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A 2D Stokesian approach to modeling flow induced deformation of particle laden interfaces Nader Laal-Dehghani, Rajesh Khare, and Gordon F. Christopher Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02448 • Publication Date (Web): 06 Sep 2017 Downloaded from http://pubs.acs.org on September 12, 2017

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A 2D Stokesian approach to modeling flow induced deformation of particle laden interfaces Nader Laal Dehghani, Rajesh Khare and Gordon F. Christopher

Abstract: A Stokesian dynamics simulation of effect of surface Couette flow on the microstructure of particles irreversibly adsorbed to an interface is presented. Rather than modeling both bulk phases, the interface, and particles in a full 3D simulation, known interfacial interactions between adsorbed particles are used to create a 2D model from a top down perspective. This novel methodology is easy to implement and computationally inexpensive, which makes it favorable to simulate behavior of particles under applied flow at fluid-fluid interfaces. The methodology is used to examine microstructure deformation of monodisperse, rigid spherical colloids with repulsive interactions when a surface Couette flow is imposed. Simulation results compare favorably to experimental results taken from literature, showing that inter-particle forces must be one order of magnitude greater than viscous drag for microstructure to transition from aligned particle strings to rotation of local hexagonal domains. Additionally, it is demonstrated that hydrodynamic interactions between particles play a significant role in the magnitude of these microstructure deformations. Keywords: Stokesian dynamics, hydrodynamic interactions, particle-laden interface, microstructure deformation, Interfacial particle simulation, shear flow

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Introduction Background Colloidal stabilized Pickering emulsions have a wide range of applications in cosmetic, food, and pharmaceutical industries. Irreversibly adsorbed particles on drop interfaces provide mechanical protection and surface viscoelasticity, which reduces drop coalescence and deformation. These changes in turn affect bulk emulsion stability and viscoelasticity. 1, 2, 3 Because of this growing industrial use, there has been an increased study of the behavior of interfacial particles to achieve a better understanding of Pickering emulsions’ properties. Interfacial viscoelasticity of a particleladen interface and the mechanisms that dictate it are of importance because interfacial viscoelasticity affects a wide range of emulsion properties at various length scales.4, 5, 6 Understanding what mechanisms dictate interfacial viscoelasticity of a particular system would provide a useful tool in designing Pickering emulsions in the future. It is currently understood that the rheology of these interfaces is determined by the resistance of particles to flow, which is governed by the interfacial microstructure and strength of attraction between particles.7, 8, 9, 10 Therefore, a better understanding of how interfacial microstructures form and respond to would improve our ability to engineer Pickering emulsions. The relative magnitude of attractive to repulsive forces between particles at the interface is the primary determinant of interfacial microstructures at a particle-laden interface. These interfacial interactions are more complex than in bulk due to the interface modifying hydrodynamic interactions and introducing new interparticle forces.6 In particular, the dominant dipole-dipole interfacial repulsion is due to charge dissociation around the particle surface in the aqueous phase with a secondary contribution from residual charge in the non-polar phase. The nonuniform distribution of charges causes formation of dipoles through particles perpendicular to the interface and particles repel one another.11, 12 The dominant attractive capillary forces are caused by overlapping interfacial deformation stemming from undulation of the contact line around the perimeter of the particles trapped at the interface. The meniscus around the particles can be expanded to a superposition of capillary multipoles. For colloids, the leading term in interfacial deformation is a quadrupole, which occurs due to surface roughness and chemical heterogeneity. The relative orientation of particle quadrupoles governs Capillary interactions between the 2 ACS Paragon Plus Environment

