Advanced thermodynamics

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

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

CrosslinkingCrosslinking Physical

crosslinking

• Ionic hydrogel

Chemical and Physical

crosslinking• Cross-l inking without chemical reaction• ionic interaction, hydrogen

bonding, antigen-antibody interaction, supramolecular association

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

• 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

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

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=

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

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 −

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− − − + +

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 −

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 = − −

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

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

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

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

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

The predicted phase separation The predicted phase separation is entropic in originis entropic in origin

Phase separationPhase separation

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

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-

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

BundlesBundles

For long chains the bundle formation is characterized by slow kinetics

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

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