18th European Symposium on Computer Aided Process Engineering – ESCAPE 18
Bertrand Braunschweig and Xavier Joulia (Editors)
© 2008 Elsevier B.V./Ltd. All rights reserved.
Brownian Dynamics and Kinetic Monte Carlo
Simulation in Emulsion Polymerization
Hugo F. Hernandez,a Klaus Tauera
a
Max Planck Institute of Colloids and Interfaces, Am Mühlenberg 1, Golm 14476,
Germany
Abstract
In this work, a generalized structure of a multi-scale dynamic simulation model for
emulsion polymerization is presented. A simplified version of this multi-scale structure,
a multi-scale kinetic Monte Carlo - Brownian Dynamics simulation, is used to study the
competition between the capture of radicals by polymer particles and radical reactions
in the aqueous phase in a seeded semibatch emulsion polymerization. In this case, the
system is simulated by a kinetic Monte Carlo technique where the kinetic information of
radical absorption is supplied by a lower scale Brownian Dynamics simulation model.
The kinetics of radical capture by polymer particles is determined by Brownian
dynamics simulation using a Monte Carlo random flight algorithm. The multi-scale
model developed may be used to optimize the conditions for avoiding secondary
particle nucleation in any particular emulsion polymerization system.
Keywords: Brownian Dynamics Simulation, Emulsion Polymerization, Kinetic Monte
Carlo simulation, Multiscale Modeling.
1. Introduction
Radical emulsion polymerization is a widely used technique for the synthesis of
polymers for a variety of industrial fields ranging from coatings and adhesives to
biomedical applications (Tauer, 2004). Emulsion polymerization is preferred over other
polymerization techniques especially because of the higher molecular weight of the
polymers obtained, the low viscosity of the latex, the increased safety and productivity
of the reaction, and for being friendlier to the environment (Daniel, 2003). However,
good product quality control and batch to batch homogeneity are very difficult to
achieve because radical emulsion polymerization is a highly complex dynamic process
in which several simultaneous and usually competitive colloidal (aggregation,
coalescence), chemical (radical generation, polymerization, termination, chain transfer)
and physical events (diffusion, nucleation, swelling) occur at very different time scales
and dimensions. These events take place, for instance, at rates ranging from about 100 to
109 s-1 and involving entities of very different length scales, such as ions and molecules
(< 1 nm), macromolecules (1 – 10 nm), polymer particles (10 nm – 1 m) and monomer
droplets (>1 m). Additionally, emulsion polymer systems may present a marked
spatial heterogeneity on the macroscopic scale (imperfect mixing), making it very
sensitive to temperature and composition profiles inside the reactor. For all these
reasons, it is clear that a very precise representation of the process is only possible if
different simulation techniques –such as Brownian Dynamics simulation, kinetic Monte
Carlo stochastic simulation, Molecular Dynamics simulation, Computational Fluid
Dynamics, etc.– are integrated into a multi-scale simulation approach (Fermeglia and
Pricl, 2007; Kevrekidis et al., 2004; Chen and Kim, 2004).
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H. F. Hernandez and K. Tauer
2. Multiscale modeling and simulation of emulsion polymerization
A general representation of the different scales in emulsion polymerization is presented
in Figure 1. At least seven different characteristic scales can be considered for multiscale modeling of emulsion polymerization. At each level, different simulation
techniques can be used depending on the type of information required.
Figure 1. General multi-scale structure of emulsion polymerization
The atomistic scale considers the electronic interactions between different atoms of the
same or different molecules. These interactions are determined by the corresponding
solution of the Schrödinger equation. Therefore, Quantum Mechanics (QM) can be used
to obtain valuable information about the system, such as activation energies and
reaction rate coefficients, long-range potential parameters (such as the Lennard-Jones
parameters) and interaction parameters (such as the  Flory-Huggins parameter). At the
molecular scale, the information obtained by QM can be used to determine transport
properties (viscosity, diffusivity, thermal conductivity), chemical potentials and
interfacial tensions. For this purpose, techniques such as Molecular Dynamics (MD)
simulation or Monte Carlo (MC) simulation can be used. At the macromolecular scale it
is possible to use a coarse grained (CG) representations of the structure of matter
considering groups of atoms as single units, and it is also possible to use the mean field
(MF) approach, by assuming continuity for the most abundant chemical species. This is
the case of water in the aqueous phase of an emulsion polymerization, or monomer
inside the polymer particles. Kuhn lengths and gyration radii of the polymer segments
can be calculated and the conformation and arrangement of macromolecules in the
aqueous phase and inside the polymer particles can be determined. Transport properties
of small molecules inside polymer particles and transport properties of macromolecules
in the continuous phase can be determined using Lattice or Off-lattice chain models. At
the colloidal scale Brownian entities ranging from 1 nm to 1 m are defined, and the
kinetics of colloidal events such as collision, aggregation, flocculation, adsorption, etc.,
can be evaluated. The frequent collisions of small molecules from the continuous phase
are responsible for the random motion, and therefore, for the diffusion of Brownian
entities. This effect can be represented by Brownian Dynamics (BD) simulation which
is based on the solution of Langevin’s equation of Brownian motion. At this level,
Population Balances (PB) are also useful for incorporating polydispersity effects. At the
Brownian Dynamics and Kinetic Monte Carlo Simulation in Emulsion Polymerization
microscopic scale, an integration of chemical, colloidal and hydrodynamic events is
performed using the kinetic Monte Carlo (kMC) technique, also known as Stochastic
Simulation Algorithm (SSA). The main condition for this integration is the assumption
of perfect mixing inside the reference volume. At the mesoscopic scale, mixing
hydrodynamic effects are dominant. In this case, Finite Element Modelling (FEM)
techniques such as Computational Fluid Dynamics (CFD) can be used. These methods
are able to calculate composition and temperature profiles inside the reactor. Finally, the
macroscopic scale, which in this case is the emulsion polymerization reactor, is
modelled basically with Ordinary Differential Equations (ODE) according to the laws of
conservation of mass and energy (and momentum if necessary).
