PreprintPDF Available

Particulate suspension coating of capillary tubes

September 2022

DOI:10.48550/arXiv.2209.14871

Authors:

Langqi Xing

Northeastern University

Alban Sauret

University of Maryland, College Park

Preprints and early-stage research may not have been peer reviewed yet.

The displacement of a suspension of particles by an immiscible fluid in a capillary tube or in a porous media is a canonical configuration that finds application in a large number of natural and industrial applications, including water purification, dispersion of colloids and microplastics, coating and functionalization of tubings. The influence of particles dispersed in the fluid on the interfacial dynamics and on the properties of the liquid film left behind remain poorly understood. Here, we study the deposition of a coating film on the walls of a capillary tube induced by the translation of a suspension plug pushed by air. We identify the different deposition regimes as a function of the translation speed of the plug, the particle size, and the volume fraction of the suspension. The thickness of the coating film is characterized, and we show that similarly to dip coating, three coating regimes, liquid only, heterogeneous, and thick films, are observed. We also show that, at first order, the thickness of films thicker than the particle diameter can be predicted using the effective viscosity of the suspension. Nevertheless, we also report that for large particles and concentrated suspensions, a shear-induced migration mechanism leads to local variations in volume fraction and modifies the deposited film thickness and composition.

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Particulate suspension coating of capillary tubes

D.-H. Jeong, aL. Xing, aJ.-B. Boutin, aand Alban Sauret a∗

September 30, 2022

Abstract

The displacement of a suspension of particles by an immiscible ﬂuid in a capillary tube or in a porous media is a canonical

conﬁguration that ﬁnds application in a large number of natural and industrial applications, including water puriﬁcation,

dispersion of colloids and microplastics, coating and functionalization of tubings. The inﬂuence of particles dispersed in the

ﬂuid on the interfacial dynamics and on the properties of the liquid ﬁlm left behind remain poorly understood. Here, we study

the deposition of a coating ﬁlm on the walls of a capillary tube induced by the translation of a suspension plug pushed by air.

We identify the diﬀerent deposition regimes as a function of the translation speed of the plug, the particle size, and the volume

fraction of the suspension. The thickness of the coating ﬁlm is characterized, and we show that similarly to dip coating, three

coating regimes, liquid only, heterogeneous, and thick ﬁlms, are observed. We also show that, at ﬁrst order, the thickness of

ﬁlms thicker than the particle diameter can be predicted using the eﬀective viscosity of the suspension. Nevertheless, we also

report that for large particles and concentrated suspensions, a shear-induced migration mechanism leads to local variations in

volume fraction and modiﬁes the deposited ﬁlm thickness and composition.

1 Introduction

The displacement of a suspension of particles by another im-

miscible ﬂuid (air or liquid) in conﬁned geometries and porous

media is an important process involved in many industrial and

natural situations. For instance, such processes are encountered

in enhanced oil recovery,1inﬁltration of liquid in porous me-

dia,2the transport and pollution by microplastic in soils, 3,4 the

intermittent dispensing of liquids, and microﬂuidics5,6 among

other situations. This conﬁguration can also be leveraged to de-

velop new coating processes to functionalize tubings.7Besides,

a common approach to model multiphase ﬂow in porous media

is to consider the simpliﬁed system made of capillary tubes. 8

Therefore, a fundamental understanding of the dynamics ob-

served in this conﬁguration for a large variety of ﬂuids encoun-

tered in practical applications is required.

