Modeling of vinylidene fluoride heterogeneous polymerization in supercritical carbon dioxide

Article abstract of DOI:10.1021/ma0504522

The heterogeneous polymerization of vinylidene fluoride in supercritical carbon dioxide has been investigated experimentally, and the obtained results have been interpreted through a detailed kinetic model. The comparison between model predictions and experimental data indicates the presence of two reaction loci: the continuous supercritical phase and the dispersed polymer phase. However, the presence of two reaction loci is not the result of the thermodynamic partitioning between two phases, but rather a kinetic effect. This in fact occurs because part of the radicals generated in the continuous phase, which are driven by thermodynamic equilibrium to diffuse to the dispersed phase, are actually terminated in the former before they can reach the latter. This provides a quantitative explanation for the bimodal molecular weight distributions often measured experimentally for this system.

Full text of DOI:10.1021/ma0504522

7150
Macromolecules 2005, 38, 7150-7163
Modeling of Vinylidene Fluoride Heterogeneous Polymerization in
Supercritical Carbon Dioxide
Philipp A. Mueller, Giuseppe Storti, and Massimo Morbidelli*
Swiss Federal Institute of Technology Zurich, ETHZ, Institut fu¨r Chemie- und
Bioingenieurwissenschaften, ETH-Ho¨nggerberg/HCI, CH-8093 Zurich, Switzerland
Marco Apostolo
Solvay Solexis S.p.A., World Headquarters, Viale Lombardia 20, I-20021 Bollate (MI), Italy
Roland Martin
SOLVIN S.A., Solvay Research & Technology Center, Rue de Ransbeek 310, B-1120 Brussels, Belgium
Received March 3, 2005; Revised Manuscript Received May 23, 2005
ABSTRACT: The heterogeneous polymerization of vinylidene fluoride in supercritical carbon dioxide has
been investigated experimentally, and the obtained results have been interpreted through a detailed
kinetic model. The comparison between model predictions and experimental data indicates the presence
of two reaction loci: the continuous supercritical phase and the dispersed polymer phase. However, the
presence of two reaction loci is not the result of the thermodynamic partitioning between two phases,
but rather a kinetic effect. This in fact occurs because part of the radicals generated in the continuous
phase, which are driven by thermodynamic equilibrium to diffuse to the dispersed phase, are actually
terminated in the former before they can reach the latter. This provides a quantitative explanation for
the bimodal molecular weight distributions often measured experimentally for this system.
1. Introduction
nation of these findings,3 other mechanisms have been
considered. In particular, the presence of a chain
transfer to polymer reaction3,4 has been shown to
account for the breadth of the MWD but not for its
bimodality, while imperfect mixing in the case of
continuous polymerization4 has been found not to
significantly affect the MWD.
In this work we intend to investigate this particular
aspect and, more in general, to provide a complete
description of the kinetics of this system. The detailed
kinetic model mentioned above has been extended to the
VDF precipitation polymerization in scCO2, and its
predictions have been compared with the results of
polymerization experiments carried out at different
pressure and monomer concentration values. It is worth
mentioning that, due to the complexity of the system,
the corresponding detailed models include inevitably a
very large number of parameters. Thus, to obtain a
reliable model, a specific effort has been made to
estimate most of the model parameters a priori, i.e.,
from independent sources, minimizing any direct fitting
to the experimental polymerization data.
In our previous contributions,1,2 a detailed model for
dispersion polymerization in supercritical carbon dioxide
(scCO2) was developed and applied to the case of methyl
methacrylate (MMA) polymerization. The main features
of this model with respect to similar models previously
reported in the literature are (i) it accounts for two
reaction loci, the supercritical continuous phase and the
dispersed particles, and (ii) it includes the detailed
description of the interphase mass transport of the
active polymer chains as a function of their length.
In the case of MMA dispersion polymerization in
scCO2, it was found that the amount of polymer chains
produced (i.e., terminated) in the continuous phase is
negligible compared to that produced in the polymer
particles. This is due to the fact that the interphase
transport of radical chains from the continuous to the
dispersed phase is much faster than the bimolecular
termination within the former. Consequently, the radi-
cal chains which are predominantly produced in the
continuous phase diffuse into the particles before they
terminate. These findings were confirmed experimen-
tally, in particular through the measured molecular
weight distributions (MWD) which were found to be
monomodal and made of chains grown in the dispersed
phase.2
2. Experimental Section
All experiments have been carried out at Solvay Research
& Technology Center in Brussels using a 2 L stainless steel
autoclave with maximum operating pressure of 39.5 MPa at
150 °C. The reactor was equipped with a jacket connected to
a heating/cooling circulator (-40 to 150 °C), magnetically
coupled stirrer with measured and controlled speed accom-
modating different agitator types, pressure transducer, Pt-100,
and Inconel rupture disk connected to a blow-down tank. A
45 °C pitched blade agitator with upward pumping action
was used at a rotational speed of 300 rpm. The reactor and
A different kinetic behavior has been reported in the
case of vinylidene fluoride (VDF) polymerization in
scCO2, in particular resulting in very broad, bimodal
MWDs. Although the presence of two polymerization loci
with different reaction rates could be an obvious expla-
* To whom correspondence should be addressed: e-mail
massimo.morbidelli@chem.ethz.ch; phone +41-1-6323034; Fax
+41-1-6321082.
10.1021/ma0504522 CCC: $30.25 © 2005 American Chemical Society
Published on Web 07/06/2005

