POLYMER PHASE EQUILIBRIUM CALCULATION SEYED ABBAS ... - …
Transcript of POLYMER PHASE EQUILIBRIUM CALCULATION SEYED ABBAS ... - …
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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* = ^i^iPf + ^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^^^i^^^) (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
$. = volume fraction of component i
T = absolute temperature
T = gas constant
^ °-^ v«^ = partial molar volumes of component 1 and 2 1 2 ^p phases a, 3 respectively
h', ', "hp = partial molar enthalpy of component 1 and 2 1 /^ in phases a, 3 respectively
31na a The ( L ) term in equation (39) can be evaluated using the 3$^^ 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>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
^i.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
^i.i T. + A*i a $2 (^2*^2^ 1 ^ ^ r ^ r
^i:OL TrB\/r-a ~3x /ra -^sf-.o. ~B' L (h^-h^)(v^V2^) - (h|-h^)(v^v^) _
'2
-L. (1 -1) -2x(l - V^ (43)
23
where
A$ 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
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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
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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).
5. Flory, P. J., J. Chem. Phys., 9, 660 (1941).
6. Flory, P. J., J. Chem. Phys., 10, 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).
17. Flory, P. J., J. Chem. Phys., 10, 51 (1942).
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,<I> ^ ^
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,<i> " aT p,<i> ^ '
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 ($) yields
TEXAS TECH LIBRARY
50
sg^
3T P,<i> aT p,$
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<I>, 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 3<I>i m 2 3<I>i
(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<I> + (l-<^^) - ^2^r^/r2) - '^3(1^/^2^
+ (Xi2^2 ^ Xl3^3^'^2 ' ^3^ - X23(^i/^2^V3^
56
57
^2-^2° " ^'^ t^"^2 ^ ( - 2 " ^i('^2/^i) - ^3(Vr3)
+ (X2i^l ^ X 2 3 V ( ^ 1 " ^ 3 ^ -Xi3(r2/r^)^i^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
T = absolute temperature
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*^ENXL 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^^ASE 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^^ESSURE OF CO'-^PONENT I TS( I ) = C H A P ACTFRISTIC TEMPE'^ATURE '" F COMPONENT I TH( I ) = Fr)UCFO TEMPFPA^URE '"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'^iPQNENT 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 )=^^OLECULAR 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