When a wetting liquid plug is pushed in a capillary tube by

an immiscible ﬂuid, e.g., air, it deposits a thin ﬁlm of thickness

hon the wall of the tube.9–13 For a homogeneous Newtonian

liquid of dynamic viscosity ηfand interfacial tension γpushed

by air, the formation of the coating ﬁlm is governed by the

competition between viscous and surface tension forces through

the capillary number Ca = ηfU/γ, where Udenotes the velocity

of the moving air/liquid interface at the rear of the plug. The

thickness hof the liquid ﬁlm deposited on the wall of a tube

aDepartment of Mechanical Engineering, University of California,

Santa Barbara, California 93106, USA

∗asauret@ucsb.edu

of radius Ris a function of Rand Ca only. 12,14 If the ﬂuid

is partially wetting, i.e., its contact angle is θ > 0, the ﬁlm

dynamics is more complex as the contact line motion plays a

role in determining the deposition patterns. 15,16

When solid particles are dispersed in a liquid, the usual in-

terfacial dynamics and rheological approaches used to describe

capillary ﬂows can often fail when the lengthscale of the cap-

illary object, here the liquid ﬁlm, becomes comparable to the

diameter of the particles. This peculiar dynamics of suspension

has been considered in various conﬁgurations such as the forma-

tion of droplets,17–22 the stability of jets,23 the fragmentation

of suspension sheets,24 the motion of contact lines,25 and dur-

ing the formation of thin ﬁlms.26 The challenge arises from the

fact that for capillary ﬂows of suspensions, in addition to the

ﬁlm thickness h, an additional lengthscale enters into the prob-

lem: the particle diameter d. This complexity when using a

suspension of non-Brownian particles dispersed in non-volatile

liquids has been considered during the formation of a thin ﬁlm

of suspension in a dip-coating process. 26,27 The dip-coating of a

substrate consists of withdrawing a solid body initially dipped

in a liquid bath so that it emerges covered with a thin layer

of liquid.28,29 The thickness hof liquid deposited on a plate

withdrawn at a velocity Ufrom a bath of Newtonian ﬂuid of

viscosity ηfand surface tension γfollows the Landau-Levich-

Deryaguin law (LLD): h= 0.94 ℓcCa2/3(for Ca <10−2) where

Ca = ηfU/γ is the capillary number and ℓc=pγ/(ρ g )is the

capillary length.29–33 The dip coating with suspensions of non-

Brownian particles has been shown to exhibit diﬀerent coating

compressed

air

Three-way

valve

L L

Syringe

High speed camera

xfront

xrear

(a) (b)

(c)

(d)

compressed

air

syringe

High-speed camera

three-way

valve

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p

Figure 1: (a) Schematic of the experimental setup used to study the displacement of a suspension plug and the deposition of a

coating ﬁlm in a circular capillary tube of diameter D= 2 R. The capillary tube is connected to the atmosphere on one end,

and the suspension plug is pushed by applying a constant pressure diﬀerence ∆p. (b) Example of an experiment showing that

the translating plug leaves behind a coating ﬁlm of thickness h. The time goes from left to right, and the scale bar is 2 mm. (c)

Example of measurement of the velocity of the front ufand of the rear urof the plug from the time evolution of their respective

position xfand xr(shown in inset). (d) Resulting coating thickness rescaled by the radius of the capillary h/R calculated using

the measurements presented in (c) and Eq. (2). The solid line is given by Eq. (3). Inset: Coating thickness has a function of

the velocity of the rear of the plug ur.

regimes depending on the particle diameter dand the coating

thickness.26,27,34 At small capillary number Ca, corresponding

to very thin ﬁlms (h≪d), no particles are entrained, and only

a liquid ﬁlm coats the plate. At intermediate capillary numbers

leading to h≲d, a heterogeneous coating made of clusters of

particles arranged as a monolayer is observed on the substrate.

Finally, at large capillary numbers where h > d, the thickness of

the entrained ﬁlm is captured by the Landau-Levich-Derjaguin

law using the eﬀective viscosity of the suspension η(ϕ), where

ϕis the solid fraction of particles in the suspension. A similar

behavior has also been reported for the dip coating of cylindri-

cal substrates when accounting for the change in the LLD law

due to the curvature of the substrate.35 The role of the particle

diameter, compared to the ﬁlm thickness, has allowed the devel-

opment of new capillary ﬁltration36 and sorting methods34,37

leveraging the dip coating platform.

