Kinetic Theories for Complex Fluids
Qi Wang
Department of Mathematics, Interdisciplinary Mathematics
Institute, & NanoCenter at USC
University of South Carolina
Columbia, SC 29208
Research is partially supported by grants from NSF-DMS, NSFCMMI, NSF-China
Collabrators in the work discussed in the
presentation
•
Biofilm:
– Tianyu Zhang, Montana State University
– Nick Cogan, Florida State Univ
– Brandon Lindley, University of South Carolina
•
Polymer-particulate nanocomposites:
–
–
–
–
Greg Forest, Univ of North Carolina at Chapel Hill
Ruhai Zhou, Old Dominion Univ.
Guanghua Ji, Beijing Normal University, PRChina
Jun Li, Nankai University, PR China
Applied and Computational Mathematics at USC
• A new applied and computational mathematics program has been
launched at USC in 2009.
• The aims of the program are (i) to implement modern applied and
computational mathematics curriculum in graduate training, (ii). To
support science and engineering education and research on the campus
of USC, (iii). to foster interdisciplinary research across various
disciplines in science, engineering and medicine.
• The research areas in the ACM includes: computational mathematics,
modeling and simulation of complex fluids/soft matter, computational
biology, cellular dynamics, geo-fluid dynamics, climate modeling,
wavelet analysis, imaging sciences, approximation theories, etc.
• More details can be found at
http://www.math.sc.edu/~qwang/USC_applied_math.htm
Opportunities in ACM at USC
• Graduate students: graduate students can pursue PhD track in applied
and computational mathematics supported by TA and RAs. Students
will be able to learn courses across various disciplines pertinent to their
research areas.
• Faculty: Three new faculty members in bio-mathematics related to
biofabrication will be hired in the next three years. This year the
opening is a senior-level full professor position. This is supported by a
$20M NSF EPSCOR-TRACK II grant (2009-2014). Intensive
interaction with MUSC center on biofabrication and bioengineering at
Clemson is expected.
• Postdoctoral training at Interdisciplinary Mathematics Institute.
Outline
• Introduction to kinetic theories for mesoscopic
dynamics
• Example 1: kinetic theories for biofilms
• Example 2: kinetic theories for polymer
particulate nanocomposites
• Conclusion
“Definition” of Complex Fluids or Soft Matter
• “A fluid made up of a lot of different kinds of stuff”; defining feature
of a complex fluid is the presence of mesoscopic length scales in
addition to the macorscopic scales which necessarily plays a key role
in determining the properties of the system. (Gelbart et al, J. Phys.
Chem. 1996).
• Complex fluids are also known in the physics community as the soft
matter, the matter between fluids and ideal solids. “Soft condensed
matter is a fluid in which large groups of the elementary molecules
have been constrained so that the permutation freedom within the
group is lost.” (T. A. Witten, Reviews of Modern Physics, 1998)
• Common feature in complex fluids/soft matter: “mesoscopic scale
morphologies, dynamics and physics dominate the material’s
macroscopic properties.”
• Examples: polymer solutions, metls, gels, surfactant solutions such as
micellar solutions and microemulsions, colloidal suspensions such as
ink, milk, foams, and emulsions, blood flows, biofilms, mucus, and
muscles, cytoplasma, etc.
Modeling approaches
Small length and time scale. Computational models: MD simulations,
Monte Carlo Simulations, Ab Initio computations, discrete mechanical models,
etc. These are microscopic models. (Computational intensive.)
Intermediate length and time scale. Kinetic theories, multi-scale kinetic theories, Coarse
grain models (Dissipative particle methods), Bownian dynamics, Lattice Botzmann
