Wormlike Micelle Rheology Xueming Tang#, Weizhong Zou#, Taraknath Mandal #, Fang Yuan #, Mike Weaver+, Peter Koenig‡, and Ronald G. Larson# # Department of Chemical Engineering, University of Michigan, Ann Arbor, MI 48109 + Analytic Sciences, The Procter & Gamble Company, Mason, OH 45040 ‡ Computational Chemistry, Modeling and Simulation, The Procter & Gamble Company, West Chester, OH 45069

Sponsors: Procter and Gamble, NSF

Xueming Tang Weizhong Zou Taraknath Mandal Fang Yuan

Peter Koenig Mike Weaver

The Challenge of Multiple Length and Time Scales in Surfactant Solutions

Entangled Network

http://www.ifnh.ethz.ch/vt/research/projects/vivianel 1023 degrees of freedom!

sodium dodecyl sulfate

1 nm

10 nm

100 nm 1 mm

10 cm

A B

C D

E

Increase surfactant/salt concentration

Micellar solutions: the salt curve citronellol

linalol

limonene

vanillin

10% SLES No additives

Cumene Linalool

DPG

larger aggregates form“bicelles” larger aggregates form cylinders

Snapshots of micelles with 100 SDS molecules

GROMOS45A3 OPLSAA CHARMM36 OPLSUA GROMOS53A6

Liquid crystal phase formed in small

region, 4 O share 1 Na+ Intermediate Head groups are spread out

:S

:O

:CH2, CH3

:Na+, Cl-

Sodium-oxygen condensed patch

CTAC in NaSal: Stable Threadlike Micelle

Equal molar

Sal-/CTA+

Packing density: nL≈21~27/nm

Experiments: 21~28/nm

Head group: 2D hexagonal lattice

Hydrophobic tail: radially oriented

3.28M NaCl

Ordering of Atoms

Umbrella Sampling

density

PMF

z = r, radial coordinate

mi

d 2vi

dt2= f

i

molecular dynamics (MD) simulations:

Video from student Kyle Huston

Weighted Histogram Analysis Method (WHAM)

1st window 20th window

ΔG0 SDS C12E5

-RTln(CMC) (experiment)

8.8 kT 13.6 kT

Atomistic Simulation* 9.7 kT 13.1 kT

0

2

4

6

8

10

12

14

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

PM

F (

kT

)

R (nm)

C12E5

SDS

Potential of Mean Force

1. Kumar; et al. J. Comput. Chem. 1992, 13, 1011

2. Yuan; et al. Larson Langmuir 2015, 31, 1336

*SDS Micelle Aggregation Number: n = 60

*C12E5 Micelle Aggregation Number: n = 54

n= 60 SDS surfactants

Fang Yuan

EO head group

𝑃𝑀𝐹 = ∆𝜇𝑛0= 𝜇𝑛

0 − (𝜇𝑛−10 + 𝜇1

0)

Micelle Size Distribution

09:53:19 Time

(inferred from potentials of mean force)

C16E22

Yuan and Larson, JPC B (2015)

∆𝜇𝑛0= 𝜇𝑛

0 − (𝜇𝑛−10 + 𝜇1

0) 𝑋𝑛 = 𝑋𝑛−1𝑋1𝑒𝑥𝑝 −∆𝜇𝑛

0

𝑘𝐵𝑇

Micelle Size Distribution

09:53:19 Time

(inferred from potentials of mean force)

C16E22

Yuan and Larson, JPC B (2015)

∆𝜇𝑛0= 𝜇𝑛

0 − (𝜇𝑛−10 + 𝜇1

0) 𝑋𝑛 = 𝑋𝑛−1𝑋1𝑒𝑥𝑝 −∆𝜇𝑛

0

𝑘𝐵𝑇

09:53:19

Scission free energy From coarse-grained Martini MD –WHAM simulations

CTAC/NaCl; CTAC/NaSal

viscosity of CTAC/NaSal, Vinson et al. J. Phys. Chem. 1992, 96:474

R R = molar ratio of salt/surfactant

viscosity of CTAC/KBr, Khatory et al. Langmuir 1993 9:1456

∆𝐺𝑠𝑐𝑖𝑠𝑠(𝑅 = 0.6) = 27 𝐾𝐵𝑇

∆𝐺𝑒𝑛𝑑(𝑅 = 0.6) = 13.5 𝐾𝐵𝑇

Length scales in threadlike micelles

Beth Schubert, PhD thesis, U. Delaware, 2003

Persistence Length

Average Micelle Length

Mesh Size

d = 2rcs

Time scales: micelle breakage time, reptation time

Entanglement Length

Reptation: Edwards, De Gennes, trep ~L3

Micelle breakage & Re-joining: M.E. Cates (1989): tbreak ~1/L

new tube size micelle break

Micellar relaxation dynamics

Slide from Peter Koenig, P&G

Breakage Rejoining

Reptation

Contour length fluctuations

CR (Double reptation)

