February 2003
D1. Modeling of Polymer Phase Transitions
This project will: investigate dynamic models for phase transitions from liquid to other
states, such as glass, gel, and crystal for polymers, polymer mixtures, and colloids.
The models, based on population balances (distribution kinetics), will be applied to
spinodal decomposition, gelation, nucleation, growth and ripening, and glass
transition.
Primary Faculty co-Advisors
Benjamin J. McCoy, Chemical Engineering, (polymer reaction kinetics, phase
transition dynamics, and complex phenomena)
Gudrun Schmidt, Chemistry, (nanotechnology, complex molecular self-assemblies,
smart fluid applications)
Off-campus Participant: George Roberts, Department of Chemical Engineering, North
Carolina State University. Professor Roberts has published extensively on
polymerization kinetics, particularly for reactions in environmentally safe solvents,
such as carbon dioxide.
Technical Proposal: Macromolecular Phase Transition Dynamics
Background and Introduction
The dynamics of phase transitions have been a topic of intensive research, and
many of the governing principles are fairly well understood. Comprehensive and
systematic approaches are still needed, however, to unify and systematize the
understanding of its many facets. A broad area of study, phase transition dynamics
embraces first-order transitions, including nucleation, growth and coarsening of particle
(liquid-solid) or droplet (vapor-liquid) size distributions. A variation on this process is
spinodal decomposition, or spontaneous and barrierless condensation. A transition that is
not first-order is the glass transition from liquid to solid glass, characterized by the
absence of nucleation. The gel transition, occurring for example during aggregation and
intermolecular crosslinking of macromolecules, is another process that is of current active
investigation because of its importance in polymer processing. These phenomena are all
characterized by size-distributed heterogeneities, usually manifested as different sized
clusters or macromolecules. The kinetics and dynamics of size distributions, whether for
particulates or macromolecules, can be mathematically described by population balance
modeling. Population balance models represent a powerful approach to quantifying
complex chemical and engineering phenomena. The proposed research concerns the
application of this technique to phase transition kinetics and dynamics involving
polymers, polymer mixtures, and colloids.
Phase Transition Dynamics
The three steps of a first-order phase transition are nucleation (either
homogeneous or heterogeneous), growth, and Ostwald ripening (coarsening). In spinodal
decomposition, the phase change is spontaneous, with no energy barrier for nucleation.
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Homogeneous nucleation can be described via classical nucleation theory, such that
volume and surface energies combine to form a free energy barrier that must be
overcome to form a nucleus of critical size. The nucleation step is difficult to observe
experimentally, due to the very small size of the critical nuclei, but it dictates the rest of
the growth process. For polydisperse colloid suspensions, simulations have shown that
the free energy barrier may increase, after going through a minimum, as the density
increases [1]. Thus, a slowdown in nucleation rate at high density may be related to an
increase in the free energy barrier, not a slow-down in the kinetics as fluid motion
becomes sluggish. This same simulation relates the probability of formation of a critical
nucleus to the polydispersity of the suspension, such that polydisperse suspension may
not crystallize at all, even when compressed [2]. According to classical nucleation
theory, the free energy barrier variation with polydispersity is due to an increase of the
interfacial free energy, (if the chemical potential difference between the solid and liquid
states is assumed constant). Turnbull has suggested that the interfacial free energy should
be proportional to the latent heat of fusion, but the simulation indicates that when the
latent heat becomes zero, the interfacial free energy, and thus the free energy barrier, are
non-zero [2]. In the case of zeolite A crystal growth at room temperature, an amorphous
gel formed prior to nucleation occurring [3]. This type of behavior is also not explained
via classical nucleation theory. So, the classical theory of homogeneous nucleation does
not appear to account for all observed or modeled behaviors, and thus another approach
to explaining this critical step in phase transition is needed.
Various mechanisms by which growth of a nucleus may occur have been proposed,
which range from solution-mediated transport to hydrogel transformation processes [3].
The zeolite A experiments mentioned above reported that crystals grew at the expense of
the amorphous agglomerates, resulting in crystals that were similar in size to the
amorphous precursor particles. When the solution was heated above room temperature,
however, much larger crystals resulted, presumably assembled from material drawn from
solution [3]. So, two different growth mechanisms are seen in the same solution,
depending on the temperature at which the crystallization is carried out. These two
mechanisms result in crystals of different sizes and different properties. Thus, a single
explanation for growth that applies to all phase transitions is unlikely to be correct, and a
theory that allows for different mechanisms in competition with one another is desirable.
