Interfacial polycondensation—Modeling of kinetics and film properties
Sunil S. Dhumal, Shrikant J. Wagh , A.K. Suresh
Department of Chemical Engineering, Indian Institute of Technology, Bombay 400076, India
a b s t r a c t
Keywords:
Reverse osmosis membranes
Thin film composite membranes
Interfacial polycondensation
Thermodynamics of phase separation
Polymer properties
Interfacial polycondensation (IP) is an important technique used in the encapsulation of a variety of
active ingredients and synthesis of thin film composite membranes. The present work seeks to advance
our understanding of the mechanisms underlying the reaction, phase separation and film formation
in this process, and hence, of how the film properties are influenced by preparation conditions. The
model presented here incorporates all the essential physicochemical processes at a fundamental level
through simple phenomenologies: ionic equilibria in the aqueous phase, resistances due to external mass
transfer, diffusion through polymer film, interfacial reaction, thermodynamics of phase separation, and
formation of a coherent film. The model has been tested against the data previously communicated [S.J.
Wagh, Studies in interfacial polycondensation. Ph.D. Thesis. IIT Bombay, 2004; S.J. Wagh, S.S. Dhumal, A.K.
Suresh, An experimental study of polyurea membrane formation by interfacial polycondensation, Journal
of Membrane Science, submitted for publication] on polyurea microcapsules. The influence of the model
parameters and preparation conditions, on the properties of the polymer and film and their development
during reaction, have been studied. The study provides important insights into the process and should
help in designing synthesis methodologies to suit the application.
1. Introduction
Interfacial polycondensation (IP) is a technique of wide applicability for encapsulation of active ingredients (say for controlled
release or containment), enzyme immobilization [1], and synthesis of thin film composite membranes (say for applications such
as RO) [2]. IP offers the possibility of rapid production of polymers,
under normal conditions of temperature and pressure, in an almost
ready-to-use form. The mechanistic aspects of the process are, however, not well understood because of the difficulties in following
the fast kinetics and the need to account for the interplay of several equilibrium and rate processes in any comprehensive modeling
effort. As a result, only empirical information exists, and even this,
only for particular systems, on how synthesis conditions (such as
solvent used, concentrations employed, interfacial area available)
affect the polymer film properties. Clearly, properties such as film
thickness, molecular weight and its distribution, and the degree of
crystallinity have an important bearing on the functional attributes
of the product, and a predictive capability on how synthesis conditions influence such properties would go a long way in designing
processes to deliver desired product characteristics. The present
work is an attempt in this direction.
IP reaction involves a step growth polymerization between two
monomers, each dissolved in one of a pair of immiscible phases. The
reaction occurs at, or in a thin region adjacent to, the interface of the
two immiscible phases, and the polymer product, being insoluble
in both the phases, accumulates as a film at the surface of contact
between the phases. Morgan [3] has described the salient features
of the IP technique in detail for the preparation of films, fibers and
coatings. While the exact locale of the reaction is not established in
all cases, the balance of evidence is in favor of the organic side of the
interface [3–6]. Mechanistically therefore, IP can be considered as
a process of heterogeneous mass transfer with chemical reaction,
further complicated by the simultaneous occurrence of polymer
phase separation and film formation.
Table 1 summarizes the literature, on the modeling of the IP
process. The table shows that in general, the different physicochemical rate and equilibrium processes which have been considered
are some or all of the following: (i) ionic equilibria for the aqueous phase monomer, (ii) transport of the aqueous phase monomer
and/or the organic phase monomer from bulk phases to the site of
reaction, (iii) the reaction between the two monomers, and finally,
(iv) the phase separation of the formed oligomeric species. As for
modeling the film formation, three different approaches can be
seen. In the first, the reaction is assumed to occur at the interface (initially between the two liquid phases, and later between
759
Table 1
Summary of literature on modeling of IP
Sr. no.
Author
Rate/equilibrium processes considered
Diffusion
Kinetics
1
2
Pearson and Williams [7]
Sirdesai and Khilar [8]
Second order reaction
Second order reaction
No
No
3
4
Yadav et al. [9]
Janssen and Nijenhuis [10]a
Second order reaction
Diffusion control (no reaction kinetics)
No
No
5
Yadav et al. [11]
Second order reaction at the interface
No
6
Karode et al. [12]
Diffusion of diol through polymer film
Diffusion of diamine through swollen polymer and through
pores separately
Diffusion of diamine through polymer film
Diffusion of triamine through top layer and sub layer
separately
(i) Ionic equilibria in aqueous phase; (ii) mass transfer of
diamine from bulk to interface; (iii) diffusion of
unprotonated diamine through polymer film
(i) Mass transfer of both monomers from bulk to interface;
(ii) diffusion of diamine through polymer film
MW and PD
7
Karode et al. [13]
(i) Mass transfer of both monomers from bulk to interface;
(ii) diffusion of diamine through polymer film
MW and PD
8
Kubo et al. [14]a
(i) Mass transfer of triamine from bulk to interface; (ii)
diffusion of unprotonated triamine through polymer film
Diffusion of triamine through polymer film
Diffusion of both amine and acid chloride through formed
polymer gel
Diffusion of triamine through polymer film along with
byproduct HCl diffusion
(i) Detailed kinetics, reaction in a zone;
(ii) phase separation by spinodal
decomposition
(i) Detailed kinetics, reaction in a zone;
(ii) phase separation by nucleation and
spinodal decomposition
Second order reaction
Second order reaction
Second order reaction
No
No
Second order reaction
No
9
10
Ji et al. [15]a
Freger and Srebnik [16]
11
Bouchemal et al. [17]a
a
Model predictions: MW,
PD and crystallinity
No
Tri-functional amine is used instead of bi-functional along with the bi-functional isocynate or acid chloride.
the already-formed film and organic phase) and the entire polymer formed is assumed to form the film [10,11,14]. In the second,
the reaction is assumed to occur in a reaction zone which lies on
the organic side of the interface mentioned above, and the polymer
formed is excluded from the reaction zone as it forms; the reaction zone gets pushed into the organic phase as the film grows
[12,13,15,17]. In the third, the reaction is assumed to occur in a
steady reaction zone having a finite thickness, in which the polymer forms and accumulates (the viscosity in the zone increasing
as a result), ultimately taking on a gel-like form [16]. In the first
two approaches, the film thickness is explicitly calculated, increases
with time and increases the diffusion resistance with time. In the
third approach, a film thickness is calculated based on the polymer
concentration in the reaction zone, and the diffusion of monomers
is modeled as taking place through a gel-like structure.