Pointer algorithm: simulation of ensemble of micelles

G(t) = GN m2(t) +Rouse + bending modes

Zou, W. and Larson, R.G. J Rheology 58 (2014) 681-721.

Including Bending Modes

bending modes

From Oelschlaeger et al., Langmuir, 2009  

G''(w) µ (iwtp)3 4

Tk

li

l

TkB

b

p

pp

p

B

3

4343

,)(2

)(

tt

s

LBIMG

LBREG

15

)()(,

15

)()(

2

2

sD

t

BD

 

lp = bK /2

Morse & Macintosh

diffusing wave spectroscopy

Micelle Parameters from Rheology

09:53:19 Time

V º tbr / trep

From neutron scattering or molecular simulations

obtain from Diffusing Wave Spectroscopy

Key parameters from mechanical rheometry

Result and analysis o Results

Average fitting deviation < 5%

Parameters Mechanical

data DWS data

Combined

data

115 105 115

2.49 1.16 1.82

1.45 1.60 1.59

1.35 1.41 1.36

4 4 4

155 161 153

116 114 112

fit mechanical only

Fit DWS only

Fit both mechanical and DWS

Effect of Persistence Length lp

09:53:19 Time

Effect of Breakage Rate,

09:53:19 Time

V º tbr / trep

Micelle Parameters obtained from Pointer Algorithm

09:53:19

branching

SLE1S

Scission Free Energy from Rheology

ave. aggregation no. <n> ave. micelle length <L>

SLE1S/CAPB

< 𝐿 > ∝ 𝑒𝑥𝑝 Δ𝐺𝑠𝑐𝑖𝑠𝑠/2𝑘𝐵𝑇

Δ𝐺𝑠𝑐𝑖𝑠𝑠

M = mass of surfactant; rs = density of surfactant; d = micelle diamter, NA = Avogadro’s number

Δ𝐺𝑠𝑐𝑖𝑠𝑠= 24 - 28 kBT

Constrained diffusion model

1

2

3 4

Diffusion

Current Work: Adding Branches to Simulation Method

xi = displacement

Constrained diffusion model

1

2

3 4

Diffusion

Current Work: Adding Branches to Simulation Method

xi = displacement

Constraint

Constrained diffusion model

1

2

3 4

Diffusion Constraint

23

Kirchhoff Circuit model

Current Work: Adding Branches to Simulation Method

xi = displacement

1 0 0 0 0 0 −1

0 1 0 0 0 0 −1

0 0 1 0 0 0 −1

𝑎41 0 0 1 0 0 0

0 𝑎42 0 0 1 0 0

0 0 𝑎43 0 0 1 0

−𝑎41 −𝑎42 −𝑎43 0 0 0 0

∆𝑥41 ∆𝑥42 ∆𝑥43 𝐹1 𝐹2 𝐹3 𝐹4

=

𝑏41 𝑏42 𝑏43 0 0 0 0

𝑎𝑖𝑗 =∆𝑡

𝜁𝑖𝑗, 𝑏𝑖𝑗 =

2𝑘𝐵𝑇∆𝑡

𝜁𝑖𝑗𝑛𝑖𝑗

8 Modelling of branched micelles: micelle architecture

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Breakage

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Recombination

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Fuse the two lists

7

Modeling of Branched Micelles: Budding Process

Add buds

Rejoining

Sprouting Remove un-sprouted buds

Short Sprouts

Constrained diffusion

reptate

reptate

reptate

Additional relaxation

Budding process

ς𝑏𝑑 =𝜏 𝑏𝑢𝑑𝜏 𝑏𝑟

Nucleation process Buds = nucleation seed

0%

20%

40%

60%

80%

100%

0.010.390.771.151.531.912.292.673.053.433.814.194.574.95

System composition

We

igh

t p

erc

en

tag

e

Time (s)

