Supplementary material to Dynamic stratification in drying films of colloidal mixtures

Andrea Fortini,1, 2, ∗ Ignacio Martın-Fabiani,1 Jennifer Lesage De La Haye,3 Pierre-Yves Dugas,3 Muriel

Lansalot,3 Franck D’Agosto,3 Elodie Bourgeat-Lami,3 Joseph L. Keddie,1 and Richard P. Sear1, †

1Department of Physics, University of Surrey, Guildford GU2 7XH, United Kingdom2Theoretische Physik II, Physikalisches Institut, Universitat Bayreuth,

Universitatsstraße 30, D-95447 Bayreuth, Germany3Laboratoire de Chimie, Catalyse, Polymeres et Procedes Universite Claude Bernard Lyon 1, France

EVOLUTION OF DENSITY PROFILES

The evolution of the stratified structure during evap-oration and of the related density profiles is shown inFig. 1.

STRATIFICATION AT DIFFERENT SIZERATIOS AND INITIAL VOLUME FRACTIONS

Snapshots of the final configuration obtained in simu-lations of binary mixtures with size ratio dl/ds=2, and14 are shown in Figure 2. The formation of a top layerdepleted of large particles is clearly visible in both cases.

In Fig. 3 we show snapshots of the top regions of the fi-nal configurations for initial volume fractions η0=0.1 (a)η0=0.2 (b), and η0=0.4 (c), obtained with an evaporationvelocity vev = 0.1. The layer with only small particles isvisible in the final film layer regardless of the initial vol-ume fraction, but as the initial particle density increase,the segregation mechanism becomes less efficient.

DETAILS OF THE SIMULATION METHOD

We run computer simulations of mixtures of sphericalparticles in a simulation box with dimensions Lx×Ly×H.Periodic boundary conditions are used in the x- and y-direction, while in the z-direction the box is delimited atthe bottom by a hard substrate, and at the top by a softwall, which models an air-water interface. We do not ex-plicitly simulate the solvent water molecules but describethe motion of the colloidal particles using Langevin dy-namics, which includes Brownian diffusion effects and ne-glects hydrodynamic flow. The equation of motion for aparticle i with diameter di, mass mi at position ri is

miri = −∑i<j

∇Uij(ri, rj)− ξiri + δFi , (1)

where ξi = 3πνdi is the friction coefficient and ν isthe viscosity of the solvent. The term δFi is a randomforce sampled from a Gaussian distribution with width√kBTξi, where kB is the Boltzmann constant and T the

temperature. The Langevin dynamics was carried outusing the LAMMPS package [1] with a time step dt =

0.001 τsB = 0.0025 t0, where τsB = d2s/D

s0 is the Brow-

nian time of the small particles, and t0 is the standardLennard-Jones unit of time in LAMMPS. We also used afriction parameter ξi = 100 di and a temperature of 40,both in standard Lennard-Jones LAMMPS units.

The interaction energy models screened charged par-ticles, i.e., the interparticle potential energy uij betweenparticles i and j, either small or large, of diameters diand dj is a short range Yukawa interaction

uij(r)/kBT =

{ε

kBTe−κ(r−σ) r < rc

0 r ≥ rc(2)

where r is the centre-to-centre particle distance, ε is thecontact energy, and κ determines the steepness of thepotential. These two parameters influence the overallsoftness of the potential. In this simulation, we chooseε/kBT = 25 and κds=20. The distance σ = (di + dj)/2and the cut-off is rc = (di + dj)/2 + ds.

The binary mixture is modeled by small particles withdiameter ds = 1 and mass ms = 1 and large particleswith diameter dl = 7ds and mass ml = d3

lms. Becauseof the soft interaction between particles, it is possibleto define an effective diameter of the spheres using theBarker-Henderson relation [2]

deff = d+

∫ ∞d

(1− exp[−uij(r)/kBT ])dr , (3)

which gives the distance where the repulsive interactionis of the order of 1 kBT . For our parameters, the effec-tive diameters are deff

l = 7.19 and deffs = 1.19, for large

and small particles, respectively. The effective size ratiotherefore is deff

l /deffs =6.04. The model neglects deforma-

bility and coalescence of the particles, which can occurwhen film forming particles are used in experiments.

The interaction Uiw(z) between particle i with diame-ter di and the hard substrate at the bottom is modeledby

Uiw(h)/kBT =

εwkBT

(dih

)12

h < di/2

0 h ≥ zc(4)

where εw/kBT = 100 determines the strength of the re-pulsive interaction and h is the distance of a particle fromthe substrate.

2

ab

cde

a b c d e

f

Figure 1. Evolution with time of the density profiles of the small particles, for a mixture with size ratio dl/ds = 7 and numberratio Nr = 151. (a)-(e) are snapshots at times t = 7.5 × 102τB , 6.6 × 103τB , 1.25 × 104τB , 1.83 × 104τB and 2.4 × 104τB ,respectively. The corresponding density profiles are plotted in (f); ρs is the number density of small particles. The top surfaceis at z/ds = 0 and is at the right.

