

(~oi ~3\ 2 2

d<|) 3 . = 0 (36)

This final result may be simplified by noting that the coefficients

of the d$, and d$, terms are related by the Gibbs-Duhem equation for

component activities, i.e., for a binary system of phase TT:

* ; ( •

3lna TT 1

-)

3$ 1 IT M,p m 2

T 3lna^

+ I ^ ( !-) 3(|) 1

7T a,p = 0 (37)

where m = v^. (ratio of the pure component molar volumes).

The derivation of equation (37) is discussed in Appendix B.

Therefore by using equation (37) and applying it to equation (36)

we get

16

^(h^-h^)(v^-v^) - (h^-h^)(v^-v^)-^

RT

dT

r^t. oi/~a ~3 a/~a ~B\ -\ 2 ( 2 P 1 (^rn^

<l>- a

3lna a 1 ( ^ )

3$ 1 a M,p 1

a

' " $ 2 ^ V ^ 2 ) + ni^/(v^-v^)

$, 3

31na 3 3 ( — r - ) T d$; = 0 3$ 1

3 'T,p"n (38)

where

m = v«, = ratio of pure component molar volumes '2/v 1

$. = volume fraction of component i


T = gas constant

^ °-^ v«^ = partial molar volumes of component 1 and 2 1 2 ^p phases a, 3 respectively

h', ', "hp = partial molar enthalpy of component 1 and 2 1 /^ in phases a, 3 respectively

31na a The ( L ) term in equation (39) can be evaluated using the 3$^^ T,p

Flory-Huggins equation for solvent activity:

In a. In $^ + (1- I) <l>2 + X^2

Then

31na a 1 ( ~)

3$ 1 a M,p

- ^ - (1 - ) - 2X (l-*i°) (40)

17

which can be substituted into equation (39).

Algorithm for Solution of the Differential Equation

The algorithm used to solve the differential phase equilibrium

equation (39) is shown in Figure 1.

We used Euler's numerical method of analysis to solve the

equation. The iteration is continued until 'l*,' is equal to 0.995

(which makes ^,^ equal 0.005). Using this method, we assume an initial

value for temperature. Also, we let

r~ A. oLf-oL ~3

fCT,*^") = $3°(v^-vf) + m t^°(v°-vg)

_(h^-h^)(VV^) - (h^-h|)(vv^)

*2 u *i ^-(1 -i)-2x(l-*/)

a letting a < <!>, < b

which in our case a and b are 0 and 1 respectively. Now to approxi

mate the equation we shall consider the interval 0 < ^ < 1. We

will now partition this interval into n subintervals of equal

length.

Let ($l")i = ( i )o + i X A^^^

where A$^" = [b - ($^^)^] / n

We shall then denote the corresponding values of T = 1{^^ ) at the

points (i )^ by

18

1. Assume an initial value of

T and <^^^

2. Determine V ^ V2 using

equation (A29)

3, Assume an approximate value

for \J , V2

3: 4. Determine 9.2^ ^-io ^sing

P-j*2 " ^XP (a+b/T) and

X. 12 = p^ +P2 - 2p

*

12 T

5. Calculate T-, , T^ using T. = T/T.

Calculate v,, sj^, solving

equation (6) by Newton's

method

No

10. Calculate v „ using P *

^TSp 1 ISp

11. Calculate density using

p = 1/V3P

9. Calculate

new v-j, V2 by

V. = v . - FV/FPV >T\

8. Differentiate

FV to get FPV

Figure 1. Algorithm for the solution of the differential phase equilibrium equation.

1

19

12. Calculate V p V2 using

16

V,. - M./P.

I 13. Assume a constant value

for 4) 3

^^ 1 - ^?

14. Calculate x? using

(f) and V^, V2

X^ = 1 - x^

3 15. Calculate 00? using

X? and M-j, M2

3 _ 1 3 ^2 " ^1

— B Calculate 4* using 3 * '*

'^r ^Isp' ^2sp \J/^ = 1 _ vi/3

^2 ' *1 V

17. Set

(*?)i+l = (<^?)i^K Z

18. Using initial value of (4>-|)Q calculate

(4>?)-| "fJom step 16, and then calcualte

x^, w^, H' , X2, (1)2, H'2, as we did in

steps 14, 15, 16

19. Calculate T of mixture *

using T = T/T ^ —

Fiqure 1, Continued.

20

20. Calculate p of mixture

using p = p/p*

21. Assume a first approximation

for V of mixture to be

V = V, + V2

Y. 22. Calculate v of mixture,

solving equation (6) by

Newton's method

±. 26. Calculate x using

equation (21)

27. Calculate a using

equation (A28)

T 28. Calculate F,, h2

in a and 3 phases

using equation (A26)

29. Calculate V^ in

a, 3 phases using

equation (A23)

I

No

25. Calculate new v

by V = V - FV/FPV

24. Differentiate FV

to get FPV

J

Figure 1. Continued

21

29. Calculate Denomenator and numerator

(HO) of equation (39) and also cal-31 na^

culate ( -) 3$ a 'T,P

30. Calculate 2 31na a

T i+1

= T . .^arNumerator (H0)-,/RT^^, '" ^

$« 3$i

.a 31. Is $, greater than or

equal to 0.994 No

Yes

STOP

Figure 1. Continued.

22

T^ = T[($^").]

Using this notation we can now approximate the equation by an algebraic

equation. To approximate the derivative at the point ( i ').: by

the quotient

î.l - i dT

A$ a d$ a

*r = (*i"'i

We are simply approximating the slope of the curve T = T{^-^ ). Now

the slope of the curve T = T($^°') at ^ ^ = (O^^'). must be f[($^°')p (T.)]

Thus, we have the following algebraic result

V l _ l i . f[(, a).,(T.)] (41)

A$ 1

The value of f [(<l'-j°) • ,(T.)] is the slope of the tangent line to the

curve. Equation (41) can be recast in the form

T.^^ = T. + A^^^x f [($^«)., (T.)] (42)

or

î.i T. + A*i a $2 (^2*^2^ 1 ^ ^ r ^ r

î:OL TrB\/r-a ~3x /ra -^sf-.o. ~B' L (h^-h^)(v^V2^) - (h|-h^)(v^v^) _

'2

-L. (1 -1) -2x(l - V^ (43)

23

where

A$ 1^ = l./k and ($^°^). = i X A$ "

in the iterating i runs from 0 to k. Equation (42) can be solved using

a high-speed computer. If we let i = 0 then,

T^ = T Q + A$^^ X f [($^^)^, (T^)] (44)

By knowing T Q and ($I")Q, T^ can be obtained from equation (44) and

from

(*Pl = (*i")o.*A*/

We can then calculate (< i°')i. and so on, by letting i run from zero

to k.

