Model of linear viscoelasticity Future research · 03/11/2010 · Introduction Model of linear viscoelasticity Numerical experiments Future research An extension of the Rouse theory

IntroductionModel of linear viscoelasticity

Numerical experimentsFuture research

Rheology control by branching modeling

Volha Shchetnikava

Department of Mathematics and Computer ScienceTU EINDHOVEN

November 3, 2010

Volha Shchetnikava Rheology control by branching modeling



Project description

Project description

The targets of the project are:

To predict the topology of members ofensembles of arbitrary branched polymers on thebasis of a reactor model (i.c. ldPE)

To predict the linear and the non-linearrheological response of branched polymer melts(i.c. ldPE)




Project description

Low-density Polyethylene

LDPE is a highly branched structure characterized by:

Broad molecular weight distribution

Both long and short side chains are present

Irregularly spaced branches

Long chain branching has a tremendous effect on the rheology

Transition from short to long chain branching at Me

Exhibit ”strain hardening” in uniaxial extensional flow

Exhibit ”strain softening” in shear flow




Project description

Arbitrary branched polymers

Macromolecules are described by graphs (trees) and representedas:

Vertices - branch points and ends of branches (arms)

Weight of the edge - molecular weight of the strand

The adjacency matrix of a weighted graph




Project description

Ensemble of arbitrary branched polymers

We specify a collection of a large number of branched molecules byintroducing parameters:

α indicate the molecular species 1, . . . ,Ns

i label the various molecules of the species α 1, . . . ,Nα

Example:

100 molecules of species 1, 2000 molecules of species 2, 1000molecules of species 3




An extension of the Rouse theoryMacroscopic stress in the system

Bead-spring structure

Portion of the molecule is replaced by entropic ”spring”

Mass of the chain is concentrated in ”beads”

Rn is the location of the nth bead

Nmn - number of the Kuhn segments of length b in theportion of the molecule

Spring constant between beads n and m is 3kBTNmnb2





The Langevin equation

The individual bead is driven by the sum of four forces:

1 fn - potential force, force from neighboring beads

2 f fr - friction force

3 fr - fluctuating force

4 fex - external force

Equation of motion for the nth bead in the absence of inertia

fnn + f frn + frn + fexn = 0






To start with consider the potential energy of the system when thesprings are Gaussian

U =∑

segments

K

2Nmn[Rn − Rm]2

where Rn(t) = (Xn(t),Yn(t),Zn(t)) and K = 3kBT/b2

We can rewrite the potential energy as

U =K

2

N∑n=1

N∑m=1

AnmRn · Rm

The connectivity matrix A is always symmetric with real elements





Example of the connectivity matrix

A =

12 0 −1

2 0 0 00 1

3 −13 0 0 0

−12 −1

343 0 0 −1

20 0 0 1

4 0 −14

0 0 0 0 1 −10 0 −1

2 −14 −1 7

4






Thus, the potential force is equal to

fnn = − ∂U∂Rn

= −KN∑

m=1

AnmRm

The friction force is proportional to velocity

f frn = −ζ dRn

dt

The fluctuating force is characterized by the following 2 moments:

< frn >= 0

< f rnα(t)f r

mβ(t′) >= 2ζkBT δnmδαβδ(t − t ′)






The Langevin equation of motion for the nth bead becomes

ζdRn

dt+ K

N∑m=1

AnmRm(t) = frn(t) + fexn (t)

The flow field in the system is represented as

fexn (t) = ζVn(t)

In the case of a shear flow

Vn(t) = (κ(t)Yn(t), 0, 0)

where κ(t) is the shear rate






The Langevin equation in the Cartesian coordinates

ζ(dXn(t)

dt− κ(t)Yn(t)) + K

N∑m=1

AnmXm(t) = f rxn(t)

ζdYn(t)

dt+ K

N∑m=1

AnmYm(t) = f ryn(t)

ζdZn(t)

dt+ K

N∑m=1

AnmZm(t) = f rzn(t)

The set of these equations represents Brownian motion of coupledoscillators





The Langevin equation in normal coordinates

The Langevin equation expressed in normal coordinates

ζ(dXp(t)

dt− κ(t)Yp(t)) + KλpXp(t) = f r

xp(t)

ζdYp(t)

dt+ KλpYp(t) = f r

yp(t)