Interestingly, the ﬂuid mechanics and ﬂow topology underly-

ing the dip coating process share many common features with

the ﬂow of a plug of liquid in a tube, including the presence

of a stagnation point governing the coating ﬁlm and the evo-

lution of the ﬁlm thickness hwith the capillary number Ca.38

As a result, the entrainment threshold of isolated particles in

the coating ﬁlm during the dip coating of a plate is consis-

tent with the entrainment threshold in a capillary tube.36,39

However, the relevance of the diﬀerent coating regimes and the

properties of the coating ﬁlm remain more elusive for a plug of

particulate suspension at a moderate volume fraction pushed

by an immiscible ﬂuid in cylindrical capillary tubes. Yet, such

an approach could allow coating the inner wall of tubes, giving

them some surface properties.7,40 It would also lead to a bet-

ter description of multiphase ﬂows that involve particles and

interfaces in porous media.

In this article, we focus on the intermittent ﬂow of a plug of

non-Brownian and neutrally buoyant suspension of monodis-

perse spherical particles in cylindrical capillaries. We experi-

mentally characterize the deposition of the coating ﬁlm during

the process and illustrate how the nature of the coating depends

on the velocity Uof the plug of suspension for diﬀerent particle

diameters d. We ﬁrst present our experimental methods in sec-

tion 2. Section 3 reports our general observations. We consider

in section 4 the nature of the coating ﬁlm in more detail, and

we investigate the role of the particle diameter and of the vol-

ume fraction of the suspension. We also discuss the important

role of shear-induced migration to rationalize our observations.

2 Experimental methods

2.1 Experimental setup

The experimental setup, presented in ﬁgure 1(a), consists of

a cylindrical glass capillary tube connected to compressed air,

and to a syringe. The glass capillary tube (Vitrocom) is 30 cm

long and has an inner diameter D= 2 R= 1.5 mm. The results

with diﬀerent diameters can be rescaled with the diameter of

the capillary tube so that we focus here on one diameter and

vary the other parameters. The only constraint is that the

diameter of the particles should be small compared to the di-

ameter of the tube. 39 Similarly, the viscosity of the interstitial

ﬂuid can be rescaled through the capillary number and will not

modify the dynamics, 26 therefore we kept the same interstitial

ﬂuid throughout the study.

The suspensions are made of spherical polystyrene particles

(Dynoseeds TS, Microbeads) of diameter 20 µm≤d≤250 µm

dispersed in a high-density silicon oil (Sigma Aldrich). The

silicone oil has a dynamic viscosity at 20oCof ηf= 0.11 Pa s,

an interfacial tension γ≃25 ±2 mN m−1and a density ρp=

1058 kg m−3. The silicone oil perfectly wets both the capillary

tube and the particles. 18 The densities of the particles range

between ρp= 1056 kg m−3and 1062 kg m−3depending on the

batch considered. A mechanical stirrer is used when initially

preparing the suspension and during a series of experiments to

disperse the particles in the silicone oil. The average density of

the polystyrene particles is close enough to the density of the

interstitial ﬂuid, and the capillary tube is placed vertically, so

that the suspensions can be considered neutrally buoyant over

the timescale of one experiment, typically from a few seconds to

a few tens of seconds. The volume fraction of the suspension,

ϕ=Vp/Vtot, where Vpand Vtot are the volume of particles

and the total volume of suspension, respectively, is varied in

the range 0.05 <ϕ<0.25. In this study, we did not consider

larger volume fractions as the pressure required to translate the

plug at a signiﬁcant speed would be too large for our system. In

addition, for volume fractions larger than ϕ= 0.35 we observed

the apparition of an unstable ﬁnger at the front of a suspension

plug, which could modify the dynamics.41

To visualize the motion of the plug, we use a 20 ×20 cm

backlight LED Panel (Phlox), and the dynamics is recorded

with a high-speed camera (Phantom VEO 710L) equipped with

a macro-lens (Nikkor 200mm). Additional photos of the coating

ﬁlms are taken with a Nikon D7200 camera and a 200 mm

macro lens. After each run, the capillary tube is thoroughly

cleaned with Isopropanol (IPA, Sigma Aldrich), rinsed with DI

water (Millipore), and dried with compressed air before being

reused.