Experimental evidence [3,5,6,11,18–21] points to a strong influence of the conditions employed in the preparation on the nature
and properties of the film that forms. However, most of the models [7–11,14–17] focus on the kinetics and the variation of film
thickness with time, and do not attempt to predict quantitatively
the polymer properties as a function of process parameters. Properties such as molecular weight, polydispersity and crystallinity
affect important characteristics of the polymer [20–22] such as
mechanical properties, viscosity, ease of processing, permeability.
Indeed, in their work on encapsulation, Yadav et al. [22] observed
an order-of-magnitude variation in the permeability of the capsule
wall because of variations in crystallinity. Karode et al. [12,13] were
the first to consider the detailed kinetics in their models and hence
predict the molecular weight distribution. Their models however,
have not been adequately tested against experimental data. For one,
they were developed for the unstirred nylon 6–10 system and may
require modifications for other systems. Even for the nylon system, the model assumes, as does much of the earlier literature,
that the solvent effect is explained solely by the partitioning of the
aqueous phase monomer. Recent work in this laboratory [5,6] has
shown the effect to be much more complicated, with solvent properties such as polarity playing a significant role. Properties such
as crystallinity, while experimentally shown to be dependent on
preparation conditions [11,19], have not so far been modeled.
As already remarked, the fast kinetics of membrane formation
makes it difficult to monitor the reaction and the development of
structural attributes during IP. Tracking the reaction via reactant
consumption or polymer film thickness has been attempted. The
latter is particularly difficult in microencapsulation studies since
the film is extremely thin and the amount of polymer formed,
miniscule. Yadav et al. [9] used an on-line pH probe to follow
the consumption of the aqueous monomer, since in their system,
the reaction does not produce any species that changes the pH
of the system. Chai and Krantz [23] proposed the techniques of
light reflectometry and pendant drop tensiometry to follow the
development of film thickness and rigidity. While these techniques
have potential, they would need considerable refinement before
quantitative information on kinetics can be obtained from them.
Interpretation of kinetic data is another area that requires care.
Since the overall mechanism of reaction involves physical transport and chemical reaction, issues of transport limitations and
controlling regimes should be considered in estimating the kinetic
parameters (reaction rate constant and/or diffusivity). While such
considerations have often not been employed, an exception is the
work of Yadav et al. [11] in which explicit criteria are established
and used for regime identification, albeit with a simplified kinetic
model. Table 2 gives a summary of the literature on experimental
methods and parameters estimated.
With the above background, the present work describes a comprehensive modeling framework of unstirred IP for microcapsule
formation, by extending the work of Karode et al. [13]. In addition to kinetics, the model predicts the evolution of film thickness,
mass crystallinity and MWD with time. Our recently reported
experimental data on the synthesis of polyurea membranes in two
solvents [5,6] have been used to test the model and estimate model
parameters.
2. Experimental
Experiments on the IP reaction producing polyurea microcapsules were reported in our earlier work (Wagh et al. [5,6]).
The studies employed hexamethylene-1,6-diamine (HMDA) as the
760
Table 2
Summary of literature: experiments and parameter estimation
Sr. no.
Author
Parameter estimated from model
Experimental data used for parameter estimation
1
2
3
4
5
6
7
Pearson and Williams [7]
Yadav et al. [11]
Karode et al. [12]
Karode et al. [13]
Kubo et al. [14]
Ji et al. [15]
Bouchemal et al. [17]
Rate constant and diffusivity
Rate constant and diffusivity
Diffusivity
Rate constant and nucleation rate constant
Rate constant and diffusivity
Diffusivity
Rate constant, diffusivity, porosity, swelling rate, etc.
Fractional conversion of diisocyanate
Fractional conversion of diamine from kinetic and diffusion control experiment
Polymer film thickness variation with time
Polymer film thickness variation with time
Fractional conversion of triamine
Polymer film thickness variation with time
Polymer film thickness variation with time
aqueous monomer and hexamethylene-1,6-diisocyanate (HMDI) as
the organic monomer, the reaction being conducted at the dropcontinuous phase interface of a oil-in-water dispersion to produce
microcapsules a few microns in diameter. Experiments were conducted over a range of monomer mole ratio (R), moles of limiting
monomer (nL ), phase volume ratio (Vd /Va ) and organic solvents. The
on-line fast pH-probe technique of Yadav et al. [9] was employed to
follow the kinetics. The polymer wall of the capsules was recovered
and characterized with respect to intrinsic viscosity (as a measure
of molecular weight) and mass crystallinity (WAXD). All reactions
were carried out at room temperature (29–30 ◦ C), and repeat runs
were carried out so that statistically significant conclusions could
be arrived at. Details are available in [5,6].
sidering a small section of the aqueous–organic interface with the
formed polymer film separating the phases at some intermediate
stage of the reaction, the concentration profiles of the monomers
in the film and adjacent regions are shown schematically in Fig. 1.
The rate and equilibrium processes considered in the model are
clear from the figure. The aqueous phase monomer (in practice,
usually with an amine function) undergoes protonation in solution
according to
−A + H+ −AH+
(1)
so that the monomer A–R–A exists in unprotonated, singly protonated and doubly protonated forms, their relative abundance given
by the ionic equilibria
H+ A − R − AH+ H+ A − R − A + H+
(2)
3. Theory
H A−R−AA−R−A+H
The model developed in this work takes that of Karode et al.
[13] as the starting point. The model has been extended to nonbuffered systems with the incorporation of ionic equilibria in the
aqueous phase, and extended to enable predictions of crystallinity.
While the model is developed in general terms, with the available
data in mind, its applicability to the polyurea system, and in the
production of self-supported films, is the specific point of focus in
the following description. In particular, the kinetics would need
appropriate changes when the monomer functionality is different
from two, as is often the case with IP films for RO applications.
The system considered is the microencapsulation of an organic
phase dispersed as drops of uniform size in an aqueous phase, by the
interfacial polycondensation reaction between an aqueous phase
monomer A–R–A and an organic phase monomer B–R –B. For the
polyurea system, these are, respectively HMDA and HMDI. Con-
The concentration of the unprotonated form A0a in the bulk aqueous
phase is thus related to the total concentration AT of the monomer
by the following relation [11]
+
A0a =
+
AT
AT
,
=
f (h)
(1 + h/Ka2 + h2 /Ka1 Ka2 )
(3)
(4)
where Ka1 and Ka2 are the equilibrium constants of the reactions
(2) and (3), respectively. Since the reaction occurs in the organic
phase, transport of only the unprotonated form is considered. Two
rate processes – external mass transfer and diffusion through the
(swollen) polymer film – transport the monomer to the reaction
zone, which is assumed to be of thickness ‘ε’, and located on the
organic side of the interface (see [13] for details of the concept of a
reaction zone). Clearly, if the pH of the external (aqueous) phase is
controlled (say through the use of a buffer), the unprotonated form
Fig. 1. Schematic diagram showing the different regions and physicochemical processes considered in the modeling of interfacial polycondensation. Note the regions in
which external mass transfer, molecular diffusion, and reaction kinetics operate and the use of the partition coefficients to treat phase equilibria for HMDA at the interfaces.