ζdZp(t)

dt+ KλpZp(t) = f r

zp(t)

λp are eigenvalues of the connectivity matrix A (always real)

The random force fr

p(t) has moments

< fr

p(t) >= 0

< f rpα(t)f r

qβ(t ′) >= 2ζkBT δpqδαβδ(t − t ′)





Nondimensionalization

We introduce dimensionless variables:

x = xb

t = t Kζ

fr

m(t) = frm(t) 1Kb

κ(t) = κ(t) ζK

σαβ = σαβbK





Nondimensionalization

The dimensionless Langevin equation in normal coordinates

dXp (t)

dt− κ(t)Yp (t) + λpXp (t) = f r

xp (t) (1)

dYp (t)

dt+ λpYp (t) = f r

yp (t) (2)

dZp (t)

dt+ λpZp (t) = f r

zp (t) (3)

The variance of the random force

< f rpα(t)f r

qβ (t ′) >=2

3δpqδαβδ(t − t ′)





Contribution to the macroscopic stress from one polymer t.

The macroscopic stress is

σαβ = − 1

V

∑n

< fnαRnβ > −Pδαβ

fnα = −KN∑

m=1

AnmRmα

Using normalized coordinates the stress becomes

σαβ =K

V

∑k

λk < RkαRkβ > −Pδαβ

In the case of shear flow Skxy (t) =< Xk (t)Yk (t) >

σxy =b3

V

∑k

λkSkxy (t)





Contribution to the macroscopic stress from one polymertype

By averaging the linear combination of the Langevin equations, weget

dSpxy (t)

dt= −λpSpxy (t) +

1

2κ(t) < Y 2

p (t) > (4)

We know that

Yp (t) =

∫ t

−∞e− t−t′

τp f ryp (t ′)dt ′

Thus

< Y 2p (t) >=

1

3τp






The resulting equation is

dSpxy (t)

dt= − 2

τpSpxy (t) +

1

3κ(t)τp

The solution has the following form

Spxy =1

3

∫ t

−∞τpe− 2(t−t′)

τp κ(t ′)dt ′






Equation for the stress:

σxy (t) =

∫ t

−∞G (t − t ′)κ(t ′)dt ′

Here G (t) is given by the following :

G (t) =b3

3V

∑p

e− 2t

τp





Macroscopic stress of the system

After Fourier transforming we obtain the dynamic moduli

G ′(ω) =b3

3V

N∑p=2

(ωτp)2

1 + (ωτp)2

G ′′(ω) =b3

3V

N∑p=2

ωτp1 + (ωτp)2

Note τ1 =∞






The shear stress of the entire system is the sum of the shear stresscontributions of the separate molecules

σsysxy =

b3

3V

Ns∑α=1

Nα

nα∑k=1

λαk < Xαk Y α

k >

The total number of Kuhn segments in the whole system is

Ntot =

NS∑α=1

Nα

nα∑m=1

nα∑n>m

Nαmn






The relaxation modulus of the system

Gsys (t) =1

3Ntot

Ns∑α=1

Nα∑p

e− 2t

ταp

The storage modulus and loss modulus reads:

G ′sys(ω) =1

3Ntot

Ns∑α=1

Nα

N−1∑p=1

(ωταp )2

1 + (ωταp )2

G ′′sys(ω) =1

3Ntot

Ns∑α=1

Nα

N−1∑p=1

ωταp1 + (ωταp )2




H-polymer




Decomposition of G”




Star-polymer








Arbitrary branched polymer








Future research

Introduce finite extensibility of the segments by consideringFENE-P segments

Introduce entanglements in the system by consideringanisotropic friction and Brownian forces in the spirit of theBird-Deaguiar encapsulated dumbbell model

Develop an (approximate) constitutive model for the arbitrarybranched polymer ensemble

Finalize and test the implementation of the model

Study the influence of molecular topological complexity on thelinear rheology of arbitrary branched polymer ensembles (suchas compare to BoB model)




Acknowledgment

I would like to thank:

Prof.dr. J.J.M. Slot

Oleg Matveichuk

DPI


Documents

Model of linear viscoelasticity Future research · 03/11/2010 · Introduction Model of linear viscoelasticity Numerical experiments Future research An extension of the Rouse theory