Results

The computational algorithm shown in Figure 1 was used to pre

dict the polymer phase equilibria for the following binary systems:

a) polyisobutylene and benzene

b) polyisoprene and methyl ethyl ketone

The pure component characteristic parameters used for polymer

and solvent are listed in Tables I and III. The physical constants

used are listed in Tables II and IV. For the benzene/polyisobutylene

system an overall polymer volume fraction of 0.54 at 312.6°K was used

to determine the phase equilibrium in order to illustrate the use of

the algorithm for design calculations. Equilibrium calculations were

25

"zn UJ CO

CQ »—I CL.

on o

en LU I— tu 2 :

• — 1

UJ - J OQ < : h-

< : 0 .

« j • - H

»— 0 0 1—«

on

o

DC

o

\— LU

o ex. s : o CJ

I UJ on

s-= j

«tj cn-o £- C (U (U as en cx.on Z2 E

CO

o • r -4-> CO 0) •r- i-t- 3 E <D to •> 0 +-> V) •»: ^-^ U CU D - I— fO S- ns S- Q - O

x :

0 •f— <D 4-> S -CO 73

• r - 4-> s- ro CU S- -

+ J CL * CJ Q . H -fO E S- CU rtJ H -

. C CJ

E • 1 —

> r— <v

: ^

O CU Q-• 1 - S- -K t/> ^- o > • 1 - C_) u I CU -o ex. i-

C/) fO

C71

CU CO

0 0 0 i n •—

0

0 0 0 0

1 1

0

OJ as ID

CX) I —

o o r^ cxD CO r-^

LD as

en

in CU

o CU ex.

en

CU c CU

'>> 4-> r j

JD 0 CO

•r— >> 0

O -

r— CU c: CU N

c CU

CO

N • M •r— C CO zs ro S -

D -

- o c «r5

s -(U c: c o

CO

<D CJ i . zs o

C/0

ro 4-> rt3

Q

26

TABLE II

MOLECULAR WEIGHTS AND PHYSICAL CONSTANTS FOR PIB/BENZENE

Solvent Molecular Weight - 78 g/gmole

Polymer Molecular Weight - 40,000 g/gmole

Initial Temperature - 312.6 °K

Initial Solvent Volume Fraction - 0.46

Gas Constant - 1.987 cal/gmole °K 3

Pressure - 0.0242 cal/cm

Total Mass - 1.0 gram

23 Avogadro's Number - 6.03 x 10

Solvent Volume Fraction in Second Phase - 0.15

^Data Source: Eichinger and Flory (12).

27

UJ

D -»—I

on o

UJ

UJ

QO < :

t o I—r err UJ I— CJ

<:

UJ

o a. s: o o I

UJ ct:

=3 4-> "a &- C CU CU <o to O L D : : ZD E CU

o +-> to

• 1 —

i . CU +-> CJ <o

a s- CO =3 to » to * CU ex. s-

S- cs_ 03

E CJ

r— <T3 ( J

o

o •r - CU tO U

•I— 4-»

CU S -+ J CU -K I— CJ Q _ l — CU (O E ^ S- CU «3 h -

o o O LO

1 i n CVJ

o o O vo

1

o t—

r— KO

0 0 o

« >

CJ

CJ CU ^ •r- S- * M - O > c n •r- CJ •*>>. o I CU CO CU - o E E

cn (d r—

nr o

to CU

•r— o ex.

cn

o o U3 CTi ' ^ I D

CM CO cr>

c» LO

c o CU

CU c CU

i -

o to r—

•r- > ,

4 J

N 4->

to 3 to s-

•o

CU c c o

CQ

CU a s-r3 o

C/0

ro +J «T3

Q

O CU

28

TABLE IV

MOLECULAR WEIGHTS AND PHYSICAL CONSTANTS FOR PIP/ME

Solvent Molecular Weight - 72 g/gmole

Polymer Molecular Weight - 26,500 g/gmole

Initial Temperature - 298 °K

Initial Solvent Volume Fraction in Phase 1 - 0.387

Gas Constant - 1.987 cal/gmole °K 3

Pressure - 0.0242 cal/cm

Total Mass - 1.0 gram 23

Avogadro's Number - 6.03 x 10

Solvent Volume Fraction in Second Phase - 0.15

29

also performed for the polyisoprene/methyl ethyl ketone system with

an overall polymer volume fraction of 0.613 at 298°K. In both systems

the volume fraction of the solvent in heavy phase was assumed to be

15 percent. The results of these calculations are shown in Figure

2 and Figure 3. These calculations have been done in a constant

pressure of 1 atm. This is a good assumption at low pressures because

we are calculating partial molar volume at constant pressure. At a

high pressure the assumption of constant pressure is not true. The

determination of pt^ was based on the data of Bonner and Prausnitz

(1) for A (binary parameter).

Bonner and Prausnitz (1) published the values of the binary

interaction energy density (p^2) " ^ twenty binary polymer/solvent

solutions at different temperatures. Their data show that the

binary interaction energy density is a function of temperature be

cause of the appearance of A in p^2 equation. Since they had only

two data points available for the desired two systems we assumed an

Arrhenius' plot of the logarithm of p-^^ versus the inverse of

absolute temperature to calculate the temperature dependency of

P* 12*

lnp*2 = a + b/T

using this plot with the method of least squares, we obtain

a = 4.792 b = 4.333 (for PIB/Benzene)

a = 4.882 b = 6.19 (for PIP/MEK)

30

0) (0

04

P

lO

rO

CJ

CJ

I o 00 OJ

ro OJ

• H

U 0) B > i

O

O

c o

•H +J O (d

0)

o >

Mo 3dnivd3drai

03

>

u 4J n3 V 0) CU

0) +J

0)

o Id u •

0)

• 0) CN N

c • (1)

•H \ EM CQ

H

o

31

h

I UJ

is S o ^ I

b

rfi o - ^ -

E ^

^ ^

o 00 fO

o N ro

O CD ro

O lO ro

O ^

ro

O ro ro

O OJ

ro

O ro

O O ro

O a O 00 OJ

ro OJ

0) CO rd

Si

•H

u 0)

o 04

o c o

.H o (d

0)

CJ o

CO >

U

:i

<d

D4

£ 0) 4J

Ti 0) 4J <d

O i H «d u

ro

>lo 3dniV^3dlM31

Ci4 a , M

M O

32

The results shown in Figure 2 and Figure 3 illustrate the

equilibrium results which can be expected. The light phase is es

sentially polymer free and the heavy phase contains only a small

amount of the solvent. This is what we would expect. Another ex

pected result is that by increasing the amount of polymer in heavy

phase the enthalpy of the solution increases significantly. Note

that Figures 2 and 3 are based on the initial temperature chosen.

By changing the value of the initial temperature, a new set of plots

can be drawn.

nnan^n^H^MBI

33

Conclusions

1. One can use this method to predict a phase

diagram of a binary polymer-solvent system

by varying the constant value of <l>, and

obtaining a set of isopleths.

