Mesoscopic simulations of the rheology of entangled wormlike micelles

Edo Boek(1)

Johan Padding(1,2,3)

Wim Briels(3)(1) Schlumberger Cambridge Research, UK(2) University of Cambridge, UK(3) University of Twente, NLacknowledgments: V.Anderson, J.Crawshaw, M.Stukan, J.R.A.Pearson (SCR)

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oil-responsive surfactant fluids

+ +

+

++ +

+ + +

+

wormlike micellesvisco-elastic network of

wormlike micelles

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

Shear Rate (s-1)

Vis

cosi

ty (P

a.s)

40 oC (104 oF)

70 oC (158 oF)

90 oC (194 oF)

130 oC (266 oF)

150 oC (302 oF)

+oil

spherical micelles or micro-emulsions

+salt

hydraulic fracturing

other applications: food products, personal care (shampoo, …)

CH3–(CH2)7

C C

HH

(CH2)11–CH2–N–CH3

CH2–CH2–OH

CH2–CH2–OH

+

—Cl

EHACerucyl bis-(hydroxyethyl)methylammonium chloride

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available REoS are inadequate

0 10 20 30 40 50

100

102

104

Inst

anta

neou

s sh

ear

stre

ss /

Pa

100 120 140 160 180

0 10 20 30 40 50

100

102

104

Inst

anta

neou

s sh

ear

stre

ss /

Pa

100 120 140 160 180

0 10 20 30 40 50

100

102

104

Inst

anta

neou

s sh

ear

stre

ss /

Pa

100 120 140 160 180

= 1 s

= 10 s

= 100 s

Step up down shear rateStep up in shear rate

Time / s

Time / s

Time / s

J0

0

1 2τ τ D λ DG

d 1 k τ : Ddt λ

Bautista-Manero:

Inst

anta

neou

s she

ar st

ress

/ Pa

Time / s

= 1

= 100

= 10

• Problems:

1. poor fit to transient data (Anderson et al. 2006)

2. extensional viscosity (Boek, Pearson et

al., JNNFM 126, 39-46 (2005)

3. normal stresses

0 0 J 1

0

parameters G , , , λ,λ , ...( ), , kλ determined from steady state expt

solvent

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predictive multi-scale simulation model:chemistry to rheology

• Level 1:Microscopic Molecular Dynamics (MD) yields mesoscopic properties

• Level 2:Mesoscopic (Brownian Dynamics) simulation model yields rheological properties

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mesoscopic simulation model (1/4)• each unit (red sphere)

represents the midpoint of one persistence length lp– conservation of mass

• the endpoints (blue spheres) of the WLM are found by extrapolating from the first / last bonds– orientation of “monomer”

must be traced explicitly

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mesoscopic simulation model (2/4)

• Bonded interaction:

• Mesoscopic property input:– Persistence length lp– Elastic modulus K– Scission energy Esc

– Activation barrier Ea

212b p sc

p

K r l El

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mesoscopic simulation model (3/4)• Brownian Dynamics

(overdamped) of rigid rods of dimension lp x d in a solvent of viscosity s

• Additional mesoscopic input:– Solvent viscosity s

1

1

1

2

ln / ˆ ˆ ˆ4

S

B

p

s p

t t t t t t

k T t t

l dt t t

l

r r F r

r r

I u u

Total systematic force on unitAnisotropic random displacement and friction which depend on rod orientation

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mesoscopic simulation model (4/4)• Charge interactions are ignored

– Uncharged or charged system with small screening length.• Excluded volume interactions are ignored

– WLMs as long thin threads. No spontaneous nematic phase.• Uncrossability of threadlike wormlike micelles is treated by

TWENTANGLEMENT

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mechanical properties from MD simulation of worm-like micelle

• lp = 30 nm• d = 4.8 nm• K = 2 nJ/m• J.T. Padding, E.S. Boek and W.J. Briels, J. Phys.: Condens. Matter 17, S3347–S3353 (2005).

• solvent is water: s = 10-3 Pa s

• experimentally Esc = 20-50 and Ea = 10-25 kBT– scission-recombination extremely rare!– preliminary results with Esc = 17 kBT

• 12 kBT + 2.5 kBT ln (lp / d)– and lower Ea (1.5 kBT)

pl

d

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llpp = 30 nm = 30 nm

Ly = 340 nm

example: 8% EHAC + 3% KCl

• Typical simulation:– Total 4.000 – 32.000

persistence length units

– Box size 300-600 nm– Average worm contour

length O (m)– Computational speed:

0.1 – 1 ms/week on one 2.8 GHz Pentium 4 processor

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linear rheology shear relaxation modulusshear relaxation modulus

(measured from equilibrium(measured from equilibriumstress fluctuations)stress fluctuations)

, , ,

0

1

xy xyB

i j iji j

VG t S t Sk T

S r r FV

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non-linear rheology• impose constant shear rate between upper and

lower face of the periodic box• do not assume affine solvent flow field

– instead, let solvent reactto flow velocity of wormlike micellarmaterial

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transient stress• usually large

1st normal stress difference

• overshoots in all transient stresses

• 2nd normal stress difference has a positive overshoot before becoming negative

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shear thinning• average length of WLM

decreases with shear rate

• average breaking time decreases with shear rate: opposite effect from

• viscosity decreases rapidly with shear rate

1

1break L

c L

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simulation and experiment – shear viscosity

8% EHAC

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references• J.T. Padding and E.S. Boek, ``Evidence for diffusion controlled recombination kinetics in model wormlike micelles’‘,

Europhysics Letters 66, 756-762 (2004).

• J.T. Padding and E.S. Boek, ``The influence of shear flow on the formation of rings in wormlike micelles: a nonequilibrium molecular dynamics study'‘, Phys. Rev. E 70, 031502 (2004).

• E.S. Boek, J.T. Padding, V. Anderson, P. Tardy, J. Crawshaw and J.R.A. Pearson, ``Constitutive Equations for Extensional flow of wormlike micelles: Stability analysis of the Bautista-Manero model'', J. Non-Newtonian Fluid Mech. 126, 39-46 (2005).

• J.T. Padding, E.S. Boek and W.J. Briels, ``Rheology of wormlike micellar fluids from Brownian and Molecular Dynamics simulations'', J. Phys.: Condens. Matter 17, S3347–S3353 (2005).

• V. Anderson, J.R.A. Pearson and E.S. Boek, ``The rheology of worm-like micellar fluids'', in Rheology Reviews 2006, D.M. Binding and K. Walters (Eds.), British Society of Rheology, 217-255 (2006).

• E.S. Boek, V. Anderson, J.T. Padding, W.J. Briels and J. Crawshaw, submitted for publication (2006)