A very important aspect of multi-scale modeling is the processing and exchange of
information between the different scales. A lower-scale model requires information
about the state of the system (temperature, velocity, composition, etc.) which is
determined at a higher scale, while at the same time the upper-scale model requires
parametric and structural information of the system obtained at a lower scale. Therefore,
top-down and bottom-up information exchange procedures must be clearly defined
(Chatterjee and Vlachos, 2006; Maroudas, 2000; Broughton et al., 1999). In the topdown procedure, a suitable grid decomposition method based on the distribution of
states of the corresponding system scale must be used, while in the bottom-up
procedure, the integration of the lower-scale results must be performed. In the next
section, an illustrative example of multiscale modeling in emulsion polymerization is
presented consisting only of two scales, the microscopic and the colloidal scales.
3. Application of multiscale modeling to semibatch seeded emulsion
polymerization
The system considered for this example is the semi-batch surfactant-free emulsion
polymerization of vinyl acetate in the presence of a monodisperse polystyrene seed
latex, initiated by a water-soluble initiator (potassium persulfate) at 80°C. A multiscale
model is developed to simulate the molecular weight distribution of polymers in the
aqueous phase during polymerization in the presence of radical-capturing polymer
particles. For a monomer-starved feed addition policy, it is possible to assume that the
monomer concentration in the aqueous phase is constant, that the monomer
concentration in the polymer particles is low enough to give rise to a high viscosity
inside the particles (therefore negligible radical desorption rates), and that the particles
grow at the rate of monomer addition.
3.1. Microscopic scale: Kinetic Monte Carlo (kMC) simulation
The kMC simulation technique is based on the SSA introduced by Gillespie (1976). The
basic idea of this method is to randomly simulate competitive events (such as chemical
reactions) based on their frequency of occurrence. The kMC method is a very good
alternative to the classical deterministic rate equations especially when very infrequent
events are considered or when a very low number of reactants are present. The kMC
method has been successfully used to describe polymerization processes (Nie et al.,
2005; Tobita et al., 1994; Lu et al., 1993). In the present example, the following
competitive events are considered: Initiator decomposition in the aqueous phase, radical
capture by polymer particles, and propagation and termination by recombination
reactions in the aqueous phase. Additional events such as chain transfer reactions or
termination by disproportionation can be easily included in the formulation of the kMC,
but they are not considered in this example for simplicity. The key state variables at the
microscopic scale are the chain length distribution and concentration of polymer in the
3
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H. F. Hernandez and K. Tauer
aqueous phase. The kinetic coefficients used for the chemical reactions were taken from
the literature (Ferguson et al., 2002). The capture rate coefficients are periodically
calculated by lower-scale Brownian Dynamics simulations of the system.
3.2. Colloidal scale: Brownian Dynamics (BD) simulation
The BD simulation method is based on the solution of Langevin’s equation for
Brownian motion, which contains a random force term caused by the frequent random
collision between molecules. BD has been successfully employed for simulating a wide
variety of systems including molecules, colloidal particles and macromolecules, and it
has been previously used for studying radical capture in emulsion polymerization
(Hernández and Tauer, 2007). Radical capture kinetic coefficients can be easily
estimated with BD simulation, by determining the average time required by a radical
generated in the aqueous phase to enter a polymer particle. When the size of the
polymer particles changes above a certain tolerance value in the kMC simulation, a BD
simulation of radical capture under the new conditions is triggered. The capture rate
coefficients for every chain length of the radicals are calculated and given back to the
kMC model, to continue the upper-scale simulation.
3.3. Multiscale kMC-BD simulation
3.3.1. Simulation conditions
In Table 1, the full set of conditions and parameters used during the multiscale kMCBD simulation is presented. In all cases the final polymer volume fraction was 20%.