Coarsening or ripening, the last step in a first-order transition, involves the dissolution of
small crystals with their mass going to larger crystals, which continue to grow. In
essence, the small crystals give up their molecules so that the larger crystals increase in
mass. The theoretical basis for this behavior is the Gibbs-Thomson relation, in which
smaller particles have a relatively larger interfacial energy and are thus more soluble than
larger particles. Once the smaller particles shrink to the critical nucleus size, they
become thermodynamically unstable and disintegrate [4 & 5]. Conventional ripening
theories, such as the Lifshitz, Slyozov, Wagner (LSW) theory and modifications [6]
account for denucleation indirectly, through the mass balance. Although the LSW theory
is the foundation for understanding Ostwald ripening, the form of the particle size
distribution predicted by LSW theory does not match experimental results in several
cases [6]. The distribution kinetics theory [6], based on a population balance approach,
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improved on the LSW model by directly accounting for denucleation and interactions
between particles, producing a model that is valid at all times from zero to infinity.
Another type of phase transition is the glass transition, in which a metastable liquid
becomes a highly viscous, molecularly disordered glass when cooled or compressed
rapidly, such that nucleation and crystallization do not occur. This type of phase
transition is also seen in colloidal suspensions. Clusters (i.e. heterogeneous structures) are
frequently used to explain this phenomenon, and competition between disordered
clustering and ordered crystallization can lead to glass formation. Experimental
observation of clusters in supercooled colloidal fluids has been reported [17]. Thus, a
model that allows for competing cluster processes is clearly applicable. The distribution
kinetics theory has been successfully used to model the viscosity and dielectric relaxation
time dependence on temperature for glassformers [15].
Gelation represents an irreversible phase transition, in that once the gel state is reached,
the polymer will not return to the sol state without a chemical change in the solution,
such as a pH modification. Flory developed statistical models to explain how polymer
gelation occurs through cross-linking. A novel distribution kinetics approach, in which
two distribution variables are considered simultaneously, molecular weight and number
of cross-links, yields a general model for gelation [9 & 10]. Cross-linking also occurs in
polymer-clay nanocomposites [18], and gels of these compounds are of interest for their
unique properties. In polymer-clay composites, bonds exist between polymer and clay,
as well as between polymer molecules. The properties of polymer-clay nanocomposites
can be influenced by application of a shearing force, because the bonds between clay and
polymer are weaker than bonds between polymers. The colloidal and rheological
properties of polymer-clay gels can potentially be modeled by a population balance
approach similar to that applied to gelation.
The process of spinodal decomposition, a non-first order phase transition, has many
applications in materials science and engineering, including polymer blending. For some
spinodal decompositions, the initial segregation phase of the decomposition is fast, on the
order of a few milliseconds. For two liquid phases separating from one miscible liquid
phase, experiments have demonstrated that following the initial separation, small microdomains grow by diffusion and coalescence [7]. Experiments in which the phase
transition is retarded by considering only very shallow quenches below the critical
temperature demonstrate that these domains are at local equilibrium with each other, and
they grow first by diffusion, and then by convection [7]. Similar results are obtained in
very viscous polymer blends [8]. Simulations indicate, however, that deeply quenched
liquid mixtures are never at local equilibrium, because segregation and growth occur
simultaneously [7]. The last stage of a spinodal decomposition phase transition for a
liquid mixture involves coarsening of the phase-separated droplets. During this last
stage, once the droplets are large enough that buoyancy dominates surface tension effects,
the mixture separates by gravity. Thus the mechanisms that dominate a spinodal
decomposition phase transition depend on the process. The process and the competing
mechanisms underlying the phase transition should both be considered in a mathematical
model simulation.
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Proposed Research
In this work, we propose to model the dynamics of polymer phase transitions, including
spinodal decomposition, glass transition, nucleation, growth and ripening, and gelation.