761
Table 3
Reaction intermediates (A–: NH2 –; B–: NCO–; R: (CH2 )6 ; R : (CH2 )6 for the present
polyurea system)
Notation
Species
Structure
–X–
A0
B0
C0
An
Bn
Cn
Repeat unit
HMDA
HMDI
Oligomer
Oligomer
Oligomer
Oligomer
−[NH − OCNH − R − NHCO − NH − R]−
A−R−A
B − R − B
A − R − NH − NHCO − R − B
A − R − Xn − B
B − R − Xn − B
A − R − Xn − NH − NHCO − R − B
Table 4
Generalized reaction scheme for the formation of polyurea oligomers
Reaction
Values of m and n
Rate
Rate constant
A0 + B0 → C0
Am + Bn → Cm+n
Am + Cn → Am+n+1
Bm + Cn → Bm+n+1
Cm + Cn → Cm+n+1
m, n = 0
m, n ≥ 0
m, n ≥ 0
m, n ≥ 0
m, n ≥ 0
ki A0 B0
kp1 Am Bn
kp2 Am Cn
kp3 Bm Cn
kp4 Cm Cn
ki = 4k
kp1 = 4k
kp2 = 2k
kp3 = 2k
kp4 = 2k
is a constant fraction of AT , and consideration of ionic equilibria
is not crucial. This corresponds to the approach taken by Karode
et al. [12,13]. If however, as in the polyurea system, the pH of the
aqueous phase is not held constant, ionic equilibria in the aqueous
phase must be considered.
As a result of reactions in the reaction zone, three types of
oligomeric species form in general, depending on the nature of the
end-groups as shown in Table 3. The possible reactions among the
oligomers are shown in Table 4. Reactions are treated using the
equal reactivity hypothesis as in Karode et al. [13], but the group
reaction rate constant is assumed to be a function of the organic
solvent used [5,6].
Solution thermodynamics governs the concentration at which
any given oligomer starts to come out of solution. Fig. 2 shows a
typical phase diagram for an oligomer Ymr . Two modes of phase
separation, viz., nucleation and spinodal decomposition are considered in the model. Depending on the temperature of reaction, there
is a volume fraction ˚Ybn
(the lower binodal concentration), above
mr
which the oligomer is unstable with respect to phase separation
into a polymer-rich phase (with the upper binodal composition)
and a polymer-lean phase (with the lower binodal composition).
When the composition is between the binodal and the spinodal
limits (the ‘metastable’ region), this phase separation occurs by
nucleation. Classical nucleation theory is used to calculate the size
of the nuclei that form. The nuclei coalesce and form a film when
their total projected area becomes equal to the interfacial area. If the
volume fraction of the oligomer in solution continues to increase
and hits the spinodal curve at some stage, an instantaneous ‘spinodal’ decomposition occurs, with the concentration of the oligomer
returning to the lower binodal value, and the excess polymer coming out of solution at the upper binodal composition and adding
to the film thickness. During a spinodal decomposition event, any
existing (uncoalesced) nuclei in the reaction zone are also swept
into the film.
In order to predict the crystallinity of the phase that separates
out and ultimately forms a film, we assume that the rate of phase
separation determines the time available for crystallization while
the kinetics of crystallization determine the time required for crystallization. The crystallinity achieved would be determined by how
these two time constants compare. A simple phenomenology is
therefore proposed in this work. The kinetics of crystallization is
assumed to be fast with respect to nucleation, but slow with respect
to spinodal decomposition, so that the nucleated species have the
maximum crystallinity allowed by the structure (Xmax ) while the
spinodally decomposed material is completely amorphous. The
degree of crystallinity of the film can therefore be calculated as a
weighted average of crystallinities of the material phase-separated
by the two mechanisms.
The quantitative details of the above physical picture are presented in the following sections.
3.1. Kinetics of monomer consumption and oligomer formation
At any time when the polymer film has built up to a thickness ıt ,
the rate of consumption of monomer A in the bulk aqueous phase
is given by (subscripts a, s, p and r stand, respectively for aqueous
phase, organic phase, polymer and reaction zone)
−
DA0 aI dAT
= kLA0 aI A0a − A0ap =
KA0ap A0ap − KA0sp A0r
dt
ıt
(5)
where kLA0 is the external mass transfer coefficient in the aqueous phase and the K’s are partition coefficients which describe the
phase equilibria at interfaces (see Fig. 1). Solving Eq. (5) for A0ap we
obtain
A0ap =
kLA0 ıt /DA0 KA0ap A0a + KA0sp /KA0ap A0r
1 + kLA0 ıt /DA0 KA0ap
(6)
Hence,
−
dAT
= keff aI A0a − KA0sa A0r
dt
(7)
where the effective transport coefficient keff is given by
keff =
kLA0
(8)
1 + kLA0 ım /DA0 KA0ap ıt /ım
The rate of consumption of monomer B in the bulk organic phase
is given by
−
dB0s
= kLB0 aI
dt
V
a
Vs − Vr
(B0s − B0r )
(9)
The mass balance equations for the monomers in the reaction zone
are given by
dA0r
k = eff A0a − KA0sa A0r − ki A0r B0r
ε
dt
nmax
Fig. 2. Schematic representation of phase diagram for a typical oligomer (UCST:
upper critical solution temperature).
− kp1 A0r
m=1
nmax
Bmr − kp2 A0r
m=0
Cmr
(10)
762
dB0r
kLB0
Amr − kp3 B0r
Cmr
=
(B0s − B0r ) − ki A0r B0r − kp1 B0r
ε
dt
nmax
nmax
m=1
m=0
(11)
While, in theory, chains of all possible lengths need to be considered, a maximum chain length nmax has been introduced above in
order to achieve closure in numerical computations.
