advanced thermodynamics
TRANSCRIPT
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Role of cross-links in bundle Role of cross-links in bundle formation, phase separation and formation, phase separation and
gelation of long filamentsgelation of long filaments
Ortal LeviDepartment Of Chemical EngineeringJuly 2013
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Branched structures and Branched structures and networksnetworks
• Exists in many physical, chemical and biological systems• Chemical systems- chemical hydrogels• Physical systems- physical hydrogels, wormlike
micelles/micro emulsions, dipolar fluids
• Applications- medical industry (drug delivery and healing) , food industry
Physically cross linked networks
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CrosslinkingCrosslinking Physical
crosslinking
• Ionic hydrogel
Chemical and Physical
crosslinking• Cross-l inking without chemical reaction• ionic interaction, hydrogen
bonding, antigen-antibody interaction, supramolecular association
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Hydrogel FabricationHydrogel FabricationChemical hydrogels
Physical hydrogels
Hydrogen bonding
hydrophobic interaction
crystall inity
stereocomplex formation
ionic complexation
Covalently crosslinked
Noncovalently crosslinked
Thermoset hydrogels
Thermoplastic hydrogels
Volume phase transit ion
Sol-gel phase transit ion
Reliable shape stabil i ty and memory
Limited shape stabil i ty and memory
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• Presenting a phase diagram of a generic system of cross-linked equilibrium chains in terms of two independent transitions in the system
• Prediction of thermodynamic and structural behavior of solutions of long cross-linked filaments.
Research ObjectivesResearch Objectives
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The theoretical modelThe theoretical modelThe system-
•A grand-canonical ensemble- network in equilibrium with a reservoir of ends and junctions
•junctions and ends are viewed as “thermal defects” of the system whose “ground state” is an assembly of infinite linear chains
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Assumptions-
•No specific interactions between the monomers, except excluded volume-dilute solution
•Mean Field Approximation
•Sparse junctions and ends-
The theoretical modelThe theoretical model
( , )j eφ φ φ=
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The theoretical modelThe theoretical modelParameters-
-the monomer density
j
e
j
e
c
φεε
ρφφ
-junction energy
-end energy-cross links density
-end cup molecule density
-number of branching points
-number of free ends points
relative to the bond energy between two monomers in the chain
Per unit volume
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Mathematical development-•We start with the grand canonical potential per unit volume-
•The probability of bond breaking- two end formation•The probability of collision of two ends- • a factor of the microscopicall qualities of ends-Chain flexibility and effective collision surface area
The theoretical modelThe theoretical model
21( , , , , ) (1)
2j e j e e jφ µ µ ε ε φ φ φΩ = − −
Excluded volume
2( )/e e Te µ εφ −
2 21e aφ −
1a −
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Mathematical development-•In equilibrium-P(break)=P(create)
•For an f component junction formed from f-2 ends and an internal monomer- energy cost for breaking a junction and creating f-2 ends-
The theoretical modelThe theoretical model
( )/ 1/21 (2)e e T
e a e µ εφ φ−=
( 2)( ) lne e j j ff T aε µ ε µ− − − + +
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Mathematical development-•Probability of junction break-up:
•Probability of collision of f-2 ends with an internal monomer-
•In equilibrium• coefficient of the microscopical freedom of the junctions, including the configurational entropy of the bonds and monomers in the junction
•
The theoretical modelThe theoretical model
( )/ ( 2)( )/j j e eT f T
j fa e eµ ε ε µφ − − − −
2 21
f fe aφ φ− −
( )/ /2 (3)j j T fj fa e
µ εφ φ−=fa −
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Mathematical development-
• Grand canonical potential-
The theoretical modelThe theoretical model