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particles. 13, 14 The most intensive method to simulate colloidal particles at an interface with an underlying flow, is to create a full 3D simulation that attempts to model bulk phases, particles, and the interface. There are a number of such methodologies in the literature that take this approach inclduing molecular dynamics,15 lattice boltzman,16 diffuse-interface field approach,17, 18

or phase field dynamics.19, 20 The difficulty in general with these methods is the overall

complexity of particle-laden interfaces, which is computationally expensive to model without using limiting assumptions, small modeling volumes, and/or non-trivial coupling of various methods.21, 22 There has been increased focus on creating 2D simulations with a top down view to create less compuitationaly expensive simulations. For instance, the melting of a 2D crystal via increasing temperature using molecular dynamics with a LJ potential has shown the existence of a hexatic phase between crystal order and fluid structure.23 Brownian dynamics, combined with a simple hydrophobic attractive interaction, has been used to understand aggregation.24 The dynamics and assembly of particles at liquid interfaces interacting through a range of repulsive power law potentials has been investigated via molecular dynamic simulations.25, 26 However, these attempts use interfacial interactions that do not conform to known physical models of interfacial particles. This is sometimes due to their focus on colloidal crystal defect dynamics and/or understanding the behavior of 3D systems using a reduced dimensionality. Therefore, these models fail to capture the more complex physics seen in particle laden interfaces. In addition to direct particle interactions, solvent-mediated hydrodynamic interactions need to be accounted while simulating interfacial systems; however, currently the effect of hydrodynamic interactions has not been studied extensively.27, 28, 29 There are few studies focusing on pairwise particle hydrodynamic interactions at fluid-fluid interfaces theoretically. Among those few, Vassileva et al. 30 use an approximate expression for hydrodynamic resistance for glass particles at oil-water interface with viscosity ratio of one. In Vassileva’ s study, the single particle drag coefficient is multiplied by the drag coefficient for two particles in a continuous phase and the results are in good agreement with experimental observations. Boneva et al. 31 use the same approach for submillimeter-sized glass particles at oil-water interface of viscosity ratio one and compare their experimental results with theoretical prediction. 31 They find reasonable agreement, although the experimental drag is larger than theoretical prediction and they note that 3 ACS Paragon Plus Environment

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the larger drag could be due to the effect of the meniscus around the particles. Additionally, Bleibel et al. 32, 33 use bulk-like hydrodynamic interactions to study colloids trapped at an interface between two fluids where inter-particle interactions are long-range and their results are in good agreement with light scattering observation data for particles at a fluid interface and also Stokesian dynamics simulations. In the present work, a 2D Stokesian simulation methodology is used to model shear deformation of colloidal microstructures trapped at a fluid-fluid interface using physically accurate models of both interfacial inter-particle interactions and hydrodynamic forces. This methodology is used to investigate the effect of inter-particle interactions versus shear forces on microstructural deformations at a fluid-fluid interface with an underlying surface Couette flow. Imposing a range of shear rates and surface concentrations, the nature of the microstructure regimes and the effects of hydrodynamic interactions on these regimes are characterized. Simulation Approach To accurately simulate particle-laden interfaces as proposed, the basic physics of particles at interfaces will dictate a number of decisions in creating the model. First, particles become trapped at an interface due to the dominance of interfacial energy over Brownian thermal energy.1 Therefore, particle movements perpendicular to the interface are suppressed. Furthermore, interfacial particles’ Reynolds number are on the order of 10-4 to 10-6, allowing particle inertia to be neglected.. Therefore, the basic simulation will model surface flows as Stokes flows and thus Reynolds number is identically set to be zero for all the simulations. Because of these assumptions, we can use the following simple equations to find a particle's, i, position vector, x, and orientation, ϕ, ∗ ∆ + ∆ + ∆ = +

(1)

+ ∆ = + ∗ ∆ + ∆

(2)

< . > = 2 ∆ < . >

(3)


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where V is velocity, ω is angular velocity, ∆t is time step, ∆ is Brownian displacement and

∆ is Brownian rotational motion that are random values with Gaussian distribution functions

whose average values are zero and whose variance-covariance are 2 ∆ 34 where Kb is Boltzmann constant, T is temperature, and µ is viscosity.

To find V and ω, Stokes equations for particles in suspensions can be used due to the inertialess particles. Stokes equations using generalized mobility matrix for particles in bulk are as below 35:

=

! " !

+ ∑% $

&& %

'& %

&' %

'' ( .

