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Calculations have been carried out, based on Flory–Huggins solution theory, to analyze the behavior of the ternary nonsolvent–solvent–polymer phase diagram for typical membrane-forming systems. Consideration is given to the behavior of the spinodal

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Modeling of Asymmetric Membrane Formation.
11
The Effects
of
Surface Boundary Conditions
L.
YILM Z
and
A
J.
McHUGH*
Department
o
Chemical Engineering University
o
Illinois
Urbana
Illinois
618 1
Synopsis
n
analysis is carried out to evaluate the effects of alternate surface boundary conditions on the predictions of
our
previously developed (Part
I)
pseudobinary diffusion model
for
membrane formation by the phase inversion process. Attention is addressed to a comparison
of
concentration profiles in the quenched film
for
a constant flux interface (CF) condition and a mass transfer rate
(MT)
interface condition. numerical algorithm is developed to handle the
MT
condition based on an explicit, finite difference marching method. Comparison
of
concentration profiles with those obtained earlier for the CF boundary condition show that since results for both cases are very similar, either condition can be used in concentration profile calculations. Changes in bath conditions will mainly affect membrane formation through the changed solvent/nonsolvent flux ratio during quenching.
INTRODUCTION
The field of membrane science and technology has undergone an enormous growth since the discovery that asymmetric skinned structures can be formed by the so-called phase-inversion process involving casting
a
solution of a polymer in a solvent (or mixture of solvents) followed by quenching in a nonsolvent bath. The widespread application of such membranes has gener- ated numerous studies on the conditions and possible mechanisms for the formation of the characteristic morphologies during and after phase inversion.2-8 Despite this high level of activity, however, relatively few attempts have been made to quantify and/or systematize the information in terms
of
models for the structure formation. Therefore, in a recent paperg (to be referred to
as
art
I)
we presented a systematic modeling approach emphasizing both the thermodynamic and kinetic aspects of phase inversion during the quench period.
A
pseudobinary diffusion formalism
w s
developed and used in conjunction with an analysis of the ternary phase diagram behavior of typical membrane forming systemdo
to
evaluate the phase sep- aration characteristics. The advantages, predictive power, and potential for further use of this approach were also discussed in detail.
n
important feature of the pseudobinary formalism is that the mixing rule decouples from the diffusion equations, thus allowing one to easily superpose composition paths on the ternary phase diagram and study these indepen- dently of the concentration development
as
a
function of time and distance in
*Author to whom correspondence should be addressed. Journal of Applied Polymer Science, Vol.
35, 1967-1979 1988) 988
ohn
Wiley Sons, Inc.
CCC
0021-8995/88/071967-13 04.00
1968
YILMAZ AND McHUGH
the
film.
s
discussed in
art
I, however, choice of the appropriate surface boundary condition for the latter calculation
is
one of the more complicated and controversial issues of the mass transfer m~deling.l - ~ n
this
paper
this
aspect of the problem
is
discussed in more detail by assessing both the nature and degree of effect of alternate boundary conditions on the computed concentration profiles. In such a fashion one can also gain some feel for the importance of specific recipe conditions on the membrane formation char- acteristics. Likewise, since changes in the bath conditions (i.e., by stirring or introducing convective current effects) can affect both the surface boundary condition and the solvent-nonsolvent flux ratio
k ,
a
knowledge of one (e.g.,
BC
effects) should enable determining the probable role of the other, thereby improving the predictive capacity of the pseudobinary model. Since control of the bath-side conditions is relatively easy, such information should also aid in the design of desired membrane structures. We begin with a brief discussion of the pseudobinary model and the associated alternate surface boundary conditions, followed by a detailed analysis of the concentration profiles.
PSEUDOBINARY
DIFFUSION
EQUATIONS
The pseudobinary diffusion model
starts
from the assumption that, due to the highly entangled nature
of
the casting solution, the polymer mass flux
n3
will be negligible compared to that of the nonsolvent,
nl,
or solvent,
n,.
Thus by defining liquid density and mass fractions
wi
on a polymer-free basis, the nonsolvent flux equation becomes
n
=
-pDvW,
Wl(nl
n,)
(1)
In
this
expression,
D
is
a concentration-dependent phenomenological diffusion coefficient and overbars are used to signify a polymer-free basis. Combination with the equations of continuity
v
- nl
n,)
ap
_
t
v
enl
ap
t
3)
leads to the following equations for diffusion in the
z
direction (measured into
the
film relative to the bath interface position,
z
=
0):
kpD
wl
=
_-
a
t
:z
kw,
a,
5)
where
k
=
k
and
k
=
-n,/n,.
Since experimental
observation^^ ^
have shown that the skin forms very quickly
s
a thin layer at the surface, our
MODELING OF ASYMMETRIC MEMBRANE FORMATION.
I1
969
calculations are necessarily restricted to short times and distances from the interface. Thus, taking to be independent of position, the previous set of equations reduces to
s
mentioned, the question of appropriate initial and, most especially, boundary conditions needed to obtain a well-posed set of equations is
to
some extent a controversial aspect of the modeling. In the absence of an evapora- tion step, the
first
four of these would be straightforward and can be written
s
follows:
p
at
0, 0 and
s
o
for all
0
wl=wli att=O,z>Oandasz+ coforallt>O where subscripts refer
to
constant initial values. In most cases the initial nonsolvent composition would in fact be zero, i.e.,
li
0.