For higher oligomers, the balance equations have to account for
the possibility of phase separation. A general mass balance equation
for any oligomeric species Ymr within the reaction zone thus has the
form
dYmr
bn
= rYmr − u Y mr rYmr ppt
dt
(12)
for
m = 1 to nmax
m = 0 to nmax
Ymr < Y mr
sp
for Ymr = Amr and Bmr
for Ymr = Cmr
bn
vs ıp − ıs
m = 1 to nmax
m = 0 to nmax
Ymr ≥ Y mr ,
sp
bn
for Ymr = Amr and Bmr
for Ymr = Cmr
bn
Ym ˚ Ymr
(14)
MYm
The function u is the step function
u(x) = 0
for x < 0
(15)
u(x) = 1
for x ≥ 0
(16)
and serves to switch on the nucleation mechanism for phase separation in Eq. (12) when the composition is between the binodal and
spinodal envelopes, and switch it off otherwise.
rYmr , the net rate of generation of any Ymr species by reaction
in the reaction zone, is given for the different types of oligomeric
species considered, by the following expressions
m−1
nmax
A(m−n−1)r Cnr − kp1 Amr
n=0
nmax
Bnr − kp2 Amr
n=0
Cnr ,
n=0
m = 1 to nmax
(17)
nmax
m−1
B(m−n−1)r Cnr − kp1 Bmr
n=0
Anr − kp3 Bmr
Cnr ,
(18)
rC0r = ki A0r B0r − kp2 C0r
Anr − kp3 C0r
n=0
m
n=0
n=0
Bnr − kp4 C0r
C(m−n−1)r Cnr − Cmr
Bnr + kp4
n=0
Cnr
kp2
(19)
nmax
Anr
n=0
nmax
,
m = 1 to nmax
2Ymr
fvYmr
(22)
where Ymr is the interfacial energy between the nucleus and the
surrounding lean phase and fvYmr is the free energy of coagulation
per unit volume. Kamide et al. [27] provide equations to calculate these quantities. The above equation shows that the closer the
composition is to the binodal, the larger the critical nucleus size.
Clearly, the probability of formation of such large nuclei as calculated on or very close to the binodal curve should be negligible. In
order to account for this, in this work, we disallow nucleation if
the nucleus size as calculated above is more than the thickness of
the reaction zone. Thus, for a critical nucleus to form, we should
have
(23)
dVXYmr
bn
= km ˚ Ymr
dt
m = 1 to nmax
m = 0 to nmax
for Ymr = Amr and Bmr
for Ymr = Cmr
(24)
Cnr
n=0
n=0
nmax
+ kp3
m−1
nmax
n=0
A(m−n)r Bnr + kp4
nmax
RCNYmr = −
The rate at which the nucleated phase grows in volume is modeled
as [28]
n=0
m = 1 to nmax
nmax
3.2.1. Phase separation by nucleation
The treatment of nucleation in this work is based on [13] with
additional features that allow a prediction of the crystalline fraction. Classical nucleation theory may be used to calculate the critical
nucleus size of any oligomer Ymr as
2RCNYmr
≤1
ε
nmax
n=0
(21)
where ıp and ıs are the solubility parameters of an oligomer
and solvent, respectively. The ıp values are calculated for
each oligomeric species based on a group contribution method
[26].
In the above equation, the function Y bn
mr shows the distance of the
instantaneous composition from the binodal envelope:
Y mr = Ymr −
2
RT
(13)
for
rCmr = kp1
The events leading to phase separation and the mechanisms
of phase separation considered in this work were described earlier with reference to the phase envelopes of Fig. 2. In this
work, the Flory–Huggins theory has been used to calculate binodal and spinodal curves for each oligomer at the temperature
of reaction [24]. The solution is assumed to be sufficiently
dilute, and interactions between oligomers sufficiently small, that
the curves for each oligomer can be calculated as in a binary
solution of that oligomer in the solvent. The oligomer–solvent
interaction parameter , is estimated by the following equation
[25]:
= 0.34 +
Ymr = Y mr
rBmr = kp3
3.2. Phase separation and film formation
and
rAmr = kp2
In order to avoid double counting in the second last term of the Eq.
/ n.
(20), the kp4 should be kp4 /2 if (m − n − 1) =
(20)
where km is a nucleation rate constant (assumed the same for
all species here). As described earlier, this material is assumed to
achieve the maximum crystallinity allowed by the structure [29],
and the X in the subscript on the LHS denotes this. While the phaseseparating material at any stage can either form new nuclei or
deposit on the existing nuclei causing their growth, we assume
here, as do Karode et al. [13], that the nucleation rate is sufficiently
high that growth can be neglected.
Eqs. (22) and (24) allow us to calculate the number of critical nuclei (NCNYmr ) formed per unit volume, at any time for any
763
oligomeric species Ymr
dNCNYmr
=
dt
3
1
3
RCNY
Vr
mr
4
m = 1 to nmax
m = 0 to nmax
dVXYmr
dt
for Ymr = Amr and Bmr
for Ymr = Cmr
(25)
Once the nucleation rate is known, the rate of precipitation by
nucleation (rYmr ppt ) is calculated as follows
rYmr ppt =
m = 1 to nmax
m = 0 to nmax
km
bn
Y mr
Vr
for Ymr = Amr and Bmr
for Ymr = Cmr
At the spinodal decomposition event, any nuclei dispersed in the
reaction zone are also assumed to be swept into the film so that
phase separation starts afresh after the spinodal event. In similar
fashion, at the end of the reaction, all the remaining nuclei from
the reaction zone are assumed to be incorporated into the film. The
addition of crystalline material to the film in this fashion has to be
accounted for in the calculation of crystallinity.
The total polymer film thickness (ıt ) at any time, based on contributions from both nucleated and spinodally phase-separated
material, is given by
ıt = ıtX + ıtNX
(32)
(26)
The nuclei generated are randomly dispersed in the reaction zone.
These nuclei are assumed to coalesce and form a film when
nmax
2
NCNYmr Vr RCNY
≥a
mr
(27)
m=0
The
polymer film thickness (ıtX ), based on nucleated material (VX =
VX ), as a function of time, is then calculated as
ıtX =
MC0r dp ˛s
n 6p
Vd
L
(33)
(28)
where the calculation accounts for swelling through the swelling
index ˛s .
3.2.2. Phase separation by spinodal decomposition
sp
When ˚Ymr ≥ ˚ Ymr , i.e. the volume fraction of any species
equals or exceeds the lower spinodal limit, the oligomer precipitates out spontaneously by spinodal decomposition. The polymer
precipitated thus is considered to be completely amorphous, and
its amount (in any time step) can be calculated as follows. For any
oligomer whose concentration crosses the spinodal limit, a balance
across the spinodal decomposition event gives
bn
˚Ymr − ˚ Ymr
bn
(29)
bn
˚ Ymr − ˚ Ymr
where the NX in the subscript on left side denotes that this polymer
is non-crystalline. Summing such contributions over all oligomers
which achieve the spinodal concentration in a given time interval
gives the incremental volume of amorphous polymer added to the
film in that interval
⎛
VNX − VNXold = Vr ⎝
bn
˚ Amr
bn
bn
˚ Amr − ˚ Amr
+
bn
˚ Bmr
bn
bn
˚ Bmr − ˚ Bmr
⎞
+
bn
˚ Cmr
bn
˚ Cmr − ˚bn
C
⎠
where VNXold is the volume of amorphous polymer precipitated by
spinodal decomposition mechanism up to the current time step.
