Particle Deposition Kinetics of Colloidal Suspensions in

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Particle deposition kinetics of colloidal suspensions in microchannels at high ionic strength Cesare Mikhail Cejas, Fabrice Monti, Marine Truchet, Jean-Pierre Burnouf, and Patrick Tabeling Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b01394 • Publication Date (Web): 12 Jun 2017 Downloaded from http://pubs.acs.org on June 18, 2017

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Particle deposition kinetics of colloidal suspensions in microchannels at high ionic strength Cesare M. Cejas,∗,† Fabrice Monti,† Marine Truchet,† Jean-Pierre Burnouf,‡ and Patrick Tabeling† †Microfluidics, MEMS, Nanostructures Laboratory, CNRS Gulliver UMR7083, Institut Pierre Gilles de Gennes (IPGG), ESPCI Paris, PSL Research University, 6 rue Jean Calvin, Paris 75005 France ‡Departement des Sciences Pharmaceutiques, Sanofi Recherche, 13 quai Jules Guesdes, BP 14 Vitry-sur-Seine 94403 France E-mail: [email protected] Abstract Despite its considerable practical importance, the deposition of real brownian particles transported in a channel by a liquid, at small Reynolds numbers, has never been described at a comprehensive level. Here, by coupling microfluidic experiments, theory and numerics, we succeed to unravel the problem in the case of straight channels at high salinity. We discover a broad regime of deposition (“van der Waals regime”), in which particle-wall van der Waals interactions govern the deposition mechanism. We determine the range of existence of the regime, for which we calculate the concentration profiles, retention profiles, and deposition kinetics analytically. The retention profiles decay as the inverse of the square root of the distance from the entry and the deposiA tion kinetics is given by the expression, S ≈ ( 2.1kT ξL )1/2 , where S is a dimensionless

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deposition function, A is the Hamaker constant, and ξL is a dimensionless parameter characterizing fluid flow properties. These findings are well supported by numerics. Experimentally, we find that the retention profiles behave as x−0.5±0.1 (where x is the distance from the channel entry) over three decades in scales, as predicted theoretically. By varying the flow conditions (speed, geometry, surface properties, concentration), so as to cover four decades in ξL , and taking the Hamaker constant as a free parameter, we accurately confirm the theoretical expression for the deposition kinetics. Operating in the van der Waals regime enables control of the deposition rates via surface chemistry. From a surface science perspective, working in the van der Waals regime enables to measure the Hamaker constants of thousands of particles in a few minutes, a task that would take much longer time to perform with standard AFM.

Keywords clogging, van der Waals, microfluidics, particle deposition, colloids, Hamaker constant, particle-wall interaction

Introduction Despite its apparent simplicity, the problem of deposition of brownian spherical particles moving through a channel, at small Reynolds numbers, represents a complex physical situation even in the simplest geometries. Close to walls, particles are subjected to a variety of effects: drag exerted by the flow, 1,2 rotation, 3 drift, hindered diffusion, 4–6 adhesion, 7 and electrostatic forces. 8 The multiplicity of forces acting on the particles in the near vicinity of the walls complicates the description of their trajectories, whether they are captured by the walls or transported downstream. The question has important practical consequences in channels, porous media or filters, since particles adsorbed to the walls represent nucleation centers for the growth of aggregates. 9 When the deposited particles are firmly anchored to

