Modeling of Molecular Transfer in Heterophase Polymerization

Feature Article

Modeling of Molecular Transfer inHeterophase Polymerization

Hugo F. Hernandez, Klaus Tauer*

Mass transfer across interfaces greatly determines the kinetics of heterophase polymerization.All molecules in the system can cross any interface as long as they possess enough energy toovercome barriers such as interfacial tension, chemical potentials, etc. Two main groups ofmathematical approaches have been used tomodel mass transfer: macroscopic deterministicand molecular stochastic. Macroscopic modelingmay use fundamental laws, thermodynamicexpressions and empirical or semi-empiricalequations. Molecular models are based on thediscrete character of nature and include stochasticsimulation, BD or MD. Special emphasis is placedon themost relevant molecular transfer processesobserved in free-radical emulsion polymerization.

1. Introduction

The importance of polymeric materials for our modern

world and daily life is unquestioned. Synthetic polymer-

ization processes are complex, involving a huge variety of

molecular species of different size and sometimes also

differentmolecular shapes and compositions. Heterophase

polymerization is a generic term which describes poly-

merization reactions under non-homogeneous conditions

with respect to physical and chemical properties of the

reaction mixture. This means the existence of gradients

such as in density and chemical composition and conse-

quently, the coexistence of different phases. In this type of

polymerization processes, the transfer of molecules from

one phase to another takes place simultaneously with the

chemical reaction events. One of the most representative

types of heterophase polymerization is the free-radical

H. F. Hernandez, K. TauerMax Planck Institute of Colloids and Interfaces, Am Muhlenberg 1,14476 Golm, GermanyFax: þ 49 331 567 9502; E-mail: [email protected]

Macromol. React. Eng. 2009, 3, 375–397

� 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

emulsion polymerization process which will be considered

exclusively.

In a typical ab initio batch emulsion polymerization a

slightly soluble monomer is dispersed in water, forming

monomer droplets. These droplets may be stabilized

against coalescence using a surface active agent (surfac-

tant). The initiator isadded to thesystemandthenthechain

reactions start either in the aqueous or in the monomer

phase or in an interfacial layer, depending on the parti-

cular properties of the initiator. As polymerization pro-

ceeds, phase separation occurs giving rise to the formation

ofpolymerparticles. Thereafter thepolymerizationkinetics

is additionally influenced by vigorous transfer of all kind of

molecules between the different phases. The reliable

modeling of these mass transfer processes is crucial for

design, engineering, optimization, and control of both

the polymerization processes and the properties of the

polymers.

The intention of this paper is to describe and to

analyze the most relevant techniques used to model mass

transfer in heterophase polymerization, including classical

deterministic procedures but also molecular modeling

strategies.

DOI: 10.1002/mren.200900016 375

H. F. Hernandez, K. Tauer

376

The paper is organized as follows. In Section 2, a more

detailed description of radical heterophase polymerization

processes ispresented. Section3describes themost relevant

methods used to model mass transfer between phases. In

Section 4 some of the most relevant transfer events

occurring in free-radical emulsion polymerization are

analyzed and discussed considering the differentmodeling

strategies. Finally, in Section 5 besides some concluding

thoughts possible future trends in modeling of mass

transfer processes in heterogeneous systems are briefly

discussed.

Figure 1. General representation of a heterogeneous system con-taining continuous phases (A, B, C, F) and segregated phases (a, b,c, d, e). A particular example of this system is an ab initio free-radical emulsion polymerization with deficient stirring. In thiscase, phases A and a correspond to the volume of gas, phases Band b are mixtures of monomers containing dissolved oligomers,phases C and c are aqueous solutions of salts (initiator molecules,ionic surfactants, buffer, etc.) and other soluble components,phases d and e are additional segregated phases like for examplepolymer particles and micelles, and phase F is the solid phase ofthe metallic wall of the reactor, which corresponds to the heattransfer area. The subscript of segregated phases is the identi-fication number of each individual droplet or particle in thesystem.

2. General Picture of HeterophasePolymerization

Homogeneous processes are restricted to solution or bulk

polymerizations, where the polymer is soluble in the

monomer phase. Examples of heterogeneous polymeriza-

tion are gas-phase polymerization, dispersion polymeriza-

tion, suspension polymerization, miniemulsion polymer-

ization, microemulsion polymerization and emulsion

polymerization. All these systems are characterized by a

continuous phase containing segregated or dispersed

phases. It is convenient to distinguish phases by their

characteristic length scales. Clearly, the characteristic

lengths of continuous phases are much larger than those

of segregated phases. Two phases in direct contact with

different compositions are characterized geometrically by

an interface and energetically by an interfacial tension.

A very general representation of a heterophase poly-

merization system involving many different phases is

presented in Figure 1. An ab initio batch monomer-flooded

polymerization in a stirred tank reactor with four

continuous phases (denoted by capital letters) and also a

variety of segregated phases (denoted by lower case letters

and numbers) contains different interfaceswith either two

or three phases in contact. Note, this is the most

complicated and challenging scenario for heterophase

polymerization and hence, useful for all the following

considerations. Obviously, the largest interfaces and hence

the most important for transfer of energy and matter are

those between segregated and continuous phases scaling

with the ratio between the overall amount and the

characteristic length of the dispersed phase. However,

regarding the productivity of the reactor, the interface

between the reactor wall and the interior is crucial because

via this interface the polymerization heat is removed. For

instance, the maximum rate of polymerization, for iso-

thermalsemi-batchorcontinuouspolymerizationprocesses,

is determined by the maximum heat transfer through the

reactor wall, which is reduced by fouling due to the

deposition of polymer. Also, hot cooling is frequently



applied to remove reaction heat for which the gas phase of

the reactor plays an important role.

Segregated phases such as particles, drops, and bubbles

can be present in one or more continuous phases.

Depending on the particular conditions, monomer drops

and polymer particles are present in the continuous

aqueous phase but also water drops and polymer

particles (if cross-linked or insoluble) can be present in

the monomer phase (if not properly dispersed or in very

large monomer droplets). Special cases are multiple

emulsions where either oil-in-water emulsion drops are

dispersed in a continuous oil phase or vice versa.

Different phases exist as a result of differences in the

attractive and repulsive forces acting on each molecular

species. This balance of forces is influenced by various

factors such as temperature, pressure, chemical reactions,

andexternal forces (e.g. gravity, centrifugal forces, electrical

forces, magnetic forces, etc.). Whether or not phase

separation occurs is the result of a competition between

energetic and kinetic factors. The energetically favored

state, determined by physical and chemical potentials, can

only be realized if the mobility under the particular

conditions allows it. This principle is the basis of particle

DOI: 10.1002/mren.200900016

Modeling of Molecular Transfer in Heterophase . . .

Klaus Tauer (born 1951) studied chemistry at theFriedrich Schiller University Jena from 1970–1977and received his PhD on ‘‘Investigations of spectralsensitization of photoconductivity of poly(arylenevinylene) polymers’’ in 1978. In September 1977 hestarted working on heterophase polymerizationsat the Institute for Polymer Chemistry of theAcademy of Sciences of the former GDR in Teltow-Seehof. His research interests were kinetics,mechanisms, and mathematical modeling ofemulsion polymerization of vinyl chloride, whichwas also the topic of his habilitation 1987. SinceJanuary 1992 he works as a scientist at the MaxPlanck Institute of Colloids and Interfaces in thedivision of Colloid Chemistry. His current researchinterests are mechanism and modeling of hetero-phase polymerization, particle nucleation andreactive surfactants as well as block copolymersand block copolymer latex particle via hetero-phase polymerization.Hugo F. Hernandez Garcıa received a mastersdegree in chemical engineering from the Univer-sidad Nacional de Colombia in Medellin in 2004.Starting in 2001, he worked in the chemical indus-try at Andercol S.A. in Medellin, Colombia, wherehe held different positions including processengineer, six sigma black belt and research &development engineer. In 2006 he started hisdoctoral studies at the Max Planck Institute ofColloids and Interfaces/University of Potsdam(Germany) working under the supervision ofKlaus Tauer in the department of Markus Anto-nietti in the field of multi-scale simulation ofheterophase polymerization. In 2008 he obtainedthe degree of Dr. rer. nat. in the field of colloidchemistry from the University of Potsdam. Cur-rently, he holds a post-doctoral research fellow-ship from Andercol S.A. for working in the field ofheterophase polymerization at the Max PlanckInstitute of Colloids and Interfaces.

aIn fact, we can detect the presence of dispersed phases of a certainsize by the naked eye because of Tyndall’s effect, but we cannotobserve them in the same way we observe macroscopic drops of oilin water.

morphology development, where a particular phase

separation in a mixture of different polymers is achieved

by the careful control of both thermodynamic and kinetic

factors.[1]

Thermodynamics acts in the direction that the chemical

potential of any species is the same in any phase. In other

words, there isadriving force thatpushesall components in

a given system to be present in all phases. As the chemical

potential on the colloidal scale is size dependent, also the

chemical composition of colloidal objects is size dependent

at equilibrium. Consequently, non-monodisperse polymer

particles in the same system may have similar but not

identical compositions.

The mobility of the molecules in the system is not only

important during the phase separation but it remains



important during all stages of a polymerization. The

application of the formalism of classical equilibrium

thermodynamics to a given experimental system requires

equilibrium and hence, time independent considerations.

Thermodynamic equilibrium is characterized in a rigorous

way by simultaneous vanishing of all fluxes and forces.[2]

This means for a polymerization reaction a state where

thermal, mechanical, and chemical equilibrium is reached.

During steady state several properties do not change with

time such as conversion, composition, or average particle

size during continuous polymerizations. In this sense a

steady state, though it is in average time-independent, is

not at thermodynamic equilibrium as fluxes are not

vanished. A system at ’dynamic equilibrium’ is character-

ized by reversible processes where forward and backward

reactions occur with the same rate. This is the case for

swelling equilibrium where molecules continuously move

from one phase to the other in balanced amounts that the

chemical potentials on either side in average are not

changed. However, this equilibrium consideration holds

only for long period of times and large areas or volumes.