Note that the summation in Eq. (30) is taken over only those
oligomers which precipitate by a spinodal mechanism in the time
interval under consideration.
On spinodal decomposition, the concentration of the oligomer in
bn
the reaction zone immediately drops to ˚ Ymr and the film thickness
increases. The part of the film thickness (ıtNX ) due to spinodally
phase-separated material at any time is given by
VNX ˛s
aI Va
The most important feature of this model is its ability to predict
the crystallinity, molecular weight distribution and polydispersity
of the polymer at any reaction time. As only the polymer precipitated due to nucleation will contribute to the crystallinity, the
crystallinity of the polymer precipitated out of solution is given
by
XW = Xmax pXYm VXYmr
pXYm VXYmr +
(34)
pNXYm VNXYmr
whereXW and Xmax are, respectively the instantaneous and maximum mass crystallinity. The density ratio of crystalline to
amorphous polymer is between 1.1 and 1.17 for most of the polymers like polyamides, polyethylene, PET, etc. [26]. As data for
polyurea are not available, the density ratio of crystalline to amorphous polymer is taken as 1.13 and pNX is calculated from a
group contribution method [26] as a density of the phase-separated
oligomer.
The number average and weight average molecular weight of
the precipitated polymer, in the film and dispersed nuclei in the
reaction zone, at any time is given by
MN =
W
N
(35)
MW =
Q
W
(36)
(30)
mr
ıtNX =
ım =
3.3. Polymer properties
VX ˛s
aI Va
VNX = Vr
where now the ıtX includes any contributions arising from the
spinodal decomposition events as well. The maximum value of the
polymer film thickness (ım ) is given directly from the stoichiometry
of the reaction [11], by
(31)
Table 5
Parameters held constant in model simulations
Parameter
Value
Source
dp
kLA0 , kLB0
Ka1
Ka2
KA0ap
KA0sa (CH)
KA0sa (CCl4 )
nmax
Xmax
˛s
ε
1.2 × 10−6 m
5 × 10−3 m/s
10−9.83
10−10.93
0.375
320
53
100
0.5
1.6
1 × 10−8 m
Wagh et al. [5,6]
Calculated (Sh = 2)
Dean [31]
Dean [31]
Yadav [32]
Wagh et al. [5,6]
Wagh et al. [5,6]
–
Van Krevelen [29]
Yadav [32]
Karode et al. [12]
764
where
W=
n
max
VXAmr pXAm + VXBmr pXBm
nmax
+
m=1
N=
n
max VXA
m=0
mr
pXAm
MAm
+
VXBmr pXBm
MBm
Q =
+
+
mr
pXCm
n
max VNXA
MCm
+
m=0
VXAmr MAm pXAm + VXBmr MBm pXBm
VNXAmr pNXAm + VNXBmr pNXBm
mr
nmax
+
VXCmr MCm pXCm
n
max
+
m=0
m=1
3.4. Solution of model equations
The set of 9nmax + 7 odes (Eqs. (7), (9)–(12), (24), (25)), with the
initial conditions
B0r = B0s =
pNXAm
MAm
The polydispersity of the polymer is determined from the ratio of
MW and MN . The variation with time, of the total nucleated volume
(VX ) and spinodally phase-separated volume (VNX ), thus gives an
idea of how the properties of the polymer film vary with time, based
on the above equations.
AT = AT0 ,
+
A0r = C0r = VXC0r = NCNC0r = 0,
Amr = Bmr = Cmr = VXYmr = NCNYmr = 0
for m ≥ 1
VNXCmr pNXCm
m=0
m=1
m=1
0
B0s
,
nmax
m=1
nmax
VXC
m=1
n
max
VXCmr pXCm
n
max
(37)
was solved using the subroutine Livermore Solver for Ordinary Differential Equations (LSODE) [30]. The choice of nmax is based on a
satisfactory closure of material balance. The external mass transfer coefficients are calculated on the basis of a Sherwood number
of 2, since the drops are sufficiently small in size. Values of these
and other model parameters used in the computations are given in
Table 5.
4. Results and discussion
4.1. Polymer solution thermodynamics
Before proceeding to a prediction of kinetics and film properties with the model, we generate the polymer phase diagrams
for each oligomeric species using the Flory–Huggins theory [24].
The relevant thermodynamic parameters for the four solvents used
by Wagh et al. [5,6] are tabulated in Table 6, and typical phase
envelopes, generated for the oligomer A10 in cyclohexane and pxylene using these parameters, are shown in Fig. 3. It may be
expected that the ‘better’ solvents (with lower values of ) will
allow oligomers to grow to longer lengths and hence result in higher
molecular weights, but as Fig. 3 shows, the differences are negligible
at the temperature of interest (302–303 K). These considerations
therefore suggest that kinetics could have an important role to play
in controlling molecular weights. Calculations also show that, even
in the ‘better’ solvents, for all but the shortest chain lengths, the
lower and upper binodal limits are close to 0 and 1 in volume
fraction terms.
+
VNXBmr pNXBm
MBm
nmax
VNXC
+
mr
pNXCm
MCm
m=0
VNXAmr MAm pNXAm + VNXBmr MBm pNXBm
nmax
+
VNXCmr MCm pNXCm
m=0
The range of volume fractions over which the system is
metastable (and hence produces crystalline nucleates) is of interest in the context of the present theory. Calculations show this
interval (a) to be small in general, (b) to decrease rapidly as chain
length increases, and (c) to widen somewhat with an increase in
the solvent solubility parameter. These considerations are relevant
as far as the likelihood of spinodal decomposition is concerned, and
hence for the precipitation of amorphous polymer.
4.2. Estimation of kinetic parameters from data
The model parameters which need to be estimated by fitting
experimental data are the reaction rate constant (k), diffusivity
(DA0 ) and nucleation rate constant (km ). While one or both of the
first two parameters (depending on the extent of diffusion limitations) control the rate of reaction, the last parameter mainly
influences the film properties such as molecular weight and crystallinity. k and km would be expected to vary with the solvent. The
diffusivity (DA0 ) through the polymer film, while basically a function of polymer structure, can also depend on the solvent since the
structural attributes of the film in different solvents could be different. Thus, it is necessary to estimate all three parameters separately
for each solvent.
Preliminary simulations confirmed the expectation that the process kinetics were little influenced by km values. Hence a parameter
estimation strategy is followed in which k and DA0 values are estimated from the kinetic data, with km being held at some reasonable
value. Based on these k and DA0 values, km is estimated by matching the experimental crystallinity with the model prediction. Since
k and DA0 estimation involves fitting a pair of parameters with a
non-linear model, it is important to start with good initial guesses.