Method. These are mesoscale models. (Hopefully, computational manageable.)
Large length and time scale. Continuum models, multiscale continuum models,
reduced order models, etc. These are macroscopic models.
(Computational less expensive.)
(Phase space or configurational space) Kinetic theory for
complex fluids
(Doi & Edwards, 1986, Bird et al., 1987)
Mesoscopic description: dynamical distribution of “model”
molecules. The transport equation is the Smoluchowski
equation or kinetic equation.
Coupling via macroscopic velocity, velocity
gradient, moments of the distribution
Macroscopic description: mass, momentum, and energy
balance equations.
Conservation equations
Constitutive equations
Kinetic equations: mesoscopic transport equations
Smoluchowski equation for mixtures
• Smoluchowski equation can be extended to account for active
materials, live materials such as active filaments, sperms, virus,
bacteria, and reactive materials in multi-species environment. The
generic form of the equation in this system is given by
@f i
@t
_ i ) + gi
= ¡ r ¢( xf
Where fi is the pdf or ndf for species i and gi is the source or reactive
term for the species. Conservative properties or conditions are
imposed on fi and gi, which may be algebraic or integral form. The
source terms come from the decomposition or decoupling of the pdf
(ndf) into independent pdf (ndf) fi, i.e. f=f1 fn.
Coupling to the macroscopic transport equations
Smoluchowski equation for pdf at mesoscale
Balance equations for mass, momentum, energy at the macroscopic scale
•
•
•
•
The coupling with the macroscopic mass, momentum, and energy transport is
achieved via the stress constitutive equation. The viscous part of the extra
stress and the elastic part of the extra stress is calculated separately.
The viscous part of the extra stress is done semi-phenomenologically by fluid
dynamics and/or ensemble averaging.
The elastic part of the extra stress is done using the variational principle or the
virtual work principle for equilibrium dynamics. For nonequilibrium dynamics
(like active systems), averaged forces per unit area have to be calculated using
ensemble averages (Kirdwood, Briels & Dhont, etc.).
Two examples of kinetic theories are given in the following to elucidate the
formulation: biofilms and polymer particulate nanocomposites.
Example 1: Biofilms
What are Biofilms?
•
Biofilms are ubiquitous in nature
and manmade materials. Biofilm
forms when bacteria adhere to
surfaces in moist environments by
excreting a slimy, glue-like
substance called the extracellular
polymeric substance (EPS). Sites
for biofilm formation include all
kinds of surfaces: natural materials
above and below ground, metals,
plastics, medical implant
materials—even plant and body
tissue. Wherever you find a
combination of bacteria, moisture,
nutrients and a surface, you are
likely to find biofilms.
In a pipe
Plaque on teeth
In a creek.
In a membrane
Where do biofilms grow?
•
•
Biofilms grow virtually everywhere, in
almost any environment where there is
a combination of moisture,
nutrients, and a surface.
This streambed in Yellowstone
National Park is coated with biofilm
that is several inches thick in places.
The warm, nutrient-rich water
provides an ideal home for this
biofilm, which is heavily populated by
green algae. The microbes colonizing
thermal pools and springs in the Park
give them their distinctive and unusual
colors.
Staph Infection (Staphylococcus aureus biofilm) of the surface of a
catheter (CDC)
Giant, Mucus-Like Sea Blobs on the Rise, Pose Danger
National Geographic News, October 8, 2009
Biofilm expansion, growth and transport process in flows
Characteristics of Biofilms: dynamic, cellular structure and
signaling, and gene expression
• Biofilms are complex,
dynamic structures