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the surfaces, they serve as precursors for the formation of arches 10 that induce clogging. Particle deposition phenomena are challenging to understand and they play important roles in many sectors (e.g. filtration, 11 oil industry, pharmaceutical industry). For these reasons, it has been investigated for decades. In this domain, reviews 12,13 are rare; references 14–16 nonetheless provide useful information on the subject. Continuum theories worked out in the early part of the last century have led to establish analytical solutions that specify, for simple geometries, mass fluxes towards the walls, 17,18 under the questionable assumption that particles are pointwise. Although often taken as references, these solutions may be misleading, because they do not take into account the near wall hydrodynamics at the particle scale, nor the particle-wall interactions at the nanoscale, that most often contribute to deposition kinetics. Colloidal filtration theory (CFT) proposes simplified equations for describing the deposition process but they are, in essence, phenomenological. 11,19,20 Over the last decade, a number of numerical investigations have been undertaken either from a continuum (advection diffusion equations) or discrete (Langevin equations) standpoint. 21–24 Most of these studies, although informative, concern specific situations from which it is difficult to establish a general picture. On the experimental side, with the development of microfluidics, documented studies have been carried out in the last years. 23–25 Mustin et al 21 and Unni et al 22,23 analyzed the deposition kinetics of colloidal polystyrene (PS) spheres in microchannels with flows driven by pressure gradients and electrokinetic forces. Using microfluidics, Wyss et al 24 studied general clogging phenomena while Sauret et al 25 probed steric effects on clogging. A similar study was reported using an impinging jet configuration. 26,27 Progress made in instrumentation and computation have led us to a point where it should be possible to construct new rationales in describing deposition phenomena. Since several mechanisms come into play at micro- and nanoscales, there should exist different regimes of deposition, associated to different laws. In the important case of real particles, where sizes are larger than the characteristic scales of adhesion or electrostatic interactions, no attempt has been done yet to establish such a rationale. From an engineering viewpoint,

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this is unfortunate because this jeopardizes the possibility to investigate complex situations of practical interest (porous media, filters, blood vessels,..) with right lines of thought. Here, we focus on the particular case of high salinity, where electrostatic forces are fully screened. We use microfluidics for the experimental investigation and Langevin equations for the numerical modelling. We consider one of the simplest geometries that may be conceived (straight microchannels), for which we could use well controlled surface chemistries, precisely imposed flow conditions along with excellent optical observation windows, enabling the analysis of thousands of particles at the individual level. In the course of the study, we figured out a new regime of deposition, unnoticed thus far, which could represent the first building block of a general description. We also point out interesting consequences of the present work.

Materials and Methods Microfluidic Chip Preparation The microfluidic chip (Fig. 1a) consists of a series of parallel channels between an entry and an exit reservoir, 2mm long with reservoir height range, hres = 25 − 40 µm. The dimensions of the channels range in the following intervals: 10µm ≤ h ≤ 40µm, w = 100µm, and 400µm ≤ L ≤ 3mm (Fig. 1b). In all experiments, the channels are always fabricated 150µm apart to eliminate cake formation between adjacent channels and to ensure that each channel acts independently from one another. At L 0, it reads: η C(ξ, η) ≈ q A ξ η 2 + 2.1kT

(5)

The solution reveals that a thin depleted zone grows along the downstream direction. Its √ adimensional thickness δ (defined by the point where concentration is divided by 2) is q A δ(ξ) = 2.1kT ξ. In non adimensionalized variables, this leads to: ϑ(x) = rδ =

r

AxD 2.1kT Ur

(6)

This regime, being dominated by van der Waals forces can be called “van der Waals regime” (VdW). One characteristic of this regime is that the Hamaker constant plays a crucial role and therefore, it is possible to control the deposition kinetics by modifying the surface chemistries of the wall and the particles. This remark has far-reaching practical consequences. It is also interesting to note that, although the process is deterministic, the growth of the depleted boundary layer is diffusive-like, with an apparent diffusion constant have been anticipated from inspection of Eq. 3. 31 10


AD , 2.1kT

a result that could

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The VdW regime ends when the first term of the RHS of Eq. 1 becomes comparable to the second, i.e. when diffusion comes into play. By working out Eq. 1, one finds that, in A . kT

terms of orders of magnitude, this occurs when ξ ∼

The VdW regime is valid for large

attractive forces or channels of moderate lengths, in the sense, A kT

ξL

a regime dominated by diffusion takes

place. One important feature of diffusive regime is that the deposition kinetics no longer depends on A. This implies that, in such a regime, it is not relevant to work out the surface chemistries for attempting to control the adsorption kinetics. Here we concentrate on VdW regime. Under continuous injection at rate

