Mathematical Modeling and Simulation of Vinylidene Fluoride

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Article pubs.acs.org/IECR

Mathematical Modeling and Simulation of Vinylidene Fluoride Emulsion Polymerization Prokopis Pladis,† Aleck H. Alexopoulos,† and Costas Kiparissides*,†,‡,§ †

Chemical Process and Energy Resources Institute, CERTH and ‡Department of Chemical Engineering, Aristotle University of Thessaloniki, P.O. Box 472, 541 24 Thessaloniki, Greece § Department of Chemical Engineering, Petroleum Institute, Abu Dhabi ABSTRACT: In the present study a comprehensive model is employed to describe the dynamic behavior of an industrial scale reactor for the emulsion polymerization of vinylidene fluoride (VDF) under different initiator and chain transfer agent (CTA) addition policies. A comprehensive kinetic model combined with a thermodynamic and a particle population model were simultaneously solved to calculate the VDF addition rate, reactor pressure, molecular weight properties, and the particle size distribution of polyvinylidene fluoride produced in a semibatch emulsion polymerization reactor. The proposed emulsion polymerization model takes into account the gas−liquid equilibrium of VDF to determine the aqueous phase VDF concentration and the changes in operating pressure. The effects of particle crystallinity on radical entry and coagulation rate were also considered. The computational model results (cumulative monomer feed and feed rate, reactor pressure, molecular weights, and particle size distribution) are found to be in agreement with available experimental data.

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INTRODUCTION Fluorinated polymers and their copolymers are versatile macromolecules exhibiting a wide range of physical and enduse properties ranging from thermoplastic to elastomeric and from semicrystalline to totally amorphous. In general fluorinated polymers display enhanced durability and chemical resistance as well as low flammability, dielectric constant, and surface energy. These specialty polymers have found many applications in the traditional construction and chemical industries, as well as a great number of high-tech applications in automotive and aerospace, optical, and microelectronics industries. In 2011, the global market for fluoropolymers was estimated at $7.25 billion with an expected annual growth rate of 5.8%.1 Polytetrafluoroethylene (PTFE) represents 58% of the world production (by weight), while polyvinylidene fluoride (PVDF) represents 21%. Other commercially available fluoroelastomers include chlorotrifluoroethylene (CTFE), hexafluoropropene (HFP), 1-hydro-pentafluoropropene (HPFP), and perfluoromethyl vinyl ether (PMVE). PVDF shares many of the desired characteristics of fluoropolymers (e.g., improved thermal and oxidative stability and good resistance to many chemicals and environmental factors). It should be noted that PVDF exhibits greater strength as well as wear and creep resistance than PTFE and FEP. Moreover, PVDF displays strong piezoelectric properties and has a higher dielectric constant in comparison to other fluoropolymers.2 PVDF is commonly used in architectural coatings and insulation, in chemical processing and semiconductor manufacturing, and numerous other high-tech applications (e.g., electrodes in lithium batteries, etc.). It has a melting temperature of ∼170 °C which is less than the typically encountered polymer processing temperatures (e.g., 200−260 °C). Because of its unique combination of mechanical and electrical properties as well as its processability and low-cost (relative to other fluoropolymers), PVDF has become the second largest in volume fluoropolymer manufactured after PTFE. © 2014 American Chemical Society

PVDF is commercially produced via emulsion and suspension polymerization at pressures of 10−300 atm and temperatures of 10−130 °C. The free-radical emulsion polymerization of vinylidene fluoride (VDF) is carried out in the presence of a stable fluorinated surfactant, using peroxide or persulfate initiators. Suspension polymerization is conducted in an aqueous medium in the presence of a colloidal dispersant such as a hydroxy cellulose.3 Dispersion/precipitation polymerization of VDF in supercritical CO2 has also been reported4−6 but no full-scale commercial processes have been developed so far. Emulsion polymerization is an important industrial process commonly employed in the manufacture of particulate polymer dispersions.7−9 Emulsion polymerization is a highly complex process involving multiphase polymerization kinetics coupled with multiphase thermodynamic equilibria and heat/mass transfer phenomena occurring at different spatial and temporal scales.10−12 Traditionally, emulsion polymerization is considered to proceed through a series of polymerization stages which are characterized by the state and availability of the monomer (e.g., the presence of dispersed monomer droplets, presence of monomer-swollen polymer particles, etc.). At the start of polymerization (stage I), particles are formed through nucleation, and the monomer primarily exists as a separate phase in the form of monomer droplets dispersed in the continuous aqueous phase. Stage I polymerization continues till the end of nucleation. In stages I and II monomer is transferred from the dispersed monomer phase via the continuous aqueous phase to the polymerization loci (e.g., monomer-swollen particles). Stage II continues till the monomer droplets are Special Issue: John Congalidis Memorial Received: Revised: Accepted: Published: 7352

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Figure 1. Schematic representation of the various phases present in the semibatch emulsion polymerization of VDF.

depleted. Finally, in stage III, polymerization is sustained by the absorbed monomer in the polymer phase till its complete consumption.10 At the molecular scale, emulsion polymerization is characterized by the pertinent free-radical polymerization kinetics (i.e., chain initiation, propagation, termination reactions, etc.), polymer chain formation and crystallinity. Equilibrium thermodynamics and mass-transfer processes determine the molecular species concentrations (e.g., monomer, macroradicals, etc.) in the various phases (i.e., monomer droplets, aqueous phase, and polymer particles) present in the system. At the particle-scale, the process is dominated by particle nucleation and growth, particle stabilization via the adsorption of surfactant molecules, and monomer and radicals transfer between the polymer particles, the continuous aqueous phase, and the monomer droplets. The emulsion polymerization of VDF does not follow the typical stages (I, II, and III) of the classical emulsion polymerization process since there is not a separate monomer phase in the form of emulsified droplets dispersed in the aqueous phase. Instead, the VDF monomer is present in the overhead gas phase from which it is transferred to the polymerization loci via the continuous aqueous phase. Consequently, assuming insignificant entrainment of the gas phase, no separate VDF droplets exist in the aqueous phase (see Figure 1). The VDF polymerization rate and PVDF molecular weight properties are controlled by the selected monomer, initiator, and chain transfer agent (CTA) feed policies.13,14 In fact, the VDF feed rate controls the reactor pressure which in turn affects the monomer partitioning and concentrations in the different phases (i.e., gas, aqueous, and polymer). The initiator can be supplied to the polymerization medium continuously or in discrete “shots” to affect the primary radical production rate and thus the polymerization rate and productivity. A CTA is usually employed to control the molecular weight, and, therefore, the PVDF melt viscosity. Typical chain transfer agents employed include acetone, tert-butanol, isopropyl alcohol,15 and ethyl acetate.16