Under such conditions the uncritical use of equilibrium

thermodynamics should be done, if so ever, cautiously,

because this can lead to obscuration.[3]

The identification of different phases in the system

depends strongly on the scale at which the system is

observed. Observation with a resolution much lower than

the characteristic scale of the segregated phases leads de

facto to the suggestion of a homogeneous system. For

example, small particles of liquid or soliddispersed inwater

like inmilk or latex cannot bedetected geometrically by the

naked eyea and the system appears to be one single

continuous phase as interfaces are not visible. However, if

the same system is observed with a powerful microscope,

the segregated phases are clearly to be seen. On the other

hand, if the resolution is too high and the magnification

reaches almost molecular dimensions, small local fluctua-

tions incompositionmightbe interpretedasbeingdifferent

phases.Under suchconditionapolymer solutionwill notbe

regardedasonesingle continuousphasebutas composedof

pure polymer or polymer/solvent complexes segregated in

a continuous pure solvent phase. This means that all

multicomponent systems observed at a close-to-molecular

scalewill be regarded as heterogeneous systems. Therefore,

homogeneouspolymerizationsystemscouldbe regardedas

only particular cases of the more general heterophase

polymerization systems.

The free-radical polymerization mechanism can be

described by the following reactions which can take place

www.mre-journal.de 377


378

in all phases:

Macrom

� 2009

Initiator decomposition : I!kd 2R� (1)

Initiation : R� þM!ki P�1 (2)

Propagation : P�i þM!kp

P�iþ1 (3)

Termination by combination : P�i þ P�j !ktc

Diþj (4)

Termination bydisproportionation :(5)

P�i þ P�j !ktd

Di þMj

Chain transfer : P�i þ T!kfTDi þ T� (6)

The nomenclature used in reactions (1) to (6) is the

following:

kx: C
hemical reaction rate coefficient of reaction x
I: In
itiator molecule
R�: P
rimary radical
M: M
onomer molecule
Mj: P
olymer molecule with chain length j and one
available double bond

Pi�: P
olymer radical with chain length i
Di: D
ead polymer with chain length i
T: C
hain transfer agent. T can represent the monomer
(chain transfer tomonomer), the samepolymer chain

(backbiting), a different polymer chain (branching),

or any other molecule present in the system with a

labile hydrogen atom (e.g. surfactant, solvent, etc.)

T�: R
adical derived from the chain transfer agent.
Simultaneously with the chemical changes physical

events take place. It is convenient to describe these phase

transfers by first order kinetic coefficients.

Transfer from a continuous phase to a segregated phase

dispersed in that continuous phase (absorption or entry):

CX !ka;Xyi

Cy;i (7)

Transfer from a segregated phase dispersed in a

continuous phase to that continuous phase (desorption

or exit):

Cy;i !k0;yiX

CX (8)

Transfer from a continuous phase to another continuous

phase:

CX !kc;XZ

CZ (9)

ol. React. Eng. 2009, 3, 375–397

WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

In Equation (7)–(9), the following nomenclature is used:

ka, k0, kc: mass transfer rate coefficients for absorption,

desorption and transfer between continuous phases,

respectively.

CX: Molecule C in continuous phase X

Cy,i: Molecule C in the i-th segregated phase of y

It is important to notice that the molecular transfer

between two different segregated phases is not possible

because intermediate steps througha continuousphase are

always required. In general, phase transfer events can only

take place between phases in direct contact. The condition

for steady state can be expressed as follows:

ka;Xyi C½ ��X¼ k0;yiX C½ ��yi (10)

kc;XZ C½ ��X¼ kc;ZX C½ ��Z (11)

where the brackets and superscript ‘‘�’’ indicate concentra-

tions and steady state, respectively. Equation (10) corre-

sponds to the steady state between a continuous phase (X)

and a phase segregated in that continuous phase (yi).

Equation (11) expresses the steady state condition between

two continuous phases in direct contact (X and Z). Notice

also that at steady state:

C½ ��XC½ ��yi

¼ k0;yiX

ka;Xyi¼ KXyi (12)

C½ ��X� ¼ k0;ZX ¼ KXZ (13)

C½ �Z ka;XZ

where KXyi and KXZ are the partition coefficients between

the phases X and yi, and X and Z, respectively. The partition

coefficients are constant for a particular set of conditions,

and as can be seen,will be affected by the same factors that

influence the kinetics of phase transfer in the system (e.g.

temperature, composition, pressure, etc.). More details

about partition coefficients will be given in Section 3.

There are two thermodynamic effects determining

the transfer of molecules between phases. The first is

the entropic effect caused by the random motion of the

molecules in all directions. This random motion is

responsible for the uniform local concentration observed

in perfectly mixed systems. The second is the enthalpic

effect caused by different attractive and repulsive forces

acting on the diffusing molecule. These forces can either

facilitate or impede phase transfer of a given molecule.

In principle, every single molecular species present in

the system can be transferred from one phase to

another. However, local attractive and repulsive forces

between the molecules, mainly at the interfaces (e.g.

interfacial tension), determine the easinesswithwhich this

transfer takes place.

DOI: 10.1002/mren.200900016


Ina typical free-radical emulsionpolymerizationprocess,

primary radicals can be transferred from the continuous

aqueous phase to any segregated phase (radical absorption

or radical entry), or can be transferred from the segregated

polymer particles to the continuous aqueous phase (radical

desorption or radical exit); monomers can be transferred

from themonomer droplets to the polymer particles trough

the continuous aqueous phase (monomer swelling);

surfactant molecules can be transferred from the contin-

uous phase or micelles to the surface of the polymer

particles through the continuous phase (surfactant adsorp-

tion); monomer and water can be transferred to the vapor

phase (evaporation); oxygen andnitrogenmolecules canbe

transferred from the vapor phase to the aqueous phase (gas

absorption), etc.

As a result of the entropic and enthalpic effects, the

concentrations of each type of molecule in each phase are

usuallyverydifferent. For this reason, the individual kinetic

reactions (1)–(6) take place in each phase with different

rates depending on the particular composition. Since the

composition of each phase is determined by individual

mass transfer events (7)–(9) they strongly influence the

kinetics of heterophase polymerization.

A clear example of the influence of mass transfer on the

kinetics is the rate of polymerization in free-radical

emulsion polymerization. Neglecting polymerization in

the continuous phase, the rate of emulsion polymerization

can be approximated quite accurately by the rate of

monomer conversion inside the polymer particles:[4]

Macrom

� 2009

rp ¼kp M½ �pnNp

NA(14)

rp is the rate of polymerization, kp is the rate coefficient of

propagation, [M]p is theaverage concentration ofmonomer

inside the polymer particles, n is the average number of

radicals per polymer particle, Np is the number concentra-

tion of particles in the system, and NA is Avogadro’s

constant.

Under conditions where Equation (14) is valid the

monomer concentration decisively acts twofold. Firstly, it

determines directly rp and secondly, as it is a solvent for the

polymer it influences the viscosity and controls mobility

inside the particles. High polymer or low monomer

concentration causes the apparent rate coefficient of

propagation to decrease. This decrease is the result of the

reducedmobility of themonomermolecules. The reduction

of mobility eventually reaches a critical point at which the

propagation reaction is practically suppressed. This phe-

nomenon is known as glass effect.[5] The glass effect is an

example of how molecular transfer on small length scales

(here inside the polymer particles) influences polymeriza-

tion kinetics. This effect is important under monomer-

ol. React. Eng. 2009, 3, 375–397


starved conditions at high conversion. [M]p depends on the

competition between consumption of monomer by propa-

gation and the transfer rate of monomer into the polymer

particles. The latter isquitecomplexas transferbetweenthe

monomer droplets and the continuous aqueous phase,

between the aqueous phase and the polymer particles, and

if present, between themicelles and the aqueous phase are

involved. If a steady state of [M]p is reached it is the result of

intricate interactions between the continuous phase, each

monomer droplet, the individual monomer-swollen poly-

mer particle, and, if present, also monomer-swollen

micelles. Differences in the particle number, size distribu-

tions, composition and surface chemistry of each segre-

gated phase, and in the composition of the continuous

phases (themonomers are also distributed in the gasphase)

influence the steady state. On the other hand, how fast this

state is achieved depends on the rate of monomer transfer

between the segregated phases and the continuous phase,

and in particular, on the mobility and solubility of the

monomer in the phases. An additional complicated facet

can occur when the amount of monomer transferred from

the segregated to the continuous phase is larger than the

amount of monomer that can be solubilized by the

continuous phase. This can happen locally in smaller

volume elements of the dispersion especially close to the

interface of the monomer droplets. In this context it

should be pointed out that the state of monomer in the

continuous phase is not only that of molecularly dissolved

molecules. Beside singlemonomermolecules the existence

of aggregates and droplets has been experimentally

proved.[6] Moreover, the polymer particles behave inde-

pendently of each other because of deviations in their

interface composition and internal stress, both of which

influence swelling. Clearly, such scenario is in contrast to

thermodynamic conclusions for swelling that are based on

ensembles of large numbers neglecting differences

between the particles except their size.

The next factor in Equation (14) is the average number of

radicals per particle,which is determinedby the generation

and consumption of radicals, as well as by the transfer of

radicals between the polymer particles and the continuous

phase. The transfer of radicals fromthe continuousphase to

the polymer particles is denoted as radical absorption,

radical capture or radical entry, whereas the transfer of

radicals from the particles to the continuous phase is

denoted as radical desorption or radical exit. Obviously,

the dynamics of radical transfer (radical absorption and

desorption) are also critical factors greatly influencing

the kinetics of emulsion polymerization. In addition, the

generation and consumption of radicals is also influenced

by the viscosity of the different phases, which changes in

the course of the polymerization considerably and is the

reason for mass transfer limitation effects resulting

eventually in cage and gel effects.



Figure 2. Evolution of the probability distribution function as thenumber of events considered is increased: Case a shows theuniform probability distribution of a single event; cases b andc correspond to Gaussian probability distribution obtained whenseveral individual events are considered simultaneously, thewidth of the distribution (standard deviation) is observed todecrease by increasing the number of events; case d shows aDirac’s delta function obtained in the thermodynamic limit of thesystem.

380

The last factor influencing the rate of emulsion

polymerization is the number concentration of polymer

particles. The number of particles is determined by the

competing processes of particle nucleation and coalescence

or coagulation andbothprocesses dependonmass transfer.