Modeling of Branched Micelles: Results

Total number of micelles: 2500; Average micelle size 𝑀 : 4 μm; Dimensionless breakage time 𝜍𝑏𝑟: 0.003; Dimensionless budding time 𝜍𝑏𝑑: 0.33; Micelle entanglement length 𝑙𝑒: 100 nm; Micelle persistence length 𝑙𝑝: 50 nm;

Micelle diameter 𝑑: 4 nm;

Linear micelle (35 wt%)

Lightly branched micelle (60 wt%)

Highly branched micelle (5 wt%)

Size (μm)

Nu

mb

er

of

bra

nch

ju

nct

ion

Nu

mb

er

de

nsi

ty

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

0 10 20 30 40 500

2

4

6

8

10

Micelle size

0.00%

0.30%

0.60%

0.90%

1.20%

0 3 6 9 12 15

Linear

Dangling

Length (μm)

Nu

mb

er

de

nsi

ty

Strand length distribution

Linear micelle

Dangling strand

Backbone

Average strand length 𝐿𝑠𝑡 = 2.2 μm

Number of branch junctions per micelle size 𝛽 = 0. 09 per μm

Modeling of Branched Micelles: Effect of Branching

𝑮"& 𝑮“ (𝐏𝐚)

𝝎 (𝐫𝐚𝐝/𝐬)

Decrease of viscosity by 3 fold

Storage and loss modulus

Contribution from low frequencies

Contribution from high frequencies

𝐺𝑁: plateau modulus 𝜍𝑏𝑟: dimensionless reaction time 𝛼𝑒: semi-flexibility 𝑀 : average micelle size 𝛽: branching level 𝑑: micelle diameter

Modified genetic algorithm

𝐺∗ 𝜔 = ℱ 𝐺𝑁𝜇𝛼 𝑡, 𝑀 , 𝛼𝑒, 𝜍𝑏𝑟 , 𝛽 + 𝐺𝐻 𝜔

0.001

0.01

0.1

1

10

100

1000

0.1 1 10 100 1000

𝑮"& 𝑮“ (𝐏𝐚)

𝝎 (𝐫𝐚𝐝/𝐬)

𝐺𝑁 = 132.8 Pa; 𝜍𝑏𝑟 = 0.003; 𝛼𝑒 = 2; 𝑙𝑝 = 50 nm

𝑀 = 4 μm; 𝛽 = 0.09 per μm

0.001

0.01

0.1

1

10

100

1000

0.1 10 1000

𝑮"& 𝑮“ (𝐏𝐚)

𝝎 (𝐫𝐚𝐝/𝐬)

𝑀 = 4 μm; Linear micelles

Lightly branched micelles Unbranched micelles

Coarse-Graining Complex Mixtures

Tang, Koenig, McConaughy, Weaver, Eike, Wang, Zou, & Larson, to be submitted (2015)

Obtained with assistance of COSMOtherm software (Klamt et al.), as described in

Liyana-Arachchi, Jamadagni, Eike, Koenig, and Siepmann, J Chem Phys 142 (2015)

Xueming Tang, U Mich

Peter Koenig P&G

0

5

10

15

20

-2 0 2 4 6 8

Zero

Sh

ear

Vis

cosi

ty (

Pas

)

log Pow

Impact of perfume raw materials on viscosity

Linalool

Cumene

Isopropyl myristate

Experimental viscosity and DPD result

Isopropyl myristate

Cumene Linalool

No

rmal

ized

ato

m c

ou

nt

No

rmal

ized

ato

m c

ou

nt

tails

heads

perfume

Summary

• Atomistic molecular simulations, combined with umbrella sampling (weighted histograms), can yield free energies for micellization of spherical micelles, for breakage of thread-like micelles, and potentially for branch formation

• Coarse-grained, micelle-level models, allow inference of micelle properties, such as length, breakage time, and persistence length, from rheological properties.

• The inclusion of branch formation into micelle models is now feasible.

• Surfactant additives, such as perfumes, influence micelle properties, and thereby the rheology. Molecular simulations are beginning to allow assessment of these relationships, opening the possibility of de novo design of surfactant system.

09:53:20 Time

End

09:53:20 Time