In the present study, the VDF addition policy to the reactor aimed at maintaining the reactor gas phase pressure at a predetermined value with the aid of a feedback controller (Figure 1). Near the end of polymerization, the monomer addition was terminated, resulting in a continuous decrease of the reactor pressure. The initiator employed was potassium persulfate (KPS), while a fluorinated surfactant (i.e., Forafac F1176) was used to stabilize the generated polymer particles. Finally, ethyl acetate (EA) was used as a CTA in single or multiple addition shots to control the PVDF molecular weight. PVDF is a polymer exhibiting different crystalline forms depending on the polymer manufacturing conditions. It is also well-known that PVDF crystallinity affects its toughness, mechanical strength, impact resistance, and other properties as well. PVDF produced via emulsion polymerization consists of amorphous and crystalline domains (e.g., 50−60% crystalline volume) depending on the polymerization conditions. These crystalline domains can significantly alter the effective polymerization volume (i.e., the fraction of the particle volume in which polymerization can take place) which in turn affects both the average number of radicals per particle and the VDF polymerization rate. Moreover, external particle surface morphology and surface inhomogeneity (i.e., crystalline versus amorphous particle surface domains) could result in different local particle surface charges, particle stabilization forces, and, thus, particle−particle interactions and coagulation rates. Consequently, there is a pressing need to elucidate and quantify the multiphase behavior, thermodynamic equilibrium, and molecular species partitioning in the multiphase system as well as the effect of polymer crystallinity on the overall polymerization kinetics, polymerization rate, and molecular weight developments. The present paper deals, for the first time, with the development of a comprehensive model to describe the emulsion polymerization of VDF in an industrial-scale reactor operating under different initiator and CTA addition policies. The proposed mathematical model includes a detailed kinetic mechanism of the VDF emulsion polymerization, a detailed 7353

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thermodynamic model describing the partitioning of different molecular species in the multiphase system, and a particle population balance model as well as a particle stability equation to predict the polymerization rate, molecular properties, and particle size distribution of PVDF produced in a semibatch industrial-scale reactor in terms of process operating conditions. It should be noted that previous studies on VDF included the modeling of emulsion copolymerization of VDF with HFP16 and the polymerization of VDF in supercritical CO2.5,6 However, all the previous studies were limited to the kinetic modeling of the VDF polymerization and did not provide any information on the particle size distribution (PSD) and the effect of polymer crystallinity on the polymerization rate, molecular weight developments, and particle stability in terms of initiator and CTA feed policies. In particular, the present model accounts for the time-varying concentrations of the various species (i.e., monomer, initiator, surfactant(s), and radicals) in both the aqueous and the particulate polymer phase. The equilibrium concentrations of VDF in the gas, aqueous, and polymer phases are calculated to predict the instantaneous VDF feed rate in order to maintain a certain pressure in the reactor in terms of operating conditions and polymerization recipe. The effect of polymer crystallinity on radical entry and particle coagulation rate are also considered. As a result, the model can account for timevarying particle nucleation, particle growth, and particle coagulation rates. Finally, the kinetic mechanism describing the VDF emulsion polymerization comprises a comprehensive series of reactions including initiator decomposition, polymer chain propagation, and termination by combination as well as transfer to monomer, to polymer, and to CTA reactions. In what follows, the proposed emulsion polymerization modeling approach is detailed. The kinetic mechanism is first outlined. Then, the thermodynamic model is described, followed by the calculation of radical entry/desorption rates, prediction of polymer crystallinity, particle stability, and calculation of PSD. In the final section of the paper, simulation results are shown for different polymerization scenarios, and model predictions are compared with normalized experimental data. Development of a VDF Emulsion Polymerization Kinetic Model. In previous studies, a great number of comprehensive mathematical models have been proposed to describe the dynamic behavior of emulsion polymerization and copolymerization of various monomers in batch and semibatch reactors.7,10,17−21 Kinetic Mechanism of Emulsion Polymerization. In the present study, the following kinetic mechanism was selected to describe the free-radical polymerization of VDF:16,22

Chain transfer to monomer reaction: k tm

R •n + M ⎯→ ⎯ R1• + Dn

Radical transfer to chain transfer agent reaction: k tx

R •n + X → R1• + Dn

I → 2R•

k tp

R •n + Dm → R •m + Dn

kb

R •n → R •n + SCB k tc

R •n + R •m → Dn + m

(8)

R•n ,

In the above elementary reactions, the symbols Dn denote the “live” and “dead” polymer chains of chain length “n”, respectively. It should be noted that monomer addition to the growing “live” polymer chain can be affected by two different ways, namely, the regular head to tail (h-t) configuration or the inverted head to head (h-h) structure (where the CF2 groups are referred to as the “head” and the CH2 groups as the “tail”). CH 2CF•2 + CH 2CF2 kp

→ CH 2CF2CH 2CF•2 (h‐t )

(9)

CH 2CF•2 + CH 2CF2 kp

→ CH 2CF2CF2CH•2 (h‐h)

(10)