2.2 Measurement of the coating thickness

At the beginning of an experiment, a suspension plug of volume

Vis placed at the inlet of the tube using the syringe. We ensure

that the initial length of the plug is about L∼10 cm, so that

L >> 2R. The inlet of the tube is then closed by a three-way

valve, and the plug is initially at rest. The entrance of the tube

is connected to a source of compressed air with the three-way

valve, as shown in ﬁgure 1(a). The plug is set in motion by

opening the entrance of the setup to the pressurized chamber.

The pressure is controlled using a pressure regulator (Omega

AR91-015) in the range 350 Pa <∆p < 30,000 Pa. The sus-

pension plug then translates in the capillary tube at a typical

velocity between a few mm/sto a few cm/s, as shown in the ex-

ample of ﬁgure 1(b). The Reynolds number based on the radius

of the capillary remains smaller than 0.1, and we can neglect in-

ertial eﬀects. Another relevant dimensionless parameter in this

system is the Weber number, which compares inertia with the

capillary force and is deﬁned as We = ρur2(R−h)/γ ,12 . The

Weber number remains smaller than 0.2in our experiments,

meaning that inertia eﬀects are also negligible compared to

capillary eﬀects. The length of the plug decreases over time

as the liquid is continuously deposited at the rear of the plug

on the capillary wall. As a result, since the translation of the

plug is driven by a pressure diﬀerence, the translational veloc-

ity increases during one experiment. We measure the instanta-

neous velocity and the corresponding local thickness, following

the method developed by Aussillous & Quéré for Newtonian

liquids.12

To measure the thickness of the coating ﬁlm, we rely on the

measurement of the time-evolution of the position of the front

and rear interfaces, xfand xr, respectively. 12 The positions of

the menisci are extracted from the movies using ImageJ and

custom-made Matlab routines. An example is shown in the

inset of ﬁgure 1(c). The length of the plug, L(t) =|xf−xr|

decreases over time since some suspension is left behind, coating

the wall of the capillary tube. We then compute the velocity

of the front and rear menisci, uf= dxf/dtand ur= dxr/dt,

respectively [ﬁgure 1(c)]. Since dL/dt=uf−ur, we have:12

−dV

dt



plug

=−πR2dL

dtand dV

dt



ﬁlm

=π urR2−(R−h)2,

(1)

where his the thickness of the ﬁlm and Ris the radius of the

capillary tube. Mass conservation implies that −dV /dt|plug =

dV /dt|film . Therefore, the value of h/R is obtained through

urand ufwith the relation

R= 1 −r1 + 1

dt= 1 −ruf

.(2)

Since the velocity of the suspension plug is increasing during

one experiment at constant ∆p, we are able to cover a small

range of capillary numbers Ca = η ur/γ during each experiment

at an imposed pressure. We can then obtain the correspond-

ing ﬁlm thickness h, as shown for one example in ﬁgure 1(d).

Compared to measurements of the coating thickness by direct

visualization, the advantage of this method is that curvature

eﬀects due to the cylindrical cross-section of the capillary tube

are not an issue. In addition, this method has been shown to

provide very reliable and accurate results with homogeneous

Newtonian and viscoplastic ﬂuids.12,42

2.3 Validation: coating by a Newtonian

homogeneous fluid

We ﬁrst perform the experiments with the interstitial Newto-

nian liquid only. The ﬁlm thickness left on the wall of a tube

was initially predicted by Bretherton43 in the limit of small

capillary numbers Ca = ηfU/γ ≪1, who theoretically ob-

tained h/R = 1.34 Ca2/3. The evolution of the ﬁlm thickness

for a broader range of parameters has been obtained through

experiments and numerical simulations. In particular, Aussil-

lous & Quéré12 obtained experimental results consistent with

the experiments carried by Taylor9and have shown that the

thickness of the coating ﬁlm follows the empirical law :

R=1.34 Ca2/3

1 + 3.35 Ca2/3.(3)

More recently, numerical simulations by Balestra et al.14 have

conﬁrmed this expression and extended it to immiscible liquids

of diﬀerent viscosity ratios.