Fluids consisting of a single polymer, a mixture of polymers, and colloids composed of
polymer will be considered. The modeling technique to be applied is distribution kinetics,
or population balance modeling. In general terms, phase transitions can be regarded as
reaction-like mechanisms in which molecules cluster together, or break apart, at various
rates, resulting in a distribution of cluster sizes that changes with time until equilibrium
(or steady state) is reached. The rates of these reaction-like processes may depend on
such variables as cluster size, temperature, and pressure. Population balance equations
can be written for the clusters as well as for monomer, and mathematically transformed to
moment equations, allowing calculation of moments representing molar concentration,
number average molecular weight, and polydispersity. The various moments can be
related to properties of the fluid, such as viscosity in the case of the glass transition, or
supersaturation in the case of nucleation and growth. For gelation by aggregation
accompanied by inter- or intramolecular crosslinking, the number of crosslinks c-an also
be incorporated into the population balance equation [9 & 10].
Previous work in population balance modeling has demonstrated that distribution kinetics
for nucleation, reversible growth, and reversible aggregation processes can be used to
model such phase transition behaviors as cluster growth by monomer addition, cluster
dissolution by monomer dissociation, classical nucleation by monomer addition,
denucleation by monomer dissociation, ripening, vitrification, and induction time for
precipitation due to low nucleation rate [11]. This earlier work, which assumed that the
associated rates for each process were functions only of temperature, has been extended
for the metastable phase transition case to include rates dependent on the cluster mass [12
& 13]. Further extensions accounted for the Gibbs-Thomson effects on equilibrium
solubility, phase transition energies, interfacial energies, and the critical nucleus size for
denucleation [14]. For the glass transition, only temperature dependence has been
modeled to date [15].
For a first-order condensation phase transition, nucleation may occur via a homogenous
route, in which a free energy barrier must be surmounted to form a critical nucleus, or a
heterogeneous route, in which nucleation sites allow monomer deposition. Formation of
critical nuclei can be viewed as a series of monomer-addition reactions. For the
homogeneous route, the rate of nucleation depends on the size of the free energy barrier.
This free energy barrier can be related to the supersaturation, which is in turn given by
m(0)/m(0)eq, the molar concentration of monomer/molar concentration of monomer at
equilibrium. For heterogeneous nucleation, the same reaction scheme can be assumed to
form the nuclei, but without a free energy barrier-dependent rate. Once the nuclei are
formed, they grow via monomer addition. Growth via monomer addition can be modeled
similarly to the nucleation scheme above, using a “reaction” of the form,
M(xm) + C(x)  C(x + xm),
where M is monomer, C represents clusters, and x represents size of the cluster. Since
this process is random, the resulting clusters vary in size, yielding a cluster size
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distribution. Clusters may also coalesce and break up, which can be described via a
process of the type,
C(x') + C(x)  C(x + x').
In the ripening stage, due to the reduction in interfacial area as clusters increase in size,
large clusters continue to grow and small clusters dissolve. This process can be modeled
similarly to the growth stage by introducing the Gibbs-Thomson relations. Population
balances on monomer, M, and clusters, C, can then be written, in terms of the rates, as
well as any source/sink terms in the balance. For example, cluster denucleation would
appear as a sink in the cluster balance, but would be a source term in the monomer
balance, since monomers result when the clusters denucleate. Given a model of this
form, expressions for the reaction rates must be determined, and the results tested against
experimental data for polymer condensation. Recent results [4, 5, 6, 12, 14] need to be
extended to polymers. An additional validation of the model is the comparison with
molecular-level models of phase transitions, such as lattice-Boltzmann models [16]. In
addition, our model has not yet been applied to polymer mixtures that phase separate
either below the upper critical solution temperature or above the lower critical solution
temperature. For a mixture, the phase separation occurs via the same processes as
described above, so that a similar model may be applicable. Recent experiments describe
a distribution of droplets of varying sizes that grow with time [8], indicating that
distribution kinetics has potential to accurately model this phenomenon. As polymer
colloids have a particle size distribution, a population balance model may also prove
useful in describing the properties of such a solution, as well as its ability to crystallize.