•Gene and Cellular
Structures
Modeling Challenges
• Basic mathematical models for the growth of the biofilm colony
should account for the properties of the EPS, nutrient
distribution/transport/consumption, bacterial dynamics, solvent
interactions, etc.
• Additional features can include: intercellular communication, signaling
pathways, impact of the gene expression; drug interaction with the
biofilm components, especially, the bacterial microbes….
• Existing models: low-dimensional models, diffusion limited
aggregation for patterns, discrete or semidiscrete model coupled with
local rules or automata, viscous fluid models or multi-fluid continuum
models.
• Disadvantages of the multifluid models: how to imposed initial and inflow/out-flow boundary conditions for each velocity in multi-fluid
models? Numerical methods in multi-dimensions may be difficult.
• Our approach: one fluid, multi-component modeling, to
systematically include more components. Advantages: an averaged
velocity is used and the material is treated as incompressible. (See
Beris and Edwards, 1994).
A two-component kinetic model for biofilms
Schematic of the model
EPS
network
Bacterium
Solvent
A kinetic theory formulation
Another model (nonseparable model, diffusive
stress)
T he Sm oluchow ski equat ion f or t he e®ect ive p olym er is
@
1
Ã+ r ¢( v Ã) = r ¢( ¸ r ¹ Ã) + r q¢ ( r q¹ Ã) ¡ r q( ( r v n + ( a¡ 1) D n ) ¢q Ã) + G n :
@t
³
T he st ress const it ut ive equat ion is given by
d¿n
¿n
2´ n
¸2
¡ W n ¢¿n + ¿n ¢W n ¡ a[ D n ¢¿n + ¿n ¢D n ]+
=
D n+
r ¢( Án r ¿n ) ;
dt
¸1
¸1
¸ 1 Án
a³
w here ¸ 1 = 4»º ³k T is t he relaxat ion t im e, ´ n = 4»
is t he E P S
B
p olym er viscosit y, ¸ 2 = ¸4»³ :
T he viscous st resses are
¿s = 2´ sd s; ¿ps = 2´ psD n :
T he m obilit y is assum ed as
¸ = ¸ 0 Án ( 1 ¡ Án ) :
Boundary conditions
Stability of a constant steady state
The mixing energy and growth rates
a3<0
a3>0
Numerical method and simulation issues
• A 2nd order projection method is devised to solve the momentum
transport equation for the average velocity.
• A second order semi-implicit solver is developed to solve the
generalized Cahn-Hilliard equation based on GMRES method.
• A second order Crank-Nicolson scheme is used to solve the nutrient
transport equation.
• A first order streamline upwind scheme along with a high order RK
scheme is used to solve the constitutive equation for the polymer.
• The interface between the biofilm and the solvent is defined as
{x|Án(x,t)=0+}.
Single hump growth up to t=300
Biofilm growth
Sheared thick biofilm colony vs thin collony
Shedding in shear
Viscoelastic behavior
Oscillatory shear
References
•
•
•
•
•
•
T. Y. Zhang, N. Cogan, and Q. Wang, “Phase Field Models for Biofilms. I.
Theory and 1-D simulations,” Siam Journal on Applied Math, 69 (3)
(2008), 641-669.
T. Y. Zhang, N. Cogan, and Q. Wang, “Phase Field Models for Biofilms. II.
2-D Numerical Simulations of Biofilm-Flow Interaction,” Communications
in Computational Physics, 4 (2008), pp. 72-101
T. Y. Zhang and Q. Wang, Cahn-Hilliard vs Singular Cahn-Hilliard
Equations in Phase Field Modeling, Communication in Computational
Physics, 7(2) (2010), 362-382.
Q. Wang and T. Y. Zhang, “Kinetic theories for Biofilms”, DCDS-B, in
revision, 2009.
Brandon Lindley and Q. Wang, Multicomponent models for biofilm flows,
submitted to DCDS-B, 2009.
Q. Wang and T. Y. Zhang, Mathematical models for biofilms,
Communication in Solid State Physics, submitted 2009.
Example II: Polymer-particulate
nanocomposites
What are polymer nanoparticle composites?
•
The polymer nanoparticle composites are mixture of polymer matrix with
nanosized particle fillers. They may share the properties of both components
or even develop new ones.
Nanoparticles
dispersed in
polymer matrix
•