Table 6
Thermodynamic data for A10 oligomeric species
Solvent
Cyclohexane
CCl4
p-Xylene
Toluene
ıs (cal/cm3 )1/2
vs (cm3 /mol)
˚sp
˚bn
˚ bn
8.2
108.04
3.3486
0.00809
1.8708 × 10−23
0.9848
8.6
97.1
2.5413
0.01016
1.064 × 10−17
0.9622
8.8
123.44
2.8436
0.01125
1.37 × 10−16
0.9731
8.9
106.3
2.3743
0.01212
9.80 × 10−15
0.9535
The binodal and spinodal limits shown refer to a temperature of 303 K.
Fig. 3. A typical phase diagram for an oligomer: temperature–volume fraction profile of A10 species in cyclohexane and p-xylene.
765
In this work, we obtain such initial guesses using the simpler model
of Yadav et al. [11].
Yadav et al. [11], using lumped second order kinetics at the interface, obtain for the time t required for the reaction to proceed to any
extent,
t =
p
kLA0 aI
+
ln H + K1 (H − 1) +
pM˛nL Va a
DA0 KA0ap p
where
dp
6Vd
K 2
2
2
fD (H) +
H2 − 1
pdp KA0sp
6kKA0ap AT0
⎧
⎨
1
(R − 1)H p + 1
ln
+ K1
fK (H) =
R
⎩ (R − 1)p
H
+ K2
H
fK (H)
(38)
Hp
dH
(R − 1)H p + 1
1
⎫
⎬
H p+1
dH
(R − 1)H p + 1
⎭
(39)
1
and
fD (H) =
ln H + K1 (H − 1) +
+
1 p−1
K 2
K1
H
−p+1
2
2
H −1 +
1
−1
−
2−p
H −p − 1
p
K2 H
−p+2
−1
(40)
where H stands for the hydrogen ion concentration (h) normalized
with respect to its initial value, and is a function of conversion. The
three terms on the right in Eq. (38) quantify the contributions of
the three rate processes in series: external mass transfer, diffusion
through the polymer film and interfacial reaction.
From our reaction times and the values of kL in Table 5, external
mass transfer is unlikely to be the rate controlling process. Linearity (or lack of it) of a plot of time vs. fK (H) or fD (H) indicates the
controlling mechanism between the remaining two, and gives the
corresponding rate parameter.
We therefore estimate k and/or DA0 based on Eqs. (39) and (40).
While Yadav et al. [11] assume the reaction to be at the interface, in
the present model, we have a zone of thickness ‘ε’ adjacent to the
interface in which all reactions take place. The volumetric reaction
rate constant (k) required for the present study is therefore estimated based on the surface reaction rate constant just by dividing
it with reaction zone thickness, i.e. k = k /ε, where the constant k
now contains a partition coefficient to account for the fact that the
reaction is on the organic side of the interface.
A representative plot of fK (H) vs. time, for the data of Wagh et
al. [5,6] is shown in Fig. 4. The plot is linear up to approximately
82% conversion, suggesting kinetic control cover almost the entire
conversion range. Kinetic control is also suggested for most of their
Fig. 4. Linearity of fK (H) vs. time plot: data of Wagh et al. [5,6] with cyclohexane
solvent. Numbers shown on the lines indicate conversion at the last point considered.
data, as Wagh et al. [5,6] point out, by the strong dependence the
data show on organic-side parameters such as HMDI concentration
and solvent type. While the lines do not show an intercept of 0, this
is because of a brief initial period of adjustment occasioned by the
manner in which the reaction is started [5,6]. Based on the slopes
of these plots, the reaction rate constant (k ) has been estimated.
Beyond the period of kinetic control, there is a region of mixed
control, before diffusion control takes over, if at all. In most of the
experiments whose data are used here [5,6], diffusion control is not
seen till upwards of 99% conversion. Where possible, an estimate
of DA0 has been obtained based on linear plots of fD (H) vs. time at
large conversions.
Values of the parameters estimated as above are shown in
Table 7 for two different organic solvents, cyclohexane and CCl4
(bracketed values). By using these estimates as initial guesses, the
best-fit values of k and DA0 for the present model are found by
minimizing the root mean square residual. These values are also
tabulated in Table 7.
Morgan [3] estimates a rate constant value in the range 102 to
106 m3 /kmol s for homogeneous polycondensation. According to
Janssen and Nijenhuis [10], the polycondensation rate constant was
in between 104 and 106 m3 /kmol s for TDC and DETA system. A typical value for the diffusivity of a molecule such as HMDA through a
swollen polymer would lie in the range 10−11 to 10−15 m2 /s [9,12].
Comparison of such values with the values of fitted model parameters shows a reasonable agreement of the present estimates with
expectations from the literature.
Having determined k and DA0 , the nucleation rate constant (km )
has now to be estimated for both the solvents, by fitting the experimental crystallinity data [5,6]. This was done by selecting 3–4
random samples with each solvent so as to cover the entire range
Table 7
Estimated model parameters in two solvents: cyclohexane and CCl4 (values for CCl4 are shown in brackets)
Parameter
Reaction rate constant (m3 /kmol s)
Diffusivity, DA0 (m2 /s)
Nucleation rate constant km (m3 /s)
a
Surface reaction rate constant (m4 /kmol s).
Expt. run
S51 (T31)
S52 (T32)
S53
S51 (T31)
S52 (T32)
S53
Model
Yadav et al. [11]a
Modified
Present model
7.92 (0.41) × 10−4
5.03 (0.44) × 10−4
4.15 × 10−4
1.09 (0.29) × 10−11
1.3 (0.31) × 10−11
9.8 × 10−12
–
7.92 (0.41) × 104
5.03 (0.44) × 104
4.15 × 104
1.09 (0.29) × 10−11
1.3 (0.31) × 10−11
9.8 × 10−12
–
1.5 (0.085) × 104
1.2 (0.1) × 104
1.0 × 104
1.0 (0.5) × 10−11
5.0 (5.0) × 10−12
5.0 × 10−12
2.05 (0.588) × 10−5
766
Fig. 5. Conversion–time behaviour with cyclohexane solvent (R = 4.0, Vd /Va = 0.1,
0 = 2.0 kmol/m3 ): comparison of model with experinL /Vd = 0.5 kmol/m3 and B0s
ment.
of crystallinity observed experimentally. The fitted values of km for
both the solvents are shown in Table 7.
4.3. Comparison of model predictions with experimental data
Figs. 5 and 6 show a comparison between the predicted and
experimental conversion–time behaviour for two solvents, under
otherwise identical conditions. The variability in conversion measurements, estimated separately in repeat runs, was found to vary
from a minimum of 0.2% to a maximum of 3.5%. The fit is therefore
generally satisfactory, and especially so up to about 80% conversion for both the solvents (cyclohexane and CCl4 ) with a root mean
square residual of 0.033. Because of the weak influence of diffusion resistance in most of these experiments, diffusivity is poorly
estimated. Further, it is possible that the effective diffusivity of the
film varies during reaction, as the work of Yadav et al. [22] shows
the permeability of the capsule wall to be a strong function of the
crystallinity. It is possible to have a crystallinity-dependent diffusivity in the model formulation, but the resulting complexity is not
justified by the data available.
