Kinetic Theories for Complex Fluids

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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, NSF-CMMI, NSF-China

AAAA
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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

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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.ed
  u/qwang/USC_applied_math.htm

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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.

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Outline

• Introduction to kinetic theories for mesoscopic
  dynamics
• Example 1 kinetic theories for biofilms
• Example 2 kinetic theories for polymer
  particulate nanocomposites
• Conclusion

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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 materials 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.

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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.)
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(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.
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Conservation equations
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Constitutive equations
The constitutive relations need detailed
information about the microstructures of the
material system.
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Kinetic equations mesoscopic transport equations

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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

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. ff1? fn.
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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.

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Example 1 Biofilms
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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
  materialseven 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
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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.

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Staph Infection (Staphylococcus aureus biofilm)
of the surface of a catheter (CDC)
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Giant, Mucus-Like Sea Blobs on the Rise, Pose
Danger National Geographic News, October 8, 2009
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Biofilm expansion, growth and transport process
in flows
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Characteristics of Biofilms dynamic, cellular
structure and signaling, and gene expression

• Biofilms are complex, dynamic structures

• Gene and Cellular Structures

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Modeling Challenges

• Basic mathematical models for the growth of the
  biofilm colony should account for the properties
  of the EPS, nutrient distribution/transport/consum
  ption, 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 in-flow/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).

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A two-component kinetic model for biofilms
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Schematic of the model
EPS network
Solvent
Bacterium
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A kinetic theory formulation
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Another model (nonseparable model, diffusive
stress)
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Boundary conditions
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Stability of a constant steady state
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The mixing energy and growth rates
a3lt0
a3gt0
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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 xÁn(x,t)0.

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Single hump growth up to t300
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Biofilm growth
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Sheared thick biofilm colony vs thin collony
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Shedding in shear
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Viscoelastic behavior
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Oscillatory shear
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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.

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Example II Polymer-particulate nanocomposites
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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.
• 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.

Nanoparticles dispersed in polymer matrix
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Microstructure in polymer clay nanocomposites
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Isotropic and Nematic phases in Boehmite in
polyamide-6 (Picken et al, Polymer, 2006)
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Rheological Functions in PSDMHDI (Zhao et al,
Polymer, 2005)
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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)

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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
m1

Rodlike
Discotic
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Free energy functional
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Linear elastic springs (Rouse chain)
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R5
R3
R1
R4
R2
m
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Incompressibility constraint The system is
assumed incompressible so that the volume
fractions add up to unity
fVs f dm1, where V is
the volume of the nanoparticle. This constraint
is upheld at every material point x. A Lagrange
multipliers method is then used to derive the
fluxes of the NP and the flexible polymer host in
the inhomogeneous regime, respectively.
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Kinetic model
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The rest of the equations
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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 mmimm.
• Key material parameters rf semiflexibility,
• N strength of excluded volume
  interaction,
• b a1-a2 strength of
  polymer-nanorod interaction. bgt0 promotes a
  perpendicular orientation between the nanorod and
  the polymer chain blt0 favors a parallel
  alignment between them.

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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.
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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.
• f90.

Top Pe0 Bottom Pe0.1.
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Transient shear stress
Experimental comparison transient viscosity
(PBTMMT, Wu et al, Euro Polym. J, 2005)
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G, G,
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Spatial-temporal structure formation in
inhomogeneous flows (1-D).
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Dynamical smectic structure in inhomogeneous flows
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References

• 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 polymer-particulate 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.

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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.