The fraction of the polymer chains exhibiting an inverted (h-h) configuration has been measured by high resolution 19F and 1H nuclear magnetic resonance (NMR) spectroscopy and ranges from 3.5 to 7.1%.23−26 In Table 1, literature data regarding the Table 1. Reported Values on the Degree of Structural Defects for a Number of PVDF Samples ref

h−h (%)

comments

Herman et al., 1997 Herman et al., 1997 Herman et al., 1997 Russo et al., 1993

5.5 7.1 4.0−6.5 3.6

Russo et al., 1993

4.1

Russo et al., 1993

4.5

Schneider et al., 2001 Schneider et al., 2001 Schneider et al., 2001 Schneider et al., 2001 Schneider et al., 2001 Sajkiewicz et al., 1999

5.5 4.8 4.7 4.2 3.9 5.2

(2)

Commercial product, Xn = 300 oligomer oligomer, suspension polymerization oligomer, solution polymerization in acetone oligomer, solution polymerization in ethyl acetate oligomer, solution polymerization in methyl acetate Kynar 500 Kynar 1000 Kynar 740 Solef 1001 Kureha PVDF Kynar 880 N (Mw = 400.000, Mn = 149.000)

(3)

degree of structural defects for a number of PVDF samples are summarized. These head-to-head or tail-to-tail defects are influenced by the polymerization conditions and temperature. However, the effect of polymerization temperature on the

Chain propagation reaction: kp

R •n + M → R •n + 1

(7)

Termination by combination reaction:

Chain initiation reaction: ki

(6)

Intramolecular transfer reaction (formation of short chain branching):

(1)

R• + M → R1•

(5)

Chain transfer to polymer reaction (formation of long chain branching):

Initiator decomposition: kd

(4)

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pertinent rate constants on monomer conversion, polymer molecular weight, and temperature. More details can be found in Appendix A and in the original paper of Keramopoulos and Kiparissides.28 Dynamic Molar Species Balance Equations. On the basis of the proposed kinetic mechanism described in the previous section, a set of ordinary differential equations can be derived to describe the conservation of the various molar species in the aqueous, polymer, and emulsion phases.

number of head−head defects has not been reported in the open literature. In the present work, this effect was taken into consideration based on proprietary data. Figure 2 illustrates the

Initiator concentration in the aqueous phase: dC Iw F C w dVw = I − I − RIw dt Vw Ve dt

(16)

Monomer concentration in the emulsion phase: dCm F C dVe = M − m − Rp dt Ve Ve dt

(17)

Chain transfer agent concentration in the emulsion phase: Figure 2. Effect of polymerization temperature on the percent headto-head structural chain defects and percent PVDF crystallinity.

dCx F C dVe = x − x − Rx dt Ve Ve dt

effect of temperature on the % head-to-head defects and PVDF crystallinity. Note that the degree of PVDF crystallinity is influenced by both the fraction of chain structure defects as well as by the amount of short chain branches. Kinetic Rate Functions. Following the original developments of Congalidis et al.,17,27 the net production rate of the various molecular species of interest can be derived. It is important to point out that although the polymer particle phase is considered as the main locus of polymerization, monomer consumption in the aqueous phase is also included in the present model. Thus, the total monomer consumption rate in the emulsion phase, Rp, will be given by the sum of the monomer consumption rates in the polymer phase, Rpp, and the aqueous phase, Rwp .

(18)

Similarly, one can derive dynamic molar species balance equations for the electrolyte, C yw , and surfactant, C s , concentrations in the aqueous phase, as well as dynamic balance equations for the respective leading moments of “live” and “dead” total polymer chain length distributions and the concentration of long chain branches (LCB). A more comprehensive derivation of the pertinent molar species equations can be found in the original publications of Richards and Congalidis17,27 on the kinetic modeling of emulsion copolymerization and in the work of Kammona et al.21 The time variation of the total emulsion volume and the respective volumes of the aqueous and polymer phases can be calculated by the following differential equations. Volume of the emulsion phase:

Rate of initiator consumption: RIw = kdC Iw

⎛ ⎞ dVe 1 1 = Q F + ⎜⎜ − ⎟⎟MVDFR pVe dt ρm ⎠ ⎝ ρp

(11)

Total rate of monomer consumption: R p = R pp + R pw

Volume of the aqueous phase:

(12)

dVw = QW dt

Rate of monomer consumption in the particulate phase: R pp = (k pCmp + k tmCmp)R p•(Vp/Ve)

(13)

dVp

(14)

dt

Rate of chain transfer agent consumption: R x = k txCxpR p•(Vp/Ve)

(20)

Volume of the polymer particle phase:

Rate of monomer consumption in the aqueous phase: R pw = (k pCmw + k tmCmw )R w• (Vw /Ve)

(19)

= MVDFR pVe −

Vp dφp φp dt

(21)

where φp is the polymer volume fraction in the monomerswollen polymer particles which is calculated using the Sanchez−Lacombe EoS (see the thermodynamics section of the paper). QF and QW denote the respective volumetric flow rate of the monomer and water feed streams. All other symbols are explained in the nomenclature section. Finally, the volume of the gas phase will be given by the difference of the total reactor volume minus the volume of the emulsion phase.

(15)

where R•p and R•w are the total number of radicals in the particulate and water phase, respectively. Diffusion Controlled Reactions. Diffusion-controlled termination, propagation, and initiation reactions were related to the well-known phenomena of gel, glass, and cage effects, respectively. A comprehensive free-volume multicomponent model was employed to calculate the functional dependence of 7355

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Volume of the gas phase: Vg = Vreactor − Ve

(22)

Development of a Population Balance Model. Calculation of the Particle Size Distribution. In the present study, a population balance equation (PBE) model was developed to calculate the dynamic evolution of the latex particle size distribution (PSD) in the emulsion polymerization of VDF. Accordingly, a number density function, f(V,t), representing the number of particles in a differential volume size range, V to V + dV, per unit reactor volume was introduced to describe the dynamic evolution of the PSD. Following previous developments,21,29,30 one can write the following PBE to describe the dynamic evolution of the number density function n(V,t) in the reactor.