%

∑ )* " ∑ +*

+$

,& ( : ./ ,'

(4)

where ∑ 0% and ∑ % are the net force and torque acting on particle i due to particle j, µij is the

generalized mobility matrix, Cti and Cri are shear disturbance tensors, ./ is the strain rate tensor and / and / denote velocities of the underlying flow at the position of particle i. Simulation Methodology Parameters In order to finish the Stokesian dynamics simulation, we need to establish the hydrodynamic interactions, inter-particle forces and torques, parameter space, and data quantification methodologies. These are outlined below.

Hydrodynamic Interactions In this study, the viscosity ratio of the two fluids are assumed to be one and it is assumed that particles’ equators are located at the interface. However, in the future, we will be able to adjust hydrodynamic interactions to allow for not equi-viscosity and a range of contact angles using the methodology outline, but leave that to future studies. In literature, an approximate expression for pairwise hydrodynamic resistance of two particles approaching at a fluid-fluid interface has been modeled as the product of the interfacial drag of an isolated sphere multiplied by the resistance of approaching spheres in an infinite medium, 36 1

>

01'23 = 6567 8 ' , : ; S*

>S*

J' '* K >M*

L

(7)

L

(8)

L

(9)

The other dominant force acting on the particles is dipole-dipole repulsion. As charged particles attach to the interface between a polar and a non-polar fluid and the inter-particle distance is much greater than the Debye screening length, a dipolar electric field is created by each particle, which is perpendicular to the interface. This dipolar electric field, which is due to non-uniform distribution of charges around the particle surface, is mediated through the non-polar phase and causes particles to repel one another. The repulsive interaction between the particles is modeled with the following potential and resultant force, 11, 12

@W = 0W =

BX BX * YZ[ >\*

]BX BX * YZ[ >M*

^1 = 45_`Ja, bBc K7 ] sin] a

(10)

(11)

(12)

Where ^1 is effective dipole moment, d% is particle distance, bc is non-polar fluid dielectric

constant, _ is the electric charge density, `Ja, bBc K is a known dimensionless function, r is

particle radius, a is the contact angle and bBc = bB /bc is the ratio of the dielectric constants of the particle and the non-polar fluid. 7 ACS Paragon Plus Environment

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Van der Waals interactions can play a role for study of particle-particle interactions. However, these short range attractive forces are negligibly small when particles are charged and far apart due to strong repulsions.38 Therefore, they are ignored in the current simulation. Finally, a soft sphere potential with a L-8 dependence (@d~

=

>g

+ h for L < 2r is used to ensure

no direct particle overlap during particles’ movements. The constant c is used in the potential equation to assure continuity of potential for all inter-particle distances. Non-dimensionalization The nominal particle diameter, do, the nominal capillary force o1n\

jlmL

ijklmL 1n

, and capillary time scale

are used to non-dimensionalize all equations: 0pB2' = −

'̅L '̅*L >pS*

cos 2∆% + 0 ∗ .

0pBT' = −

'̅L '̅*L

= pA2B = − Y

>pS*

>pM*

sin 2∆%

'̅L '̅*L >pM*

'̅\ '̅*\ rst\ u rst\ u*

(13)

(14)

sin 2∆%

(15)

&& w &' w ppp& pppp pppp pppp pppp ∆wv = xyz ∗ J

ppp / v + ,v : ./ K + ∑|Jv{ . 0{ + v{ . { K} × ∆ ̅ + ∆

(16)

'& w '' w pppp ppp' pppp pppp ∆wv = xyz ∗ J ppppp pppp / v + ,v : ./ K + ∑|Jv{ . 0{ + v{ . { K} × ∆ ̅ + ∆

(17)

< ̅ . ̅ 0 > = 7 ̅ ∆̅ < p . p 0 >

(18)

'

Where the notations are in dimensionless forms: 7̅ = 1 is the dimensionless particle radius, n