Macromolecules, Vol. 38, No. 16, 2005
Vinylidene Fluoride Heterogeneous Polymerization 7151
Table 1. Operating Conditions and Recipes of the
Experimental Runs
different times. The amount of polymer produced was evalu-
ated by gravimetry, and the complete MWD was measured
by gel permeation chromatography (GPC). These measure-
ments were performed by preparing solutions of PVDF at 1 g
L-1 in N,N-dimethylformamide (DMF, HPLC quality grade)/
0.1 mol L-1 LiBr under heating (60 °C) and under stirring for
30 min. A Waters Alliance 2690 separation module equipped
with Waters µ-Styragel HR type columns (2 × HR-5E, 1 × HR-
2) and a Waters 2410 refractive index detector were used at
40 °C, flow rate of 1 mL min-1, and injected sample volume
100 µL. The molecular weight analysis is based on the
universal calibration procedure with PMMA narrow standards.
The viscosity-molecular weight relationship for PVDF has
been checked by the independent determination of the molec-
ular weight using static light scattering in N,N′-dimethyl-N,N′-
trimethylene urea as the most suitable solvent. The Mark-
Houwink constants used are K ) 1.32 × 10-4 dL g-1, a ) 0.674
(PMMA), K ) 5.63 × 10-5 dL g-1, and a ) 0.803 (PVDF).
DEPDC
VDF
T
[°C]
F
p0
[MPa]
run
[mol L-1
]
[mol L-1
]
[kg L-1
]
1
2
3
4
5
5 × 10-3
5 × 10-3
5 × 10-3
5 × 10-3
5 × 10-3
3.1
1.0
6.2
3.1
3.1
50
50
50
50
50
0.79
0.79
0.79
0.65
0.89
20.4
20.4
20.4
13.3
33.2
the different lines could be vented, evacuated, and purged with
nitrogen. CO2 and VDF were fed to the reactor in liquid form
by means of cooled, plunger-type metering pumps. The supply
tanks were placed on precision balances allowing for the
control of the quantities (precision of 1 g). Samples were
extracted from the reactor into a 25 mL stainless steel high-
pressure cylinder by expanding a small amount of reaction
mixture through a filter to low pressure so as to stop the
polymerization immediately. The unit was equipped with a
supervisory control and data acquisition system.
3. Model Development
2.1. Materials. VDF was obtained from the Solvay plant
Tavaux (F) and used as received. CO2 grade 4.8 was used as
received. Diethyl peroxydicarbonate (DEPDC) was synthesized
in a 500 mL three-necked flask equipped with thermometer,
dropping funnel, and reflux condenser. The synthesis proce-
dure was the following: 100 mL of demineralized H2O was
cooled to about 5 °C by an ice/water bath. Under stirring, 12
mL of ethyl chloroformate and 6.64 g of a 30% H2O2 solution
were added to the flask. Next, 24 mL of a 5 N NaOH solution
was added slowly via the dropping funnel so that the temper-
ature did not rise above 10 °C. The reaction mixture was
allowed to stir for another 10 min at about 5 °C. Afterward,
50 mL of precooled 1,1,1,3,3-pentafluorobutane (T ≈ 5 °C) was
added under increased stirring speed. The organic phase was
separated after 5 min in a separation funnel and collected in
a graduated Schott glass. The water phase was extracted a
second time with 46 mL of precooled solvent. The collected
organic phases were mixed and completed to 100 mL at 5 °C
(concentration: approximately 10 g/100 mL). The analysis for
active oxygen was done by dissolving 100 mg of FeCl3 in 10
mL of water (catalyst solution) and 10 g of KI in 40 g of water
(20% solution). 100 mL of acetic acid was mixed with 100 mL
of 2-propanol. Under stirring the two solutions were combined
under N2 atmosphere for 30 min. 1 mL of the DEPDC solution
was added to 20 mL of the KI solution, and afterward 2 drops
of the FeCl3 solution were added. After 15 min stirring at room
temperature, the free iodine was titrated with the Na2S2O3
solution (0.05 N). The same procedure was used for the blank
reagents. Obtained yields in peroxide were typically between
89% and 92%.
2.2. Reaction Procedure. In a typical polymerization
reaction, a weighted quantity of the DEPDC initiator solution
(100 g L-1 in 1,1,1,3,3-pentafluorobutane, synthesized by the
procedure described above) was injected by a syringe in the
open reactor, cooled to -5 °C. The solvent was then completely
evaporated by applying vacuum, followed by five nitrogen
purging and vacuum cycles to eliminate all traces of air. The
stirrer was turned on, and part of the total amount of CO2
(80%) was then pumped as liquid to the autoclave, followed
by the feeding of the liquid VDF. Next, the reactor was heated
under stirring to the reaction temperature (50 °C). Once the
temperature was stabilized at the set point, an additional
quantity of CO2 was pumped in to reach the desired pressure.
Finally, after a predetermined pressure drop, the autoclave
was cooled to 40 °C, and the pressure was isothermally
released through a heated control valve. Residual VDF and
CO2 were eliminated by applying vacuum for 3 h.
As mentioned above, the general model for heteroge-
neous polymerization in supercritical CO2 developed and
applied to the dispersion polymerization of MMA in
previous studies1,2 has been used. Therefore, only a
short description of the model is reported below, with
special attention to the few changes requested by the
specificity of the VDF precipitation polymerization.
3.1. Model Assumptions, Kinetic Scheme, and
Interphase Equilibria. The main characteristics of the
model are as follows: (1) Two reaction loci are consid-
ered: the polymer-rich dispersed phase and the CO2-
rich continuous phase. (2) Low molecular weight species
(solvent, initiator, and monomer) undergo very fast
transport between the phases and are therefore as-
sumed to be at equilibrium conditions at all times. For
larger molecular weight species the model describes the
kinetics of the interphase mass transport. (3) A chain
length dependent partition coefficient for polymer chains
between continuous and dispersed phase is considered.
(4) The process of particle formation or nucleation is not
simulated, and a constant number of particles is as-
sumed throughout the entire process. This is because
the nucleation period where the particle number changes
significantly is expected to be very short in these
systems.5,6 It is worth noting that no stabilizer is used,
and as a consequence, a polymer powder made of
irregular fragments of different sizes is typically found
experimentally. However, since the model needs to
estimate the specific particle surface area, ap, as a
function of conversion, X, the following relation has been
used:
1/3
Xf
X
f
a )
ap
(1)
( )
p
f
where ap is the ap value at X ) Xf and is used as an
adjustable parameter. As discussed later in detail, this
relation is compatible with spherical polymer particles
in constant number. (5) The crystalline part of the
polymer is assumed to be impermeable to all species and
therefore neglected when evaluating interphase parti-
tioning of the various species.
Experiments at three different initial monomer concentra-
tions as well as three different densities (pressures) were
carried out. A summary of operating conditions and recipes is
given in Table 1.
2.3. Analytical Techniques. To monitor the time evolution
of the reaction, samples were withdrawn from the reactor at
The kinetic scheme considered for VDF polymeriza-
tion includes, besides the usual initiation and propaga-
tion reactions, bimolecular termination both by combi-