2. Calculation shows the extreme sensitivity of

partial molar volume to the values chosen for

the characteristic parameters.

34

Recommendations

1. Determine polymer-polymer interaction parameters

in order to calculate the phase equilibrium for

a ternary system.

2. Apply the differential method using the values

for the volume fraction in two phases at con

stant temperature.

3. Design an experiment to measure the liquid-

liquid phase equilibria in order to verify the

differential method.

4. Calculate the polymer phase equilibrium at the

high pressures and compare to the one at low

pressures.

35

Literature Cited

1. Bonner, D. C , and J. M. Prausnitz, A.I.Ch.E. J., 19, 943 (1973)

2. Cheng, Y. L., Ph.D. Dissertation, Texas Tech University (1976).

3. Flory, P. J., Principles of Polymer Chemistry, pp. 495-594, Cornell Univ. Press, Ithaca, N.Y. (1953).

4. Flory, P. J., Thermodynamics of High Polymer Solution, J. Chem.

Phys., ]0_, 51 (1942).



7. Flory, P. J., Disc. Faraday Soc, 49 , 7 (1970).

8. Flory, P. J., J. Amer. Chem. Soc., 87 , 1833 (1965).

9. Huggins, L. M., J. Chem. Phys., i, 440 (1941).

10. Modell, M. and R. C. Reid. Thermodynamics and its Applications, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1974).

11. Prigogine, I., The Molecular Theory of Solutions, North Holland Publishing Co., Amsterdam (1957).

LIST OF REFERENCES

1. Beret, S., and J. M. Prausnitz, A.I.Ch.E. J., 21, 1123 (1976).

2. Bondi, A., Physical Properties of Molecular Crystals, Liquids,

and Gasses, Wiley, N.Y. (1975).

3. Bonner, D. C., and J. M. Prausnitz, A.I.Ch.E. J., 19, 943 (1973).

4. Bonner, D. C , J. Macronol. Sci. - Revs. Macromol. Chem., C13(2), 263-319 (19757:

5. Bonner, D. C , N. F. Brockmeir, and Y. L. Cheng, Ind. Eng. Chem. Process Design and Development, 13, 437 (197471

6. Bonner, D. C., D. P. Maloney, and J. M. Prausnitz, Ind. Eng. Chem. Process Design and Development, 13, 91 (1974).

7. Bonner, D. C , and J. M. Prausnitz, J. Polym. Sci. Polym. Phys. Ed.,

12_, 51 (1974).

8. Cheng, Y. L., Ph.D. Dissertation, Texas Tech University (1976).

9. Dodge, B. F., Chemical Engineering Thermodynamics, McGraw-Hill

Company, Inc., N.Y. (194471

10. Ehrlich, P., J. Polym. Sci.: Part A, 3_, 131 (1965).

11. Eichinger, B. E., and P. J. Flory, Trans. Faraday Soc., 61, 2053 (1968).

12. Eichinger, B. E., and P. J. Flory, Trans. Faraday Soc., Part 2, 61, 2053 (1968).

13. Eichinger, B. E., and P. J. Flory, Macromolecules, 1, 285 (1968).

14. Flory, P. J., Principles of Polymer Chemistry, pp. 495-594, Cornell Univ. Press, Ithaca, N.Y. (1951).

15. Flory, P. J., Thermodynamics of High Polymer Solutions, J. Chem.

Phys., 10., 51 (1942).

16. Flory, P. J., J. Chem. Phys., 9 , 660 (1941).


18. Flory, P. J., Disc. Faraday Soc, 49, 7 (1970).

36

37

19. Flory, P. J., J. Amer. Chem. Soc., 87_, 1833 (1965).-

20. Huggins, L. M., J. Chem. Phys. , 9, 440 (1941).

21. Huggins, L. M., J. Phys. Chem. , 46 , 151 (1942).

22. Huggins, L. M., J. Amer. Chem. S o c , 86 , 3535 (1964).

23. Meyer, K. H., Natural and Synthetic High Polymer, Interscience Publishers, Inc., N.Y., pp. 582-595 (1942).

24. Miller, A. R., Th_e Theory of Solutions of High Polymer, Clarendon Pren., Oxford (1948).

25. Modell, M., and R. C. Reid, Thermodynamics and Its Applications, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1974).

26. Prausnitz, J. M., Molecular Thermodynamics of Fluid Phase Equilibria, Prentice-Hall, Inc., N.Y. (1959).

27. Prigogine, I., The Molecular Theory of Solutions, North Holland Publishing Co., Amsterdam (1957).

28. Prigogine, I., and R. Defay, Chemical Thermodynamics, Longmans,

Green, London (1954).

29. Van Laar, J. J., Z. Phys. Chem. , 72_, 723 (1910).

30. Yamakawa, H., Modern Theory of Polymer Solutions, Harper and Row, N. Y. (19717::

31. Yamakawa, H., S. A. Rice, R. Corneliusen, and L. Kotin, J. Chem. Phys., 38, 1759 (1963).

APPENDIX

A. DETERMINATION OF PARTIAL MOLAR VOLUME

AND PARTIAL MOLAR ENTHALPY

38

DETERMINATION OF PARTIAL MOLAR VOLUME

The partial molar volume expression can be obtained by using the

Flory's reduced equation of state (1).

?c- -1/3 2 ^ = J^ _ _1_

- V 3 T " -If V ' -1 vT (Al)

Differentiating both sides of equation (AL) with respect to N. at

constant T,p, N.,., yields

T 3N. T,p,Nj. J 3iM. p,T,Nj " -2 dH- T,p,Nj

, l/3v"^^^ (v^^^-1) - l/3v"^^^ (v^^^) av (v^/3-l)2 3N. T,p,Nj

1 3v 1 9T ^ - 2 - dH. T,p,Nj ^ -2 8N. T,p,N^ (A2)

in which N. = number of moles of component i and p, T, v are the

reduced pressure, temperature and molar volume respectively.

After simplification and combining terms, one has

3v 8N. T,p,Nj

1 f 3v2/3 (v^/3.i)2 ;;2^J

aN

.1/3 V

. V jp. i ^'P'Nj T(v^/3-l) T 3N. T,p,N.

(A3)

39

40

4 ^

at this point we need to determine -J- r M and -^ T r. M SN. T,p,N. 3N^ T,p,N.

and this can be done as follows:

at Determination of 3N. T,p,Nj

Knowing the definition of T and differentiating both sides with

respect to N. at constant T,p, N.,. yields

T = ^ (A4)

9 - _I_ Hi (A5) M T T,p,Nj " ^^2 aN. T,p,Nj ^ '

in which T* is the characteristic temperature of the mixture and

it can be determined using the Flory's expression

T* = , P" , (A6) T ^iPf 2P2

T* T* h '2

Assuming the denominator to be equal to some constant A* and again

differentiating with respect to N- at constant T,p,Nj yields

a ^ T* 12i

aT* _ . i ! W : ^ i ! ! i : ^ - '^^ (A7) arr T,p,N. ' A*

I <J

in which

D* = ! l ! i . ! # (A8) h '2

and

41

ao^ 3Ni T,p,Nj

afi 3N. T,p,N.