The experimental data for a capillary tube of radius R=

750 µmare reported in ﬁgure 2. The error bars for all ex-

periments reported in this study are typically around ±5% of

the mean value. The experimental results match very well the

prediction given by Eq. (3) in the entire range of capillary num-

bers considered in this study, ensuring that our experimental

approach and methods are correct. Besides, from the beginning

to the end of an experiment, the evolution of the ﬁlm thickness

always follows the theoretical prediction, and there is thus no

inﬂuence of the history of the system for a Newtonian ﬂuid. We

shall see later that adding particles modiﬁes this observation.

Figure 2: Dimensionless ﬁlm thickness h/R as a function of the

capillary number Ca for the silicone oil without particles (pure

ﬂuid, ϕ= 0). The solid line is given by Eq. (3), and the circles

are the experimental measurements. Each color corresponds to

one experiment at a given inlet pressure ∆p(indicated in the

legend). Inset: dimensional thickness of the coating ﬁlm hfor

varying velocities of the rear meniscus ur.

3 Translation of a suspension plug

3.1 Phenomenology

We now consider the presence of neutrally buoyant and non-

Brownian particles dispersed in the interstitial liquid at a vol-

ume fraction ϕ. We report examples of the outcome of exper-

iments performed with diﬀerent particle diameters and trans-

lation velocity urat the same volume fraction ϕ= 0.19 in

ﬁgures 3 and 4. Similar to the dip coating conﬁguration, 27,36

we observe three distinct regimes. (i) At very low velocities

and relatively large particle diameters, no particles are present

in the coating ﬁlm. The coating layer is too thin to let the

particles be entrained in the ﬁlm, and the particles are trapped

in the meniscus which acts as a capillary ﬁlter.36,44 This ﬁrst

regime is referred to as the liquid only regime, corresponding

to h≪d, and is illustrated in ﬁgure 3(a). (ii) At larger ve-

locity and relatively moderate particle diameters, the particles

begin to be entrained in the coating layer as illustrated in ﬁgure

3(b). The particles tend to form clusters due to the capillary

attraction between them to decrease the liquid/air interfacial

deformation.26,45 Because of the capillary interactions between

particles and the relatively low volume fraction of the suspen-

sion (ϕ= 0.19 here), the distribution of the particles in the

coating layer is heterogeneous with regions of the tube inner

wall covered by a liquid ﬁlm only, whereas other regions are

covered by a ﬁlm with a monolayer of trapped particles that

form clusters. The particles are randomly dispersed in the ﬁlm,

and the number of particles per surface area increases with the

translation velocity ur. The velocity threshold between this

heterogeneous regime where h∼dand the liquid-only regime

depends on the particle diameter. (iii) Increasing further the

velocity urand for small enough particle diameters, a second

threshold is reached. As reported in ﬁgures 4(a)-(b), we observe

(a)

(b)

Figure 3: Examples of coating ﬁlms obtained at small capillary

numbers for suspensions of volume fractions ϕ= 0.19 show-

ing (a) the "liquid-only" regime where the ﬁlm formed is much

thinner than the particle diameter and free of any particle, here

for d= 250 µmand ur∼0.3 mm/s, and (b) the "heterogeneous

ﬁlm" regime for d= 80 µmand ur∼0.8 mm/sand where the

coating ﬁlm is thinner than the particle diameter but still al-

lows some particles to be entrained and deposited on the outer

wall of the capillary tube. The time goes from top to bottom,

and the experiments are performed in a capillary tube of di-

ameter D= 1.5 mm. The arrow in (a) indicates the direction

of translation of the interface, and the time goes from top to

bottom. Scale bars are 1 mm. Movies are available in supple-

mentary materials.