ϕQ , vP

the total

number of deposited particles reads (for h/r >> δ(ξL )): NA (t) ≈ =

h 2

h 2

r ϕQt − r vP

r ϕQt − r vP

Z

∞ 0

r

A ) dη 1 − C(ξL , η, kT

A ξL 2.1kT

(8)

in which we recall that ξL =

LD Ur r2

(9)

From this expression, the retention profile p(x), i.e the number of particles deposited at the wall between x and x + dx, reads:

p(x) =

1 dN K = 1/2 NA dx x

11


(10)

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in which N (x) =

h 2

r ϕQt − r vP

ADx 2.1kT Ur r2

1/2

(11)

and K is a normalization constant. This expression differs from Leveque 17 and Levich 18 laws for describing deposition phenomena in channels. It also differs from classical CFT, where concentration of particles are described by an exponential decay with distance. 19 For the rest of the paper, we will work with a function S defined as:

S=

The function

r h −r 2

h 2

− r vp NA (t) r ϕQt

(12)

S is in fact the collector efficiency, a central quantity in engineering litera-

ture. 11 According to Eq. 11, one has:

S=

r

A ξL 2.1kT

(13)

in the VdW regime and should depend only on ξL in the diffusive regime.

Numerics In order to test the theory numerically, we carried out Langevin simulations for the channel flow depicted in Fig. 2a, restricting ourselves to the case w/h >> 1. The Langevin model describing the situation is defined by the following equations: 32

x(t) ˙ = γU (z) + βx δ(t) y(t) ˙ = βy δ(t) +

z(t) ˙ = βz δ(t) +

Ar dβy D − µβy D dy 6(y − r)2

dβz Ar D − µβz D dz 6(z − r)2

12


(14)

(15)

(16)

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Here, x, y and z are the particle coordinates, U (y, z) is the unidirectional flow speed βx , βy , βz are factors expressing the effect of the confinement on the diffusion coefficient of the particle, µ is the fluid viscosity, and δ(t) is a zero mean step random function with √ amplitude 2Dτ (where τ is the discrete incremental step used to calculate the trajectories). The expressions for βx , βy , βz are provided in SI-3 and U (y, z) is given in 33 in the form of a Fourier series. Concerning the boundary conditions, we impose that, if the particles are at less than some distance from the wall, they irreversibly stick on it (we checked that the choice of this distance - in the range 0.1-1nm - is not critical). A Fig. 2b shows the evolution of the quantity S as a function of kT at fixed ξL . Fig. 1b q A 1 > ξL , S is indistinguishable from 2.1kT ξL (see red lines in Fig. 2b). shows that, when kT

this agrees well with theory. In the opposite case, S is found independent of

A , kT

as expected

from the theoretical argument given in the preceding section. Concentrating now on the VdW regime, it is interesting to note that the particle trajectories z(x) can be calculated analytically. One finds:

(z(x) − r)2 ≈ z(0) − r)2 −

AD x 2.1kT Ur

(17)

in which z(0) is the particle altitude at the channel entry x = 0. Particles feel the attraction field as they travel downstream, deflect their trajectories accordingly for A > 0, leading to a van der Waals “rain” on the wall z = r under a strong “wind” sustained by the principal stream. There is a critical altitude, zC , defined by:

zC − r =

r

AD 2.1kT Ur

(18)

in which all particles injected below zC are captured by the walls. With this expression, the deposition kinetics can be calculated, leading to expressions identical to Eq. 8. Fig. 2c and Fig. 2d compare calculated concentration fields, retention profiles, and deposition kinetics with theory. The concentration profiles are well reproduced by Eq. 5.