) dp = 1 is the dimensionless inter-particle distance, 0p = ijklmL / 1 is the dimensionless force, >

n

n

& p = ijklmL is the dimensionless torque, ̅ = 1 is the dimensionless displacement, ̅ = is the +

dimensionless time where =

o1n\ jlmL

n

, p/ = 1 !z is the dimensionless underlying flow velocity

n


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where yz is the applied shear rate, w/ = z is the dimensionless angular velocity of the underlying W flow, ,̅ & = 1 and ,̅ ' = , ' are the dimensionless shear disturbance tensors, .p/ = !z is the

n

dimensionless strain rate tensor and ̅ && = =

=/ko1n

, ̅ '& = ̅ &' + =

P

'' = L , ̅

=/ko1n

PP

=/ko1n\

are

the dimensionless mobility tensors, and , which 8 = 1, is the corrector coefficient for mobility

and shear disturbance matrices Three dimensionless parameters are created from this scheme that describe the system physics. The first, 0∗ =

BT 'TBc 2B2' 'AT

=

kL L 1n\ YlmL Z[

(19)

is the ratio between dipole repulsion and capillary attractions between interfacial particles. As F* increases, repulsion between particles becomes the dominant interaction. The next, z

\

mT2' 'AT o1 yz ∗ = 2B2' 'AT = jlmnL

(20)

is the ratio between shear forces and inter-particle forces acting on particles. For large yz ∗ , the shear forces are dominant over inter-particle interactions. Finally, 7 =

'c2c 'AT 2B2' 'AT

=

+

ijklmL

(21)

is the ratio between Brownian force versus capillary force acting on particles, and is used to consider the Brownian motion of particles at the interface. Obviously, these three parameters in experimental conditions cannot be selectively set, but in simulations, the values are set independently. Parameter Selection The basic simulation domain is a square (50 x 50 particle diameters), in which periodic boundary conditions are placed in all directions, using sliding brick scheme to account for the surface Couette flow.39 A constant number of monodisperse particles with 7̅ =1/2 and θ =90° are used in each domain with surface concentrations ranging from 5% to 35%. It is known that colloidal


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particles at an undisturbed interface form a hexagonal lattice configuration in equilibrium40 and in order to shorten simulation time, the initial microstructure is made a perfect hexagonal lattice. In fact, the code was validated by observing the motion of randomly placed, repulsively dominated particles in the absence of underlying base flow forming a nearly perfect equilibrium hexagonal microstructure, indicating the simulation captures the basic physics of particle laden interfaces for repulsive particles. An interfacial Couette flow is applied to the interface; interactions are made repulsive by setting F* = 1000 in all simulations implying that repulsion between particles are 1000 times stronger than attraction between them. yz ∗ is varied between 0.01 and 10 to simulate different applied

shear rates. To reduce computational costs, a cut-off length, dpA& over time for Hydrodynamic (solid line). and non-Hydrodynamic (dashed line) interaction simulations. As more shear cycles happen, more defects appear in the monolayer and non-Hydrodynamic interaction simulations appear to become more irreversible compared to Hydrodynamic

interaction simulations. (a) Surface concentration of 15% and yz ∗ =1. (b) Surface concentration of 23 ACS Paragon Plus Environment

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25% and yz ∗ =1. (c) Surface concentration of 35% and yz ∗ =1. (d) Surface concentration of 25% and yz ∗ =0.05.

Figure 13: Phase diagram (yz ∗ vs. Surface concentration) of results for Hydrodynamic Interaction simulations showing the transition from rotation to slip flows. (a) Phase space is determined by visual inspection and four distinct regimes are shown in this figure: slip flows (triangle), rotation (diamonds), stall (square), and transition between these regimes (circle). (b) the mean of vorticity field magnitude is plotted for each simulation.