The question of alternate boundary conditions
at
the bath-film interface is considered in the next section.
7)
8)
- -
SURFACE BOUNDARY CONDITIONS
The quench period mass transfer model developed by Cohen et al. ne- glected mass transfer in the bath by assuming that the surface of the film instantaneously equilibrates with the bulk composition of the bath. Thus surface concentrations were taken
s
constant during the quench process. For
a
number of reasons this is a very problematic assumption. First of all, there are experimental observations that contradict this assumption. For some cases Cabassd5 observed accumulation of a solvent layer in the bath next
to
the film surface, and in other cases he observed a convective flow
of
solvent emanating from the polymer film. Both of these observations imply bath side mass transfer. In addition, a number
o
studies6s'5p16 have indicated that mass transfer on the bath side of the interface can have a strong influence on the structure of the formed membrane. For example, asymmetric structures can result when
a
nonsolvent liquid is used in the quench bath while, if the film is quenched in
a
nonsolvent vapor environment, the result is a completely porous membrane.16.17 Elimination of this assumption then leaves two equally plausible choices, either
o
which can be justified qualitatively on the basis of experimental observations for different membrane forming systems. The
first
choice (used exclusively in
art
I) is that of constant flux at the surface and is based on the above-mentioned st~dy,'~ ndicating that, for many systems, a fast, convective flow of solvent occurs into the bath resulting from the density difference between the solvent and nonsolvent. Thus the constant flux condi- tion (CF) can be written
s
n,J,=,
const (94
n21z=o
const (9b) and
1970 YILMAZ AND McHUGH
These
relations thus imply that
k’lz,o
will be .constant, and, since, by assumption,
k’ z)
=
IZ’I =~
one has that the flux ratio will be constant throughout the diffusion path. Hence
eq
6)
can be integrated, subject to eqs. 7) and
8),
to yield the following important relation between the liquid density and
k
1
kw,c)p,
p=
1
kw
We note
that
eq.
lo), which has been obtained without having to solve eqs.
4)
and
5),
directly serves
as
the basis for construction of composition paths on
the
ternary phase diagram
as
given in
art
I.
The second and equally plausible surface condition
is
based on qualitative experimental observations reported by Strathmann and co-workers,16, sug- gesting
the
existence of a mass transfer boundary layer on the bath side.
This
condition referred to
as
MT) can be formulated
as
follows: where
Pbath
is
the bath side liquid density, subscript
b
refers to bath side conditions,
and
the
ki
are
mass transfer coefficients. Using the fact that
W
W
=
1
eads to the following
=
k’lz=o
=
const
n21z=o
nllr=O
kl
_
Hence one finds that the flux ratio will be constant for
this
case
as
well and
the
relation between liquid density and flux ratio, i.e.,
eq
lo), remains unchanged
as
do the predictions and conclusions based on the superposed ternary pr~files.~ hus the differences between the two boundary conditions will be shown in the various component profiles
as
a function of time and position in the film. To
ssess
this
effect, one needs to solve the defining diffusion equation.
CONCENTRATION PROFILES FOR
MT
BOUND RY
CONDITION
Since
the
flux ratio
2
remains constant,
eq
10) can be substituted in eq.
4)
to yield
the
following diffusion equation for constant
D
=
Do
1
wl
1
kw
at
Combination of eqs. l), lo), and lla) gives the following expression
or
the
MODELING
O
ASYMMETRIC MEMBRANE FORMATION.
I1
1971
surface boundary condition: where
pi*
1
KW,,) .
Together with eqs.
7)
and
(8),
one then has a properly posed equation set to solve for the nonsolvent concentrations
as
a function
of
time and position. Introducing the transformation variable
f
KG,/ ~
Kw,
k~,J 1
kwli),
a dimensionless time,
r
K,olb)2t/Do
and
a
dimensionless distance
x
z/Lo
with
o
being the initial film thick- ness, leads to the following equation set to be solved: where Although the mathematical definition of differs here from that used for the constant flux condition in
art
I,
we note that its physical significance
as
a ratio of external to internal mass transfer resistances remains the same. Introduction
of
the MT condition produces an important mathematical conse- quence, however, in that since eq.
18)
is nonlinear, analytical solution is no longer p~ssible, ~~~~ ecessitating use of a numerical integration scheme.
SOLUTION
SCHEME
To
handle the semi-infinite boundary condition,
an
explicit finite difference marching method has been selected.
se
of
an implicit (or semi-implicit) method would require assigning an arbitrary condition for infinity, and
this
position would have to be changed with progression in time. Likewise, the selection of such a condition needs to be checked by repeated calculation
o
concentration profiles at the Same time value for different positions assumed
as
infinity, which makes the treatment more difficult. Since
an
explicit method is used, the
+x
direction
c n
be left open.21 Calculation of the concentration profile at time is simply terminated when
f x,
t
becomes smaller than
a
preselected fixed value without the need to
know
x,
which is a function of time, beforehand.
It
was
decided to employ a variable grid in which increments in space,
Ax,
depend on both time and position. The reason for choosing a time-dependent discretization is that, due to the developing concentration field with time,
Ax
can be increased
at
large times without jeopardizing the numerical accuracy.

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