( )/( )/2 1/2 /21
1( , , ) / (4)
2j je e
TT fj e fT a e a e µ εµ εφ µ µ φ φ φ−−Ω = − −
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Free energyFree energy• The relation between free energy and the grand canonical
potential-
(5) Legendre Transformf µφ=Ω+ →
21 1( , , ) / (ln 1) (ln 1) ( / ln ) / ln ln (6)
2 2 2j e j j e e j j f e e j e
fF T T a Tφ φ φ φ φ φ φ φ φ ε φ ε φ φ φ φ= + − + − + − + − −
Excluded volume
Free energy of “ideal gas” of junctions and ends
Energy cost due to the network constrained-reduction of entropy
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free energy free energy TotalTotal
( , , ) ( , , ) ( ) ( )totj j e c j e eF c F F c Fφ φ φ φ φ φ ρ φ= + − + −
Free energy of unbound cross l inkers
Free energy of unbound ends
ψψ
Both molecules as an ideal solut ion-
( ) ( ) (ln 1)e cF F Tψ ψ ψ ψ= = − 1
ln(1 )
ψψ ψ− →−
=
/ /1/2 1/2
/ //2 /2
/ (1 )
/ (1 )
e e
j j
T Te
T Tf fj f f
e e
ca e a e
ε ε
ε ε
φ ρ φ φ
φ φ φ
− −
− −
= +
= +
totF Is minimizes to f ind the end and junction equil ibrium density-
Densit ies vary with
, ,cφ ρ
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Density as a function of cross linkers-•Strong cross linkers-
•Therefore-
Total free energyTotal free energy
( 0, 1)jj T
εε < =
j cφ → • Most of the cross linkers are in the junctions
• Junctions tend to form in low temperatures
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Total free energyTotal free energyDensity as a function of cross linkers-
•Weak cross linkers-
•Therefore- •The total free energy with :
( 0 , 1)jj T
εε > ?
• Most of the cross linkers are in the solution
• In low temperatures the number of junctions 0 ,j eφ φ
Junctions reduces free energy
Ends reduces free energy
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Phase separationPhase separation• Can be caused by high cross linkers density, low temperature
• Requires the matrix of second derivatives of to be positive-definite
• This condition defines the spinodal and the critical point for density-
totF
sc
Dilute phase Dense phase
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The predicted phase separation The predicted phase separation is entropic in originis entropic in origin
Phase separationPhase separation
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3 ( 0)jf ε= ¬ >
( )sc c φ>
4, 0 , 1
0.05 , 0.005
j ee
c
εε εε
ρ
< =
= =
4 /
4
0.015,
10 / 3 , 0.005
Tf
e
a eε
ε ρε
=
= =
Strong crosslinking
4 ( 0)jf ε= ¬ <
Phase DiagramPhase Diagram
Weak crosslinking
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Phase separation and gelationPhase separation and gelationGelation/ Percolation transition• A connected network spanning the entire system-thus
dependent on concentration alone• The transition- Continuous Topological (structural) Un thermodynamicOccurs when-
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BundlesBundles • Definition-rigid chains (rods) with connective cross linkers
in a parallel structure- nematically aligned bundles of chains
• Favorable formation in low temperatures-entropy driven• Occurs when - the free energy of the bundle is
lower than that of the isotropic networkb iF Fp
Transitional entropy of the bundles
Reduction in rotational entropy
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BundlesBundles
For long chains the bundle formation is characterized by slow kinetics
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ConclusionsConclusions This model predicts three transitions-1.Phase separation-dense network and spares network2.Gelation transition-infinite network spanning the entre system3.Bundles- nematic phase-parallel crosslinked chains•Strong crosslinking-most cross linkers are in the junctions, junctions don’t break it low temperatures•Weak crosslinking-low junction density compared to crosslinkiners density, almost non existent in low temperatures
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Legendre TransformLegendre Transformהטרנספורם מעביר את הפונקציה לפונקציה חדשה התלויה בנגזרת החלקית •
לפי המשתנה הבלתי תלוי של הפונקציה הישנה.
)נבצע את המעבר:• , , ) ( , , )j e j eFφ µ µ φ φ φΩ →
( , , ) ( , , ) ( , , )j e j e j e j e j j e ej e
F φ φ φ φ µ µ µ µ φ µ µ φ µ φ µµ µ
∂Ω ∂Ω= Ω − − = Ω + +∂ ∂
eφjφ