7152 Mueller et al.
Macromolecules, Vol. 38, No. 16, 2005
nation and by disproportionation as well as chain
transfer to polymer:
for deviations from the arithmetic mean of the param-
eter υ/. The volume fraction φi is defined as
ij
kdj
9
8 2I•j
kIj
98 R1,j
nc
ωi
ωj
initiation:
Ij
φi )
(4)
∑
/
/
/
/
(
)
/
j)1
Fi υi
Fj υj
I•j + Mj
where ωi is the mass fraction of component i in the
mixture, and F/i and υi/ are the characteristic mass
density and close-packed molar volume of component i,
respectively. The mixing rule for the characteristic
interaction energy, ꢀ/mix is given by
kpj
98 Rx+1,j
propagation:
termination:
Rx,j + Mj
kfpj
Rx,j + Py,j 8 Px,j + Ry,j
ktdj
Rx,j + Ry,j
8 Px,j + Py,j
nc nc
1
ꢀ/mix
)
φ φ ꢀ/ υ/
i j ij ij
(5)
∑ ∑
ktcj
/
i)1 j)1
Rx,j + Ry,j 8 Px+y,j
υmix
with
Note that, when two subscripts are given, the first
one (x or y; x, y ) [1, ∞]) indicates the chain length and
the second one the phase (j ) 1 for the continuous phase
and j ) 2 for the dispersed phase). The same kinetic
scheme, but with different values of the kinetic rate
constants, is applied to both phases.
ꢀ/ij ) (ꢀ/ii + ꢀj/j)1/2δij
(6)
where ꢀ/ii and ꢀj/j are the characteristic mer-mer inter-
action energies for components i and j and δij is another
empirical binary parameter. The number of sites oc-
cupied by a molecule of the mixture, rmix, can be
calculated from the following mixing rule:
About the description of the interphase equilibrium
partitioning of the low molecular weight species, the
Sanchez-Lacombe model7,8 was used for monomer and
solvent, while for the initiator a constant partition
coefficient was used.1
nc
3.2. Equations. The basic model equations are the
mass balances of the low molecular weight species
(monomer, solvent, and initiator) and of the active and
terminated polymer chains, which are reported in the
following. The meaning of all variables is detailed in
the Notation section. Symbols in square brackets have
to be intended as molar concentrations and without
brackets as numbers of moles.
Low Molecular Weight Species. The material
balances of these species are the same as those reported
in ref 1, while the equations describing the interphase
partitioning have been changed as follows.
When using the Sanchez-Lacombe equation of state
(EOS) to model phase behavior and partitioning of a
specific system, three characteristic parameters, ꢀ*, υ*
and r, are needed for both pure components and
mixtures. While the values of these quantities for the
pure components are usually evaluated by fitting inde-
pendent experimental data of some suitable single-
component thermodynamic property (cf. refs 7 and 8),
for the mixture appropriate mixing rules are typically
used. In this work, the mixing rules proposed by
McHugh and Krukonis9 for a system of nc components
have been selected:
φi
1
)
(7)
∑
rmix
ri
i)1
where ri is the number of sites that a molecule of species
i occupies in the lattice. The interphase partitioning of
monomer and carbon dioxide has been evaluated by
equating the corresponding chemical potential in each
phase, µi, estimated through the following relationship:9
nc
ri
2
µi ) RT ln φi + 1 -
+ ri -F˜
φ υ/ ꢀ/ -
j ij ij
∑
(
)
/
{
[
]
[
rmix
(
j)1
υmix
nc
ꢀ/mix
φ υ/ + ꢀ/mix + RTυ˜ (1 - F˜) ln(1 - F˜) +
∑
j
ij
[
]
)
j)1
nc
F˜
ln F˜ + pυ˜ 2 φ υ/ - υ/
(8)
∑
j
ij
mix
]
}
)
ri
(
j)1
where F˜ ) 1/υ˜ is the reduced density. The Sanchez-
Lacombe EOS is applied to each phase and, combined
with the constraint of constant reactor volume (Vreactor
) V1 + Vtot), provides the volumes of the two phases,
Vj, and th2e overall pressure, p. Note that in agreement
with assumption 5 the volume of the polymer phase,
nc nc
υ/mix
)
φ φ υ/
(2)
∑ ∑
i
j
ij
i)1 j)1
V2 ) Vtot - Vc2ryst, used in all calculations is actually
2
only the amorphous part of the total polymer volume.
The crystal fraction was assumed to be constant during
the reaction and set to 65% as obtained experimentally
by DSC.
with
υ/ii + υ/jj
υ/ij )
ηij
(3)
2
Population Balance Equations (PBE) for Active
and Terminated Polymer Chains. The equations
used in this work have been modified with respect to
where ηij is an empirical binary parameter accounting

Macromolecules, Vol. 38, No. 16, 2005
Vinylidene Fluoride Heterogeneous Polymerization 7153
the previous ones2 in order to include the chain transfer
to polymer reaction:
able experimental polymerization runs and to enhance
model reliability. All the parameter numerical values
and the corresponding sources are listed in Table 2
while their derivation is discussed in the following.
4.1. Kinetic parameters: Continuous Phase. The
following Arrhenius-type expressions have been used in
order to account for temperature and pressure depend-
encies of the kinetic rate constants:
dRx,j
) kpj[Mj][Rx-1,j]Vj(1 - δ(x - 1)) +
dt
2fjkdj[Ij]Vjδ(x - 1) - (kpj[Mj] +
∞
(ktdj + k ) [R ])[R ]V - K A ([R ] - [R/ ]) -
∑
tcj
y,j
x,j
j
x,j
p
x,j
x,j
y)1
E
RT
∞
∞
k(T,p0) ) A exp -
(13)
(14)
(
)
kfpj[Rx,j]Vj
y[P ] + k x[P ]V
[R ] (9)
∑
∑
y,j
fpj
x,j
j
y,j
y)1
y)1
∆V#
RT
k(T,p) ) k(T,p ) exp -
(p - p )
0
(
)
0
x-1
∞
dPx,j
dt
1
) ktcj
2
[R ][R
]V + k [R ]V [R ] -
j tdj x,j j y,j
∑
∑
y,j
x-y,j
y)1
y)1
where A is a preexponential factor, E the activation
energy, ∆V# the activation volume, and p0 the reference
pressure, usually equal to the pressure at which the
values of k are available.
∞
Kx,jAp([Px,j] - [P/x,j]) + kfpj[Rx,j]Vj
y[P ] -
y,j
∑
y)1
∞
Propagation. No value could be found for the VDF
propagation reaction in the literature. However, for the
aqueous microemulsion copolymerization of VDF/HFP
(hexafluoropropylene), Apostolo et al.10 reported an
kfpjx[Px,j]Vj
[R ] for j ) 1, 2 and x ) [1, ∞] (10)
y,j
∑
y)1
where δ(x - x0) indicates the Kronecker delta function,
defined as equal to 1 for x ) x0 and 0 otherwise.
It is worth noting that, in the frame of this model,
the only required information about the polymer phase
morphology is the overall interphase surface area,
which, assuming equal spherical polymer particles, is
overall propagation rate constant, kp of 2100 L mol-1
s-1 at 80 °C and 2.2 MPa. Even though evaluated under
different conditions, it is quite reasonable to use the
same value in supercritical CO2 since it has been
reported that the solvent effect on the propagation rate
coefficient of different vinyl monomers (styrene and
methyl methacrylate) is practically nonobservable.11
Moreover, Beuermann et al.12 measured a kp difference
between polymerization in bulk and in supercritical CO2
less than a factor 2. The selected value was defined as
the average of the propagation rate constants, kpA and
kpB, of the two radical species corresponding to mono-
mers A (VDF) and B (HFP):
2
given by Ap ) 4πrp Np. In the following we use the final
value of the polymer particle specific surface area, afp,
as an adjustable parameter and estimate the number
of particles under the assumption of spherical geometry
as follows:
Xfm0VDFFPVDF2apf 3
Np )
(11)
36π
where Xf is the final conversion (at which afp has been
obtained), m0VDF the total initial monomer mass, and
FPVDF the density of the polymer. During the reaction
Np is assumed constant in time, and since the dispersed
phase volume V2 obviously increases, we can back
compute the increase in time of rp and of the specific
surface area as reported by eq 1, and from this the value
of the total particle surface area which is the only
morphological parameter in the model:
kpA[A] + kpB[B]
kp )
(15)
[A] + [B]
Applying the pseudo-kinetic approximation, kpA and kpB
are given as follows:
k
kpA ) kpAAPA + pBB PB
(16)
(17)
rB
afp3
k
Xfm0VDF
(12)
kpB ) kpBBPB + pAA PA
Ap )
2
rA
ap
Note that as a consequence of assumption 4 above, very
small values of Ap would lead to an accumulation of
polymer chains in the supercritical phase such as to
promote homogeneous nucleation that is however not
accounted for in this model.
More details about the numerical solution of the
resulting system of mixed algebraic-differential equa-
tions can be found in our previous work.2
where Pi is the probability of having a radical chain with
end group of type i, kpAA and kpBB are the homopolym-
erization constants, and rA and rB are the reactivity
ratios. For the system VDF/HFP the latter are equal to
2.9 and 0.12, respectively.13 Note that this set of values
is different from that we used in a previous paper,10 and
it has been selected because evaluated more recently.
Moreover, it is well-known that HFP does not homopo-
lymerize which means that kpBB , 1. Thus, the overall
propagation rate constant for this system can be ap-
proximated by the following expression:
4. Parameter Evaluation
The reliable evaluation of the large number of model
parameters is always a critical issue when developing
a detailed kinetic model. In this work, whenever pos-
sible, these have been estimated from independent
experimental data to minimize the fitting of the avail-
1 - xA
kp ≈ kpAAPA xA +
(18)
(
)
rA