Pf

_ ' l

P2

^2 (A9)

This will then give

aT aN. T,p,Nj.

T 3p* + JL 3D* T*A* aN. T,p,N. A* W: T,p,N. (AlO)

Determination of aN. T,p,N.

Knowing the definition of p and differentiating both sides with

respect to N. at constant T,p, N. ,. yields

p = p/p*

_aL aN. T,p,Nj

_2_ ap: p*2 aN. T,p,N^

(All)

in which p* is the characteristic pressure of the mixture and it

can be determined by using Flory's expression

p* = ^/p*-, + 'i'2 P*2 + 2^l^2Pl2 (A12)

Since m, = 1-^, (A13)

an' 1 aN. T,p,N^

34*2 aNT T,p,Nj

(A14)

This yields

42

ap* 3N. T,p,Nj.

ay^

3 N 7 T,P,N J

2p^*4'^-2p2*H'2+2p*2(l'2-¥^) (A15)

Now by substituting (AlO) and (All) back into equation (A3) and making

all necessary simplifications, we can solve for 3v 3N. T,p,Nj-

3v 3N. T,p,Nj.

ii aN. T,p,Nj

? ^* * t.

-.1/3

Tp T*A*(v'^'^-l)

^1/3 aA^ (^V3.^)^, aN. T,p,N.

^ + .3~2/3/~l/3 Tx2 .2-. _ 7 3v (v -1) V T_ (A16)

Now using the definition of v.

V = v/v*, (A17)

in which v is the volume per segment and v* is the hard-core volume

per segment. Differentiating (A17) with respect to N^ at constant

T,p,N. yields w

av aN. T,p,Nj.

1 3v V' 3N. T,p,N^

(A18)

or

3v W: T,p,Nj. = V

* ii aN. T,p,Nj

(A19)

43

In this case then the total volume can be defined as the total segment

times the volume per segment or

V = (N^r^ + N2r2)v (A20)

Then by differentiating (A20)

\3N.y T,p,Nj. = V. = r i (SNJ T,P,NJ.

Using equation (A19)

* 3v

(A21)

(A22)

where v^ is the partial molar volume of component i and r. is the

number of segments per molecule of i.

Equation (A16) can be substituted in equation (A22) to get the

partial molar volume in terms of characteristic parameters and sub-

stituting |Ei ^_p^^ , (|^)T^P^N.' ^* "^ ' " ^et • J • J

V. = r i^ VaN,/ T,p, N (2p*$^-2p*^2 - 2p*2(^2''^l'^

P_ + ] L" t ,~2/3,.l/3 Tx2 ~2i ' 3v (v -1) V J

vp* Tp

.1/3 V

-1 /3 1 V -1

/T,pf *2P2\

-1/3 fP* PA

, ^2P2\ (A23)

44

According to Flory (1)

r, M,v*

K '- W^ (A24) 2 V 2 s p

One must arbitrarily fix either r or r2 to determine the other. We

have set 4, equal to unity.

ay^ "aNT T,p,N. ^ " ^ determined by using the segment

J

fraction definition

i i

Partial molar volume can be calculated for either component by using

equation (A23).

DETERMINATION OF PARTIAL MOLAR ENTHALPY

The partial molar enthalpy of each component can be obtained

from the enthalpy of mixing since,

T- _ aAHm "i " 3N. T,p,Nj

According to Flory (2) for the small pressures the enthalpy of mixing

is given by

AHm = rNv* [$^P*(v:[^-v"^) + <^2P2(^2^-^'^) ^ ^102^12^'^^

45

or differentiating with respect ti N. at constant T,p,N. we can J

obtain the partial molar enthalpy from the following equation:

where

h. = p^v^ (v'^-v""') + (aT/v)(f.-f)/f

+ (v| x^2/^) (1 •" ' ) n-e.)' (A26)

1/2 ^ 2 " P* "" (h/^2^P2 " 2(s.,/s2)"" Pf2 (A27)

a = 3v^/3-3

4T-3V1/3T (A28)

V* = H^ . vj^p (A29)

In our case here we let e. = 4" because we did let s /s-j - 1.

46

Literature Cited

1. Bonner, D. C. and J. M. Prausnitz, A.I.Ch.E. J., 29 , 943 (1973).

2. Eichinger, B. E., and P. J. Flory, Trans. Faraday Soc., 61, 2053 (1968).

APPENDIX

B. DETERMINATION OF PRESSURE AND TEMPERATURE

VARIATION OF ACTIVITY AND DERIVATION

OF GIBBS-DUHEM EQUATION FOR

BINARY MIXTURE

47

Pressure Variation

To evaluate the pressure variation of activity we can use the

definition of activity in terms of fugacity ratios

a- = f,./f/ (Bl)

wehre (°) denotes some standard state.

Now taking the natural logarithm of equation (Bl) and differentiate

with respect to pressure at constant temperature and volume fraction ($)

yield

3lna. 31nf. 31nf.° 1_ _ ]_ 1 (DO)

ap T,p " ap T,$ ' 3p T,$ ^"^^^

Now using the definition of fugacity in the integrated form in terms

of free energy

g. - g.° = RT In f./f- (B3)

differentiating with respect to pressure at constant temperature and

volume fraction ($) yields

3g. 3g,° 3lnf. alnf.° ^ 1 1 > DT L - RT ' (B4^

"3^ T,$ ' "aTT.o • ' " T T " T,<D ap T, ^ ^

using legendre theorem we know

ap T,n 1

48

49

1 p ~ T,n " "^i"

Substitute these two equations into equation (B4) and by using equation

(B2) we get

31na. v.-v.°

""ap" T,$ " RT ^ ^

where v^ is the partial molar volume of component i, v.° is the molar

volume of component i at some standard state, R is gas constant, and

T is absolute temperature.

Temperature Variation

Again using the same definition of activity in pressure variation

derivation and taking the natural logarithm of both sides and then

taking the derivative with respect to T at constant pressure and

volume fraction yield

aina, 3lnf. 3lnf.° }_ _ T_ 1 /DC)

3T p , ^ aT p, " aT p, ^ '

Now using the definition of fugacity in terms of Gibbs energy

g.-g.° = RT Inf, - RT Inf °

differentiating with respect to temperature at constant pressure and

volume fraction ($) yields

TEXAS TECH LIBRARY

50

sg^

3T P, aT p,$

alnf.°

- ^' aT

alnf.