Figure 3. Schematic representation of possible collision events between two semicrystalline PVDF particles.

semicrystalline particles can be obtained from a modification of the coagulation rate function, eq 24, by introducing a particle surface crystalline parameter, αc. Thus, the total particle coagulation rate kernel, considering all possible collision events (see Figure 3), becomes:

ϑf (V , t ) ϑ(G(V )f (V , t )) + ϑt ϑV = S(t ) +

∫V

V /2

⎡ (1 − αCi)(1 − αCj) (1 − αCi)αCj βij = β0ij⎢ + WAAij WACij ⎢⎣

β (V − U , U )f (V − U , t )f (U , t ) d U

min

− f (V , t )

∫V

Vmax

β (V , U )f (U , t ) d U −

min

f (V , t ) d V V dt

+

(23)

where β(U, V) is the coagulation rate between particles of volumes U and V, G(V) denotes the particle growth rate, and S(t) is the particle nucleation rate due to either homogeneous and/or micellar particle nucleation. The coagulation rate kernel βij was determined using the Fuchs modification of the Smoluchowski equation.31 2 4k T 1 (ri + rj) βij = B 3μ Wij rri j

αCi(1 − αCj) WCAij

+

αCiαCj ⎤ ⎥ WCCij ⎥⎦

(25)

where β0ij is the fast coagulation rate kernel (eq 24) for Wij = 1) and αCj is the surface crystalline fraction of a particle with size rj. Additionally, if the surface crystallinity fraction, αc, is assumed to be size independent, then eq 25 becomes ⎡ (1 − α )(1 − α ) (1 − αC)αC α α ⎤ C C βij = β0i⎢ +2 + + C C⎥ WAAij WACij WCCij ⎥⎦ ⎢⎣

(24)

(26)

where ri, rj are the particle radii, μ is the continuous phase viscosity, kB is the Boltzmann constant, and T is the reactor temperature. The stability ratio Wij represents the ratio of the actual particle aggregation rate over that of a completely unstable system (i.e., fast coagulation kinetics).31 Calculation of the Coagulation Rate Kernel. As mentioned before, PVDF particles consist of amorphous and crystalline domains. This means that the external particle surface morphology (i.e., crystalline versus amorphous surface domains) could affect the local concentration of particle surface charges and absorbed surfactant as well as the local particle viscoelastic properties (e.g., particle stickiness), thus influencing particle− particle interactions and particle coagulation rate. To describe the coagulation rate of such semicrystalline particles, a surface morphology function, expressed in terms of a particle surface crystalline parameter, was introduced. A schematic representation of the possible particle collision events between the different semicrystalline particles is depicted in Figure 3. As can be seen, when two semicrystalline particles (i.e., with both amorphous and crystalline regions) collide, the following possible collision events can take place: amorphous−amorphous, amorphous−crystalline, and crystalline−crystalline. The respective Fuchs’ stability ratio for the three possible particle collisions is denoted by WAA, WAC (=WCA, due to symmetry) and WCC, respectively. The value of the stability ratio will depend on the local particle surfactant concentration (i.e., amount of surfactant molecules adsorbed on the crystalline and amorphous surfaces of the two colliding particles). The coagulation rate between

It should be noted that as a first approximation, the value of the parameter αc can be set equal to the bulk phase ratio of the crystalline polymer. Calculation of the Particle Growth and Particle Nucleation Rates. The particle growth rate, G, is a sizedependent function given by:32 ⎡ Rp ⎤ ⎛ ⎞ p ⎥ tanh⎜ r ⎟ G = n ̅ (r )⎢ ⎢⎣ NAφpρp ⎥⎦ ⎝ rFH ⎠

(27)

where n̅(r) is the average number of radicals per particle of size r, ρp is the polymer density, and rFH is the Flory−Huggins radius. In general, the particle nucleation rate in an emulsion polymerization reactor will depend on the type and concentration of the surfactant, the initial initiator concentration, the polymerization temperature, and the ionic strength of the medium as well as the presence of micelles and the monomer solubility in the aqueous phase. In general, particle nucleation can proceed by the homogeneous and/or the micellar nucleation mechanism. In this work, the surfactant concentration was always less than the critical micelle concentration, CCMC, and, thus, particle nucleation occurred solely by the homogeneous nucleation mechanism. Accordingly, the homogogenous particle nucleation rate was calculated by the following equation:17,33 R hom 7356

1 − jcr R tw R ew ⎞ Vw w ⎛ = NA RI ⎜⎜1 + w + w ⎟⎟ Ve ⎝ Rp Rp ⎠

(28)

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where jcr is the value of the critical length of the polymer chains formed in the aqueous phase. Note that polymer chains with a chain length larger than jcr will precipitate out to form a new particle nuclei of volume V1. Thus, the minimum particle size, V1, can be expressed in terms of jcr according to the following equation:

V1 = jcr Cmw /ρp NA

the total rate of radical desorption from latex particles, total rate of entry into particles, and termination in the aqueous phase, respectively. Assuming that the quasi-steady-state approximation for the “live” radical chains in the aqueous phase holds true, the above equation becomes RIw + R d − R ew − R tw = 0

(29)

The average number of radicals per particle, n,̅ will be given by the Stockmayer−O’Toole solution to the Smith−Ewart recursion relation:35

and assuming that all nuclei are formed with a volume of V1 the nucleation term of eq 23 is obtained from