7154 Mueller et al.
Macromolecules, Vol. 38, No. 16, 2005
Table 2. Model Parameter Values and Sources
units
parameter
reference
Kinetic Parameters: Continuous Phase
f1,0 ) 0.6
reference 20
Ad1 ) 6.3 × 1016
Ed1 ) 132
s-1
reference 20
kJ mol-1
cm3 mol-1
MPa
reference 20
reference 20
reference 20
reference 10, this work
references 14,15, typical value
references 14-16, typical value
reference 10
∆V#d1 ) 0
p0,d1 ) 27.6
Ap1 ) 6.4 × 105
Ep1 ) 15
L mol-1 s-1
kJ mol-1
cm3 mol-1
MPa
∆V#p1 ) -25
p0,p1 ) 2.2
Ap/
Ep/
) 6.5 × 109
) 70
L1/2 mol-1/2 s-1/2
kJ mol-1
cm3 mol-1
MPa
reference 18, this work
reference 18, this work
references 14-16, 19, typical value
reference 18
xt,1
∆V#t1 ) 15
xt,1
p0,p/
) 23.4
xt,1
kfp1/kp1 ) 1.0 × 10-6
fitted
Kinetic Parameters: Dispersed Phase
f2,0 ) 0.6
reference 20
Ad2 ) 6.3 × 1016
Ed2 ) 132
s-1
reference 20
kJ mol-1
cm3 mol-1
MPa
reference 20
∆V#d2 ) 0
reference 20
p0,d2 ) 27.6
Ap2 ) 6.4 × 105
Ep2 ) 15
reference 20
L mol-1 s-1
kJ mol-1
cm3 mol-1
MPa
reference 10, this work
references 14,15, typical value
references 14-16, typical value
reference 10
∆V#p2 ) -25
p0,p2 ) 2.2
Ap/
Ep/
) 6.5 × 109
) 70
L1/2 mol-1/2 s-1/2
kJ mol-1
cm3 mol-1
MPa
reference 18, this work
reference 18, this work
references 14-16,19, typical value
reference 18
xt,2
∆V#t2 ) 15
xt,2
p0,p/
) 23.4
xt,2
k
fp2/kp2 ) 1.0 × 10-6
fitted
Free Volume Parameters
D0 ) 7.66 × 10-4
cm2 s-1
J mol-1
cm3 g-1 K-1
cm3 g-1 K-1
K
by fitting data from reference 36
by fitting data from reference 36
by fitting data from reference 36
by fitting data from reference 35
by fitting data from reference 36
by fitting data from reference 35
references 33, 34
references 47, 48
references 33, 34
references 47, 48
E ) 8.21 × 102
K1,VDF/γ ) 1.04 × 10-3
K1,PVDF/γ ) 6.22 × 10-5
K2,VDF - Tg,VDF ) -0.6
K2,PVDF - Tg,PVDF ) 330
V/VDF ) 0.690
K
cm3 g-1
cm3 g-1
cm3 g-1
cm3 g-1
K-1
V/CO ) 0.589
2
V/PVDF ) 0.565
ref
V
) 0.231
FH,CO2
R
) 8.76 × 10-4
references 47, 48
CO2
ê
ê
) 0.484
references 33, 37
VDF/PVDF
) 0.284
references 33, 37
VDF/CO2
Thermodynamic Parameters
VDF
ꢀ* ) 2,632
J mol-1
by fitting data from reference 35
by fitting data from reference 35
by fitting data from reference 35
υ* ) 7.76
r ) 6.03
CO2
cm3 mol-1
ꢀ* ) 2,536
υ* ) 4.41
r ) 6.60
J mol-1
reference 49
reference 49
reference 49
cm3 mol-1
PVDF
ꢀ* ) 6,652
υ* ) 20.16
J mol-1
cm3 mol-1
g cm-3
reference 39
reference 39
reference 39
F* ) 1.920
binary interactions
δVDF/CO ) 0.925
by fitting data from reference 40
fitted
2
δVDF/PVDF ) 0.820
δCO2/PVDF ) 0.925
by fitting data from reference 40
by fitting data from reference 40
assumed
η
η
) 0.838
) 1.000
VDF/CO2
VDF/PVDF
ηCO2/PVDF ) 0.840
initiator partitioning
KDEPDC ) 1.0
by fitting data from reference 40
this work
oligomer partitioning
R1 ) -0.04
by fitting data from reference 45
by fitting data from reference 45
reference 45
R2 ) -0.2
m11 ) 4.04 × 10-1
m27 ) 9.25 × 10-2
particle surface area
afp ) 0.27
reference 45
m2 g-1
fitted
where xA is the molar fraction of monomer A in the
reaction mixture. Under the conditions reported in ref
10, a value of 3870 L mol-1 s-1 is estimated for the VDF
homopolymerization constant, kpAA. Using an activation
energy Ep ) 15 kJ mol-1 (typical for vinyl halogens14,15
and an activation volume ∆V#p ) - 25 cm3 mol-1
)