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The insert of Fig. 2c shows a universal representation of the concentration profiles, using the variable ζ =

η , δ(ξ)

in which δ(ξ) is the thickness of the depletion layer (Eq. 7). With

this variable, the concentration field becomes C(ζ) ≈ √ ζ2

ζ +1

. The insert of Fig. 2c shows

that concentration profiles, calculated at various distances from the channel entry, collapse on such a function. The retention profiles and deposition kinetics shown in Fig. 2d are also in excellent agreement with Eq. 10 and Eq. 11 respectively. From this study, we conclude that theory and numerical simulations are in excellent agreement.

Experiments We perform multiple experiments in each case to increase statistics and carry out image treatment to only show the particles that are captured by the walls. We track single particle sticking events and measure the following quantities: (1) particle sticking positions, x, defined as the distance between the immobile particle and the channel entry; (2) time intervals, △t∗ , between successive sticking events; and (3) the number of particles adsorbed as function of time, NA (t). Fig. 3 shows typical (treated) images of PS particles deposited on the horizontal walls for hydrophobic and hydrophilic surfaces for different values of

ξL . A/kT

The images have been

treated in such a manner that the walls are shown as white and the immobile particles are represented as black spots. In Fig. 3a, at t ∼ 6s after particle injection, we observe a significant difference of NA (t) between hydrophobic and hydrophilic cases. These particular experiments have been performed in h = 20µm, w = 100µm, and L = 400µm. Given the relatively larger size of the particles (d = 5µm), the channel height-particle size ratio is h/d = 4. At this ratio, we are still able to observe particle sticking events taking place in both the floor (bottom) and ceiling (top) of the channel in the z direction. Here, we are in VdW regime where van der Waals interactions dominate. A higher value of Hamaker constant, A, is found for (generally attractive) interaction between PS-hydrophobic walls, whereas a lower value of A is found 14


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for (generally repulsive) interaction between PS-hydrophilic walls. As expected from theory and numerics (Fig. 3a), wettability is crucial in the VdW regime, where it controls A. However, it does not play any role in the diffusive regime, as shown in Fig. 3b, in which we used particles of d = 500nm. These particular experiments have been performed in h = 10µm, w = 100µm, and L = 400µm. Given the smaller size of the particles (d = 500nm), the channel height-particle size ratio is h/d = 20. At this ratio, it is difficult to simultaneously observe particle sticking events in both the floor (bottom) and ceiling (top) of the channel in the z direction and thus images in Fig. 3b only demonstrate immobile particles observed on the floor. Certain particles, especially those too close to one another, may appear bigger but careful comparison with original images indeed confirm separate particles. In this figure, ξL A/kT

> 1. As expected, at t ∼ 25s after particle injection, we find that NA (t) is comparable

regardless of surface treatment. The value of

ξL A/kT

presented here is given as a range since

the surfaces of the PS particles used in this regime are carboxylate-modified. With A being sensitive to interaction distance, surface properties, composition, and morphology, a change in the value of A could be possible. Studies have indeed reported a range of A values for carboxylate-modified PS in glass. 34,35 Here, the presented range of

ξL A/kT

reflects the limits

when 2.0·10−21 J ≤ A ≤ 8.0·10−21 J. Studying the diffusive regime will be done in a subsequent paper. In this paper, we only focus on the VdW regime at early times of deposition. The number of particles, NA (t), deposited on the walls in conditions where

ξL A/kT

< 1 is

shown in Fig. 4a for PS-hydrophobic walls and Fig. 4b for PS-plasma treated walls. Both Fig. 4a and Fig. 4b show that in the first seconds after particle injection is performed, NA ∝ t. At later times, NA (t) tends to level off (SI-4). Visual observations indicate that this saturation occurs when walls are densely filled, suggesting that interparticle interactions play a role in the deposition process. The description of this situation stands outside the scope of the paper. We instead focus on early times, in which the deposition process involves isolated particles and naked surfaces. Fig. 4a and Fig. 4b show plots of average NA (t) over multiple runs obtained in the

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NA number of adsorbed particles

(a)

(b)

time (s)

time (s)

(c)

-0.5

PS-hydrophobic walls

(d)

-1

log S

Hamaker constant, A (J)