S = R hom/V1

(30)

n ̅ (r ) =

Owing to the lack of experimental data on the critical value of VDF, jcr, its value was obtained in terms of V1, which was treated as a model parameter. It was found that the value of jcr varied from 10 to 40 in terms of the selected experimental conditions. It should be noted that, using an empirical correlation proposed by Gilbert10 and a value for the saturated monomer concentration in the aqueous phase equal to 10−1 mol/L, the calculated value of jcr for VDF was found to be equal to 20. Radical Entry and Radical Exit Rates. The generated radicals can move from one phase (i.e., aqueous) to another (i.e., polymer particles) via the physical mechanisms of radical entry and radical desorption. Following the developments of Richards and Congalides,17 the total radical entry rate into the polymer particles will be given by R ew =

Ve R w• NAVw

∫r

rmax

ke(r ) f (r ) dr

min

ξ=4

v=

∫r

rmax

kd(r ) n ̅ (r ) f (r ) dr

min

(32)

(39)

Vw + αkd(r ) n ̅ (r ) Ve

(40)

k tc 2vpNA(1 − αc)

(41)

where α is a radical fate parameter. Equilibrium Thermodynamic Considerations. To calculate the VDF partitioning in the three-phase system (gas, aqueous, and polymer) as well the monomer feed rate into the semibatch reactor in order to maintain the pressure at a specified value, a suitable equation of state (EoS) model is needed. In the present work, two different EoS models were employed for the calculation of the phase behavior of the threephase system. In particular, the Sanchez−Lacombe (S-L) equation of state was employed for the calculation of the equilibrium concentration of VDF in PVDF while the BeattieBridgeman cubic EoS was used to determine the reactor pressure. The S-L EoS is one of the simplest statistical mechanics thermodynamic models that are capable of describing the phase behavior of a multicomponent (monomer(s)−polymer) system. Following the original developments of Sanchez and Lacombe,36−38 the general EoS is written as 10

(33)

(34)

Cwm/Cpm,

where Kwp = is the monomer partition ratio between the aqueous and polymer phases. The minimum value of the desorption rate constant, kd,min = kd1 + kd2Cx (where kd1 is equal to 0.02 s−1 and kd2 is equal to 1.5 l·mol−1·s−1) was introduced to take into account the increased desorption rate of the small length “live” polymer chains generated via the ethyl acetate chain transfer reaction. Radical Balances in the Aqueous and Polymer Phases. The dynamic molar balance equation for the total concentration of the “live” polymer chains in the aqueous phase, R•w, will be given by • 1 d(R w ·Vw ) = RIw + Rd − R ew − R tw Vw dt

(38)

kd(r ) c t (r )

c t (r ) =

The desorption rate constant, kd, was expressed in terms of the diffusion coefficients of the growing polymer chains in the aqueous, Dw, and polymer phase, Dp, according to17,34 kd = kd,min + 3Dw K wp/[r 2(1 + Dw K wp/2Dp)]

ρ (r ) c t (r )

ρ = keR w•

(31)

where the parameter F is an absorption efficiency factor. The total radical desorption rate can be calculated by summing up the individual particle desorption rates in terms of the PSD, f(r). Ve NAVw

(37)

Note that ρ(r) and ct(r) denote the respective first-order kinetic rate constant for radical entry into the particles and radical termination coefficient and they can be calculated by

10,33

Rd =

ξIv(ξ) 4Iv − 1(ξ)

where the order, ν, and the argument, ξ, of the modified Bessel function Iν(ξ) are size-dependent variables defined by the following equations:

The particle radical entry rate coefficients were calculated by the following equation:10,33 ke(rj) = (1 − αc)4πrjFNADw /W1j

(36)

ρ ̅ 2 + P ̅ + T̅ [ln(1 − ρ ̅ ) + (1 − 1/r )ρ ̅ ] = 0

(42)

where P̅ = (P/P*), ρ̅ = (ρ/ρ*) and T̅ = (T/T*) denote the respective reduced pressure, density, and temperature of a pure component. T, P, and ρ are the absolute temperature (K), pressure (bar), and the density (kg/m3) while T*, P*, and ρ* are the respective characteristic parameters of the pure component. For a polymer chain, the number of sites(-mers), r, occupied in the lattice can be related to its molecular weight, M, according to the following equation:

(35)

where the terms RIw, Rd, Rew, and Rtw denote the net rate of primary radical production via the decomposition of initiator,

r = P*M /(RT *ρ*) 7357

(43)

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The characteristic parameters for VDF and PVDF (i.e., T*, P*, and ρ*) were calculated using experimental data reported in the open literature. More specifically, the VDF characteristic parameters were estimated using 30 density and vapor pressure experimental values (i.e., obtained at different temperatures, 40−60 °C, and pressures, 50−350 bar) reported by Mears et al.39 and 36 experimental density values for PVDF.40 A nonlinear parameter estimator was used to fit the model predictions to the selected experimental data. The values of the fitted parameters appearing in the S-L EoS model are shown in Table 2.

It should be noted that the emulsion polymerization of VDF was carried out at a pressure near the supercritical conditions of VDF (i.e., in the vicinity of the triple-point region). It was found that the Sanchez−Lacombe EoS overpredicted the reactor pressure. Thus, an alternative EoS model, the Beattie−Bridgeman cubic EoS proposed by Mears et al.,39 was used to calculate the reactor pressure. The Beattie− Bridgeman is given by

P=

T* (K)

P* (bar)

ρ* (g/cm3)

VDF PVDF

330 657

2624.7 6489

1.311 1.768

(44)

where

Table 2. Estimated Values of Critical Model Parameters in Sanchez−Lacombe EoS component

RT (V + B) − A V2

⎛ ⎛ α⎞ b⎞ A = A0⎜1 − ⎟B = B0 ⎜1 − ⎟ ⎝ ⎠ ⎝ V V⎠

(45)

The numerical values of the Beattie−Bridgeman EoS model parameters for VDF, calculated by Mears et al.,39 are shown in Table 3. Thus, in the present study, the Beattie−Bridgeman

In Figures 4 and 5, model predictions obtained by the S-L EoS are compared with the experimental data of Mears et al.39

Table 3. Estimated Values of Model Parameters in the Beattie−Bridgeman EoS39 A0

α

B0

b

V(L/mol)

9.661 3.619

0.05639 −0.07062

0.1976 −0.0265

0.04839 0.7453

0.17−0.65 0.09−0.13

EoS was employed to calculate the reactor pressure in the VDF emulsion polymerization.