Table 1. Parameters used for the multiscale kMC-BD simulation of semi-batch seeded emulsion
polymerization of vinyl acetate
Parameter
Value
Parameter
Value
Temperature (°C)
Seed volume fraction (%)
Seed polymer density (g/ml)
Poly(vinyl acetate) density
(g/ml)
Monomer feed rate (mol/ls)
Initial initiator
concentration (mol/l)
Propagation rate coefficient
(l/mols)
Primary radical molar
volume (l/mol)
Monomer unit molar
volume (l/mol)
Water viscosity (cP)
80
0.01 - 10
1.044
1.15
Simulation volume (l)
Seed particle diameter (nm)
Water density (g/ml)
Aqueous monomer
concentration (mol/l)
Initiator efficiency
Initiator decomposition rate
coefficient (s-1)
Termination rate coefficient
(l/mols)
Primary radical molar mass
(g/mol)
Monomer unit molar mass
(g/mol)
Particle size tolerance for
BD simulation triggering
1x10-14
10-500
0.972
0.3
3x10-4
1x10-3
1.29x104
0.046
0.075
0.355
0.9
8.6x10-5
1.13x1010
96.16
86.09
5%
3.3.2. Simulation results
Figure 2 shows an example of the multiscale integration in the kMC-BD simulation.
Whenever the particle size is increased above the tolerance value of 5%, a BD
simulation is triggered and the capture rate coefficients are updated. This can be seen in
Figure 2 as “jumps” in the value of the rate coefficient. In this example, an increase of
almost one order of magnitude in the capture rate coefficient between the start and the
end of the simulation is observed.
Brownian Dynamics and Kinetic Monte Carlo Simulation in Emulsion Polymerization
5
Capture rate coefficient, kc (l water/part.s)
1e-10
1e-11
BD simulation updates
1e-12
0
2000
4000
6000
8000
10000
Time, t (s)
Figure 2. Periodic determination of the capture rate coefficient of primary radicals by BD
simulation. Seed particle size: 100 nm; seed volume fraction: 0.1%.
5
100
Number average chain length, Ln
Weight average chain length, Lw
Polydispersity index, PDI
Max. chain length, Lmax
Seed volume fraction: 1%
1000
Number average chain lenght, Ln
Weight average chain length, Lw
Polydispersity index, PDI
Max. chain length, Lmax
1
Seed particle size: 100 nm
100
15
Lmax
3
Ln, Lw, PDI
20
10
Lmax
Ln, Lw, PDI
4
25
10
10
2
5
1
0
100
200
300
400
Particle diameter, dp (nm)
500
0.1
600
1
0.001
0.01
0.1
1
10
Seed volume fraction (%)
Figure 3. Number average chain length (Ln), Weight average chain length (Lw), Polydispersity
Index (PDI) and Maximum chain length (Lmax) of the radicals formed in the aqueous phase for
different particle size and volume fraction of the initial seed. The dashed lines represent best fit
curves.
The effects of the competition between aqueous phase propagation and radical capture
by polymer particles are clearly seen in Figure 3. The chain length of the polymer
formed in the aqueous phase increases as the initial seed particle size increases (for a
constant volume fraction) or the volume fraction of the initial seed decreases (for a
constant particle size). That is, the degree of polymerization in the aqueous phase
increases as the rate of radical capture decreases. An additional effect observed is the
increase in the polydispersity of the polymer formed in water as the rate of radical
capture decreases. This result reflects the fact that smaller radicals are captured by the
particles more easily than the larger ones. Therefore, the larger radicals can grow even
further in the aqueous phase increasing the polydispersity of the system.
For the particular case considered, radical propagation in the aqueous phase is
practically suppressed when initial seed particles smaller than 80 nm at seed volume
fractions higher or equal than 1% are used. When radical propagation is unavoidable, it
is possible to determine the extent of secondary particle nucleation in the system using a
suitable nucleation model, like for example, the Classical Nucleation Theory (Becker
and Döring, 1935; Tauer and Kuhn, 1995) or a modification thereof (Hodgson, 1984).
The nucleation model requires the knowledge of the concentration (obtained from the
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H. F. Hernandez and K. Tauer
kMC-BD simulation) and the solubility of all the nucleable species. Thus, it is possible
to use the kMC-BD multiscale simulation model to optimize the conditions required for
avoiding secondary nucleation in any given emulsion polymerization system.
4. Conclusions
Multiscale dynamic simulation methods have been shown to be potential tools for the
investigation of complex systems, in particular, of emulsion polymerization processes.
In this paper, a kMC-BD simulation was presented as an example of the integration of
different scales in the modeling of semi-batch seeded emulsion polymerization
processes. The results obtained in the simulation can be used to find optimal conditions
for the suppression of secondary nucleation in emulsion polymerization.
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