= ^' 3T

p.* - ^ '"U°

+ R Inf. 1

(B7)

using Legendre theorem

-af p,n = • i = - V - ( s)

o u o

-if- p.n = - Si° = - V - (B9)

Substituting equations (B8) and (B9) into equation (B7) and simplfying

gives

P 3lnf. p 3lnf.° (g.-g.°) - (h.-R,°) = RT^ - ^ „ . - RT " . n ^

- RT In f./f.° (BIO)

Using equation (B3) we can see that the first term on the left is equal

to the last term on the right, so they can cancel and after rearranging

we get

31 na. ^n'f^/ I = - -^-^ (BU)

aT p,$ f-p2

where B. is the partial molar enthalpy of component i h-'' is the

51

molar enthalpy of component i at some standard state, R is gas con

stant, and T is the absolute temperature.

Remark

Using Jacobian transformation it can be shown that

391- ag^ 3p T,n

3g.

3T p,n

Gibbs-Duhem Equation for a Binary System

3p

3g . 3T

Our goal in this part is to derive Gibbs-Duhem equation with

respect to volume fraction instead of the one known with respect to

mole fraction. To do this we start with

3y. En, ^ = 0 (B12)

1 3n. w

at constant temperature and pressure we then have

31na-j 3lna2 "l " d ^ T,p "2 1 ^ T,p " °

or with r e s p e c t t o $-, we have

Slna, 3 T 3lnap a, n L _ L + n —^ = 0 B13 "l d^^ an^ n2 2 d<^^ 3n^ n2

The definition of volume fraction of component 1 is

52

$ "1^1

1 n-jV-j + n^Vp

differentiating with respect to n, at constant n^ yield

3$ 1 an, n2

"l^l an, n,v, + n2V2

n.

V^(n^v^ + n2V2) - v.j(n^v^)

(n v-j + n2V2)'

$1 $-i2

"7 "^ ^1^2 n 1

(B14)

Using equation (B14) and substitute it in equation (B13) we get

alna n

1 3lna<

+ n. 1 3$^ 2 3$ = 0

1 (B15)

Multiplying by v^/n^v^ + n2V2 yield

alna, V, alna

1 3<|) • V2 2 3$^ 2 = 0

by le t t ing m = vjv^ we then have the f inal result

$

alna, 1 3lna« 1 + i $^ _ ^ ^ = 0 1 3i m 2 3i

(B16)

53

Equation (B16) is the Gibbs-Duhem equation for a binary system

which can be applied to any phase, in this equation ,, ^ ^ ^ the

volume fraction of component 1 and 2 respectively, and m is the

ratio of pure component molar volumes.

54

Literature Cited

1. Dodge, B. F., Chemical Engineering Thermodynamics, McGraw-Hill Compnay, Inc., N.Y. (1944).

2. Modell, M., and R. C. Reid, Thermodynamics and Its Applications, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1974).

APPENDIX

C. DIFFERENTIAL METHOD TO CALCULATE

POLYMER PHASE EQUILIBRIUM FOR

A TERNARY SYSTEM

55

Introduction

A great amount of experimental work has been published on the

precipitation and fractionation of high polymeric substances but

not much work has been done on the theoretical investigation of

their phase relationships, especially for a ternary system of one

polymer and tv/o solvents. The phase relationship of any number of

components can be deduced if an analytical expression for the free

energy of mixing is available. Such expressions have been derived

for solutions of high polymer by several authors, notably Flory (1),

Huggins (2), and Guggenheim. They all use the methods of statistical

mechanics and base the derivation on the lattice model of a liquid.

Flory has made a study, based on his expression for the free energy

of mixing of the separation of a solution of high polymer into two

phases; the method is only applicable if both phases are dilute

solutions. The purpose of this appendix is to derive the phase re

lationship of systems of three components, two phases by using

Flory-Huggins equation for the activity.

Deviation

In this section we assume three components (solvent 1, solvent

2, polymer 3) and two phases (a,3). Using Flory-Huggins theory we

can write the chemical potentials for each component as follows:

MyM^'' = RT [ln + (l-<^^) - ^2^r^/r2) - '^3(1^/^2^

+ (Xi2^2 ^ Xl3^3^'^2 ' ^3^ - X23(^i/^2^V3^

56

57

^2-^2° " ^'^ t^"^2 ^ ( - 2 " î('^2/î) - ^3(Vr3)

+ (X2i^l ^ X 2 3 V ( ^ 1 " ^ 3 ^ -Xi3(r2/r^)î^3]

+ (X3i$l+X32^2'(^l ^ ^2' - ^12(^1)^1^2^

where

X -j = Flory's binary interaction parameter

^ = volume fraction of component i

\i^ = chemical potential of component i at temperature T and pressure P

r = number of structural segments/molecule

R = gas constant


The criteria of equilibrium for a ternary mixture coexisting as

a and 3 phase are (3):

pa ^ p6

(CD

,ro, ^2''^2° " . '^2-^2° S . i„a " = Ina ^

58

To obtain differential equations involving P, T, and value fraction

$^ we can take the total derivatives of the equations (Cl), (C2),

(C3) and expand all activities in terms of T, P, $. This will give

a general equation

,• 31na: 3lna'? (C4) dlnaJ = - ^ d T + - ^ . dP

' ' P,$^ ^P T,$J

31 na* ^-rr ^1

' T,P ^

where i = component and j = phase

Using temperature and pressure variation of activity from Appendix

B and substitute it in equation (C4) one gets a general equation • • •

(C5) dlna'? = - ^ ^ dT + - W ^ dP + ]- d d ^ RT^ ^ 3$ ' -P p ^

Now by writing equation (C5) for every component and in every phase

and using the equilibrium criteria to equate the activity of a com

ponent in two phases, one obtains a general equation.

Fi?-h. v-%.° 3lna^^ (C6) - - ^ d T + - ^ ^ d P + ^ d^^^

RT^ ^^ 3$^'^ T,P '

fi?-h,° v,^-v,° 3lna.^ . = . _i_^ dT + - V ^ dP T T P 1

59

^ ^ h^R^ v^-v^o alna.^ (C7) - - ^ dT + - ^ ^ dP + 1- d$,^

RT^ ^ 3$,°^ T,P 1 1

31na/

3$^^ T,P ^

Since we are dealing with three components, so we have three dif

ferential equations like equation (C7) for every component that must

be satisfied simultaneously.

The Gibbs-Duhem equation for a ternary system of the form

3lna, 31na« 31na-

1 3$n 2 d^-, 3 3$-j

where K = v,/v2 3 and M = v-j/v^, can be used to simplofy the three

differential equations but not to reduce the number of variables.

That is, the Duhem equation reduce the number of coefficients of

the d<l>, terms and, therefore, reduce the amount of physical property

data required in integration to obtain the final result.