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SIMULATION RESULTS AND DISCUSSION The developed model was employed to simulate the emulsion copolymerization of VDF in semibatch industrial scale reactors. The polymerization recipe consisted of water, VDF monomer, fluorinated surfactants (e.g., Forafac F1176) for particle stabilization, ethyl acetate as CTA (e.g., 0.2 to 4% w/w of total monomer) and KPS as initiator (4 × 10−3 to 4 × 10−2 % w/w of total monomer). The initiator and the CTA were added to the polymerization medium either in single shots or following a multi-injection policy (i.e., several shots at specific time instances). The polymerization temperature was set at 83 °C, while the reaction pressure was maintained at 85 bar. In Table 4, the values of the kinetic rate constants used in the model simulations are reported.

Figure 4. Comparison of Sanchez−Lacombe EoS model predictions with experimental VDF density measurements of Mears et al.39

Table 4. Numerical Values of the Kinetic Rate Constants Used in the Simulation of the VDF Emulsion Polymerization kinetic rate constantsa kp = 2.2 × 109 exp(−4539/T) ktd = 0.0 ktc = 5.0 × 1012 exp(−2533/T) ktm = 1.2 × 1011 exp(−9020/T) ktp = 2.0 × 103 exp(−4539/T) ktx = 5.5 × 105 exp(−4539/T) kb = 2.0 × 104 exp(−4539/T) kI = 4.56 × 1016 exp(−16860/T) (KPS)

ref this 22 this this this this this 45

work, 22 work, 22 work work work work

a

Kinetic rate coefficients in L/(mol·s); initiator decomposition rate in s−1, T in K

Calculation of the Average Molecular Weights. In the present study, the method of moments was employed to calculate the leading moments of the total number chain length

Figure 5. Comparison of Sanchez−Lacombe EoS model predictions with experimental PVDF density measurements of Zoller and Walsh.40 7358

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distributions (NCLD) of “live” and the “dead” polymer chains.21 The average molecular weights of a polymer (i.e., the number average molecular weight, Mn, the weight average molecular weight, MW, and the polydispersity index (PDI) can be calculated in terms of the leading moments according to the following equations. Mn = MVDF

μ1 + λ1 μ0 + λ 0

M w = MVDF

μ2 + λ 2 μ1 + λ1

≅ MVDF

μ1 μ0

≅ MVDF

(46)

μ2 μ1

(47)

The polydispersity index (PDI), a measure of the breadth of the MWD, is defined by the ratio of the weight-average to the number-average molecular weight. μμ M PDI = w = 2 20 Mn μ1 (48)

Figure 7. Comparison of experimental and predicted VDF mass feed rates.

Numerical Solution of the PBE. The PBE (i.e., eq 23) was numerically solved using the Galerkin FEM. The numerical procedure has been described in detail elsewhere.41,42 The stability function in eq 24 can account for electrostatic stabilization17,21 and steric stabilization effects as well.43 Reactor Model Simulations. The computational model described above (see Appendix B), was employed to simulate the dynamic behavior of a semibatch VDF emulsion polymerization reactor under different operating conditions (e.g., initiator and CTA feed policies). In Figures 6, 7, and 8,

Figure 8. Comparison of experimental and calculated reactor pressure.

time to maintain the specified reactor pressure. After the end of monomer addition policy, the reactor pressure starts decreasing, which is caused by the monomer depletion in the gas phase. The agreement of model predictions with experimental data is excellent for the entire course of the reaction. The molecular weight averages and the PSD for the same experimental run are shown in Figures 9 and 10, respectively. The predicted values of Mn and Mw are typical for PVDF44 for low CTA concentrations. The agreement between experimental and calculated PSDs (see Figure 10) is reasonable considering the complexity of the system. It should be noted that in this case there was some limited particle aggregation in the experimental system. The small differences in the shape of the predicted PSD compared to the experimental one are likely due to mixing effects in the experimental reactor that were accounted for in the model. In a recent publication,30 a multicompartment approach is described that accounts for the effect of mixing on the kinetics and PSD developments in largescale emulsion polymerization reactors. In Figure 11, experimental results on the total VDF fed mass are compared with model predictions for an initiator feed policy consisting of two single shots, namely, at time t = 0 and at t = 270 min. It is clear that the late addition of initiator (i.e., at 270 min) results in a significant increase of the polymerization rate. It was

Figure 6. Comparison of predicted and experimental total VDF mass in kilograms fed to the reactor.

model simulation results are compared with experimental measurements for the total mass of VDF fed into the reactor, the VDF feed rate, and the reactor pressure, respectively. It is clear that model predictions are in excellent agreement with the experimental data. Note that in this simulation a single shot of initiator and a single shot of ethyl acetate at t = 0 were used. The scaled VDF addition rate (i.e., the polymerization rate) varied from 0.2 and 0.25 kg/h for most of the polymerization time (i.e., 250 min). In Figure 8 the time-variation of reactor pressure during polymerization is depicted. As can be seen, the reactor pressure is kept constant while the VDF monomer is being fed into the reactor (i.e., 0 to 250 min). Note the model calculates the required monomer feed rate with respect to polymerization 7359

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presence of high ethyl acetate concentrations and the decrease of gel-effect (i.e., due to the higher mobility of “live” polymer chains). Finally, in Figure 12, the time evolution of the PSD is shown for the case of a high surfactant concentration. The model predictions

Figure 9. Time evolution of the number and weight average molecular weights of PVDF produced in a semibatch emulsion polymerization reactor.