Particle nucleation is in principle a phase transition that is

strongly influenced by the diffusion and phase transfer of

polymer or oligomer chains through the system. Particle

nucleation can be suppressed if the rate of transfer of

polymer chains from the continuous aqueous phase to

the existent polymer particles is high compared to

the rate of aggregation (by diffusion) of polymer chains

in the continuous phase.[7] On the other hand, particle

coalescence favors the formation of newparticles since due

to the decreasing overall interface the concentration of

oligomers/polymers in the continuous phase increases and

hence, the probability of nucleation also increases. Coales-

cence and coagulation requires closest contact between

particles. The most relevant mass transfer process taking

place during coalescence or coagulation is the Brownian

diffusion of the particles on the colloidal scale.

This brief consideration revealed that a correct under-

standing and an adequate description of mass transfer

events on the various length scales is crucial for a detailed

understanding of heterophase polymerization.

In a typical emulsion polymerization with about 50%

solids content and an average particle size of 100nm the

average particle number is about 1018 per liter of water.

For slightly water soluble monomers these particles

are the main reaction loci where much more than 90% of

the monomer reacts. This means that the individual

propagation reaction takes place in a space of attoliter

volume that corresponds to about 107monomermolecules.

However the number of simultaneously growing radicals

inside a single polymer particle is over the whole duration

of the polymerization rarely larger than unity. As a

consequence, modeling as a continuous flux of matter

(radicals) is no longer correct as significant stochastic

fluctuations can be important. A similar scenario is typical

for reactions in biological systems, that is, in living cells. To

treat such conditions kinetically Gillespie introduced a

method that connects traditional chemical kinetics and

stochastic approaches.[8] Under such conditions the parti-

cular composition of the reaction environment determines

the next reaction step. How efficient this procedure is has

been shown recently by modeling free-radical bulk

polymerization of methyl methacrylate up to high conver-

sions.[25]Amodifiedalgorithmfor thestochastic simulation

of chemical reactions subjected tomass transfer limitation

(imperfect mixing) has been used. This algorithm takes

into account the mixing by diffusion of the reacting

species between two consecutive reactions. The accuracy of

the algorithm relies on the precise determination of the

diffusion coefficients. But it allowed to simulate the



polymerization including cage, gel and glass effect with a

single set of kinetic parameters.

3. Methods for Modeling Mass Transfer

The modeling of mass transfer can be classified according

to the scale of observation in two different types:

(i) Macroscopic and (ii) microscopic or molecular modeling.

3.1. Macroscopic Modeling

Macroscopic modeling is based on the assumption that

the system is beyond its thermodynamic limit, that is,

that the number of molecules present in the system is so

large that the overall behavior of the system becomes

deterministic instead of probabilistic. A graphical explana-

tionof the thermodynamic limit is presented inFigure2. Let

usassumethat everysinglemolecule in thesystembehaves

according to an arbitrary probability distribution function

(a uniform distribution is used in Figure 2a). If the effect of

several independent molecules is added, according to the

central limit theorem[9] a normal or Gaussian probability

distribution function will be obtained (Figure 2b). This

effect can be easily understood considering the character-

istic functions of the probability distributions.[10] If the

number of events (or molecules) considered is increased,

the width of the Gaussian distribution (characterized by

the standard deviation) is reduced (Figure 2c). At a certain

point, the number of events considered is so large that

the standard deviation of the distribution becomes

negligible and the distribution resembles a single line,

Dirac’s delta function (Figure 2d). This critical point is

known as the thermodynamic limit of the system.

Different types of macroscopic deterministic models

have been used to describe mass transfer processes.

DOI: 10.1002/mren.200900016


These models can be subdivided in (i) first principles,

(ii) semi-empirical, (iii) equilibrium thermodynamics, and

(iv) empirical models.

3.1.1. First-Principles Modeling

First-Principles or Fundamentalmodels ofmass transfer are

basedonthedifferential equationsobtainedbyFick[11] after

following the same method employed by Fourier to

describe heat transfer. Fick’s results for isotropic materials

can be summarized in two laws as follows:

Macrom

� 2009

Ji ¼ �Dir Ci½ � (15)

@ Ci½ � 2

@t¼ Dir Ci½ � (16)

Figure 3. Schematic representation of the net flux across aninterface at different concentration as a result of moleculardiffusion. Region A has a higher molecular concentration thanregion B and therefore the flux of molecules by diffusion from Ato B is higher than from B to A. In this particular example, theconcentration in A is twice the concentration in B.

These results indicate that the total number ofmolecules

of type i crossing an imaginary interface of unit area per

unit time (Ji) is related to the diffusion coefficient of the

molecules in the medium (Di) and to the local spatial

gradient (5) of the molecular concentration (Ci); and that

the rate of change in the local molecular concentration is

related to the diffusion coefficient of the molecules and to

the Laplacian (52) of the molecular concentration, that is,

to the second partial derivative of the concentration with

respect to the position. These equations are sometimes

interpreted by assuming that molecular transfer is caused

by the difference in concentration between two regions.

However, this is just an apparent interpretation. Let us

consider the example presented in Figure 3, where two

imaginary zones (AandB) aredescribedby thedashed lines,

and themolecules are diffusing in the horizontal direction.

If the net force acting on the diffusingmolecules is zero (for

example in pure components or in ideal mixtures), the

displacement of each molecule is uniformly randomly

distributed, andtherefore, thenumberofmoleculesmoving

on each direction is the same. The net displacement is then

proportional to thedifference in thenumberofmolecules at

each side, that is, it is proportional to the gradient in local

concentration. However, the real cause of molecular

diffusion is the frequent random collisions with the

neighbor molecules and not the difference in molecular

concentration, even though as the result of the random

motion, the net flux of molecules becomes proportional to

the difference inmolecular concentration. If there is a non-

zero force acting on the molecules (for example at

interfaces), Equation (15)–(16) are no longer valid, and

chemical potentials should be considered instead of

concentrations to describe the diffusion process.

The rigorous application of Ficks equations to hetero-

phase polymerization systems involves the solution of

partial differential equations with four independent

variables (timeand the three spatial dimensions) expressed

usually in spherical coordinates centered on a representa-

ol. React. Eng. 2009, 3, 375–397


tive segregated phase. However, if the geometry of the

system is not perfectly spherical, cylindrical or planar, the

analytical solution of Fick’s equations becomes a very

difficult task. These complex geometries can be modeled

by partitioning the system into smaller regions having

simpler geometries[12] but at the expense of an increased

number of equations to be solved.

In anisotropic media, diffusion depends on the direction

in which it is measured, and the diffusion coefficient is

actually a function of the local spatial composition around

the diffusing molecule. Some common examples of

anisotropic media are crystals, textile fibers, and polymer

films inwhich themoleculeshaveapreferential directionof

orientation. In these cases, Fick’s laws remain valid but the

diffusion coefficient is now a matrix and not a scalar.

Anisotropy is especially important at interfaces since

diffusion across the interface is different than diffusion

in any other direction. For this reason, and considering that

chemical potentials should be used instead of concentra-

tions, rigorous analytical solutions of molecular transfer

across interfaces in real heterogeneous systems using

fundamental equations are very difficult if not impossible

to achieve.Numerical solutions canbeobtainedusingfinite

elements approximations, but a detailed knowledge of

anisotropic diffusion coefficients for all molecules at

interfaces is required. In practice, fundamental modeling

is used to describe diffusion at both sides of an interface,

whereas some assumptions are introduced in order to

obtain analytical solutions describing mass transfer across

the interface.



382

3.1.2. Semi-Empirical Modeling

A significant simplification of

Fick’s equations can be obtained

by assuming that the bulk of

eachphase (bothcontinuousand

segregated) isperfectlymixed. In

this case, mass transfer of a

certain molecular species is

expressed as a function of the

difference in concentration of

this species at both sides of the

interface, using semi-empirical

parameters denoted as local

mass transfer coefficients

(h)b.[13] A typical representation

of an interface is presented in

Figure 4. For this example, the

net flux of molecules across the

interface (Jj,Ab) can be calculated

as:

bLocalthe litfusionwhichcients.

Figure 4. Concentration profile of component j as a function of position at the interface betweenphases A and b. [Cj]A indicates the concentration of j in the bulk of phase A. [Cj]i

A indicates theconcentration of j at the interface on the side of phase A. [Cj]i

b indicates the concentration of j atthe interface on the side of phase b. [C ] indicates the concentration of j in the bulk of phase b. The

Macrom

� 2009

Jj;Ab ¼ hj;Ab Cj

� �i

A� Cj

� �i

b

� �(17)

j b

thicknesses of the boundary layers at each side of the interface are indicated by the symbol d. Jj,Aindicates the molecular flux of component j from the bulk of phase A to the interface. Jj,Abindicates the molecular flux of component j from across the interface. Jj,b indicates the molecularflux of component j from the interface to the bulk of phase b.

where [Cj]iX denotes the concen-

tration of the j-th component at

the interface of the phase X.