Macromolecules, Vol. 38, No. 16, 2005
Vinylidene Fluoride Heterogeneous Polymerization 7155
(typical for vinyl monomers14-16), an initial value of 2850
L mol-1 s-1 is obtained at the operating conditions of
the experimental run 1 in Table 1.
tailed treatments are available in our previous contri-
bution1 as well as in the literature:21-24
-1
DVDF,0
DVDF
1
f2,0
Chain Transfer to Polymer. The presence of this
reaction has been indicated by Maccone et al.,17 who
found branching in the case of terpolymers of VDF,
HFP, and tetrafluoroethylene (TFE). This result is
relevant to indicate the existence of branching in VDF
polymerization because this terpolymer is composed
primarily of VDF (72%) and branching on HFP and TFE
is unlikely, since these contain only C-F bonds. More-
over, Saraf et al.4 found a solid mass of polymer at
increased monomer concentrations that was not soluble
in any solvent, which indicates that branching or even
gelation can occur in the polymerization system under
examination here. In the same work a value of 2.6 ×
10-3 for the ratio between the chain transfer to polymer
rate constant, kfp1, and the propagation rate constant,
kp1, has been obtained by fitting experimental polydis-
persity index (PDI) values. This value is expected to be
quite overestimated due to the fact that this fitting was
done using a model describing the polymerization in one
phase only, while the presence of two polymerization
loci certainly contributes to increase the PDI. Since the
chain transfer to polymer reaction affects only the
broadness of the MWD, the value of this specific ratio
has been determined by fitting the experimental MWD
data obtained in this work. As discussed later, a much
f2 ) 1 -
1 -
(19)
(20)
(
)
[
]
-1
1
1
kp2
)
+
(
)
kp2,0 4πσVDFDVDFNA
kt2(x,y) )
-1
1
1
+
4πrxy(Dx,com + Dy,com + kp2[M2]a2/3)NA
k
(
)
t2,0
(21)
where DVDF is the monomer diffusion coefficient evalu-
ated in the frame of the free volume theory of Vrentas
25-27
and Duda,
σ
VDF
the Lennard-Jones diameter of the
monomer, rxy the separation at which the termination
between chains of length x and y is instantaneous, a
the root-mean-square end-to-end distance divided by the
square root of the number of monomer units in the
chain, and Dx,com the center-of-mass diffusion coefficient
for a chain of length x evaluated from the following
universal scaling law:28
)
Dx,com ) DVDFx-(0.664+2.02ω
(22)
PVDF
smaller value has been obtained, i.e., 1 × 10-6
.
being ωPVDF the polymer weight fraction. It is worth
noting that the actual values of kt2(x,y) used in this work
are the simple arithmetic averages of the two values
obtained from eq 21 using minimum and maximum
Bimolecular Termination. Charpentier et al.18
proposed an Arrhenius-type relationship for the ratio
between the propagation rate constant and the square
values of rxy (rxy ) σVDF, rigid chain limit, and rxy
)
root of the termination rate constant, kp1/ k , at
x
xt,1
t1
2ajc1/2, flexible chain limit, jc being the entanglement
spacing).29
constant density. Accordingly, they evaluated the cor-
responding “effective” activation energy Ep/
) 78 kJ
mol-1. However, to properly account also for the pres-
sure dependence, the relationship (14) has been used,
and using the above-reported activation volume for the
propagation reaction and 15 cm3 mol-1 for the termina-
tion reaction (typical value for vinyl monomers14-16,19),
a corrected value of the “effective” activation energy
The values of the kinetic rate constants for the
dispersed phase at zero conversion are set equal to the
corresponding values for the continuous phase.
Note that using eq 19 the initiator efficiency in the
dispersed phase is estimated to be about 3 times smaller
than in the continuous phase as reported above.20 This
means that, contrary to the case of PMMA,1,2 the radical
production is expected to be significant in both phases.
4.3. Mass Transfer Coefficient. The mass transfer
coefficient required in the population balance equations
(9) and (10) is evaluated according to the two-film
theory,30,31 yielding the following expressions for the
overall mass transfer coefficients referred to the two
phases:
Ep/
) 70 kJ mol-1 has been estimated. Since there
xt,1
is no evidence in the literature for a significant contri-
bution of termination by disproportionation,4 termina-
tion by combination is assumed to be the unique
bimolecular termination mechanism.
Initiator Decomposition. The first-order reaction
rate constant for the decomposition of DEPDC in scCO2
has been estimated using the values of activation
energy, Ed1, and preexponential factor, Ad1, reported by
Charpentier et al.20 and equal to 132 kJ mol-1 and 6.3
× 1016 s-1, respectively. Note that neither a significant
pressure effect nor a noticeable dependence on the
solvent type has been observed by the same authors.
About the initiator efficiency, values around 0.6 were
estimated for temperatures between 65 and 85 °C in
the same work, where no clear temperature dependency
was observed.
4.2. Kinetic Parameters: Dispersed Phase. Since
the polymer concentration within the particles remains
quite large all along the reaction (weight fraction around
0.8 in the amorphous part), diffusion limitations on the
kinetic processes have to be accounted for. In particular,
these have been evaluated using a Fickian description
of the reactant diffusion with diffusion coefficients
estimated through the free volume theory. The adopted
relationships are summarized below, while more de-
-1
mx
1
Kx,1
)
+
(23)
(24)
(
)
kx,1 kx,2
-1
1
1
Kx,2
)
+
(
)
mxkx,1 kx,2
The local transport coefficients are given by
Dx,j
kx,j
)
(25)
δj
where Dx,1 is evaluated using the correlation by Lusis
and Ratcliff,32 Dx,2 is obtained from eq 22, and δj is a
characteristic length, in this work estimated as the
particle radius. Figure 1 shows the overall mass transfer
coefficient referred to the continuous phase, Kx,1, as a
function of the chain length. It can be seen that for short

7156 Mueller et al.
Macromolecules, Vol. 38, No. 16, 2005
K1,PVDF/γ and K2,PVDF - Tg,PVDF, were obtained by fitting
the experimental temperature dependence of viscosity
provided by Solvay-Solexis35 through the modified Wil-
liams-Landel-Ferry equation (cf. ref 33). The corre-
sponding parameters for the monomer were obtained
in a similar way but adopting Doolittle’s expression for
the viscosity-temperature relationship (cf. ref 33) and
the experimental data taken from ref 36. About the ratio
of the critical molar volume of the jumping units, ê, the
expression proposed by Ju et al. was considered.37
Finally, the preexponential factor, D0, and the critical
energy required to overcome the attractive forces, E,
have been estimated by combining the Dullien equa-
tion38 for the self-diffusion coefficient of pure solvents
with the Vrentas-Duda free volume equation in the
limit of pure solvents, as proposed in ref 33.
4.5. Thermodynamic Parameters. As mentioned
above, the Sanchez-Lacombe EOS has been used to
describe the PVT behavior and the phase partitioning
of the multicomponent system under examination. This
equation requires three pure component parameters for
each component as well as two binary interaction
parameters for each component pair.
Figure 1. Overall mass transfer coefficient referred to the
continuous phase, Kx,1 (O), and local contributions of the
continuous phase, kx,1 (dashed line), and the dispersed phase,
kx,2/mx (solid line), as a function of chain length (cf. eq 23).
Pure Component Parameters. Briscoe et al.39
reported Sanchez-Lacombe pure component param-
eters of PVDF at 42 and 80 °C. Since the experimental
data considered in the present work were obtained at
50 °C (cf. Table 1), the values at the lower temperature
have been used. For CO2 the same set of values of pure
component parameters as in our previous contribution2
was considered here. About VDF, since no data are
available in the literature, the corresponding parameter
values were evaluated by fitting a set of PVT experi-
mental data obtained by Solvay-Solexis.35
Binary Interaction Parameters. The sorption data
at 42 and 60 °C reported in ref 40 have been used in
order to evaluate the binary interaction parameters of
PVDF/CO2. About the interaction between VDF and
CO2, experimental data of density as a function of
pressure at 42 and 60 °C for the binary mixture with
xVDF ) 0.103 have been used to estimate the corre-
sponding binary parameters.40 Concerning VDF/PVDF
interactions, the only information available in the
literature is a qualitative indication of very limited
monomer solubility in its polymer.3,40 Therefore, the
parameter δVDF/PVDF has been estimated by direct fitting
of the experimental reaction data while ηVDF/PVDF has
been set equal to one, thus neglecting entropic interac-
tions.
Initiator Interphase Partitioning. No data of
interphase partitioning for the adopted initiator (DEP-
DC) have been found in the literature. However, re-
ported partitioning data of dyes and a number of other
organic species between supercritical carbon dioxide and
different polymers indicate equipartitioning (i.e., Ki )
1) when the system pressure is large enough, usually
above 10-15 MPa.41,42 Since a similar behavior can
reasonably be expected in the case of DEPDC, a species
exhibiting special affinity to the polymer, the value
KDEPDC ) 1 has been used in this work.
Figure 2. Chain partition coefficient as a function of chain
length; experimental data from ref 45 (4) and fitted curve
using eq 26 (line).
chains this coefficient is dominated by the transport
within the polymer film whereas at increasing chain
length the local transport in the continuous phase gets
more and more important and is completely dominating
for chains containing more than about 60 monomer
units. Note that the behavior of the overall mass
transfer coefficient for long chains is the result of the
rapid decrease of the chain partition coefficient, mx, at
increasing x (cf. Figure 2).
4.4. Free Volume Parameters. The evaluation of
the diffusion coefficient through the free volume theory
requires the knowledge of several physical quantities,
as described in detail in ref 1. For brevity, we do not
repeat here the entire treatment, but we simply discuss
how the values of the various parameters involved,
which are summarized in Table 2, have been estimated.
For pure CO2, the same values used in our previous
work have been considered here. The pure component
parameters for VDF and PVDF have been either esti-
mated from or fitted to their pure component proper-
ties.33 In particular, the specific hole free volume, V/i ,
has been estimated from the molar volume at 0 K, which
in turn was evaluated by the group contribution method
of Sugden.34 The free volume parameters of the polymer,
Polymer Interphase Partitioning. An empirical
expression for the chain length dependent partition
coefficient in scCO2, similar to that reported for poly-
styrene (PS)43 and PMMA,44 has been used:
log mx ) log m11 + R1(x - 11)
log mx ) log m27 + R2(x - 27)
2 e x e 27
x > 27
(26)
{