In the case of spinodal decomposition, there is no barrier to the phase change, so that it
occurs spontaneously. Thus, a reaction model for a condensation would involve the same
equations as above for growth and monomer addition. Such a model could be applied to
the spontaneous separation into two phases of formerly miscible polymers upon lowering
of temperature below the upper critical solution temperature, as well as the condensation
of a single polymer upon lowering of pressure. A distribution kinetics approach appears
valid for these cases, because light-scattering experiments show clusters that increase in
size with time during spinodal decomposition.
We will explore the pressure-induced solidification of a metastable liquid or colloidal
solution (glass transition), for example, polymer solution or melt. A previous model [15]
for cluster kinetics of glass formation will be extended, via transition state theory, to
include activation volumes in the rate coefficients for monomer-cluster additiondissociation and cluster agglomeration-breakage. The glass formation kinetics are based
on the dynamics of hypothesized cluster-monomer distributions in liquids. The
heterogeneities in glassy materials have been well documented [17]. The consequences of
this hypothesis for metastable liquids that do not nucleate are interpreted in terms of
fragility plots for the pressure effect on viscosity and dielectric relaxation time. The
proposed model yields entropy relations that afford an explanation for the effect of
pressure on entropy, and provides free volume ratios that allow representation of the
viscosity. Given values of pressure at the glass transition and when the material is fluid, a
single volume-difference parameter is needed to complete the correlation.
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Previous work on gelation models [9 & 10] will be extended to polymer-clay gels and
solutions, in which bonds between polymer and clay can be broken via application of a
shearing force [18]. In the distribution kinetics model, cross-linking has been described
by the following “reactions”:
Pi(x) + Pj(x') → Pi+j+1(x + x'),
Where Pi(x) is a macromolecule with molecular weight x and i crosslinks. The above
equation describes the loss of Pi(x), whereas the gain of Pi(x) is given by:
Pi-j-1(x-x') + Pj(x') → Pi(x).
For the polymer-clay nanocomposite, in which polymer is bonded to clay and to other
polymer, the above reaction scheme would have to be expanded to include clay, and the
clay reactions must be reversible, since bonds between polymer and clay can be broken.
Once a model of the polymer-clay composite has been developed, it could be used to
explain the response of this substance to shear, as this response is believed to result from
changes in polymer-clay interactions [18].
References
1. Oxtoby, D.W. Diversity suppresses growth. Nature 2001, 413, 694 and
references therein.
2. Auer, S.; Frenkel, D. Suppression of crystal nucleation in polydisperse colloids
due to increase of the surface free energy. Nature 2001, 413, 711
3. Mintova, S.; Olson, N.H.; Valtchev, V.; Bein, T. Mechanism of Zeolite A
Nanocrystal Growth from Colloids at Room Temperature. Science 1999, 283,
958.
4. Madras, G.; McCoy, B.J. Transition from nucleation and growth to Ostwald
ripening. Chemical Engineering Science 2002, 57, 3809.
5. Madras, G.; McCoy, B.J. Denucleation Rates during Ostwald ripening:
Distribution kinetics of unstable clusters. Journal of Chemical Physics 2002, 117,
6607.
6. Madras, G.; McCoy, B.J. Continuous distribution theory for Ostwald ripening:
comparison with the LSW approach. Chemical Engineering Science 2003, 58,
2903-2909 and references therein.
7. Gupta, R.; Mauri, R.; Shinnar, R. Phase Separation of Liquid Mixtures in the
Presence of Surfactants. Industrial Engineering Chemistry and Research 1999,
38, 2418-2424 and references therein.
8. Califano, F.; Mauri, R. Drop Size Evolution during the Phase Separation of
Liquid Mixtures. Industrial Engineering Chemistry and Research 2004, 43, 349353 and references therein.
9. Li, R.J.; McCoy, B.J. Crosslinking kinetics: Partitioning according to number of
crosslinks. Macromolecular Theory and Simulations 2004, 13, 203-218.
10. Li, R.J.; McCoy, B.J. Inter- and Intramolecular Crosslinking Kinetics:
Partitioning According to Number of Crosslinks. Macromolecular Rapid
Communications 2004, 25, 1059-1063.
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11. McCoy, B.J. Distribution kinetics modeling of nucleation, growth, and
aggregation processes. Industrial & Engineering Chemistry Research 2001, 40,
5147-5154.