Note, in Equation (17) neither

the bulk concentrations in both phases nor the profile close

to the interfacematter. Additionally, assuming that there is

no accumulation of matter at the interface, the flux of

molecules across the interface (Jj,Ab) should be the same as

the flux of molecules from the bulk to the interface of the

concentrated phase (Jj,A) or from the interface to the bulk of

the diluted phase (Jj,b). Thus,

Jj ¼ Jj;Ab ¼ Jj;A ¼ Jj;b (18)

where

Jj;A ¼ hj;A Cj

� �A� Cj

� �i

A

� �(19)

J ¼ h C� �i � C

� �� (20)
j;b j;b j b j b
mass transfer coefficients have been traditionally denoted inerature with the symbol k. However, in order to avoid con-with the kinetic rate coefficients, the symbol h will be used,is generally the symbol used for local heat transfer coeffi-

ol. React. Eng. 2009, 3, 375–397


Comparing Equation (15) and (18) to Equation (20), it is

possible to observe that this semi-empirical approach is

equivalent to the first principles model as long as:

hj;X ¼ Dj;X

dX(21)

with Dj,X the diffusion coefficient of molecule j in phase X,

and dX the boundary layer at the side of phase X of the

interface. The boundary layer represents the distance

from the interface at which the local concentration

reaches the bulk concentration. Clearly, the experimental

determination of dX is not easy, and therefore, boundary

layers and local and global mass transfer coefficients are

calculated using semi-empirical expressions which are

obtained after analyzing a huge amount of experimental

concentration data under different geometries and flow

regimes.[14]

Additionally, empirical partition coefficients can be used

together with local coefficients of mass transfer in order to

facilitate the modeling of molecular transfer across

interfaces. Assuming that the net flux of molecules across

the interface is proportional to the difference between the

DOI: 10.1002/mren.200900016


actual bulk concentration of the j-th component in phase A

and the concentration that must be present in A (½Cj�bA) inorder to reach equilibrium with the corresponding con-

centration in the bulk of phase b ([Cj]b), then

Macrom

� 2009

Jj ¼ Hj;Ab Cj

� �A� Cj

� �b

A

� �(22)

where Hj,Ab is another semi-empirical parameter called the

global mass transfer coefficient. The global mass transfer

coefficient can then be related to the local mass transfer

coefficients at both sides of the interface, Equation (18)–

(20), resembling two electrical resistances in series,

according to the following expressions:[13]

Jj

Hj;Ab¼ Jj

hj;Aþ Jj

hj;b

Cj

� �i

A� Cj

� �b

A

Cj

� �i

b� Cj

� �b

(23)

1 1 mij;Ab

Hj;Ab¼

hj;Aþ

hj;b(24)

where

mij;Ab ¼

Cj

� �i

A� Cj

� �b

A

Cj

� �i

b� Cj

� �b

(25)

mij,Ab can be regarded as the slope of a straight line

intercepting the equilibrium curve at the concentrations

[Cj]ib and [Cj]b. If the equilibrium curve can be assumed to

be linear, then the variable mij,Ab will be equivalent to the

equilibrium partition coefficient (Kj,Ab).

This type of modeling is widely used to describe

macroscopic mass transfer processes across interfaces. It

requires theknowledgeofmass transfer coefficients (which

can be estimated from semi-empirical correlations) and

equilibrium distribution coefficients.

3.1.3. Equilibrium Thermodynamics Modeling

A further simplification of themass transfer process can be

madeonce steady state conditionshavebeen reached in the

system, that is, when the net flux across all interfaces

becomes zero. Under these conditions, even though the

individual fluxes of molecules are not zero, it is frequently

assumed that the system has reached thermodynamic

equilibrium. As it was introduced already in Section 2, the

most relevant thermodynamic effects are basically enthal-

pic or entropic. In ideal mixtures there are no enthalpic

effects and mixing occurs spontaneously as a result of the

entropy increase. Ideal mixtures are always homogeneous

systems. In non-ideal mixtures, phase separation can take

place and an equilibrium condition is reached when both

entropic and enthalpic contributions to the free energy of

the system exactly counteract each other. This is the

ol. React. Eng. 2009, 3, 375–397


principle used by Flory and Huggins to describe polymer

solutions,[15] which can be used in general to describe the

thermodynamics of mixtures of species with different

molecularweights. The change in free energy ofmixing can

be expressed as:

DGmix ¼ DHmix � TDSmix (26)

The Flory-Huggins approach, considering a rigid lattice

model, produces the following expression for a two-

component system:[16]

DGmix ¼ RTðN1 ln’1 þ N2 ln ’2

þ ’1’2ðr1N1 þ r2N2Þx12Þ (27)

where Ni is the number of molecules of component i in the

system, wi is the segment fraction of i component, ri is the

number of segments of the lattice per molecule of i

component, and xij is the Flory-Huggins interaction

parameter, related to the intermolecular interaction

energies as follows:

xij /2"ij � "ii � "jj

T(28)

where eij is the interaction energy between molecules

i and j.

From Equation (27), the chemical potential of mixing of

the solvent/monomer (m1) can be obtained as the partial

molar free energy of mixing:

Dm1 ¼ @DGmix

@N1

� �T;P;N2

¼ RT ln ’1 þ 1� r1r2

� �’2 þ r1’

22x12

� �(29)

Normally, the Flory-Huggins interaction parameter is

used as an adjustable, temperature- and concentration-

dependent empirical parameter.

Morton, Kaizermann and Altier applied the Flory-

Huggins approach to the swelling of latex particles in the

presence of a bulk phase of pure swelling agent considering

an additional term corresponding to the free energy

contribution of the interfaces. They obtained an equation

which is now known as the Morton-Kaizermann-Altier

(MKA) equation:[17]

Dm1

RT¼ lnð’mÞ þ ’p 1� 1

Pn

� �þ x’2p

þ2Vmg’

1=3p

r0RT(30)



384

which at equilibrium becomes

Macrom

� 2009

ln ’m ¼ 1

Pn� 1

� �’p � x’2p �

2Vmg

RTrp(31)

where wp and wm are volume fractions of polymer and

solvent inside the particle, x is the Flory-Huggins interac-

tion parameter between polymer and solvent, Pn is the

average chain length of the polymer [according to Equation

(27), r1¼ 1and r2¼ Pn],Vm is thepartialmolar volumeof the

solvent, g is the interfacial tensionbetween theparticle and

the surrounding medium, r0 is the nonswollen particle

radius and rp is the swollen particle radius. The last term in

Equation (30) and (31) represents the resistance to the

creation of new surface upon absorption of solvent by

the polymer (swelling). The origin of this resistance is the

pressure difference between the interior of the particle and

the continuous phase (Laplace pressure), which is propor-

tional to the radius of curvature and the interfacial tension

between the particles and the continuous phase. Swelling

causes an increase in the interfacial free energy and thus, it

is a self-limiting process that stops if a certain size of the

swollen particles is reached.

However, the experimental values of swelling are much

lower than described by the classical MKA equation.

Antonietti et al.,[18] for example, observed a pronounced

dependence of the swelling ratio on particle size. In order to

explain thisphenomenon, theauthorspresentedamodified

description that considered size-relevant effects using an

additional swelling pressure term, which increases with

the curvature of the particle size counteracting swelling.

Theproposed corrected equation for swellingequilibriumis

the following:[19]

lnFm ¼ 1

Pn� 1

� �Fp � xF2

p �Vm

RT

2g

rpþ DP

� �(32)

where DP is the swelling pressure.

Thermodynamic models are commonly used to describe

phase equilibrium in awide variety of systems. Theirmain

disadvantage, besides the problems arising if equilibrium

thermodynamics is applied to a polymerizing system

which is not at equilibrium (cf. section 2), is that they

cannot per se account for the dynamics of mass transfer.

3.1.4. Empirical Modeling

One final approach for the deterministic modeling of mass

transfer is based completely on experimental data. In this

case, the concentrations of a certain component in two

different phases at steady state are related by means of a

partitionor distribution coefficient,which in general canbe

ol. React. Eng. 2009, 3, 375–397


expressed as:

Kj;AB ¼ f Cj

� ��A; Cj

� ��B

� �(33)

Where Kj,AB is the partition coefficient for the species j

between the phases A and B. Very frequently, the empirical

distribution function has the following structure:

Kj;AB ¼Cj

� ��A

� �n

Cj

� ��B

(34)

where n is a parameter obtained experimentally for the

particular system. In an ideal system, the value ofn is 1, and

thus, the following ideal expression (equivalent to Henry’s

law) is obtained:

Kj;AB ¼Cj

� ��A

Cj

� ��B

(35)

Notice that Equation (35) has been already presented in

Equation (12)and (13) considering thedynamicequilibrium

of the transfer process. However, there is a very slight

difference in the interpretation of both equations. In

Equation (12) and (13), the steady state concentrations of

a component in two phases for a particular system

composition are related by the partition coefficient, which

may have different values for different compositions,

whereas in Equation (35) the same value of the partition

coefficient holds for thewhole range of compositions of the

system. In this context, the partition or distribution

coefficient is a constant parameter used to predict the

steady state concentration of a component in one phase by

knowing its concentration in the other. As expected, the

valueof this constantwill dependon theparticular external

conditions of the system (pressure, temperature, etc.). It is

possible to predict the values of the partition coefficients at

different conditions using thermodynamic expressions

similar to those introduced in the previous section.

However, it is more common to find tabulated values of

the partition coefficients obtained from experimental

measurements. Additional information regardingpartition

and distribution coefficients can be found elsewhere.[20]

Empirical models are very useful to describe mass

transfer in complex systems using relatively simple

expressions. However, their main disadvantage is their

very limited predictive capabilities.

3.2. Microscopic Modeling

In a chemical reacting system, there are twoconditions that

can make a thermodynamic limit assumption invalid:[21]

(i) that there is a small number of at least one reacting

DOI: 10.1002/mren.200900016


species in the system or (ii) that there is at least one very

infrequent reaction compared to the remaining reactions in

the system. If this is the case, the deterministic modeling

techniques become flawed and therefore a microscopic

approach must be used. Some of the most relevant

examples of microscopic modeling include: (i) Stochastic

simulation, (ii) Brownian dynamics simulation and (iii)

Molecular dynamics simulation.

3.2.1. Stochastic Simulation

Gillespie proposed a stochastic simulation algorithm (SSA)

for the generation of particular realizations of the chemical

master equation[8,21,22] describing the reacting system. This

method has been extended to consider not only chemical

reactions but also physical processes like adsorption or

desorption from surfaces.[23]

In the original SSA formulation (also known as direct

method), the timeatwhich thenext stochastic event occurs

(t) can be calculated using the following equation:

Macrom

� 2009

t ¼ � lnðjUÞPi

ai(36)

where jU is a uniformly distributed random number

between 0 and 1, and ai is the propensity function of the

i-th stochastic event (in s–1). In general, the propensity

function can be expressed as:

ai ¼ cif ðn1;n2; ::; Þ (37)

where ci is a reaction probability, and f(n1,n2,. . .) is a

function of the number of molecules in the system which

depends on the order of the reaction. For the general case of

abimolecular reactionbetweenthemoleculesAandB, ci ki

/NA V,whereki is the rate coefficientof the i-th reaction, and

the propensity function is then given by:

ai ¼kinAnB

NAV(38)

and can also be expressed as a function the molar

concentrations (C):

ai ¼ kiCACBNAV (39)

Proceeding similarly with other types of reactions, it can

be found that in general:

ai ¼ NAVkif ðCÞ (40)

The type of event taking place at time t is determined

randomly, where the probability P of choosing an i-th

ol. React. Eng. 2009, 3, 375–397


event is:

PðiÞ ¼ aiPj

aj(41)

Onemaindisadvantageof theoriginal SSAformulation is

that it is a very inefficient method for simulating stiff

chemical reactions. Stiffness is observed when at least one

reaction pathway is orders of magnitude more frequent

than the others. Several approaches, including the hybrid

stochastic method, have been proposed to overcome

stiffness using stochastic algorithms.[24]

Additionally, a common feature of the original stochastic

simulation algorithm and most deterministic methods is

the assumption of perfectly mixed reaction volumes. In

this context, it is not sufficient that a system is homo-

geneous (that is, the local concentrations are the same for

the whole volume), but also the probability of finding a

given singlemolecule at anyposition in the systemmustbe

uniform. This means that a single molecule has the same

probabilityof reactingwitheveryothermoleculepresent in

the system,which is not possible in real systems because of

mass transfer limitations.