Macromolecules, Vol. 38, No. 16, 2005
Vinylidene Fluoride Heterogeneous Polymerization 7157
Figure 3. Conversion as a function of reaction time calculated
by the model (line) and measured experimentally (0) for the
base case (run 1 in Table 1).
where the values of m11, m27, R1, and R2 have been
obtained by fitting the data reported in ref 45 (cf. Figure
2). Since these data have been measured at conditions
not too far from the ones reported in Table 1, the values
of the parameters in eq 26 have been kept constant in
all simulations, thus neglecting any temperature and
pressure effect. This last assumption has been verified
by Bonavoglia et al.45 for the system PS/CO2. Of course,
the partition coefficient of the species of length 1 (i.e.,
monomer and monomeric radical) has been computed
with the Sanchez-Lacombe EOS.
4.6. Adjustable Quantities. Even though most of the
model parameters have been estimated from indepen-
dent sources, three quantities were considered as ad-
justable parameters: the VDF/PVDF binary interaction
parameter, δVDF/PVDF, the chain transfer to polymer rate
constant, kfpj, and the final specific interphase surface
area, afp. Since the role of each parameter is different
(the first and the last ones have significant impact on
the whole system behavior, while the second one affects
only the broadness of the MWD), it was possible to get
reliable estimates of them by fitting a single specific
experiment (base case), while using all the remaining
ones to check the model predictive capabilities.
Figure 4. MWD at various conversions calculated by the
model (a) and measured experimentally (b) for the base case
(run 1 in Table 1).
values for the adjustable quantities (δVDF/PVDF, kfpj/kpj,
afp) are given in Table 2.
The conversion curve in Figure 3 reveals an accelera-
tion of the polymerization rate at increasing conversion
which indicates some kind of gel effect. In Figure 4, it
is seen that the MWD exhibits two modes: one in the
low molecular weight region, corresponding to chains
produced in the continuous phase, and the other one at
higher molecular weights originated by chains formed
in the polymer phase. The relative importance of these
two fractions changes during the reaction, the shorter
chains being predominant at low conversion. Looking
at the shape of the two modes, it can be seen that the
one at higher molecular weights is significantly broader
than the other. Moreover, as the reaction proceeds, the
higher molecular weight chains increase in average
length while the shorter remain substantially constant.
All these experimental observations are well reproduced
by the model, thus enabling us to propose the following
reaction mechanism.
The polymerization reaction proceeds in two loci: the
continuous phase and the polymer particles. At low
conversion, the first locus is dominant and low molec-
ular weight chains are predominantly produced. Since
no gel effect is operative in this phase, no acceleration
is observed in the first part of the conversion vs time
curve, and the first MWD mode is quite narrow, being
the corresponding polydispersity about 1.5. On the other
hand, as the reaction proceeds, the amount of polymer
particles increases, leading to the fast increase of the
relative importance of the second mode of the MWD,
which becomes dominant after the first hour. Since the
5. Model Validation
To check the reliability of the model developed in this
work, its results are now compared to the experimental
data. The model-experiment comparison has been
carried out in terms of conversion vs time and MWD vs
conversion. In particular, the effects of changing initial
monomer concentration on one side and density (i.e.,
pressure) on the other have been analyzed. Note that
the calculated MWDs have been obtained by accounting
for the polymer produced in both phases, Px ) ∑2 Px,j
j)1
as computed by the population balance equations (10),
and thus the plotted functions, x2Px/(∑x∞)1xPx) are pro-
portional to the experimental GPC results.
5.1. Base Case. As mentioned above, the experimen-
tal data corresponding to the base case (run 1 in Table
1) were used to evaluate the three adjustable param-
eters of the model. The parameter fitting was done in
order to get good agreement between model and experi-
mental results in terms of conversion (Figure 3) and
MWD (Figure 4) as a function of time. The obtained

7158 Mueller et al.
Macromolecules, Vol. 38, No. 16, 2005
Figure 6. Conversion as a function of reaction time predicted
by the model (lines) and measured experimentally: runs 4
(dashed, 4), 1 (solid, 0), and 5 (dotted, O) in Table 1.
Figure 5. Total particle surface area (solid) and particle
number (dashed) as a function of conversion for the base case
(run 1 in Table 1).
termination rate in the polymer particles is significantly
reduced by the gel effect, the MWD of the second mode
exhibits larger values than the first one, and its average
value increases with conversion. In addition, the con-
version vs time curve undergoes the acceleration process
typical of the gel effect. Finally, since the high molecular
weight chains exhibit larger probability to undergo
chain transfer to polymer reaction, this mode of the
MWD is significantly broader than the first one.
It is worth repeating that the obtained value for the
ratio between the polymer chain transfer reaction rate
constant and kp reported in Table 2 is about 3 orders of
magnitude smaller than the one reported by Saraf et
al.4 As anticipated, this is due to the fact that Saraf et
al. used an homogeneous model to fit the experimental
polydispersity values. These are very high because the
experimental MWD exhibits bimodality, which is how-
ever not predicted by the homogeneous model. Since in
our model bimodality is properly accounted for by the
presence of two reaction loci, the extent of branching to
polymer needed to reproduce the experimental polydis-
persities is much smaller.
About the particle morphology, a constant number of
spherical particles was assumed to be present from the
beginning of the reaction so as to reproduce a certain
value of the final specific interphase surface area
(assumption 4). Typical evolutions of the total particle
surface area and of the particle number computed under
this assumption are shown in Figure 5.
5.2. Effect of Density. About the density effect on
reaction rate and MWD, at least for the range of density
values investigated in this work, no significant trend is
found, neither experimentally nor in the model predic-
tions (cf. Figures 6 and 7). This observation is in
agreement with the findings of Hsiao et al.46 for disper-
sion polymerization of MMA in scCO2. Although as a
whole, taking also the experimental errors into consid-
eration, this result is satisfactory, a more detailed
analysis is possible. A closer inspection of Figures 6 and
7 reveals some density effects, i.e., a slight increase of
the polymerization rate and a small shift of the two
modes of the MWD toward higher molecular weights
at increasing density. This can be the result of the effect
of density on various parameters in conflict with each
other. In fact, changing the system density affects the
kinetic rate constants as well as the interphase parti-
Figure 7. MWD at 25% conversion predicted by the model
(a) and measured experimentally (b): runs 4 (dashed, 4), 1
(solid, 0), and 5 (dotted, O) in Table 1.
tioning of both monomer and CO2. Larger densities
usually correspond to larger propagation rate constant
and smaller termination rate constant (cf. activation
volumes in Table 2), which in turn leads to increased
polymerization rates and higher molecular weights.
About the effect on partitioning, higher densities in-
crease the sorption of CO2 in the dispersed phase which
results in the dilution of the latter and, therefore, in
the reduction of all diffusion limitations on the chemical
reactions. As a consequence, larger termination and
initiation rates are expected, thus resulting in shorter