Now in order to make the three equations coupled we are going

to multiply the second component equation by K and the third com

ponent equation by M and then add the three equations to get

60

{C8)

(R^-RS) + K(R«-R8) + M(R^-R8) - _ dT

RT"^

(v?-v?) + K(v^-v^) + M(v^-v^) + —'—' ~~ ^—^ dP

3lna,°^ 3lnao°' 3lna.3°' + L. + K ^ + M ^ d<D/'

3$ °" T ,P 3$ ° T ,P 3$ ° T,P

3lna,^ 3lna^^ 3lna-^ o . P,

—r ^ —r ^ ^ —r ^i ' 3$^^ T.P 3$^^ T,P 3<D ^ T,P

3lna3°'

a 3$^" T,P

Now using the Gibbs-Duhem equation and solving for

and substitute into equation (C8) gives

(C9) (Fi ^Fi ) ^ K(B^-R^) + M(R -Fi ) m- • ' ' Q _ _ _ — — ^ — — — — — — y I

{v?-v?) + K(v°-v^) + M(v^-v^) + — 1 — ! ^ ^ ^ ^ dP

+ 1 - — ^ + K 1 - -% 1 - d*, *2" K *3 *1

* i ^ 31na/ *2^ 81na2^ g - 1 - — o- + K 1 - - \ — — g - d $ / = 0

3lna " which is the final result. -;-. and the other three terms in

3*^

a

61

equation (C9) can be evaluated by using the Flory-Huggins equation.

But, we cannot process this any further because we do need to know

the polymer-polymer interaction parameter to calculate the phase

equilibrium data and there is not any published data for polymer-

polymer interaction parameter in the literature.

62

Literature Cited

1. Flory, P. J., J. Chem. Phys., 9_, 660 (1941).

2. Huggins, L. M., J. Chem. Phys., 9., 440 (1941).

3. Modell, M., and R. C. Reid, Thermodynamics and Its Applications Prentice-Hall, Inc., Englewood Cliffs, N.J. (1974).

APPENDIX

D. COMPUTER SIMULATION

63

64

I T E R A T I V E D I G I T A L SOLUTION FGP PHASE E Q U I L I B R I U M C C

C ^ D I ^ F E P E N T I A L E Q U A T I H N J d * ^ 1 : * * * * * * * * : * * * * * : * * ; ^ * t t * * » ' * i ^ * ; ^ * * ^ * * j f 3 ^ * • * * . * 4C*-! (C** * * * * « x » - * ' * ' * * * * » :

C C c C c

* ^ i r * * *

A PROGRAM DESIGNED TO EVALUATE PHASE E Q U I L I B R I U M

CALCULATION FOR A POLYVER-SCLVFNT BINARY SYSTEM ^ * * - * - * r , * * : ( c * : ^ ^ * * c ^ r * * * ; p * * 4 t A * * * * * * : t . * : * : « : ) f c * t * * * * * * - i ^ : * * * * * * * ^ * * *

l F * ^ * « ^ i c * * * * * * i r * . ^ K A i J r 4 : » : * r * » * * , : ^ * : ^ * * - * * * * ^ : ^ : f : ^ ; * * » : t * ^ ! : i j ( t t * * i : : * : * V * * : t : * *

c C C C C c c C C C C C C c C c c C C C c C C C

C C

REAL M , K A I D IMENSION

2 3 4

V S P ( 2 ) , P S { 2 ) , V H M ( 2 ) T Y ( 2 , 2 ) , W ( 2 t 2 ) , X ( 2 , 2 ) , Z ( 2 , 2 ) , V H ( 2 ) t V S ( 2 ) t T M { 2 ) , A L F ( 2 ) tTH*^(2) , H ( 2 , 2 ) , P H M ( 2 ) t V B { 2 , 2 ) , V { 2 ) , T S ( 2 ) t M ( 2 ) , V S C ( 2 ) , R 0 H { 2 )

INPUT O^TA

2

2

2

2

BSLOP=THE SLOPE OF THE ALCG(PS12) V S . I / T CURVE CURVE

AINT=THE INTERCEPT CF THE AL0G(PS12) V S . 1 /T CURVE

IVH1=A FIRST APPROXIMATION FOR V H ( 1 ) IVH2 = A F IRST APR^CX T^ AT ICN FOR V H ( 2 )

1X1=AN I N I T I A L VOLUME FR&CTION OF SOLVENT I N PHASE 1

! X 2 = A CONSTANT VOLUME FRACTION OF SOLV^^NT IN PHASE 2

VSP=CHA^ACT£RISTIC HAPD-CORF VOLUME TS = CHA3ACTERIS'^IC T E M ^ ^ E ^ A T U R E

PS=CHAPACT ERIST IC PRESSU^F TO=AN I N I T I A L TFVPPPATUFE NA=AVOGAORO NUMBFR KR=NO. OF OUTPUTS

K=NG. OF STEPS R=GAS CONSTANT P=SYSTEM PRESSURE M=MOLFCULAR WEIGHT Q=TOTAL MASS

R EAD READ READ P EAC READ READ

( 5 , 2 ) ( 5 , 3 ) ( 5 , 3 ) ( 5 , 3 ) ( 5 , 3 ) ( 5 , 3 )

K,KR Q,NA A I N T , R S L O P R , T n , P I XI , 1 X 2 , 1 VHl , ( ( P S ( I ) , T S ( I )

2 FOR^'AT ( 1 5 , I X , 1 3 ) 3 FORMAT( ^^F13 .6 )

D X = 1 . / K T=TO X( 1 , 1 ) = I X 1

DETERMINATION CF VS

IVH2 , V S P ( I ) t ' M I ) ) »I = l t 2 )

65

no 29 1 = 1 , 2 V S ( T ) = M ( I ) * V S P ( I )

29 CONTINUE C

V H ( 1 ) = I V H 1 V H ( 2 ) = I V H 2 00 17 ISTEP=1,K I F ( X ( 1 , 1 ) . G E . 9 Q 9 4 . F - A ) GO TO 28 PS12=EXP(AINT+BSLOP/^) X12 = P S ( 1 ) 4 - P S ( 2 ) - 2 . * P S 1 2

C C DETERMINATION OF TH C

DO 31 1 = 1 , 2 T H ( I ) = T / T S ( I )

31 CONTINUE C C DETERMINATION OF VH BY THE NFWTON METHOD C

DO 22 1 = 1 , 2 23 FV=( (VH( I ) ^ * ( 1 . / 3 . ) - l ) * ( P ' ' ' V H ( I ) ^ ^ 2 . / P S ( I ) + l ) ) / V H ( I )*=^

2 ( A . / 3 . ) - T H ( I ) I F ( A B S ( F V / V H ( I ) ) . L E . 1 . E - A ) GO TO 22 PPV=={P -VHd )'-=^2./PS( I ) + l ) / 3 . / V H ( I ) -^*2.<-(2.«P*=(VH(I ) + *

2 ( 1 . / 3 . ) - l ) ) /DS( I ) / V H ( I )*^( l . / 3 . ) - ( 4 . * ( V H ( I ) * ' ( 1 . / 3 . ) 3 - 1 ) * ( P * V H ( I ) * « 2 . / P S ( I ) + l ) ) / 3 . / V H ( I ) * * ( 7 . / 3 . )