a homogeneous coating regime consisting of a uniform coating

of particles that can form multiple layers on the capillary tube

wall. The suspension behaves like a homogeneous ﬂuid with an

eﬀective viscosity modiﬁed by the presence of particles. This

regime is referred to as the eﬀective viscosity regime and cor-

responds to h≥d. The velocity thresholds between the three

diﬀerent coating regimes depend on the particle diameter dand

the thickness of the coating ﬁlm h. The observations in the

present conﬁguration are reminiscent of the diﬀerent regimes

reported in the dip coating conﬁguration, 26 so that the strate-

gies and models developed for the dip coating process seem to

(a)

(b)

Figure 4: Examples of coating ﬁlms deposited on the inner walls

of a capillary tube of diameter D= 1.5 mm by a suspension of

volume fraction ϕ= 0.19 of (a) d= 20 µmparticles translat-

ing at ur∼1.8 mm/sand (b) d= 80 µmparticles translating

at ur∼3.2 mm/s. The arrow in (a) indicates the direction

of translation of the interface, and the time goes from top to

bottom. In both cases, the coating obtained is in the eﬀective

viscosity regime where a thick ﬁlm made of diﬀerent layers of

particles is obtained. Scale bars are 1 mm. Movies are available

in supplementary materials.

be directly applicable in the case of capillary tubes coating and

deposition of thin ﬁlms in porous media. However, as we shall

see in the following, despite the similarities, some signiﬁcant

diﬀerences are present in the capillary tube conﬁguration.

3.2 Thickness of the coating film in the

thick film regime

We experimentally measure the thickness of the coating ﬁlm h

for a suspension of 80 µmparticles dispersed in the silicone oil

when varying the plug velocity. As before, the experiments are

performed at constant pressure so that during an experiment

the length of the plug decreases and the velocity urincreases

so that we obtain a series of data points at varying capillary

numbers. Following the approach done for dip coating,26 in

the regime where h≥d(corresponding in the present case

to h/R ≥0.11) we consider that the suspension behaves as a

homogeneous mixture with an eﬀective viscosity η(ϕ). In other

word, the discrete nature of the particles does not matter, and

the suspension can be seen as a homogeneous ﬂuid of larger

viscosity. In this situation, we deﬁne the eﬀective capillary

number based on the viscosity of the suspension:

Caϕ=η(ϕ)ur

γ.(4)

The eﬀective viscosity of the suspension can be estimated

through diﬀerent models such as, for instance, the Maron-

Pierce correlation:46

η(ϕ) = ηf1−ϕ

ϕm−2

,(5)

with ηfthe viscosity of the interstitial ﬂuid, ϕthe volume frac-

tion of the suspension, and ϕm= 0.59 the maximal packing

fraction for the particles and ﬂuid used.22,34

In the thick ﬁlm regime, the thickness of the suspension ﬁlm

coating the inner wall of the capillary tube should be given

by:26,27

R=1.34 Caϕ2/3

1 + 3.35 Caϕ2/3.(6)

We report the dimensionless coating thickness h/R as a func-

tion of the eﬀective capillary number in ﬁgure 5(a) for a range

of experiments performed at diﬀerent ∆pand thus diﬀerent

ur. Here, we calculate the expected viscosity of the suspension

based on the volume fraction in the syringe using Eq. (5). Then

η(ϕ)is used together with the experimental measurement of ur

in Eq. (4) to calculate Caϕand then h/R using Eq. (6). The

prediction of Eq. (6) captures reasonably well the evolution of

the coating ﬁlm despite more scattering than in the dip coating

case and the capillary tube case without particles. Neverthe-

less, as expected, the experimental data cannot be captured by

the prediction based on the viscosity of the interstitial ﬂuid [see

inset of ﬁgure 5(a)]. Therefore, it seems that, at ﬁrst order, the

results and methods obtained in the dip coating conﬁguration

in terms of Caϕcan be used to predict the coating of the inner

wall of a capillary tube by a translating suspension plug. How-

ever, we also observe in Fig. 5(a) that h/R tends to get larger

than the prediction given by Eq. (6) at relatively large Caϕ.