The SSA can be modified in order to consider also

imperfectly mixed systems. In the stochastic simulation

algorithm of imperfectly mixed systems (SSA-IM)[25] the

probability for the i-th reaction being the next event is:

PðiÞ ¼ kifiðCÞPj

kjfjðCÞ(42)

The next reaction time is calculated as

t ¼ � 3

8pNAv 6maxðDÞð Þ3=2lnðjUÞP

i

kifiðCÞ

264

3752=5

(43)

where jU is a random number obtained from a uniform

distribution in the range [0,1]. The perfectly mixed volume

is calculated as

Vpm ¼ 1536p2� �1=5 �maxðDÞ

NA

lnðjUÞPi

kifiðCÞ

264

3753=5

� 6:857 �maxðDÞNA

lnðjUÞPi

kifiðCÞ

264

3753=5

(44)

In this way, it is possible to carry out stochastic

simulations of diffusion-controlled systems, such as those

where interfacial mass transfer events are involved.



386

The most important distinction between the stochastic

simulation algorithmanddeterministicmethods is the fact

that a single stochastic simulation provides only one

realization of theprocesswhereas adeterministic approach

produces the average result of a very large number of such

realizations. If the thermodynamic limit assumption is

valid, then by averaging the results of several different

realizations performed by stochastic simulation, a very

close agreement between both strategies can be observed.

However, when the thermodynamic limit assumption is

violated, different results are obtained and in this case the

stochastic realizations of the chemical master equation are

closer to reality.[21]

In radical polymerization processes, many different

reactions take place simultaneously, some of them very

frequently (such as propagation), but some others

infrequently (such as initiator decomposition), and in

addition, radical species are usually present in very low

concentrations. These features of radical polymerization

motivate the simulation of the process using efficient

stochastic methods in order to obtain a better representa-

tion of the system and to get a deeper understanding

of the polymerization process. Stochastic simulation

has been used in heterophase polymerization in the

last two decades mainly to describe molecular weight

distributions.[26]

3.2.2. Brownian Dynamics Simulation

Brownian Dynamics (BD) simulation is a numerical

technique used to solve Langevin’s stochastic differential

equation of Brownian motion in the limit of motion

relaxation of the system. By means of this method, it is

possible to determine the paths followed by Brownian

entities (molecules, colloidal particles) as a result of the

randomcollisionswith themolecules in themedium. There

are several techniques for the numerical solution of

Brownian motion.[27] One of these methods is the Monte

Carlo random flight (MCRF) algorithm.[28] In the MCRF

method, the diffusive displacement on each direction for

each molecule or particle at each time-step dt is obtained

from a normal Gaussian distribution with mean zero and

varianceffiffiffiffiffiffiffiffiffiffiffiffi2Ddt

p. In a typical MCRF simulation, the time-

step is determined by the characteristic relaxation time of

the system:

Macrom

� 2009

dt ¼ mD

kBT(45)

where D is the diffusion coefficient of the Brownian

entities, m is its mass, T is the temperature of the system

and kB is the Boltzmann constant. In the accelerated

MCRF method,[7] the Brownian motion is simulated in

ol. React. Eng. 2009, 3, 375–397


time-steps given by

dt ¼ max ad2min

D;

mD

kBT

� �(46)

where dmin is the minimum separation between the

diffusing entity and the interface and a is a proportionality

factorwhich is selectedbasedon theprobabilityof reaching

the interface during the simulated time-step dt. When the

distance to the interface is large, the computationefficiency

is improved by increasing the time step of the simulation

proportionally to d2min/D. This means that the simulated

time-step will be proportional to the squared distance but

the minimum time-step considered will be that of the

corresponding relaxation time of the momentum of the

Brownian entity. The probability of hitting the interface is

related to the proportionality factor according to the

expression a¼ (2Z2)–1, where Z is the inverse of the normal

cumulative probability of collision and corresponds a

distance expressed asnumber of standarddeviations. Thus,

in order to obtain a collision probability in the order of 10–7

for time-steps larger than the relaxation time, a value of

a¼ 0.01852 is used. This value guarantees that practically

every collision with the interface will occur within a

resolution corresponding to the momentum relaxation

time. At each time-step, the movement of the entities in

each direction (in rectangular coordinates) is calculated as:

dx ¼ jG

ffiffiffiffiffiffiffiffiffiffiffiffiffi2Dr dt

p(47)

jG is a random number obtained from a Gaussian

distribution with mean 0 and variance 1.

BD simulation is a powerful tool for describing processes

at the colloidal or molecular scale. It has very good

predictive capabilities and can be used in complex systems.

It is however, limited to a relatively small number of

colloidal particles ormolecules, depending on the available

computational power.

3.2.3. Molecular Dynamics Simulation

Molecular Dynamics (MD) Simulation is a method used to

follow the trajectories and velocities of an ensemble of

atoms or molecules subjected to interatomic or intermo-

lecular forces for a certain period of time. Although the

atoms and molecules are composed of quantum particles,

their motion can be satisfactorily described by the classical

equations of motion:[29]

dvi

dt¼ 1

mi

Xj6¼i

Fij (48)

dxi ¼ vi (49)
dt
DOI: 10.1002/mren.200900016


where vi is the velocity, mi is the mass and xi is the

position of the i-th molecule, Fij is the interaction

force between the i-th and j-th molecules, and t is the

time. Additional external or internal (mean field) forces

can also be considered.

MD simulation can also be used to model mass transfer

in a system at the molecular scale. MD offers the most

accuratepictureofthemoleculartransferprocesses;however,

it is extremely computational demanding and today, it can

only be used to simulate relatively small systems.

3.3. Multiscale Simulation

Finally, it isworthmentioning a new trend in themodeling

of complex systems, which consists in the integration of

different simulationmethods for describing thebehavior of

the system at widely different time and length scales. This

integration is usually known as multiscale simulation.

Multiscale simulation can be defined as the enabling

technology of science and engineering that links phenom-

ena, models and information between various scales of

complex systems.[30] Growth in a number of critical

technological areas, such as nanotechnology, biotechnol-

ogy and microscale systems, may be accelerated and

catalyzed by such multiscale modeling and computational

paradigm.[31] The challenge in multiscale simulation is the

seamless coupling between the various models while

meeting conservation-laws, numerical convergence, stabi-

lity and computational speed. For example, MD simulation

is a very good alternative for the determination of

molecular diffusion coefficients, which can then be used

in first-principles or in BD simulations to describe mass

transfer at larger scales.[32] In general for complex systems,

not a single method should be used to describe all the

processes taking place simultaneously at very different

time and length scales. In particular for heterophase

polymerization, multiscale simulation has already been

shown to be a valuable technique to obtain more precise

descriptions of the process.[7,33]

4. Relevant Molecular Transfer Processes inHeterophase Polymerization

In this Section, some examples of the use of different

modeling strategies for describing mass transfer in hetero-

phase polymerization, and in particular in free-radical

emulsion polymerization, are presented.

4.1. Molecular Absorption: Radical Entry

In an emulsion polymerization process the simultaneous

highmolecularweight andhighpolymerization rate is only



possible to achieve if the polymer particles contain an odd

number of low-molecular-weight radicals. In this case, at

least one single radical will always survive termination

and continue to propagate until the monomer in the

particle is depleted. The presence of an odd number of

radicals inside theparticles ispossible thanks to thetransfer

of the activity of single radicals from the continuous

phase to the polymer particle (radical capture), or from the

polymer particle to the continuous phase (radical deso-

rption). The free radicals transferred from one phase to the

other can be primary radicals (generated after the separa-

tion of paired electrons) or growing chains (formed after

the propagation of primary radicals) of any length.[34] The

phase-transfer process of a radical takes place when a

radical reaches the interface between the polymer particle

and the continuous phase and its kinetic energy is high

enough to overcome the resistance to transfer at the

interface.

Different mechanisms have been proposed as rate-

determining steps in order to describe the kinetics of

radical capture by polymer particles:

� C
ollisional mechanism. The limiting step for the
absorption of radicals is assumed to be the ballistic

collision between the radicals and the polymer particles.

In this case, the rate of radical capture is proportional to

the surface area of the polymer particles.[35] This

approach is based on fundamental principles assuming

a ballistic motion of the radicals. The rate of radical

capture in this case is given by

r ¼ pkBT

2m

� �1=2

d2pNA R½ �wNp (50)

where kB is Boltzmann constant, T is temperature and m is

the mass of the entering radical.

� D
iffusion-controlled mechanism. Given that both radi-
cals and particles are suspended in a continuous

condensed phase (water), it is likely improbable that

they can follow perfect ballistic trajectories without

colliding with water molecules, resulting in a change in

their directions and velocities. As a result of themultiple

collisions with the continuous-phase molecules, the

overall displacement of the radicals and particles is not

ballistic but diffusional. For diluted dispersions of

polymer particles, the rate of collision by diffusion is

given by Smoluchowski equation:[36]

r ¼ 2pDwdpNA R½ �wNp (51)

where Dw is the diffusion coefficient of the radicals in the

continuous phase, dp is the diameter of the particles, [R]w is

themolarconcentrationof radicals inthecontinuousphase.