VH( I ) = VH( I ) -PV/FPV GO TO 23

22 CONTINUE C C DETERMINATION OF VSC

C DO 19 1 = 1 , 2 V S C ( I ) = V H ( D ^ V S P d )

19 CONTINUE C C DETERMINATION OF ROH C

DO 20 1 = 1 , 2 ROHd ) = 1 . / V S C ( I )

2 ) CONTINUE C C DETERMINATION OF PU^E COMPONENT MOLAR VOLUME C

DO 21 1=1,2

V(I )=M(I )/ROH(I ) 21 CONTINUE

C X(l,?)=IX2 y/p 7) = !— X(l 2)

Y(l!2)=X(l,2)*V(2)/(V(l)fX(l,2)'^(V(2)-V(l)))

Y(2,2)=l-Y(l ,2)

66

C C C

C C C

C C C

C C C

C C C

W ( l » 2 J = Y ( l , 2 ) * M ( i ) / ( Y ( i , 2 ) : - M ( l ) ^ Y ( 2 , 2 ) * M ( 2 ) ) W ( 2 , 2 ) = 1 - W ( 1 , 2 )

Z ( l , 2 ) = V I ( l , 2 ) * V S P ( l ) / ( W ( l , 2 ) ^ V S P ( l ) 4 - W ( 2 , - 2 ) * V S P ( 2 ) ) Z ( 2 , 2 ) = l - Z ( l , 2 ) x ( i t i ) = x ( i , i ) + n x X( 2 , 1 ) = 1 - X ( 1 , 1 )

Y ( 1 , 1 ) = X ( 1 , 1 ) - ^ V ( 2 ) / ( V ( 1 ) ^ X ( 1 , 1 ) * ( V ( 2 ) - V ( 1 ) ) ) Y ( 2 , l ) = l - Y ( l , l )

VMl , l ) = Y ( l , l ) * M ( i ) / ( Y ( l , l ) * M ( l ) + Y ( 2 , l ) * M ( 2 ) ) W ( 2 , l ) = l - W ( l , l ) Z( 1 , 1 ) = W ( 1 , 1 ) * V S P ( 1) / (W( I , 1 ) - ^ V S P { 1 ) + W(2,1)->^VSP(2) ) Z ( 2 , l ) = l - Z ( l , l )

DETERMINATION OF THM

DO 12 J = l , 2

F F = Z ( 1 , J ) * * 2 * P S ( 1 ) 1 - 7 ( 2 , J ) * * 2*PS( 2) + 2 . * 7 ( 2 , J ) * Z ( l , J ) ^ P S 1 2 G G = Z ( 1 , J ) ^ P S ( 1 ) / T S ( 1 ) + Z ( 2 , J ) * P S ( 2 ) / T S ( 2 ) T H M ( J ) = T / F F ^ G G

12 CONTINUE

A F I R S T APPROXI^'ATION TO DETERMINE VHM

n o 18 J = l , 2 V H M ( J ) = ( V H ( l ) i - V H ( 2 ) ) / 2 .

18 CONTINUE

DETERMINAT ION OF PHM

DO 16 J = l , 2 PHM( J ) = P / ( Z ( 1 , J ) " * 2 . ^ P S ( I ) + Z ( 2 , J ) ^ *2 . * ^PS( 2)

2 + 2 . * Z ( 1 , J ) * Z ( 2 , J ) - P S 1 2 ) 16 CONTINUE

DETERMINATION OF y^f^ BY NEWTON W F T H O D

DO 15 J = l , 2 5 F V= ( ( VH M ( J ) ^ - ( I . / 3 . ) - 1) ^ ( PH V ( J ) - VHM ( J ) ^^* 2 . +1 . ) )

2 /VHM( J )^ - ^ ( 4 . / 3 . ) -THM( J ) I F ( A B S ( F V / V H M ( J ) ) . L E . 1 . E - ^ ) GO TO 15 FPV=(PHM( J)^^VH'-M J ) ^ ^ 2 . + l ) / 3 . / V H ' i { J ) t * 2 .

2 + ( 2 . * ^ PHM (J ) ^ (VHM (J )'t M 1 . / 3 . ) - l ) ) / V H ' M J ) * M 1 . / 3 . ) 3 - ( ^ . t - C V H ' - U J ) * * ( l . / 3 . ) - l ) ^ (PHM( J) ' ^ V H M ( J ) - ' - 2 . + I ) ) / 3 .

A / V H M ( J ) * - - ^ ( 7 . / 3 . ) V H M ( J ) = V M M ( J ) - F V / F P V GO TO 5

15 CONTINUE

DETERMINATION OF KAI

DD = 3 . * A L n G ( ( V H ( l j ' ^ ^ ( 1 . / 3 . ) - 1 ) / ( VH'^( 1 ) ** (1 • / 3 . ) - 1 ) ) 2 • l . / T H ( l ) ' ^ ( l . / V H ( 1 ) - 1 , / V H ' M 1) )

E E = ( M ( 1 ) * V S P ( 1 ) / R / T / V H ' M 1) )*X12

67

C C C

C C C

C C C

K A I = P S ( 1 ) * M ( 1 ) * V S P ( 1 ) / R / T S ( I ) 7 X ( 2 , 1 ) * * 2 * D D * E F

DETERMINATION OF ALFA

DO 11 J = l , 2

A L F ( J ) = ( 3 . ^ V H M ( J ) * ^ ( l , / 3 . ) - 3 . ) / ( 4 . * T - 3 . * V H M ( J ) * * 2 ( l . / 3 . ) * T )

11 CONTINUE

DETERMINATION OF PARTIAL MOLAR ENTHALPY

00 10 1=1,2 DO 11 J = l , 2 AA=( 1 + ALF( J ) * T ) ' ^ ( I - Z ( I , J ) ) t * 2 B B = V S ( I ) * X 1 2 / V H M ( J ) C C = ( A L F ( J ) * T ) * ( T H ( I ) - T H M ( J ) ) / V H M ( J ) / T H M ( J ) H ( I t J ) = PS( I ) ' ^ V S ( I ) ^ ( l . / V H ( I ) - l . / V H M ( j ) + C C ) + BB«AA CONTINUE 15

DETERMINATION OF PARTIAL MOLAR VOLUME

DO 14 J=l,2 A = 2.*PS(1)*Z(1,J)-2.*PS(2>^Z(2,J )4-2. *PS 12*( Z ( 2, J ) 2 ~Z(1,J)) B=Z( 1,J )*PS(1 )/TS{ 1)4-Z(2,J)*PS(2)/TS(2) C=VHM(J)*^(l./3.)/(VHM(J)^*(l./3.)-l) D = PS( 1) /TS( 1)-PS(2) /TS( 2) F = l./3./VHM(J)^-' (2./3. ) / {y\-W(J ) P ( 1./3. )-l)*- 2. G=VHM( J)4^PHM( J)= «2./TH '( J)/P S = PHM(J)/THM( J) U = l./VHM(J)>f^*2./THM( J) V B ( 1 , J ) = V S P ( 1 ) * M ( 1 ) ^ ^ 2 . ^ ( A * G - A ^ T H V ( j ) * C / T / B + D ^ C / B )