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Y

yz Figure 1


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Figure 2


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Figure 3


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(a)

(b)

(c)

(d)

Figure 4


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(b)

(c)

(d)

Figure 5


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Figure 6


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(b)

(c)

(d)

Figure 7


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Figure 8


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(b)

(c)

(d)

Figure 9


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(b)

(a)

Figure 10


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(b)

(c)

(d)

Figure 11


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(b)

(c)

(d)

Figure 12


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(b)

Figure 13


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References 1. Binks, B. P. Particles as surfactants—similarities and differences. Current Opinion in Colloid & Interface Science 2002, 7 (1-2), 21-41. 2. Binks, B. P.; Horozov, T. S. Colloidal particles at liquid interfaces; Cambridge University Press2006. 3. Pickering, S. U. Emulsions. Journal of the Chemical Society, Transactions 1907, 91, 2001-2021. 4. Huang, X. F.; Na, Y.; Xiong, Y. J.; Wang, X. H.; Peng, K. M. Structure Characterization of Particle Film and Its Role in Stabilizing Emulsion. Progress in Chemistry 2016, 28 (12), 1743-1752. 5. Maestro, A.; Santini, E.; Zabiegaj, D.; Llamas, S.; Ravera, F.; Liggieri, L.; Ortega, F.; Rubio, R. G.; Guzman, E. Particle and Particle-Surfactant Mixtures at Fluid Interfaces: Assembly, Morphology, and Rheological Description. Advances in Condensed Matter Physics 2015. 6. Mendoza, A. J.; Guzmán, E.; Martínez-Pedrero, F.; Ritacco, H.; Rubio, R. G.; Ortega, F.; Starov, V. M.; Miller, R. Particle laden fluid interfaces: dynamics and interfacial rheology. Advances in Colloid and Interface Science 2014, 206, 303-319. 7. Keim, N. C.; Arratia, P. E. Yielding and microstructure in a 2D jammed material under shear deformation. Soft Matter 2013, 9 (27), 6222-6225. 8. Keim, N. C.; Arratia, P. E. Mechanical and Microscopic Properties of the Reversible Plastic Regime in a 2D Jammed Material. Physical Review Letters 2014, 112 (2), 028302. 9. Barman, S.; Christopher, G. F. Simultaneous Interfacial Rheology and Microstructure Measurement of Densely Aggregated Particle Laden Interfaces Using a Modified Double Wall Ring Interfacial Rheometer. Langmuir 2014, 30 (32), 9752-9760. 10. Barman, S.; Christopher, G. F. Role of capillarity and microstructure on interfacial viscoelasticity of particle laden interfaces. Journal of Rheology (1978-present) 2016, 60 (1), 35-45. 11. Kralchevsky, P. A.; Danov, K. D.; Petkov, P. V. Soft electrostatic repulsion in particle monolayers at liquid interfaces: surface pressure and effect of aggregation. Phil. Trans. R. Soc. A 2016, 374 (2072), 20150130. 12. Park, B. J.; Vermant, J.; Furst, E. M. Heterogeneity of the electrostatic repulsion between colloids at the oil–water interface. Soft Matter 2010, 6 (21), 5327-5333. 13. Danov, K. D.; Kralchevsky, P. A. Capillary forces between particles at a liquid interface: general theoretical approach and interactions between capillary multipoles. Adv Colloid Interface Sci 2010, 154 (1-2), 91-103. 14. Oettel, M.; Dominguez, A.; Dietrich, S. Effective capillary interaction of spherical particles at fluid interfaces. Physical Review E 2005, 71 (5), 051401. 15. Günther, F.; Janoschek, F.; Frijters, S.; Harting, J. Lattice Boltzmann simulations of anisotropic particles at liquid interfaces. Computers & Fluids 2013, 80, 184-189. 16. Frijters, S.; Günther, F.; Harting, J. Effects of nanoparticles and surfactant on droplets in shear flow. Soft Matter 2012, 8 (24), 6542-6556. 17. Millett, P. C.; Wang, Y. U. Diffuse-interface field approach to modeling arbitrarily-shaped particles at fluid–fluid interfaces. Journal of colloid and interface science 2011, 353 (1), 46-51. 18. Millett, P. C.; Wang, Y. U. Diffuse interface field approach to modeling and simulation of selfassembly of charged colloidal particles of various shapes and sizes. Acta Materialia 2009, 57 (10), 31013109. 19. Aland, S.; Voigt, A. Simulation of common features and differences of surfactant-based and solid-stabilized emulsions. Colloids and Surfaces A: Physicochemical and Engineering Aspects 2012, 413, 298-302.


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