388

In this case, the rate of radical capture depends linearly on

the particle diameter. Equation (51) is the analytical

solution of the fundamental Fick’s equations applied to a

singleparticledispersed inan infinitemedium.Considering

thatnoteverycollisionbetweenradicalandparticle leads to

a radical absorption event, and that not every absorbed

radical reacts inside the particle affecting the kinetics of

polymerization, a rate-reduction factor or an absorption

efficiency factor (F) is included in the model:[37]

Macrom

� 2009

r ¼ 2pDwdpNA R½ �wNpF (52)

F represents the absorption efficiency factor that

describes the degree to which absorption is lowered

compared to irreversible capture, which can be given by

F ¼ 1

Dw

KReqDp X cothX � 1ð Þ

� �þ W 0

(53)

where X ¼ dp

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikp M½ �pþktpðn=vpÞ

Dp

sand KReq is the equili-

briumpartition coefficient of radicals between theparticles

and water, W’ is the potential energy barrier, Dp is the

diffusion coefficient for the radicals inside the particle, kp is

the propagation rate constant, [M]p is the monomer

concentration in particles, n is the number of radicals per

particle, NA is Avogadro’s number and vp is the particle

volume. A different expression for the efficiency factorwas

proposed by Nomura:[38]

F ¼kp M½ �pþktpðn=vpÞ

ko þ kp M½ �pþktpðn=vpÞ(54)

where ko is the overall radical desorption rate constant for a

particle.

� C
olloidal mechanism. Penboss et al.[39] considered that
the polymer particles are electrostatically-stabilized

polymer colloids and used the DLVO theory to determine

the rate of radical absorption by colloidal aggregation.

The resulting dependence of the rate of radical capture to

the size of the particles was found to be approximately

linear:

r ¼ pDwdr dp þ dr

� �NA R½ �wNpk exp � Em

kBT

� �(55)

where dr is the diameter of the radical, k is the reciprocal of

the Debye length and Em is the energy barrier (electrostatic

repulsion) of the capture process. This expression has been

ol. React. Eng. 2009, 3, 375–397


obtained also from Fick’s equations in infinitely diluted

dispersions, but considering stabilization only by electro-

static repulsion. It is therefore very similar to the diffusion-

controlled mechanism, Equation (52).

� P
ropagation-controlled mechanism. Maxwell et al.[40]
proposed that the rate-determining step is the growth of

the radicals in the continuous phase up to a certain

critical chain-length after which the capture process is

imminent. According to this model, only oligomeric

radicals of a critical chain length can enter the particles,

and these radicals do not participate in any other

reaction. Entry is independent of particles’ size and

charge. The rate of radical capture assuming a propaga-

tion-controlled mechanism is:

r ¼ NAkpw

NIMz�1½ � M½ �w (56)

where kpw is the propagation rate constant in the aqueous

phase, [M]w is the monomer concentration in the aqueous

phase and N is the concentration of particles in the

dispersion. By substituting the steady-state concentration

of (z–1)-mer radicals [IMz–1], the approximate expressions

for r and the initiator efficiency, fentry are:

r ¼ 2NAkd I½ �N

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikd I½ �ktw

pkpw M½ �w

þ 1

( )1�z

¼ 2NAkd I½ �N

fentry (57)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip( )1�z

fentry ¼ kd I½ �ktw

kpw M½ �wþ 1 (58)

where z� 1þ int(–23 kJ �mol–1/(RT ln[Msat]w)) is the critical

chain length for irreversible capture.

� M
olecular stochastic simulation. BD simulation has also
been used to determine capture rate coefficients.[41,42]

The ratio between the collision rate coefficient obtained

by BD simulation (kc) and the ideal rate coefficient given

by Smoluchowski’s equation, Equation (51), is defined as

Smoluchowski number (Sm):

Sm ¼ kc

2pDrdpNA(59)

At very low volume fractions of polymer particles

BD simulation predicts the collision rate coefficient

obtained with the Smoluchowski equation (Sm¼ 1),

while for concentrated polymer dispersions (volume

fractions> 0.1%) the Smoluchowski number, and there-

DOI: 10.1002/mren.200900016


fore the capture rate coefficient, presents a linear

dependencewith respect to the volume fraction of polymer

particles in the dispersion.

Macrom

� 2009

Sm ¼ yfp þ 1 (60)

where y is a dimensionless constant obtained for the

system under the particular conditions considered in the

simulation and fp is the volume fraction of particles in

the dispersion.

Similar conclusions were obtained by Rzepiela et al.,[43]

who found that the rate of aggregation of colloidal particles

obtained by simulation always exceeds the value predicted

by Smoluchowski theory for fast aggregation, and only for

volumefractionsbelowabout0.1%thediscrepancy is small.

The values of the rate of radical absorption obtained by BD

simulation under a wide range of conditions were used to

obtain the following semi-empirical expression:

kc ¼ 2pDrNA

ypNd4p

6þ dp

!(61)

Thismeans that the effect of polymer volume fraction on

collision kinetics under diffusion-controlled conditions

explains the different results obtained during the experi-

mental determination of radical capture kinetics in

emulsion polymerization.[41]

Interaction forces, interfacial tensions, the presence of

stabilizer molecules at the surface of the polymer particles

and many other physical and chemical effects may lead to

an increase in free energy during radical capture, and thus,

to the existence of an energy barrier for radical capture. In

these cases, only a fraction of the radicals collidingwith the

particles will be effectively captured (capture efficiency)

and the other will bounce back. When a radical is not

captured by a particle because of the energy barrier, the

radical will remain close to the particle surface and

therefore it will have a very high probability of hitting

the same particle again, leading to a series of multiple

collisions in a very short time before the radical goes away

from the particle surface. BD simulations show that the

effect of the magnitude of the energy barrier at different

temperatures on the capture efficiency canbedescribed ina

general way as:[42]

kc ¼kc0 ; E < E�

kc0e�ðE � E�Þ3RT ; E � E�

8<: (62)

where kc0 is the capture rate coefficient in the absence of

energybarriers, Equation (61), and E� is themagnitude of an

apparent threshold energy caused by multiple collisions

between a radical and a polymer particle.

ol. React. Eng. 2009, 3, 375–397


4.2. Molecular Desorption: Radical Exit

The process of radical desorption can be regarded as the

opposite to radical capture. In this case, a free radical is

transferred from the interior of the particles to the

continuous phase. The mechanism by which this transfer

takes place is also similar: the radical must reach the

particle surface and then it must overcome the barrier for

desorption exerted by the interface. Desorption results in a

decrease in the concentrationof radicals in theparticles and

hence causes the rate of polymerization to decrease. For

many importantemulsionpolymerizationsystems, exit is a

major (if not the only) cause of the loss of free-radical

activity inside a particle.

Ugelstad et al.[44]were the first to propose that the rate of

radical desorptionwas inverselyproportional to thesurface

of the particle, although quantitative results were not

available. Nomura andHarada,[45] aware of the necessity of

estimating quantitatively the rate coefficient for radical

desorption from the particles for the prediction of rates of

emulsion polymerization, developed the first quantitative

modelof radicaldesorptionusingasemi-empiricalmodelof

interfacial mass transfer. They continued improving this

model during more than one decade. Asua et al.[46] com-

plemented themodel of desorption by considering the fate

of the radicals in the aqueous phase. Further improvements

of the semi-empirical model have been proposed by

considering the layer of stabilizer around electrosterically

stabilized polymer particles as an additional resistance to

the desorption process.[47] The semi-empirical macroscopic

model of equilibrium radical desorption developed by

Nomura and coworkers has been widely used to describe

radical desorption in heterophase polymerization. How-

ever, the thermodynamic limit assumption involved in the

derivation of this model may not be fulfilled by a radical

desorption process in a real emulsion polymerization

system, because in this case the radicals are present in a

very low concentration inside or around the particles.

The diffusion-controlled mechanism for radical deso-

rption is widely accepted right now, but there are minor

differences regarding the mathematical treatment of the

problem leading to different expressions for the rate of

radical desorption; however, all of them predict the same

inverse dependence on the surface area of theparticles. One

of the reasons for the use of different expressions is caused

by different interpretations of the desorption process. In

order to avoid confusion, the following types of desorption

are considered:[48]

� S
imple radical desorption. It is the result of the diffusive
motion of the radicals when no reactions are considered

inside the polymer particles. The rate coefficient of

simple radical desorption (k0) is determined by the

velocity at which the radical diffuses out of the particle

and it is a function of the particle size (dp) and



390

the diffusion coefficient of the radical inside the polymer

particle (Dp):

Macrom

� 2009

k0 ¼ lDp

d2p

(63)

where l is a constant with a value of 60.

� E
quilibrium radical desorption. The equilibrium
desorption of a radical takes into account the different

solubility of the radicals between the polymer particles

and the aqueous phase. The rate coefficient of equili-

brium radical desorption (k0�) can be related to the

simple radical desorption rate coefficient by:

k�0 ¼ k0

1þ KeqDp

Dw

dw

dp

� � (64)

where Dw is the diffusion coefficient of the radical in the

aqueous phase, Keq is the partition coefficient of the radical

between the polymer particle and the aqueous phase, dw is

the thickness of the stagnant layer in the aqueous phase

and dp is the thickness of the diffusion layer in the polymer

phase.

� N
et radical desorption. The net desorption of a radical
occurs when the radical escapes the polymer particle

after surviving the competitive reactions taking place

inside. The rate coefficient of net radical desorption (K0)

is determined by:

K0 ¼ krgen þ r� �

� k�0

k�0 þ

Pi

ki;p(65)

where ki,p is the rate of the i-th reaction involving radicals

and krgen is the rate of generation of desorbing radicals

inside the particle.