Fig. 6. Conversion–time behaviour with CCl4 solvent (R = 4.0, Vd /Va = 0.1,
0 = 2.0 kmol/m3 ): comparison of model with
nL /Vd = 0.5 kmol/m3 and B0s
experiment.
Fig. 7. Model predictions of the variation of film thickness with time of reaction
(nL = 26.91 × 10−4 moles, R = 0.9415, Vd /Va = 0.0484, cyclohexane solvent).
Using the values of the parameters as fitted above, the model
has been used to predict the variation of polymer film thickness
and properties such as crystallinity, molecular weight and polydispersity during the course of reaction. The variation in polymer
film thickness with time (conversion) for cyclohexane is shown
in Fig. 7. The film thickness is shown both in absolute terms and
as a fraction of the maximum thickness possible from a stoichiometric calculation (Eq. (33)). The stepwise nature of thickening of
the film is a direct result of the discontinuous way in which film
thickness increases in the model. The figure shows that the latter
quantity reaches only up to 0.7 instead of 1.0. The stoichiometric calculation assumes consumption of the A and B monomers in equal
molar amounts. This would not hold if there is a preponderance of
(say) the B-ended species and the film is composed of low chain
length oligomers, so that the imbalance in the end-groups makes
a difference to the overall amounts of A and B monomers incorporated into the film. Thus, the final conversion of HMDA for this
run was only 77% while HMDI gets completely converted. Also, the
oligomers which remain in solution at the end of the run do not
contribute to the film thickness according to the model, although
this amount would be expected to be small enough to be generally
negligible.
We may examine the relative abundance of the different
oligomeric species as a function of time during the reaction using
the model in order to examine the above-pictured scenario in
greater detail. Fig. 8(a–c) shows the variation (with time) in the
volume fractions of A, B and C type oligomers, respectively, in cyclohexane. For purposes of comparison, the spinodal limits of some of
the oligomers featured are shown by horizontal lines in Fig. 8(b).
It is clear that B-ended species dominate at all times, and A-ended
species are present at the lowest concentrations. All the species, in
general, show the characteristics expected of reaction intermediates in a sequential reaction scheme (except B0r , a reactant), their
variations punctuated by film formation events. Because of the
dominance of B-ended species, these species also dominate the
phase-separation events during the course of the reaction. After
each spinodal decomposition event, the volume fraction of the
oligomer jumps to the lower binodal, thereby influencing the kinetics of formation of other oligomers as well. Due to this, volume
fraction profiles of both the A- and C-ended species show a drop
just after a film formation event.
As the figure shows, the trends for A-type oligomers are somewhat different from those for the other two types. There are several
factors contributing to this. Firstly, we have to consider the fact that
the availability of monomer A in the reaction zone is influenced
767
Fig. 8. Predicted volume fraction profiles of (a) A, (b) B and (c) C type of species in
the reaction zone (nL = 26.91 × 10−4 moles, R = 0.9415, Vd /Va = 0.0484).
by pH variations in the aqueous phase—as the diamine concentration in the bulk aqueous phase decreases because of reaction,
the pH goes on decreasing, lowering the amount of unprotonated
diamine present in the bulk aqueous phase (Eq. (4)), and hence
the amount of A0r present in the reaction zone. Secondly, the final
increase in the volume fraction of A-ended species is because of
the total consumption of monomer B. As the value of R is less
than 1, monomer B is completely converted and at the end as
diamine continues to diffuse into the reaction zone, the B- and Ctype oligomeric species present in solution get converted into the
A-type species.
Fig. 9. Effect of k on model predictions: (a) molecular weight, (b) polydispersity and
(c) crystallinity.
It is instructive to consider the effect of the different rate
parameters, diffusivity, reaction rate constant and nucleation rate
constant, on the film properties like weight average molecular
weight, polydispersity and crystallinity. Calculations showed (as
expected for a reaction that is reaction controlled for the most part)
that the influence of diffusivity is slight on polymer properties.
Figs. 9 and 10 show how some of the film properties of interest,
namely the (weight average) molecular weight, polydispersity and
768
Fig. 11. Effect of nL on model predictions of (a) molecular weight and polydispersity
and (b) crystallinity (R = 0.5, Vd /Va = 0.045, Va = 533 ml).
Fig. 10. Effect of km on model predictions: (a) molecular weight, (b) polydispersity
and (c) crystallinity.
crystallinity, vary with fractional conversion for different values
of reaction rate constant and nucleation rate constant, respectively for the above discussed reaction conditions in cyclohexane.
The crystallinity variation shows some initial disturbances because
of the competition between nucleation and spinodal decomposition mechanisms, and then passes through a maximum. The
molecular weight and polydispersity values are found to increase
continuously with the fractional conversion. The polymer film
formed by the IP technique is found to be highly polydisperse
in nature because of the variety of oligomeric species which get
phase-separated by nucleation and spinodal decomposition mechanism.
The reaction rate constant shows a remarkable effect on the
polymer properties as shown in Fig. 9. At low values of the reaction rate constant, the rate of formation of oligomeric species is
slow, while the rate of nucleation and phase-separate is governed
by the value of km . This leads to a lower average molecular weight
with high polydispersity as compared to the situation for high reaction rates. Because of the low rate of reaction only a few oligomeric
species cross the lower spinodal limit which leads to the high crystallinity at the start; it decreases with increase in the value of the
reaction rate constant. For a given diffusivity and reaction rate constant, the nucleation rate constant also shows a significant effect
on the polymer properties (especially crystallinity) as expected, as
shown in Fig. 10. The molecular weight is found to decrease due
to increase in nucleation rate constant. This is because high nucleation rate constants will restrict the formation of long chain length
oligomeric species and ultimately leads to a high value of polydispersity. As crystallinity is dependent on the amount of nucleated
material, which in turn depends on the nucleation rate constant, the
crystallinity increases with increase in nucleation rate constant.
We may now study the influence of preparation conditions for a
given set of parameter values. Figs. 11 and 12 show the effect of the
number of moles of limiting monomer (nL ) and phase volume ratio
769
Fig. 13. Comparison of model predicted molecular weight and intrinsic viscosity:
cyclohexane.
Fig. 12. Effect of phase volume ratio on model predictions of (a) molecular weight
and polydispersity and (b) crystallinity (R = 0.5, nL = 120 × 10−4 moles, Va = 533 ml).