We model the solvent evaporation process by a movingsoft wall, which pushes the particles toward the bottomsubstrate at constant velocity vev. The position of thesoft-wall (interface) as a function of time is defined byzint(t) = H − vevt. The interaction between the softwall and a particle i with diameter di is described bya harmonic potential, which models the Pickering effectdue to the change in interfacial free energy when particlesare trapped at a interface [3]

Ui(z)/kBT =αikBT

(z − r0 − zint(t)) , (5)

where z is the particle coordinate, and r0 determinesthe contact angle θ = arccos(2r0/di). We have chosenr0 = dl/4, ds/2, for large and small particles, respec-tively. The strength of the air-water interface attrac-tion was chosen to be proportional to the area of theparticle, i.e., αi/kBT = 1000 (di/ds)

2. Effects like the

capillary attraction between the particles trapped at theinterface or effective dipolar interactions are neglected inthis model.

DETAILS OF THE EXPERIMENT

We investigated the room temperature drying of la-tex particle mixtures experimentally. The initial colloidaldispersions were prepared by blending two acrylic copoly-mer latices with different mean particle sizes. Both typesof particles were mutually repulsive and were colloidally-stable in water initially and when mixed together.

The large particles were made of a copolymer of methylmethacrylate and n-butyl acrylate in a weight ratio of40/60. The initial solids content is 20 wt%. Theywere synthesized by radical emulsion polymerization us-ing Synperonic NP30 and sodium dodecylsulfate surfac-

3

a b

Figure 2. Snapshots of the top regions of the final configu-rations. (a) Size ratio dl/ds=2, and Nr=17. (b) Size ratiodl/ds=14, and Nr=9000.

Figure 3. Snapshots of the top regions of the final config-urations for mixtures with size ratio 7:1 and number ratioNr=150 for different initial volume fractions. (a) η0=0.1 (b)η0=0.2 (c) η0=0.4.

tants (99/1 wt% ratio) at a total concentration of 3 gL−1. Sodium persulfate (0.5 wt% relative to monomers)was used as the initiator.

The z-average diameter was determined by dynamiclight scattering (DLS) (NanoZS from Malvern Instru-ments) toDz=385 nm. These particles were labelled withfluorescent Rhodamine B (0.2 wt% based on monomers).Analysis of the supernatant after centrifugation foundthat there was no Rhodamine B in the aqueous phase.The particles were stabilized by initiator fragments anda combination of non-ionic and anionic surfactants (witha weight ratio of 99/1).

The small particles are composed of amphiphilic blockcopolymers obtained by polymerization-induced self-assembly (PISA) [4]. Controlled radical polymerization(namely, reversible addition-fragmentation chain transfer(RAFT) polymerization) of methacrylic acid (MAA) wasperformed to obtain firstly a PMAA macroRAFT agent(about 4000 g mol−1) [5], which was then chain-extended

with a mixture of n-butyl acrylate (BA) and styrene (S)(55/45 wt%) to form self-stabilized particles [6]. A z-average diameter Dz=55 nm with a dispersity of 0.064were measured by DLS. Electrostatic stabilization wasprovided by anionic sulfate groups contained in the ini-tiator. The pH of the colloidal mixture was measured tobe 3.5. At this pH, which is below the value of PMAA’spKa of 5.5, the PMAA chains at the particle surface areonly weakly ionized and are collapsed [7]. Hence, thePMAA is expected to offer neither significant steric sta-bilization nor charge stabilization.

After blending the calculated amounts of these two dis-persions to achieve the desired number ratio, we addeddeionized water in order to match the initial solids con-tent used in the simulations. The final solids contentwas always in the range of 9-13 wt.% for all disper-sions. Films of these blends were cast on glass sub-strates (18 × 18 mm2), previously cleaned with acetoneand a UV ozone treatment (Bioforce Nanosciences, modelUV.TC.EU.003).

Height and phase images of the top surface of the filmswere acquired by atomic force microscopy (AFM), usingan NT-MDT Ntegra Prima microscope with intermittentcontact. Images were analyzed using NOVA software.

A Zeiss LSM510 confocal microscope (on an Axiovert200M microscope) was used to obtain stacks of plane im-ages at different depths within the sample. A drop oflarge (750 nm diameter, purchased from Fluoresbrite)green fluorescent particles was cast on top of the driedfilms. They provided a marker for the top surface posi-tion. The green and red fluorochromes were excited usingan argon laser (488 nm) and a HeNe laser (543 nm), re-spectively. Images (132× 132 µm2) were acquired every0.5 µm in the vertical direction. Results were analyzedusing the image processing package Fiji (a version of Im-ageJ).

∗ andrea.fortini@me.com† r.sear@surrey.ac.uk

[1] S. Plimpton, J. Comp. Phys. 117, 1 (1995).[2] J. A. Barker and D. Henderson, Mol. Phys. 48, 587 (1976).[3] P. Pieranski, Phys. Rev. Lett. 45, 569 (1980).[4] B. Charleux, G. Delaittre, J. Rieger, and F. D’Agosto,

Macromolecules 45, 6753 (2012).[5] I. Chaduc, M. Lansalot, F. D’Agosto, and B. Charleux,

Macromolecules 45, 1241 (2012).[6] I. Chaduc, M. Girod, R. Antoine, B. Charleux,

F. D’Agosto, and M. Lansalot, Macromolecules 45, 5881(2012).

[7] X. Wang, X. Ye, and G. Zhang, Soft Matter 11, 5381(2015).