12. Madras, G.; McCoy, B.J. Ostwald ripening with size-dependent rates: Similarity
and power-law solutions. Journal of Chemical Physics 2002, 117, 8042-8049.
13. Madras, G.; McCoy, B.J. Dynamics of crystal size distributions with sizedependent rates. Journal of Crystal Growth 2002, 243, 204-213.
14. Madras, G.; McCoy, B.J. Growth and ripening kinetics of crystalline
polymorphs. Crystal Growth & Design 2003, 3, 981-990.
15. McCoy, B.J. Cluster Kinetics for Glassforming Materials. Journal of Physics
and Chemistry of Solids 2002, 63, 1967-1974.
16. H. Başağaoğlu, C.T. Green, P. Meakin, B.J. McCoy, Lattice-Boltzmann
Simulation of 2D Phase Transition with Ostwald Ripening, J. Chem. Phys.
(2004); submitted.
17. Ediger, M.D. Spatially Heterogenous Dunamics in Supercooled liquids and the
glass transition, Annual Review of Physical Chemistry 2000, 51, 99.
18. Malwitz, M.M.; Butler, P.D.; Porcar, L.; Angelette, D.P.; Schmidt, G. Orientation
and Relaxation of Polymer-Clay Solutions Studied by Rheology and Small-Angle
Neutron Scattering. Journal of Polymer Science: Part B: Polymer Physics 2004,
42, in press.
Number of IGERT apprentices to be recruited and probable home departments: 2
total
1, chemical engineering
1, chemistry
Consistency with the Macromolecular Education, Research & Training theme:
This project requires the students to understand polymers, and the methods used to
characterize and model the dynamics of their phase transitions. The education provided
via the courses in macromolecules will provide the foundation in polymers to pursue the
project, and the research will provide training in polymer modeling both at the
macroscopic and microscopic level.
How does the project form a vector cross-product of existing research themes by the
participants?
Existing research directions.
Dr. McCoy’s current research interests related to this project include phase transition
dynamics and distribution kinetics. He and collaborators have published numerous
papers applying these modeling techniques to polymers, such as the kinetics of
polymers in solution, polymer decomposition/degradation, polymerization reactions,
and thermogravimetric analysis of polymers. He is a full professor in chemical
engineering, holding the Gordon A. & Mary Cain endowed chair.
Dr. Schmidt’s current research interests include the synthesis, development, and
characterization of new multi-component materials with hybrid properties relevant to
polymer and colloid science. In particular, her work on polymer-clay nanocomposites
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interfaces well with Dr. McCoy’s modeling expertise, in that the data generated by her
group could be used to validate models developed by Dr. McCoy’s group. Dr. Schmidt
is an assistant professor in the chemistry department.
New research direction.
This research project directly joins the research areas of the two professors involved, by
attempting to validate models developed by Dr. McCoy’s group of novel polymer-clay
composites produced and characterized by Dr. Schmidt’s group. A synergistic
collaboration between the two groups has the potential to yield interesting new results.
How do students benefit from the team-oriented research, beyond what would be
available to them from either advisor separately?
The inter-disciplinary nature of this project allows the chemical engineering student
exposure to synthesis, development, and characterization of nanocomposite materials,
an area not normally the province of engineers, while exposing the chemistry student to
distribution-level models, a technique not normally seen in chemistry.
Briefly describe the support level available to each individual faculty or off-campus
participant (i.e., without IGERT)
Dr. McCoy is independently supported for research in related fields by the Gordon A. and
Mary Cain endowed Chair. Dr. Schmidt is the recipient of an NSF Career Award, and
thus is also independently supported for research in related fields.
Interdisciplinary strengths of the team project:
Dr. McCoy has a broad and extensive background in mathematical modeling applied to a
variety of areas in chemical engineering. Dr. Schmidt has been working on polymerclay nanocomposites for several years, and is now progressing to detailed
characterization of these materials. By joining their efforts, these novel polymer-clay
materials can be modeled, and the models validated against experimental data,
something that neither could do alone.
Commitment of faculty & off-campus participants to work side-by-side with
apprentices:
Professor McCoy, a full professor in the department of chemical engineering, promises
daily interaction with the student during the approximately one month period of the
apprenticeship. Consultations with Professor Schmidt, an assistant professor in the
chemistry department, and her group will be frequent.
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