•
Improved material properties include:

Mechanical properties e.g. strength, modulus and dimensional stability

Decreased permeability to gases, water and hydrocarbons

Thermal stability and heat distortion temperature

Flame retardation and reduced smoke emissions

Chemical resistance

Surface appearance

Electrical conductivity and energy storage

Optical clarity in comparison to conventionally filled polymers
The commonly used nanoparticles include clays, silicates, nanorods, carbon
nanotubes, metals, etc.
Microstructure in polymer clay nanocomposites
Isotropic and Nematic phases in Boehmite in polyamide-6
(Picken et al, Polymer, 2006)
Rheological Functions in PS+DMHDI (Zhao et al, Polymer, 2005)
Modeling challenges for nanocomposites
• Semiflexibility: Clays and carbon nanotubes are not completely rigid.
They are semiflexible! Modeling semiflexible ensembles is harder than
either the flexible or the fully rigid ones.
• Surface physics: Polymer nano-filler compatibility issue and its
consequence to the polymer-nanoparticle interaction.
• Hydrodynamics: Hydrodynamics of a single semiflexible nanoparticle
in solvent (viscous or even viscoelastic, Jeffreys orbit?).
Hydrodynamic interaction for a cluster of semiflexible nanoparticles.
• NP Dispersion: Dispersion of nanoparticles in polymer matrix. I.e.,
intercalated vs exfoliated.
• Recent attempts: Phenomenological continuum models consistent
with the GENERIC formalism (Grmela et al. Rheol Atca 05 for fibers,
J. Rheol. 07 for platelets)
Kinetic theory for polymer nanoparticle composites
Let f(m,x,t) be the number density function (ndf) for model
nanoparticles of spheroidal shape with axis m at location x. By
adjusting aspect ratio of the spheroid, rod shaped and platelet shaped
inclusion can be modeled. Main interest: rod or platelet.
m
||m||=1
Rodlike
Discotic
Free energy functional
Linear elastic springs (Rouse chain)
R3
R1
R2
R5
R4
m
Incompressibility constraint:
The system is assumed incompressible so that the volume fractions add up to
unity:
f+Vs f dm=1,
where V is the volume of the nanoparticle. This constraint is upheld at every
material point x. A Lagrange multiplier’s method is then used to derive the
fluxes of the NP and the flexible polymer host in the inhomogeneous regime,
respectively.
Kinetic model
The rest of the equations
Excluded volume potential and other parameters
• The volume fraction of the polymer f is a constant in monodomain.
• In shear flows, Pe the Peclet number denotes the dimensionless shear
rate.
• One bead-spring mode in the chain is adopted in the simulation.
• The excluded volume potential is approximated by the Maier-Saupe
potential
Ums=-const N kBT h mmi:mm.
• Key material parameters rf : semiflexibility,
N: strength of excluded volume interaction,
b » a1-a2: strength of polymer-nanorod interaction. b>0
promotes a perpendicular orientation between the nanorod and the
polymer chain; b<0 favors a parallel alignment between them.
Effect of b on monodomain equilibirum
•Nonzero ¯ enhances
nematic order in the
nanoparticles and the
host.
•As rf increases, the
nematic order
decreases in the
nanorod ensemble.
Sq and su are order parameters for the nanoparticle and the polymer
matrix, respectively; F is the free energy.
Effect of b on monodomain steady states
•Positive b promotes
perpendicular
alignment of the
polymer with the
nanorod.
•Negative one favors
parallel alignment.
Negative one also
improves the local
nematic order in the
nenorod dispersion.
• f=90%.
Top: Pe=0; Bottom: Pe=0.1.
Transient shear stress
Experimental comparison: transient
viscosity (PBT+MMT, Wu et al,
Euro Polym. J, 2005)
G’, G’’, ´*
Spatial-temporal structure formation in inhomogeneous flows
(1-D).
Dynamical smectic structure in inhomogeneous flows
References
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•
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M. G. Forest, Qingqing Liao, and Qi Wang, “2-D Kinetic Theory
for Polymer Particulate Nanocomposites,” Communication in
Computational Physics, 7(2), (2010), 250-282.
Jun Li, M. G. Forest, Qi Wang and R. Zhou, “Flows of polymerparticulate nanocomposites: weakly semiflexible limit,” submitted
to DCDS-B, 2009.
Guanghua Ji and Qi Wang, Structure Formation in Sheared
Polymer-Rod Nanocomposites, submitted to Journal of Rheology,
2009.
G. Forest, J. Li, Q. Wang, and R. Zhou, Stability of the steady state
in flows of polymer-particulate nanocomposites in the regime of low
volume fraction, to be submitted, 2009.
J. Li and Q. Wang, “Flows of polymer-particulate nanocomposites:
II. Kinetic predictions,” to be submitted, 2009.
Summary
• Kinetic theory provides a convenient modeling platform
for tackling complex problems arising in complex fluids
leading to models with less adjustable model parameters
• Reduced order models derived from the kinetic theory can
play an key role in directing the development of
phenomenological models.
• Computational advances can improve the solution
procedure of the kinetic models at reduced cost.
• Coupling of multispecies in mixture can be done
inherently.
• The mathematical structure of the models and solution
behavior remains wide open.