� E
ffective radical desorption. A radical is considered to be
effectively desorbed from the particle only after it reacts

in the aqueous phase. This definition accounts for

the fact that desorbed radicalswhich diffuse through the

aqueous phase may be reabsorbed by a polymer particle

and continue reacting therein, without significantly

affecting the kinetics of emulsion polymerization. The

rate coefficient of effective radical desorption is (kdes):

kdes ¼ krgen �Pw 1� Pp

� �1� 1� Pwð Þ 1� Pp

� � (66)

P

Figure 5. Examples of different possible types of ellipsoidalparticles. The shapes can be obtained by varying the shapeparameters wy and wz.

where Pp ¼ i

ki;p

k0þP

i

ki;pis the probability of reaction of the

radical inside the particles and Pw ¼

Pi

ki;w

kabsþP

i

ki;wis the pro-

bability of reaction of the radical in the continuous phase.

ol. React. Eng. 2009, 3, 375–397


Simple radical desorption rate coefficientshavealsobeen

obtained from the simulation of the Brownian motion of

radicals inside theparticlesusingavariable time-stepMCRF

method for BD simulation, similar to the method used for

estimating radical capture by polymer particles. A linear

regression of the observed simple desorption rate coeffi-

cientsyields theexpressionpresented inEquation (63),with

a value of l of 57.14, in very good agreementwith the value

of 60 predicted by theory.[48]

An additional degree of complexity in real systems is

given by the fact that polymer particles may not be

perfectly spherical.[49] For non-spherical particles, the

desorption rate coefficients can be easily determined from

BD simulations after a suitable parameterization of the

shape of the particle. As an example, the rate coefficients of

radical desorption for ellipsoidal particles is considered

(Figure 5). The surface of an ellipsoidal particle is given by

the following expression:

x2 þ ’2yy2 þ ’2

z z2 ¼ d2x (67)

where x, y and z represent the Cartesian coordinates,

wy and wz are shape parameters and dx is the radius of the

y-z cross section of the ellipsoid. The BD simulation is

performed in a similar way as for the spherical particle

case, using the following condition for radical desorption:

x2r þ ’2

yy2r þ ’2

z z2r � dx þdr

2

� �2

(68)

where xr, yr and zr are the x, y and z components of the

position of the radical respectively, and dr is the diameter of

the radical. The diameter of an equivalent spherical particle

with the same volume of the ellipsoid is:[50]

deqp ¼ 2dxffiffiffiffiffiffiffiffiffiffi

’y’z3p (69)

DOI: 10.1002/mren.200900016


The volume of all particles was kept constant, corre-

sponding to the volume of a spherical particle of 100nm in

diameter. The diffusion coefficient inside the particle was

Dp¼ 10–9 m2 � s–1. An effective particle size for the

desorption process, corresponding to the particle diameter

of a spherical particle with the same desorption rate

coefficient, can be calculated directly from Equation (63):

S¼ d2x

Figpar

Macrom

� 2009

deffp ¼

ffiffiffiffiffiffiffiffiffiffiffi60Dp

k0

s(70)

Figure 7. Effective particle sizes of ellipsoidal particles as a func-tion of the shape parameters wy and wz.

Desorption rate coefficients obtained by BD simulation

for ellipsoidal particles of the same volume but different

shape parameters are presented in Figure 6. The corre-

sponding values of the effective particle sizes for different

shape parameters are presented in Figure 7. It is observed

that the rate of simple radical desorption in spheroidal

particles is lowest (and thus, the effective particle diameter

is the largest) when the particles are perfectly spherical. As

the shape of the particle deviates from the perfect sphere,

the rate of simple radical desorption increases as a result

of the shortest distances required by the radicals to reach

the surface of the particle, which is the result of the

increased surface-to-volume ratio of the particles.

The effect of the surface-to-volume ratio on the simple

desorption rate coefficient is presented in Figure 8. The

surface of the ellipsoid is calculated solving the following

double integral:[51]

Z p

0sinf

Z 2p

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2 f

’2y

þ sin2 f

’2z

cos2 u

’2y

þ sin2 u

!vuut du df

(71)

ure 6. Simple desorption rate coefficients in ellipsoidalticles as a function of the shape parameters wy and wz.

ol. React. Eng. 2009, 3, 375–397


The simple desorption rate coefficient estimated by BD

simulation can then be fitted as a second power function of

the surface-to-volume ratio (S/V):

FiguparpoiEqu

k0 / DpS

V

� �2

(72)

BDsimulationhavealsobeenused todescribedesorption

rate coefficients in different types of core/shell particles.[7a]

In general, BD simulation can be used to model radical

desorption in particles with complex geometry.

re 8. Simple desorption rate coefficients in ellipsoidalticles as a function of the surface-to-volume ratio. Datants: BD simulation data. Solid line: data fitted usingation (72).



392

4.3. Simultaneous Molecular Absorption/Desorption:Monomer Swelling

Another important physical phase-transfer process taking

place inemulsionpolymerization is theswellingofpolymer

particles. In general, particle swelling denotes the uptake of

solvent by the polymer. With a very few exceptions (e.g.

acrylonitrile, vinyl chloride), themonomer is a good solvent

for the polymer, and therefore, it can swell the polymer

particles. Swelling takes place as a result of the diffusion

through the aqueous phase of individual molecules or

clusters ofmolecules to the surface of polymerparticles and

the subsequent absorption when the interfacial energy

barrier is surpassed. Under special conditions, polymer

particles can grow by swelling to sizes several times larger

than their original nonswollen sizes.[52]

From a thermodynamic point of view, the driving force

for the swelling of a polymer by a solvent is the free energy

of polymer-solvent mixing which has both entropic and

enthalpic components, as already mentioned in Section

3.1.3. Even if themonomer and the polymer aremiscible in

all proportions in bulk, only a limited amount of monomer

can enter a latex particle from the monomer saturated

aqueous phase. Each particle can swell only to the extent

where the free energy ofmixing and the surface free energy

change on swelling exactly compensate each other and

there is a well-defined swelling equilibrium. In emulsion

systems, the presence of emulsifier greatly lowers the

interfacial tension and allows a substantial amount of

swelling.[53]

The most commonly used method for describing

monomer swelling in latex particles is the Morton-

Kaizermann-Altier (MKA) thermodynamic approach, Equa-

tion (31). Using the MKA approach, it is possible to predict

the equilibrium concentration of monomer inside a

polymer particle as a function of the average degree of

polymerization, the partial molar volume of the monomer,

the average radius of the polymer particles, the polymer-

continuous phase interfacial tension, the monomer-poly-

mer interaction parameter and the temperature of the

system. The usefulness of the MKA equation for predicting

the monomer concentration in the particles during poly-

merization is very limited for at least two reasons. The first

is that the MKA equation assumes the presence of a free

monomer phase. In order to overcome this difficulty other

approaches have been developed. The first approach uses

partition coefficient:

Macrom

� 2009

Kp;w ¼M½ ��satp

M½ �satw

(73)

where [M]p�sat is the equilibrium concentration of mono-

mer inside the particles obtained from the MKA equation,

ol. React. Eng. 2009, 3, 375–397


and [M]satw is the saturation concentration of the monomer

in the continuous phase. Therefore, the equilibrium

concentration of monomer inside the particles can be

obtained from the concentration of monomer in the

continuous phase below saturation:

M½ ��p¼ Kp;w M½ �w (74)

As an alternative approach, Vanzo and coworkers[53]

compared the free energy of monomer-polymer mixing

inside the particles (MKAequation) to the free energy of the

monomer in the continuous phase and found the following

relationship, usually known as the Vanzo equation:

lnM½ �wM½ �satw

!¼ lnð1� ’pÞ þ ’p þ x’2p þ

2Vmg

RT rp(75)

The second drawback, and perhaps the most important,

is that theMKAequation, aswell as theVanzoequation, can

not always be safely used to predict monomer concentra-

tions inside polymer particles because (i) some of the

parameters included in the model are not constant (such

as the interaction parameter and the interfacial tension),

and (ii) not all relevant effects have been considered (for

example the swelling pressure effect[18]).

Another deterministic modeling approach consists in

using empirical equations relating the monomer concen-

tration inbothphases.Oneof themost important examples

of the use of empirical method in heterophase polymeriza-

tion has been presented by Ballard et al.[54] The general

empirical expression proposed, equivalent to Equation (34),

is the following:

M½ �wM½ �satw

¼M½ �pM½ �satp

!y

(76)

where the exponent y is an empirical parameter deter-

mined for eachmonomer-polymer system. For example, for

the methyl methacrylate/poly(methyl methacrylate) sys-

temavalueofy¼ 0.6wasobtained fromexperimental data.

A similar valuehas also beenobtained for the vinyl acetate/

poly(vinyl acetate) and the styrene/polystyrene sys-

tems.[55]

Molecular modeling methods are a fundamentally

different strategy for describing monomer swelling. From

a molecular point of view, monomer or solvent molecules

are continuouslyabsorbedbyordesorbed fromthepolymer

particles. If the rate of molecular absorption per particle is

larger than the rate of desorption, swelling takes place,

otherwise deswelling occurs. Thus, swelling (or deswelling)

is simply the result of the balance between the absorption

and desorption rates ofmolecules. At steady state, the rates

DOI: 10.1002/mren.200900016


ofdesorptionandabsorptionareexactly thesame,andthus,

there isnonet change in thenumberofmolecules inside the

particles. Since the rates of desorption and absorption

dependon thediffusion coefficients inside theparticles and

in the continuous phase, their values are crucial for the

steady-state distribution of the molecules in the system.

The effect of the diffusion coefficients on the partition

coefficient (at steady-state) in dispersions of polymer

particles has been investigated using BD simulation.[7a] It

was observed that themolecules tend to accumulate in the

phase where they present the lowest diffusion coefficient.

This is reasonable because the fastest molecules will cross

the interface more frequently causing an accumulation in

the phase of lowest mobility. The effect of the ratio of

diffusion coefficients on the steady-state partition coeffi-

cient of the system can be expressed approximately using

the following empirical expression:

Macrom

� 2009

Kpw � Dw

Dp

� �b

exp � Eabsa � Edes

a

3RT

� �(77)

where b is positive and its value depends on the particular

conditions suchas thevolumefractionof theparticles in the

dispersion, Eabsa � 0 is the activation energy of the

absorption process, Edesa � 0 is the activation energy for

the desorption process. According to the expressions

obtained for irreversible absorption and desorption of

molecules, a value of b¼ 1would be expected. However, for

the dispersion of spherical particles under the conditions

considered, it is found that b� 1.6. This deviation is caused

by the fact that a single molecule at the interface can be

absorbedanddesorbed several timesbefore diffusing to the

center of the particle or to the bulk of the continuous phase,

which is not considered by the models of irreversible

phase transfer, giving rise to different effective diffusion

paths for phase transfer. It is important to notice that under

these conditions, only when the diffusion coefficients in

both phases are identical (and in the absence of energy

barriers) the concentration of themolecular species in each

phase is the same. If there is a net energy barrier for

absorption, the molecules accumulate in the continuous

phase. Similarly, if there is a net energy barrier for

desorption, the molecules will concentrate inside the

particles.