2 * Z ( 1 , J ) * 7 ( 2 , J l / Q / N A / W d , J ) / { S + - - U ) V B ( 2 , J ) = ~ M ( 2 ) * - 2 . » ' V S ? ( 2 ) ' ^ Z ( 1 , J ) * Z ( 2 , J )'l A ' ^ G - A * T H M ( J )

2 ^ C / T / B + C'^D/B ) / N A / Q / W ( 2 , J ) / ( S i - F - U ) 14 CONTINUE

D V B 2 = V 3 ( 2 , 1 ) - V B ( 2 , 2 ) D V B l = V B ( l , l ) - V B d , 2 ) DH2 = H ( 2 , 1 ) - H ( 2 , 2 ) D H l = H ( l , 1 ) - H ( l , 2 ) DEN = nHI ' i ^0VB2-DH2-^DVBl H O = ( X ( 2 , l ) ^ D V B 2 i - V ( 2)'7'X( 1 , D ' ^ P V R l / V f 1) ) /DEN DLA = 1 . / X ( 1 , 1 ) - ( 1 . - V ( 1 ) / V ( 2 ) ) - 2 . * K A I M 1 . - X ( 1 , 1 ) ) T = T + DX*HO^R*T=^OLA=!^T/X( 2 , I ) W R d E ( 6 , 6 ) I S T E P , K A I , X ( 1 , 1 ) , T

17 CONTINUE

W R I T E ( 6 , 1 ) W R I T F ( 6 , 4 ) K , K R WRITP=(6 ,7 ) P S ( 1 ) , ' " S ( 1 ) , V S P ( 1 ) , M ( 1 ) W R I T E ( 6 , 8 ) P S ( 2 ) , T S ( 2 ) , V S P ( 2 ) , M ( 2 ) W R I T E ( 6 , 9 ) R , T G , P

C C C C C c c c c c C c c c c c c c c c c c

68

WRITE WRITE

( 6 , 2 7 ) ( 6 , 3 0 )

Q,NA

THE PROGRAM NC*ÊNXL ATURE

C C C C c c c c c c c c c C c C c c

VB( I , J ) = PARTIAL MOLAR VOLUME CF COMPONENT I IN PHASE J VSP( n = C H A R A C T E R I S T I C HARO-CORE VOLUME OF COMPONENT I VSC( I ) = S P E C I F I C VOLUME OF CC^^PONENT I VHM(J)=REOUCE0 VOLUME OF MIXTURE IN PHASE J THM( J ) =P EDUCED TE^^'PEP ATURE CF MIXTUPE I N PHASE J P H M ( J ) = R E n u C F n PRESSURE OF ^MXTURE IN PHASE J Y d , J ) = MOLF FRACTION OF CCMFGNEr.'T I IN P^ÂSE J W( I , J )=WFIGHT FRACTION OF CQ^PO'^JEN'T J J J PHASE J X d , J ) = V O L U M E FRACTION CF COMPONENT I IN PHASE J Z d , J)=SEGMCNT FRACTION OF CCMPCNENT I I N PHASE J ALF( J )= ' ^HE ' 'MAL EXPANSION COEFFICIENT IN PHASE J R G H ( I ) = D E N S I T Y GF COMPONENT I H d , J ) = P A O " f ] AL MOLAR ENTHALPY CF CO^'^PONENT I I N PHASE

P S d )=CHAPACTERIST IC P^ÊSSURE OF CO'-^PONENT I TS( I ) = C H A P ACTFRISTIC TEMPE'ÂTURE '" F COMPONENT I TH( I ) = Fr)UCFO TEMPFPAÛRE '"F cn"PO\ 'ENT I V H ( I ) = R E n u C E O VOLUME CF CC^^PONENT I VS( I )=CHARACTERISTIC MCLAR VOLUME CF CO'îPQNENT I

FORMAT( MNPUT FORMAT(• K = ' ,

1 4 7 FORMAT

2 8 FORMAT

2 9 FORMAT

27 FORMAT 30 FOR^'AT

6 FORMAT

NO. OF 1 5 , 3 X , • K P = » ,

NC. OF OUTPUTS * )

d P S ( 1 ) = « , E 1 , E 1 3 . 6 , 3 X , ' M ( 1) = (• P S ( 2 ) = S F i 3 , 6 , E 1 3 . 6 , 3 X , « M ( 2 ) = ( • R=» , E 1 3 . 6 , 3X, d 0 = ' , E 1 3 . 6 , 3 X ,

) = « , E 1 3 . 6 , 3 X , ' V S P d ) = •

STEPS AND 12)

3 . 6 , 3 X , » T S ( 1 ,5^13 .6 ) 3 X , » T S ( 2 ) = ' , E 1 3 . 6 , 3 X , « V S P ( 2 ) = » , E 1 3 . 6 ) T 0 = ' , E 1 3 . 6 , 3 X , » P = « , E 1 3 . 6 ) N A = ' , E 1 3 . 6 ) KA!» ,1 '9X,» X d , l ) • , 1 8 X , » T » ) ( I X , M STEP* , 1 2 X ,

( I 5 , 7 X , E 1 2 . ^ , 7 X , E 1 2 . 4 , 7 X , F 1 2 . 6 )

THE PROGRAM N O M E ^ X L A T U R E ( C O N T . )

BSLOP=THE SLOPE OF THE AIMT=THE INTERCEPT OF

ALCG(PS12) V S . 1/T THE AL0 r , (PS12 ) VS.

CU-VE 1/T CUCVE

28

PS12=CHARACTERISTIC BINARY INTERACTION V ( I ) = M O L A R VOLU'-'E 0"^ PLRE C C ^ ^ P C N E N T I M( I )=^ÔLECULAR WEIGHT OF COMPONENT I

DEN=nENOWINATOR F P V = D F R I V A T I V E OF VOLUME FUNCTION

FV=VOLUME FUNCTION NA = AVOGADRP NU^^BER DX = CHANGE IN VOLUME FC'ACTION

P=SYSTFM PRESSURE T = ABSOLUTE TE^^PERATURE R=GAS CONSTANT Q=TOTAL MASS

STOP END

PRESSUC'E

/ /

•7'i*5„'Vr . f

POLYMER PHASE EQUILIBRIUM CALCULATION SEYED ABBAS ... - …

Documents

Transcript of POLYMER PHASE EQUILIBRIUM CALCULATION SEYED ABBAS ... - …