Macromolecules, Vol. 38, No. 16, 2005
Vinylidene Fluoride Heterogeneous Polymerization 7159
Figure 8. Conversion as a function of reaction time predicted
by the model (lines) and measured experimentally: runs 2
(dashed, 4), 1 (solid, 0), and 3 (dotted, O) in Table 1.
chain lengths. Moreover, Saraf et al.3 reported a de-
crease of the monomer partitioning coefficient between
polymer and continuous phase at increasing densities,
from which reduced polymerization rate and, once more,
lower molecular weights would be expected. Thus sum-
marizing, changes of the density of the polymerization
system affect several factors leading to changes of the
polymerization rate and the molecular weights of the
produced chains in opposite directions. Looking at the
model predictions, it is seen that the experimentally
observed increase of the polymerization rate is repro-
duced, even though in a less distinct way (cf. Figure 6).
On the other hand, higher molecular weights with
increasing density are only predicted for the chains
produced in the continuous phase (cf. Figure 7a) whereas
the experimental MWD exhibits a shift of both modes
as shown in Figure 7b. This small discrepancy between
model and experiments could also be due to the as-
sumption of constant value of the specific polymer
particle surface area at the end of the reaction in all
experimental runs.
5.3. Effect of Monomer Concentration. Figure 8
shows conversion as a function of reaction time at three
different monomer concentrations. A nice agreement
between model predictions and experimental data is
found. It is interesting to note that, in the case of the
lowest monomer concentration, the reaction rate does
not exhibit any acceleration, thus indicating that the
polymerization is predominantly occurring in the con-
tinuous phase. As the monomer concentration increases,
the main reaction locus shifts toward the dispersed
phase, thus leading to a significant increase of the
polymerization rate due to the presence of gel effect in
the dispersed phase.
Figure 9. MWD at 25% conversion predicted by the model
(a) and measured experimentally (b): runs 2 (dashed, 4), 1
(solid, 0), and 3 (dotted, O) in Table 1.
particles is different at different monomer concentra-
tions and found to have the following values: Np ) 2.1
× 1010 at [M]0 ) 1.0 mol L-1, 6.5 × 1010 at 3.1 mol L-1
,
and 13.1 × 1010 at 6.2 mol L-1
.
6. Discussion
To determine the main reaction locus of heteroge-
neous polymerization in scCO2, we need to compare the
characteristic times of interphase mass transport and
termination. This is conveniently done through the
parameter Ω defined for each phase, j as a function of
chain length, x as follows:1,2
Kx,jAp
Ωj(x) )
(27)
∞
k (x, y)[R ]V
j
∑
tj
y,j
y)1
A further confirmation to this interpretation is given
by the results shown in Figure 9 where the calculated
final MWDs are compared to the experimental ones. At
increasing monomer concentration a second mode grows
in the high molecular weight region, as a result of the
increased importance of the reaction in the dispersed
phase. Note that this second mode becomes dominant
at the same time when acceleration appears in the
reaction rate, which is in agreement with the predicted
switch of the main reaction locus from the continuous
to the dispersed phase.
where Kx,j is the overall mass transfer coefficient with
respect to phase j and Ap the total interphase surface
area. Ω values above one indicate that chain transport
is faster than termination, which means that the active
chains can leave the original phase before they termi-
nate, thus continuing their growth and eventually
terminate in the other phase. On the other hand, Ω
values below one mean that the active chains are
terminated in the phase where they are formed, thus
leading to a segregated system. In the case of hetero-
geneous polymerization in scCO2 only the transport of
the active chains from the continuous phase to the
Since the same value of the final specific surface area,
afp, has been assumed in all simulations, the number of

7160 Mueller et al.
Macromolecules, Vol. 38, No. 16, 2005
Figure 11. MWD predicted by the model under the operating
Figure 10. Ω plot calculated at 25% conversion under the
operating conditions of runs 2 (4), 1 (0), and 3 (O) in Table 1;
arrows indicate increasing degree of polymerization.
conditions of run 1 in Table 1 at increased values of the final
specific polymer particle surface area: afp ) 0.30 m2 g-1
(dashed, 4), afp ) 0.40 m2 g-1 (solid, 0), afp ) 0.50 m2 g-1
(dotted, O); inset shows the corresponding Ω plots.
particles needs to be discussed, while the opposite
transfer is always negligible, i.e., Ω2 , 1.1 Moreover,
the behavior of Ω1(x) can be understood by looking at
the behavior of Kx,1 shown in Figure 1.
In the case of the system under examination here, the
bimodal MWDs indicate that two main reaction loci are
operative. This means that part of the radicals gener-
ated in the continuous phase terminate in the same
phase before being able to reach the dispersed phase,
as requested by their equilibrium partitioning. There-
fore, Ω1 values smaller than one are expected for the
active chains. In Figure 10 the Ω values calculated by
the model for the reactions carried out at three different
monomer concentrations (i.e., runs 1-3 in Table 1) and
at 25% conversion are shown at increasing degree of
polymerization. As expected, it is found that both Ω1
and Ω2 are always clearly smaller than one, except for
Ω1 at the largest degree of polymerization where values
very close to one are obtained. This indicates that we
are not far from the situation where the active chains
originated in the continuous phase diffuse in the
polymer particles, thus giving rise to a system where
only one polymerization locus is active. In Figure 10 the
larger Ω1 values obtained at the larger monomer
concentrations indicate that the contribution of the
continuous phase as a polymerization locus is smaller,
as it is confirmed by the smaller area of the first mode
of the MWDs shown in Figure 9.
To better illustrate this point, we have performed a
few simulations where the particle specific surface area
has been artificially increased in order to favor the
transport of the radicals in the polymer particles. As
shown in Figure 11, the first mode of the MWD tends
to disappear as soon as the Ω1 values for the larger
radicals exceed unity as shown in the inset of the same
figure.
On the other hand, if we do not allow the radicals to
diffuse from the continuous to the dispersed phase, i.e.,
we adopt the assumption of complete segregation of the
active chains, then the model predicts the reaction to
take place predominantly in the continuous phase,
leading to a MWD largely different from the experi-
mental one, as shown in Figure 12. It is seen that most
of the polymer is in this case produced in the continuous
phase (i.e., low molecular weight). This is because the
monomer is much more soluble in the supercritical
Figure 12. MWD predicted by the model under the operating
conditions of run 3 in Table 1 (dashed, same curve as the
dotted one in Figure 9a) and MWD under the same conditions
without interphase transport of active chains (solid).
phase than in the polymer phase. This observation
underlines the importance of describing in detail the
transport of the radicals to the polymer particles. If such
transport would be infinitely fast, we would obtain again
a monomodal MWD like in the case of a segregated
system, but this time centered around large molecular
weights (i.e., produced in the polymer phase) as shown
in Figure 11. The bimodality of the MWD observed
experimentally can only be reproduced by correctly
modeling the competition between diffusion and termi-
nation for the radicals moving from the continuous to
the dispersed phase. Note that in the simulations above
the dead polymer chains have been assumed not to be
segregated, but rather to diffuse immediately to the
polymer phase, so as to avoid an unrealistic accumula-
tion in the supercritical phase.
The role of chain transfer to polymer is investigated
in Figure 13 where the MWD predicted by the model
(dashed line) is compared with the one that one would
obtain in the absence of chain transfer to polymer (solid
line). As mentioned above, this reaction broadens mainly
the high molecular weight mode since the longer the
chains, the larger the impact of this branching reaction.