Figure 12. Comparison of experimental and simulated particle size distributions.

are in very good agreement with the experimentally measured final PSD. As can be seen the PSDs are narrower (in comparison to the PSD shown in Figure 10) since the system is better stabilized due to the selected surfactant addition policy that limits particle aggregation after the end of the nucleation stage (i.e., 20 min). It was found that particle crystallinity affected the radical concentration in the particles as well as the particle coagulation rate. To account for the effect of particle crystallinity on the particle coagulation rate and radical concentration, a correction term (1 − αc) was introduced into eqs 26, 32, and 41. In Table 5 Figure 10. Comparison of experimental and simulated particle size distributions in a semibatch VDF emulsion polymerization reactor.

Table 5. Effect of PVDF Crystallinity on tend, D̅ , n̅, and Mn at Constant Temperature crystallinity

tend, min

D̅ (tend), nm

0.50 0.45 0.40 0.35 0.30 0.15 0.0

230 250 270 285 245 240 220

227.8 213.1 199.1 190.6 186.4 185.5 190.8

n̅(D̅ , tend) 0.83 0.75 0.71 0.69 0.68 0.72 0.85

Mn × 106 0.272 0.272 0.272 0.272 0.272 0.273 0.274

the effect of particle crystallinity (assuming the same reaction temperature) on the time corresponding to 90% VDF conversion, tend, the mean particle diameter at tend, D̅ , the average number of radicals per particle at tend, n,̅ and the number average molecular weight at tend, Mn, is shown. It is clear that the value of D̅ increases with crystallinity due to a decrease in the particle surface coverage by surfactant molecules (i.e., decrease of adsorbed surfactant concentration) leading to a decreased particle stability. On the other hand the results for tend and n̅ exhibit a U-shape behavior (i.e., they exhibit a maximum and a minimum value, respectively) due to the combined effects of crystallinity on particle coagulation rate and the radical concentration due to changes in the particle radical entry and termination rates (i.e., eqs 32 and 41).

Figure 11. Comparison of experimental and simulated VDF feed rate for a two-injections initiator policy.

found that the polymerization rate increased with the total amount of initiator added to the reactor and decreased with the total amount of CTA that was attributed to the increased desorption rate of the small chain length radicals formed in the 7360

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increase in the radical desorption rate. It is important to observe the effect of the CTA addition policy on the weight average molecular weight (see Figure 14). As the CTA concentration increases the molecular weight decreases. It is apparent that the CTA addition policy significantly affects the molecular weight developments. For example, when the CTA is added in three shots (i.e., S3) the molecular weight remains almost constant with time. Consequently, the molecular weight of PVDF can be controlled to a large degree by the CTA addition policy. Note that the observed decrease in polymerization rate (see Figure 13) is usually corrected by additional shots of initiator during the course of the reaction.

To demonstrate the effect of CTA addition policy on the polymerization rate and molecular weight in emulsion polymerization of VDF, different addition policies were tested. In particular three different CTA addition policies were compared to the nominal case (NC) presented in Figure 7. In the nominal case a single shot of ethyl acetate (total CTA fed mass = CTA0) was added at the beginning of the reaction (i.e., t = 0 min). In case S1, a single shot of ethyl acetate of a total mass equal to 3 × CTA0 was introduced to the semibatch reactor at t = 0 min. In the other two CTA addition policies ethyl acetate was added in multiple shots. In case S2, a two-shot addition policy, (at t = 0, a CTA shot equal to CTA0 was added to the reactor and at t = 150 min, a second shot of CTA equal to 2 × CTA0). Finally, in case S3, a three-shot policy (at t = 0, 100, and 200 min, a CTA shot equal to CTA0) was applied. In Figures 13 and 14 the instantaneous VDF feed rate and the weight average molecular weights are shown for the CTA addition policies listed above.

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CONCLUSIONS A comprehensive mathematical model was presented that is capable of simulating the dynamic behavior of a semibatch VDF emulsion polymerization reactor. A comprehensive kinetic model combined with a detailed thermodynamic model and a particle population model were simultaneously solved to calculate the VDF addition rate, reactor pressure, molecular weight properties, and PSD of PVDF produced in a semibatch emulsion polymerization reactor. Successful comparison of model predictions with experimental data obtained from different lab and pilot-scale reactors proved the validity of the proposed model. It was found that the calculation of PVDF crystallinity during polymerization is a key process variable since it largely affects the effective polymer reaction volume as well as the polymerization rate, molecular weight developments, and PSD through the variation of particle coagulation kernel. In addition, it was found that the addition policy of CTA was important not only with respect to molecular weight developments but also concerning the radical concentration and polymerization rate. The observed variation of molecular weight and polymerization rate with CTA addition reflects its influence on the radical concentration and indicates the possibility of controlling the molecular properties of the product.

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Figure 13. Effect of CTA addition policies on the VDF feed rate.

APPENDIX A: DIFFUSION-CONTROLLED TERMINATION RATE CONSTANT On the basis of the modeling approach of Keramopoulos and Kiparissides,28 the termination rate constant is expressed as the sum of two terms, one taking into account the effect of diffusion of polymer chains, kdt and the other describing the socalled “residual termination”, kres t . kt = ktd + ktres

(A1)

The first term on the right-hand side of eq A1 is subsequently written in terms of the corresponding intrinsic termination rate constant, ktii, and the diffusion coefficient of the polymer chains in the polymer rich-phase, Dpe: 1 1 1 = + d kt 0 4π rtDpeNA kt (A2) where rti is the termination radius.28 According to the extended free-volume theory, Dpe can be approximated by the polymer self-diffusion coefficient:

Figure 14. Effect of CTA addition policies on the weight average molecular weight.