0

-1.5 -2 -2.5

PS-hydrophilic walls

PS-hydrophobic PS-hydrophilic

-3 -3.5 -5

-4

log (!L)

theory

counts

(e)

-3

-2

-1

0

log (!L)

norm. counts

number of counts

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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NA number of adsorbed particles

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channel length (µm)

(f)

theory

norm. channel length (µm/µm)

theory

"t* (s)

Figure 4: (a) NA (t, U ) at short times using hydrophobic walls, d = 5µm, h = 20µm, ϕ = 0.3%, L = 400µm. Points represent average and the broken lines are theoretical predictions, which already agree with numerics. (b) NA = (t, U ) at short times using hydrophilic walls, d = 5µm, h = 20µm, ϕ = 0.3%, L = 400µm. Points represent average and broken lines are theoretical predictions, which already also agree with numerics. (c) Plot of A(log(ξL )) used for all experiments. It is constant for a particular surface treatment. (d) Plot of S(ξL ) function (Eq. 18) in log-log. For all experiments involving PS-hydrophobic walls, A = 8.0±0.4·10−21 J. For all experiments involving PS-hydrophilic walls, A = 2.9±0.1·10−22 J. (e) Plot of p(x) for all experiments vs. theory with the equation y = K1 x−0.5±0.1 , where K1 is a normalization constant. Note that for larger speeds (U ≥ 10mm/s) the distributions show a decrease as we approach the channel entry, which has not been represented. (f ) (top) Plot of p(x) for all experiments vs. theory (lin-lin). (f ) (bottom) Plot of normalized experimental △t⋆ distributions, p(△t⋆ ), vs. theory with the equation y = K2 t−1 , where K2 is a normalization constant. (Legend) The symbols represent: (set 1) h = 20µm, ϕ = 0.3%, L = 400µm, and with hydrophobic walls: ( ) 0.22mm/s, ( ) 1mm/s, ( ) 5mm/s, ( ) 10mm/s, ( ) 30mm/s, ( ) 110mm/s; (set 2) h = 20µm, ϕ = 0.3%, L = 3mm, and with hydrophobic walls: (2) 0.22mm/s, (2) 1mm/s, (2)10mm/s; (set 3) h = 20µm, L = 400µm, U = 1mm/s, and with hydrophobic walls: (⋆) ϕ = 3%, ( ) ϕ = 0.3%; (set 4) h = 20µm, ϕ = 0.3%, L = 400µm, and with hydrophilic walls: () 0.22mm/s, () 1mm/s, () 5mm/s, ()10mm/s, () 30mm/s , () 100mm/s; (set 5) h = 10µm, ϕ = 0.3%, L = 400µm, and with hydrophobic walls: (△)1mm/s, (△) 10mm/s; (set 6) h = 30µm, ϕ = 0.3%, L = 3mm, and with hydrophobic walls: (3) 1mms/s, (3)10mm/s; (set 7) for h = 40µm, ϕ = 0.3%, L = 3mm, and with hydrophobic walls: (C) 1mm/s, (C) 10mm/s.

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range of early times, for h = 20µm, hydrophobic and hydrophilic walls repectively, and at different velocities (see also SI-5). In all cases, NA ∝ t. The faster the velocity, the higher the deposition rate. The full lines on Fig 4a and Fig. 4b, correspond to the theory, which we have already shown to agree well with numerics. Here, we adjust A to match experiments and theory. Theory predicts that the adjusted values of A should not depend on the speed, concentration, nor flow geometry. This is what Fig. 4c shows, throughout four decades of variation of ξL . In such conditions, the agreement between theory and experiments is therefore excellent. To summarize the results, we plot S introduced earlier in Eq. 12, defined as