Macromolecules, Vol. 38, No. 16, 2005
Vinylidene Fluoride Heterogeneous Polymerization 7161
POL) and ETH (ETHZsinternal research project “Dis-
persion Polymerization in Supercritical Carbon Dioxide”)
is gratefully acknowledged.
Notation
a ) root mean square end-to-end distance per square root
of monomer units, cm
afp ) final specific particle surface area, m2 g-1
Ap ) total particle surface area, m2
Adj ) preexponential factor of initiator decomposition in
phase j, s-1
Apj ) preexponential factor of propagation in phase j, L
mol-1 s-1
Ap/
) preexponential factor of the ratio kpj/
k
in
tj
,j
x
xt
phase j, L1/2 mol-1/2 s-1/2
[A] ) concentration of monomer A, mol L-1
[B] ) concentration of monomer B, mol L-1
D0 ) preexponential factor of diffusion coefficient (free
volume theory), cm2 s-1
Figure 13. MWD computed by the model under the operating
conditions of run 3 in Table 1 (dashed, same curve as the
dotted one in Figure 9a) and MWD computed under the same
conditions without chain transfer to polymer (solid).
Di ) self-diffusion coefficient of component i, cm2 s-1
Di,0 ) self-diffusion coefficient of component i at zero
conversion, cm2 s-1
Dx,com ) center-of-mass diffusion coefficient of a chain of
length x, cm2 s-1
It also appears that by further increasing this reaction
rate one could probably induce a new mode in the MWD,
but this would clearly occur at much larger values than
those observed experimentally.
E ) critical energy to overcome attractive forces (free
volume theory), J mol-1
Edj ) activation energy of initiator decomposition in phase
j, kJ mol-1
E
pj ) activation energy of propogation in phase j, kJ mol-1
7. Conclusions
Ep/
) activation energy of the ratio kpj/
k
in phase j,
,j
x
xt
tj
kJ mol-1
A model describing heterogeneous polymerization in
scCO2 has been applied to simulate the precipitation
polymerization of VDF. After a careful evaluation of
most of the model parameters based on independent
literature information, the remaining three were evalu-
ated by fitting one set of experimental data at given
operating conditions and then kept constant for all the
other conditions investigated experimentally. Very nice
agreement between model predictions and experimental
observations was found when changing the initial
monomer concentration. The same comparison was
satisfactory when varying the density of the system,
although a very minor effect was found on both polym-
erization rate and MWD.
Moreover, the issue of the dominant reaction locus
was analyzed. Since radicals are generated in both
dispersed and continuous phase, we have potentially
two polymerization loci. Given that low molecular
weight species are at interphase equilibrium, the first
one, i.e., the polymer particles, is always active, while
the second one can be active or not depending on
whether the active radicals generated in the continuous
phase are transported or not to the dispersed phase
before terminating. This situation, which is well de-
scribed by the dimensionless Ω parameter,1 depends on
the specific conditions occurring during the process and
cannot be predicted a priori. This implies that, to
correctly describe the relative amount of polymer pro-
duced in each reaction locus, a detailed model account-
ing for the kinetics of the transport of the active chains
from the continuous to the dispersed phase is needed.
An important conclusion is that limiting models assum-
ing interphase equilibrium partitioning or segregation
(cf. ref 1) are not appropriate in this case, and they could
result in significant mechanistic misunderstandings.
fj ) initiator efficiency in phase j
fj,0 ) initiator efficiency in phase j at zero conversion
Ij ) amount of initiator in phase j, mol
[Ij] ) initiator concentration in phase j, mol L-1
I•j) amount of activated initiator in phase j, mol
jc ) entanglement spacing
kdj ) initiator decomposition rate constant in phase j, 1
s-1
kfpj ) chain transfer to polymer rate constant in phase j, L
mol-1 s-1
kIj ) initiation rate constant in phase j, L mol-1 s-1
khp ) overall pseudo-homopolymerization propagation rate
constant, L mol-1 s-1
kpi ) pseudo-homopolymerization propagation rate con-
stant of monomer i, L mol-1 s-1
kpii ) homopolymerization propagation rate constant of
monomer i, L mol-1 s-1
kpj ) propagation rate constant in phase j, L mol-1 s-1
kpj,0 ) propagation rate constant in phase j at zero
conversion, L mol-1 s-1
ktj ) termination rate constant in phase j, L mol-1 s-1
ktj,0 ) termination rate constant in phase j at zero conver-
sion, L mol-1 s-1
ktcj ) termination by combination rate constant in phase
j, L mol-1 s-1
ktdj ) termination by disproportionation rate constant in
phase j, L mol-1 s-1
kx,j ) local mass transfer coefficient in phase j, cf. eq 25,
cm s-1
K1,i ) free volume parameter of component i, cm3 g-1 K-1
K2,i ) free volume parameter of component i, K
Ki ) partition coefficient of component i
Kx,j ) overall mass transfer coefficient of a chain of length
x referred to phase j, cf. eqs 23 and 24, cm s-1
m0i ) total initial mass of component i, g
mx ) partition coefficient of a chain of length x, cf. eq 26
Mj ) amount of monomer in phase j, mol
[Mj] ) monomer concentration in phase j, mol L-1
nc ) number of components
NA ) Avogadro number, mol-1
Acknowledgment. The financial support of BBW
(Swiss Federal Office for Education and Science, Con-
tract 02-0131, EC-Contract G1RD-CT-2002-00676, ECO-
Np ) particle number
p ) pressure, MPa

7162 Mueller et al.
Macromolecules, Vol. 38, No. 16, 2005
p0,dj ) reference pressure at which kdj has been obtained,
Pi ) probability of having a radical chain with end group
of type i
MPa
p0,pj ) reference pressure at which kpj has been obtained,
MPa
F˜ ) reduced density
Fi ) density of component i, g cm-3
F/i ) characteristic density of component i, g cm-3
σi ) Lennard-Jones diameter of component i, cm
υ˜ ) reduced molar volume
p0,p
/
) reference pressure at which kpj/
k
has been
,j
x
xt
tj
obtained, MPa
Px ) total amount of terminated polymer chains of length
x in both phases, mol
υ/mix ) characteristic molar volume of the mixture, cf. eq 2,
cm3 mol-1
Px,j ) amount of terminated polymer chains of length x in
phase j, mol
υ/ii, υi/) characteristic volume of component i, cm3 mol-1
[Px,j] ) concentration of terminated polymer chains of
υ/ij ) characteristic volume of components i and j, cm3
length x in phase j, mol L-1
mol-1
[P/x,j] ) hypothetical concentration of terminated polymer
chains of length x in phase j in equilibrium with the
corresponding bulk concentration in the other phase, mol
L-1
φi ) site fraction of component i, cf. eq 4
ωi ) weight fraction of component i
Ωj(x) ) ratio between characteristic times of termination
and interphase mass transport of a chain of length x in
phase j, cf. eq 27
rmix ) number of lattice sites occupied by a molecule in the
mixture, cf. eq 7
ri ) number of lattice sites occupied by component i
ri ) reactivity ratio
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