(Vd /Va ) on the polymer properties for the parameters of Table 7 for
cyclohexane. nL does two things: through its effect on the concentration of the limiting monomer, it controls the rate, and, through
the amount of polymer that can be produced, it controls the final
film thickness. The molecular weight and polydispersity are found
to decrease with increase in number of moles of limiting monomer
because of the increased rate of reaction which will restrict the formation of long chain length oligomeric species. The crystallinity
passes through a maximum and then decreases with increase in
moles of limiting monomer. On the other hand, the molecular
weight and polydispersity show an increase with increase in phase
volume ratio and then decrease. This may be attributed to an
increase in interfacial area available for IP reaction, which increases
the rate of formation of different chain length oligomeric species for
a given number of moles of limiting monomer and molar ratio. The
crystallinity is found to pass through a minimum with an increase
in the phase volume ratio.
Figs. 13–16 show a comparison between the model predictions
of molecular weights and crystallinity (end-of-run) values with
experimental data, for cyclohexane and CCl4 , with the fitted parameters as given in Table 7. The predictions of molecular weights have
been compared with the data on intrinsic viscosities [5,6]. In view
of the sensitivity of film properties to the value of km , predictions
are shown for two values of km .
The molecular weights predicted show a peak near about the
peak experimentally observed in intrinsic viscosities. The value of
km is seen to influence not only the values but also the trends in
Fig. 14. Parity plot of polymer crystallinity: cyclohexane.
Fig. 15. Comparison of model predicted molecular weight and intrinsic viscosity:
CCl4 .
770
icant insights into the way in which structural attributes of the
polymer film vary during reaction, and in response to preparation parameters. While the model assumes a constant diffusivity
for the entire course of the reaction, a variable diffusivity is easily
incorporated by fitting different diffusivities in different conversion
ranges.
While the underlying assumptions of the model are thus borne
out, more data, for example, on how the film properties vary with
time during the course of the reaction, will help better define the
values of the model parameters, especially those related to the
phase separation and film formation.
Nomenclature
a
aI
Fig. 16. Parity plot of polymer crystallinity: CCl4 .
behaviour as well. In this work, km has been determined only from
a few end-of-run crystallinity values, and these values have been
used as constants throughout the reaction. In reality, km may vary
during the course of the reaction and from run to run. Even if a
constant average value is to be used, such a value is better determined for each experiment from a variation in film properties with
time, the way the other two rate parameters have been determined.
It has not been possible to make such time-dependent measurements of molecular weight and crystallinity in this work. Also, a
more detailed assessment of the model would require determination of MWD’s. The difficulty here is the insolubility of polyurea in
most common solvents, but GPC with HFIP as the solvent seems to
be a possibility.
From the crystallinity parity plots (Figs. 14 and 16), in spite of
the scatter, a general agreement in the trends is seen. Since crystallinities are determined by a comparison of the area under the
crystalline peaks to the total area under the X-ray diffractogram,
some scatter should be expected in the data in spite of reproductions as described by Wagh et al. [5,6]. Further, the crystallinity at
the end of the reaction as measured includes the effect of aging
after the reaction has ended. In order to see if this makes a difference, some experiments were conducted in which the reaction
was stopped when pH had leveled off by adding HCl, and the crystallinities of such samples were compared with those determined
in the conventional way by allowing the capsules to age for a long
time in situ. It was found that the crystallinity values are lower in
case of HCl added sample by a few percentage points. In view of
the above factors, the agreement seen may be considered to lend
credibility to the general precepts on which the present model is
based.
A0a
A0ap
A0r
AT
B0r
B0s
0
B0s
dp
DA0
h
k
km
kLA0
kLB0
KA0ap
KA0sa
KA0sp
MN
MW
5. Conclusions
A comprehensive modeling framework has been proposed for
the interfacial polycondensation reaction used in the microencapsulation and in the manufacture of thin film composite membranes.
The model incorporates the salient physicochemical processes
involved in the diffusion of monomers, polymerization reactions,
phase separation and formation of a coherent membrane. The
model, with parameters fitted from kinetics and end-of-run crystallinity values, is able to predict the observed trends in the kinetics
of the reaction, the influence of preparation parameters on crystallinity, and also such important properties of the film as molecular
weight distribution and polydispersity. The model provides signif-
NCN
R
RCN
t
T
V
Vs , Vd
Xmax
XW
Ymr
interfacial area, m2
interfacial area per unit volume of aqueous phase,
m−1
unprotonated diamine concentration in bulk aqueous phase, kmol/m3
diamine
concentration
at
the
aqueous
phase–polymer interface, kmol/m3
diamine concentration in the reaction zone,
kmol/m3
total diamine concentration in the bulk aqueous
phase, kmol/m3
diisocyanate concentration in the reaction zone,
kmol/m3
diisocyanate concentration in the bulk organic
phase, kmol/m3
initial diisocyanate concentration in the bulk
organic phase, kmol/m3
diameter of microcapsule, m
diffusion coefficient of diamine through the polymer film, m2 /s
hydrogen ion concentration, kmol/m3
reaction rate constant between –NH2 and –NCO,
m3 /kmol s
nucleation rate constant, m3 /s
external mass transfer coefficient in the aqueous
film, m/s
external mass transfer coefficient in the organic
film, m/s
partition coefficient of diamine between aqueous
phase and polymer film
partition coefficient of diamine between organic
solvent and aqueous phase
partition coefficient of diamine between polymer
film and organic solvent
number average molecular weight of the precipitated polymer, kg/kmol
weight average molecular weight of the precipitated
polymer, kg/kmol
number of critical nuclei, #/m3
molar ratio of diisocyanate to diamine in bulk
critical nuclei radius, m
time, s
reaction temperature, K
volume, m3
volume of dispersed organic phase, m3
maximum crystallinity
weight-based crystallinity
concentration of oligomeric species of chain length
m, kmol/m3
771
Greek letters
swelling index
˛s
ε
reaction zone thickness, m
ım
maximum polymer film thickness, m
ıp
solubility parameter for polyurea oligomers,
(cal/cm3 )1/2
ıs
solubility
parameter
for
organic
solvent,
(cal/cm3 )1/2
ıt
polymer film thickness at time t, m
p
density of polymer, kg/m3
˚
volume fraction
polymer–solvent interaction parameter
density of Ymr species, kg/m3
Ym
Superscripts
bn
binodal, polymer-lean phase
bn
binodal, polymer-rich phase
sp
spinodal, polymer-lean phase
Subscripts
a
aqueous phase
d
dispersed phase
n, m
oligomer species number
NX
spinodal decomposition
ppt
precipitation
r
reaction zone
s
organic solvent
t
total
X
nucleation
Ymr
type of species either Amr , Bmr or Cmr
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