S=

h −1 2r

vp NA (t) ϕQt

(19)

as function of ξL , for two different wettabilities. All ξL satisfy criterion defined in Eq. 7, and therefore, all experiments lie in VdW regime. This emphasizes the broad range of existence of this regime. Fig. 4d shows that S behaves as a power law, with an exponent close to 12 , as expected theoretically using approximately the same A for each case, theory and experiments agree well. This agreement holds for four decades of variation of ξL . Fig. 4c displays A values we determined with our fitting procedure for each wetting condition displays formula (Eq. 13), taking A as a free parameter. We find the best fit procedure leads to: A = 8.0 ± 0.4 · 10−21 J (hydrophobic walls) and A = 2.9 ± 0.1 · 10−22 J (hydrophilic walls) It is unfortunately difficult to compare these values with those published in literature because they are scattered, sometimes inconsistent. 36,37 Our A value for attractive case (PShydrophobic wall) is one order of magnitude smaller than AFM measurements carried out with a similar system (5.2 ·10−20 J). 29 It is five times larger than theoretical estimates of A, based on the pairwise additivity Hamaker approach (1.4 · 10−21 J), 21 shown in SI-6. We also compare our value with that of the Lifshitz theory, 36,37 also shown in SI-6. This is a

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comprehensive theory that takes into account the properties of the intervening medium and their effect on the van der Waals attraction of two macroscopic bodies. It contains a zero frequency term that includes Debye and Keesom forces 36,37 and a non-retarded dispersion energy term. Our estimate is not far off from the value calculated from the full Lifshitz theory (6.7 ·10−21 J). However, it has been argued that since the zero frequency term represents an electrostatic interaction, the presence of strong electrolytes renders it insignificant since the charges will be fully screened. 14,21 This renders the Lifshitz equation being solely dominated by the dispersion term, 14,37–39 which is not affected by electrolytes at high frequencies. In this case and accounting for screening effects, our estimates are twice the value calculated from Lifshitz theory 36,37 (4.1 ·10−21 J). However, it has also been noticed, without much explanation, that measured value of the constant from experiments and the calculated Hamaker constant from the Lifshitz theory often differs by a factor of two. 40 In addition, it has also been reported that for certain materials, the van der Waals attractions are independent of electrolyte concentration and thus the Hamaker constant values do not differ regardless of electrolyte concentration up to 1M KNO3 39 as in the case of two mica surfaces. This suggests that for such surfaces, both zero frequency and dispersion terms contribute to the overall value of the Hamaker constant. Indeed, the concept of the Hamaker constant is a complex problem as seen from the varying degree of values and arguments reported in literature. The Hamaker constant we measure (which is in fact an effective Hamaker constant) is also sensitive to a wide variety of parameters such as molecular charge and even surface roughness. 41 We therefore conclude that our measurements stand in the middle of a (broad) range of available estimates. There is thus no inconsistency between our work and previous studies. No literature data has been found for the Hamaker constant corresponding to a system of PS-oxidized PDMS in water (PS in hydrophilic walls). This is probably due to the transient nature of plasma-treated PDMS, whose relaxation time scales, back to its original hydrophobic state, are typically shorter than the time it would take to measure the constant using standard techniques. However,

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studies have shown that for repulsive cases (PS-hydrophilic wall), the Hamaker constant is one order of magnitude less than that of attractive cases. 29 In addition, recent progress has demonstrated experiments between silica particles and oxidized silicon wafer with the latter having similar hydrophilic composition to oxidized PDMS. 42–44 Despite the predicted favorable interaction between hydrophilic silica and the oxidized silicon surface, the Hamaker constant values from AFM measurements ranged from (2.2-3.3 · 10−22 J), 42–44 in agreement with the order of magnitude that we find for our PS-oxidized PDMS (10−22 ). Our results are also thus consistent with literature in this case. The low value of the Hamaker constant is attributed to the assumption of the existence of a hydration force between silica surfaces and inorganic electrolytes, 43,44 arguing that at high electrolyte concentrations, hydrated cations can bind to the oxidized surfaces, which are heavily negatively charged, and can give rise to a repulsive hydration energy. 37 Therefore in this regard also, our microfluidic-based experiments provide results that suggest the presence of hydration forces. From the A values in Fig. 4c and the deposition function S in Fig. 4d, we can conclude that, within 1). to diffusive regime ( A/kT