Dpe = (Dp0 /M w2 ) exp[−γp(ωmV m*/ξmp + ωpV p*)/VFH] (A3)

As can be seen in Figure 13, when the total CTA amount is three times the nominal CTA fed mass (i.e., S1, S2, S3 policies), the polymerization rate exhibits a significant decrease by 25−35% depending on the addition policy due to the

where

ξmp = V m*MVDF/Vp 7361

(A4)

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Article

F = Absorption efficiency factor G(V) = Particle growth rate function, m3/s I = Initiator jcr = Critical chain length value for homogeneous nucleation kB = Boltzman’s constant, J/K ke = Radical entry rate coefficient, m3/(kmol·s) kd = Initiator thermal decomposition rate constant, s−1 ki = Chain initiation rate constant, m3/(kmol·s) kp = Propagation rate constant, m3/(kmol·s) ktm = Chain transfer to monomer rate constant, m3/(kmol·s) kts = Chain transfer to CTA rate constant, m3/(kmol·s) ktp = Chain transfer to polymer rate constant, m3/(kmol·s) ktc = Termination by conbination rate constant, m3/(kmol·s) ktd = Termination by disproportionation rate constant, m3/(kmol·s) kb = Backbiting rate constant, s−1 kd = Radical desorption rate coefficient, s−1 Kwp = Monomer partition coefficient in the aqueous and polymer phases M, m = Monomer MVDF = Molecular weight of VDF monomer Mn = Number average molecular weight Mw = Weight average molecular weight NA = Avogadro’s number, kmol−1 r = Particle radius, m rm = Micelle radius. m n̅ = Average number of radicals per particle P = Pressure, bar P* = Characteristic pressure, bar P̅ = Reduced pressure PDI = Polydispersity index R = Universal gas constant, J/kmol/K QF = Total volumetric flow rate, m3/s Qw = Water volumetric flow rate, m3/s R•, R•n = Radical RwI = Rate of initiator consumption, kmol/(m3·s) Rp = Total rate of monomer consumption, kmol/(m3·s) Rwp = Rate of monomer consumption in the water phase, kmol/(m3·s) Rpp = Rate of monomer consumption in the polymer phase, kmol/(m3·s) Rx = Rate of CTA consumption, kmol/(m3·s) R•w = Radical concentration in the water phase, kmol/m3 R•p = Radical concentration in the polymer phase, kmol/m3 Rwe = Rate of radical entry, kmol/(m3·s) Rd = Rate of radical desorption, kmol/(m3·s) Rwt = Rate of radical termination in water phase, kmol/(m3·s) Rhom = Homogeneous nucleation rate, m−3s−1 r = Particle radius, m S = Nucleation rate, m−6s−1 t = Time, s T = Temperature, K T* = Characteristic temperature, K T̅ = Reduced temperature Ve = Volume of the emulsion phase, m3 Vw = Volume of the aqueous phase, m3 Vp = Volume of the polymer phase, m3 Vg = Volume of the gas phase, m3 Vreactor = Reactor volume, m3 V1 = Minimum size of particle nuclei, m3 W = Stability ratio X = Chain transfer agent y = Electrolyte

(A5)

At very high monomer conversions, the self-diffusion coefficient of the polymer becomes very small, resulting in an unrealistically low value of kt according to eq A2. The reason is that eq A3 does not account for the mobility of the radical chains caused by the monomer propagation reaction. This phenomenon is known as “residual termination”. To account for the latter contribution to the overall termination rate constant, a residual termination rate constant, kt,res, which is proportional to the frequency of monomer addition to the radical chain end, is defined: ktres = Ak p[M]

(A6) 28

where A is a proportionality rate constant.

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APPENDIX B: OVERVIEW OF MODEL EQUATIONS The computational model (i.e., equations and parameters) that was employed to simulate the dynamic behavior of a semibatch VDF emulsion polymerization reactor, is summarized in Table B1. Table B1. Model Overview model component

equations

kinetic rate functions eqs 11−15 diffusion controlled Appendix A reactions molar species eqs 16−18 and ref 21 balances crystallinity Figure 2 volumes eqs 19−22 population balance eq 23 model coagulation rate eqs 24−26 and refs 17, 21, 43. growth/nucleation eqs 27−30 radical balance eqs 31−41 thermodynamics eq 42, 44, 45 molecular weights eqs 46−48, ref 21

■

parameters Table 4 Appendix A and ref 28

solution method/parameters in refs 41, 42 refs 17, 21, 43 refs 32, 33 refs 10, 17, 33, 34 Tables 2 and 3

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Address: Department of Chemical Engineering, Aristotle University of Thessaloniki, P.O. Box 472, Thessaloniki, 541 24, Greece. Notes

The authors declare no competing financial interest.

■

NOTATION = Concentration of species i (i = I, m, x, y, s) in the water phase, kmol/m3 Cpi = Concentration of species i (i = m, x) in the polymer phase, kmol/m3 Ci = Concentration of species i (i = m, x, y, s) in emulsion, kmol/m3 CCMC = Critical micelle concentration, kmol/m3 Dn = Dead polymer chain of length “n” Dw = Monomer diffusion coefficient in water phase, m2/s Dp = Monomer diffusion coefficient in polymer phase, m2/s f(V,t) = Number density size distribution, m−6 FI = Total initiator feed rate, kmol/s FM = Monomer feed rate, kmol/s Fx = CTA feed rate, kmol/s Cwi

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Greek Symbols

αc = Particle surface crystalline fraction α = Fate parameter β = Coagulation rate kernel, m3/s μ = Viscosity, Pa.s ρ̅ = Reduced density ρ* = Characteristic density, kg/m3 ρm = Density of monomer, kg/m3 ρp = Density of polymer, kg/m3 ρ(r) = Kinetic rate constant for radical entry into the particles, s−1 φp = Polymer volume fraction in the polymer phase Subindexes and Superindexes

A = amorphous C = Crystalline e = Emulsion phase I = Initiator m = monomer p = Polymer phase s = Surfactant w = Aqueous phase x = Chain transfer agent y = Electrolyte

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