The molecular approach appears to be a very good

alternative for the prediction of non-equilibriummonomer

concentration inside polymer particles in emulsion poly-

merization, and could be very useful in systems where the

swelling equilibrium assumption is not valid, for example,

in the industrially widely used monomer-starved semi-

batch processes. This advance could be further used to

investigate thenon-equilibriumuptakeof anyother typeof

molecule, such as primary radicals, oligomers, solvents, or

ol. React. Eng. 2009, 3, 375–397


any other compound that might be useful to modify

colloidal particles. This strategy requires as crucial para-

meters the corresponding diffusion coefficients and activa-

tion free energies. Diffusion coefficients are comparable

easy accessible both experimentally from swelling data of

bulk samples and theoretically from free volume the-

ories[56] or they can be determined using molecular

dynamics simulations.[32] Activation free energies can be

estimated from time-dependent experimental swelling

data. The big advantage ofmolecularmodeling strategies is

however the inherent ability to carry out time-dependent

simulations even in combination with chemical reactions.

4.4. Molecular Aggregation: Micellization andNucleation

One of the most challenging aspects of heterophase

polymerization is the description ofmolecular aggregation

processes such as micellization (aggregation of surfactant

molecules) and particle nucleation (aggregation of polymer

molecules). Micellization, for example, has been modeled

using both macroscopic deterministic and molecular

stochastic approaches. Aniansson and Wall[57] developed

a micellization model based on semi-empirical mass

transfer equations. The general stepwise scheme of the

micellization process considered was the following:

A1 þ A1 ,kþ2

k�2

A2

A1 þ A2 ,kþ3

k�3

A3

..

.

A1 þ As�1 ,kþ

s

k�s

As

(78)

The rate of aggregation into micelles containing s

surfactant molecules is:

d As½ �dt

¼ Js�1 � Js (79)

where

Js ¼ k�s As½ �� js � js�1ð1þ j1Þ � j1ð Þ (80)

As½ � � As½ ��
js ¼ As½ �� (81)
[As] is the concentration of aggregates containing s

surfactantmolecules, [As]� is its equilibrium concentration,

kþs and k�

s are rate coefficients of aggregation/disaggrega-

tion equivalent to the semi-empirical global mass transfer



394

coefficients, Js is thefluxofaggregatesof size s in thespaceof

cluster sizes (the number of aggregates of size s absorbing

one additionalmolecule per unit time), and js is the relative

concentration of aggregates of size s away from the

equilibrium. In this approach, molecular aggregates con-

taining the same number of surfactant molecules are

considered to be different segregated phases, and the

aggregation (disaggregation) process involves the transfer

of an individual surfactant molecule from (to) the

continuous phase.

The thermodynamic formulation of the micellization

process[58] completelyneglectsdynamicsbut considers that

thesystemreachesequilibriumwhentheGibb’s freeenergy

is at its minimum.

Macrom

� 2009

m0s þ kBT lnXs ¼ s m0

1 þ kBT lnX1

� �(82)

m0s is the standard chemical potential of an aggregate of s

surfactant molecules, and Xs is the mole fraction of these

aggregates in the solution.As it canbe seen inEquation (82),

the transfer of individual surfactant molecules towards

surfactant aggregates is driven by the differences in

chemical potential between the aggregates and the

individual surfactant molecules. This model requires a

very precise determination of the chemical potentials, as

given by Nagarajan and Ruckenstein.[59]

In addition,micellization kinetics has also beenmodeled

using classical or modern nucleation theories.[60] Nuclea-

tion andmicellization are very similar processes consisting

of the spontaneous aggregation of individual entities

(atoms, molecules or ions). However, this was not realized

during the early development of their kinetic models.

Nucleationmodels, like for example theclassicalnucleation

theory (CNT),[61] are usually obtained by integrating the

semi-empirical approach with equilibrium thermody-

namics. Nucleation is an aggregation process where only

aggregates above a certain size are stable. At this critical

size, the Gibb’s free energy of the aggregate is at its

maximum. Aggregates with sizes smaller than the critical

size have a limited probability of growing because the free

energy increases with each aggregation step. However,

the growth of subcritical aggregates is favored by the

presence of thermal fluctuations. When an aggregate

reaches the critical size, its growth becomes spontaneous

because fromnowoneach further aggregation step leads to

a decrease in the free energy. The flux of aggregates is

assumed to be determined by the free energy of formation

of an aggregate according to the Boltzmann’s distribution:

Js ¼ jþs As½ � 1� expDWs þ ln Asþ1½ �

As½ �

� �kBT

0@

1A

0@

1A (83)

ol. React. Eng. 2009, 3, 375–397


where jþs is the number of molecules absorbed from the

continuous phase by the aggregate of size sper unit of time,

andDWs is the difference in aggregationwork between the

aggregate of size sþ 1 and the aggregate of size s, which are

defined by:

Ws ¼ms � sm1

kBT(84)

ms is the chemical potential of the molecular aggregates of

size s; jþs is usually determined using Smoluchowski’s

equation, which is basically the solution of Fick’s equation

for the absorption ofmolecules by amolecular aggregate in

infinite dilution.

Smit et al.[62] proposed the use of MD simulation for

investigating the surfactant aggregation process. This

model requires the definition of potential functions for

each type of interaction: hydrophobic-hydrophobic, hydro-

phobic-hydrophilic and hydrophilic-hydrophilic, either as

covalent bonds, e.g. harmonic potentials, Equation (85), or

asvanderWaals interactions, e.g. Lennard-Jonespotentials,

Equation (86) and (87). Although computationally demand-

ing, thismethod allowed the description of the dynamics of

micellization as well as of its equilibrium properties.

Uij ¼kF rij � sij

� �22

(85)

fijðrijÞ � fijðRcijÞ rij � Rc

ij
Fij ¼ 0 rij > Rc
ij(86)

sij� �12 sij

� �6" #

fijðrijÞ ¼ 4"ijrij

�rij

(87)

where Uij is the covalent interaction between two atoms i

and j, kF is a force constant, rij is the distance between the

two atoms, sij is an interaction length parameter,Fij is the

van der Waals interaction potential between atoms i and j,

fij is a Lennard-Jones-type interactionpotential, and eij is an

interaction energy parameter.

Alternatively, a more general aggregation scheme has

been solved by Mavelli[63] using Gillespie’s stochastic

simulation algorithm. The general aggregation scheme is

the following:

Ai þ Aj ,kþ

i;j

k�i;j

Aiþj (88)

The kinetic constants were determined as follows:

kþi;j ¼ Ki;jk

�i;j (89)

DOI: 10.1002/mren.200900016


� i!j! �

Macrom

� 2009

ki;j ¼ ði þ jÞ ði þ j � 1Þ! k0 (90)

where

Ki;j ¼ij

ði þ jÞ S½ �Pði þ jÞPðiÞPðjÞ (91)

exp � "i

� �
PðiÞ ¼ kBTP
j exp � "j

kBT

� � (92)

i � nmax� �2

"i ¼ inmax

"0 (93)

[S] is the total surfactant concentration, nmax is the

maximum aggregation number considered, and k�0 and e0

are parameters of the model.

Also for ab initio heterophase polymerization, particle

nucleation can be considered as being the product of

molecular aggregation.[64,65] However, the situation is

complicated as both the concentration and the physico-

chemical properties of the aggregating species changewith

time. Moreover, the presence of segregated phases (mainly

monomer drops and latex particles) strongly influences the

nucleation process due to interactionswith the aggregates.

This interaction lowers the free energy of nucleation and

thus increases theprobabilityofparticle formation. There is

experimental evidence that this so-called ‘‘heterogeneous

nucleation’’ is not only crucial for the formation of clouds

and rain drops in the atmosphere,[66] but also for particle

nucleation in heterogeneous polymerization.[65]

5. Conclusion

Mass transfer across interfaces is one of the most common

and important events taking place in heterogeneous

systems. In particular for heterophase polymerization, it

has been observed that transfer of molecules (radicals,

monomer, surfactant, etc.) between different phases

strongly determines the polymerization kinetics. Several

different approaches have been used for modeling inter-

facial mass transport, which can be classified in two main

groups: macroscopic deterministic and molecular stochas-

tic approaches. All modeling techniques available are not

contradictory but complementary between each other,

because they are suitable for describing different aspects of

the process, possibly under different conditions. Thus, for

example, equilibrium thermodynamic models do not

describe dynamic effects but are very fast alternatives to

model final equilibrium conditions (if they are kinetically

accessible). Molecular dynamics simulations are able to

describewithmicroscopic detail the dynamics ofmolecular

ol. React. Eng. 2009, 3, 375–397


transfer processes, but they are limited only to very small

systems, usually far from equilibrium. The best modeling

approach that can be used to describe a mass transfer

process will be determined not only by the particular

conditions of the process (e.g. if the system is at the

thermodynamic limit or not), but also on the purpose of

modeling, that is, on the information expected to be

obtained from the model (e.g. predicting results under

different conditions, describing the process in a fast and

simple way, etc.).

Given that each modeling method offers certain parti-

cular advantages, future developments in mass transfer

modeling should be focused on the seamless integration of

different methods, usually at different time and length

scales (multiscale simulation). The biggest challenge in

multiscale simulationmaybe the development of accurate,

fast and efficient ab initio multiscale simulation methods;

that is, to simulate mass transfer processes from the

knowledge only of the composition of the system using

neither empirical parameters nor previous experimental

results, providing accurate information at all relevant

scales (from the molecular to the macroscopic) and using

reasonable computation power. The state of the art in

multiscale simulation, in particular of mass transfer

processes, is still far away from this goal, and therefore,

there are still plentyofpossibilities ofnewdevelopments in

this direction.

Molecular-scalemodeling approaches offer new insights

into the mechanism of heterophase polymerization and

interestingly, allow certain generalizationswhichwere not

possible with macroscopic deterministic modeling.

Acknowledgements: The Max Planck Society is gratefullyacknowledged for the financial support of this research. H. H.would also like to acknowledge Andercol SA (Colombia) for thegranting of a research fellowship.

Received: March 10, 2009; Revised: June 12, 2009; Accepted: June17, 2009; Published online: August 4, 2009; DOI: 10.1002/mren.200900016

Keywords: emulsion polymerization; molecular dynamics; mo-lecular modeling; radical polymerization; simulations; swelling

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