To the best our knowledge, this van der Waals regime of particle deposition has never been reported in literature. Established for pressure driven flow in rectangular channels, it should obviously exist in other geometries such as impinging jet. The characteristics of the VdW regime might be interesting to exploit for designing collectors and filters or optimizing transport properties of particles through porous media via surface chemistry. With AFM, measuring a Hamaker constant is a time consuming and delicate operation, that can be performed for only a very limited number of particles. Here, by inverting the problem in the way leading to Fig. 3, one may determine a Hamaker constant for thousands of particles in a few minutes, without touching them. Note that the form of the adhesion law can be checked in situ, by measuring the distribution of the adsorbed particles along the channel axis, as in Fig 4e. If exponents differ from those of the present study, the relevant Hamaker constant can still be found by adapting the theoretical expressions. Beyond the measurement per se of the Hamaker constant and the determination of the power law governing the van der Waals forces, our work has several implications. In oil reservoirs, for instance, where high salinity conditions hold, an important question is to know whether nanoparticles, advected by brine injected in the injection well, may penetrate far into the reservoir in order to enhance oil recovery. With standard conditions, we find that the maximum length ∼

A U2 kT D

(in which U is the mean speed), beyond which we enter the

diffusive regime, is in the order of micrometers. Beyond this length, surface functionalization ceases to play a role. In the diffusive regime, the maximum travel length is in the order 2

of ( UDh ), independent of particle surface functionalization. This length is on the order of millimeters for standard reservoir pore sizes, which is small compared to the distance these particles are expected to travel in practical applications (from meters to kilometers). This raises a challenge for the oil industry.

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Another example in microfluidic technology : in microfluidic devices, particles are often transported towards targets (DNA/protein samples/cells,...) for analytical purposes. The challenge is to avoid them to be captured by the walls. The present work indicates that, when the working fluid is a buffer with common salt concentrations above 100mM, depending on the Hamaker constant and flow characteristics, the length of the VdW region ranges between tens of microns and tens of millimeters. It thus becomes possible to optimize the flow conditions and the surface chemistries to operate in the VdW regime and avoid particle loss.

Author Information Corresponding Author ∗

[email protected]

Present Address Microfluidics, MEMS, Nanostructures (MMN) Laboratory, CNRS Gulliver UMR 7083, Institut Pierre Gilles de Gennes (IPGG) pour la microfluidique, ESPCI Paris, PSL Research University, 6 rue Jean Calvin, Paris 75005 France

Author Contributions C.M.C developed and performed experimental and numerical analyses, F.M. helped design experimental set-up, M.T. helped carry out experiments and improved Matlab code, J.P.B provided scientific consultation and funding, and P.T. developed theory and performed numerical analyses. Manuscript was mainly written through principal contributions from C.M.C and P.T. with inputs from the rest of the authors.

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Notes The authors declare no conflict of interests and no competing financial interest.

Abbreviations PS - polystyrene; VdW - van der Waals; PDMS - polydimethylsiloxane; NaCl - sodium chloride; AFM - atomic force microscopy.

Acknowledgement This work has been supported by ESPCI Paris, IPGG (´ equipement d′ excellence “Investissements d′ avenir” program ANR-10-EQPX-34) and Sanofi. We also thank the support of CNRS Gulliver UMR 7083 and the MMN group. We are also grateful to Antoine Nigu` es for SEM imaging and for discussions with Lyd´ eric Bocquet, Christian Fr´ etigny, and Eric P´ erez.

Supporting Information Available The Supplementary Information (SI) contains the following items and is available free of charge: • SI-1: Advection-diffusion • SI-2: Hindered diffusion • SI-3: Correction factors expressing confinement of particle diffusion coefficient • SI-4: Number of particles adsorbed as a function of time • SI-5: NA (t) in channels of different lengths and heights • SI-6: The Hamaker constant This material is available free of charge via the Internet at http://pubs.acs.org/. 24


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