Indeed, the data points at larger Caϕare obtained near the end

of an experiment when the translation velocity is the largest.

The prediction becomes larger due to the non-uniformity of the

coating ﬁlm, as we shall discuss later.

4 Discussion

4.1 Effective capillary number and appar-

ent viscosity

We have seen that Eq. (3) allows predicting the evolution of

h/R for a homogeneous liquid.12 In addition, the dip coating

experiments have established that using Caϕleads to a quan-

titative prediction of h.26,27,34 Since our measurements of h

are very accurate with the present method, we can estimate

the local viscosity ηeﬀ (ϕ)that would lead to a coating ﬁlm of

measured thickness hwhen the rear of the plug is traveling at

the velocity ur.34 More speciﬁcally, for each experimental data

point of h, we use Eqs. (4) and (6) to obtain:

ηeﬀ (ϕ) = γ

urh/R

1.34 −3.35 h/R 3/2

.(7)

Figure 5: (a) Dimensionless deposited ﬁlm thickness h/R as a function of the capillary number based on the eﬀective viscosity of

the suspension Caϕ, where the viscosity η(ϕ)is calculated with Eq. (5) with the initial volume fraction in the syringe (ϕ= 0.19).

The solid line corresponds to Eq. (6). Inset: h/R as a function of the capillary number based on the viscosity of the interstitial

ﬂuid Ca. The solid line is given by Eq. (3). (b) Eﬀective viscosity of the suspension ηeﬀ leading to a coating thickness h

estimated using Eq. (7) as a function of h. The dotted line correspond to η(ϕ= 0.19) = 0.24 Pa s calculated using the Maron-

Pierce correlation. (c) Estimated volume fraction of particles ϕeﬀ based on the Maron-Pierce correlation and obtained using Eq.

(5). The horizontal dotted line represents the initial volume fraction in the suspension, ϕ= 0.19. For all ﬁgures the experiments

are performed with particles of diameter d = 80 µmdispersed at a volume fraction of ϕ= 0.19 in the silicone oil, and the plug

is translated in a capillary of radius R= 750 µmm. Each color represents an experiment performed at one given ∆p.

In this expression, the interfacial tension γis not modiﬁed by

the presence of particles,19,34,47 Ris a ﬁxed parameter, and we

measure hand urso that we can calculate ηeﬀ(ϕ)for each data

point using Eq. (7). In addition, once we have an estimate of

ηeﬀ (ϕ), we can calculate the expected volume fraction ϕeﬀ of

the ﬁlm deposited on the wall of the capillary tube using the

Maron-Pierce correlation [Eq. (5)]:

ϕeﬀ =ϕm1−rηf

η(ϕ),(8)

with ϕm= 0.59.

We report the estimation of ηeﬀ and ϕeﬀ in ﬁgures 5(b) and

5(c), respectively. We have superimposed the volume fraction

in the initial plug, ϕ= 0.19, and the corresponding viscosity

calculated with Eq. (5). We observe that, in average, during

the entire translation of the suspension plug, the estimated vis-

cosity and the corresponding volume fraction leading to the lo-

cal coating ﬁlm are of the same order of magnitude as the initial

composition of the suspension. Note that in ﬁgures 5(b) and

5(c) one color of data point corresponds to a single experiment.

For a given experiment, there is also some dispersion around

the viscosity of the initial suspension. The same observation

can also be done with ϕeﬀ calculated from the Maron-Pierce

correlation. ϕeﬀ does not seem constant and seems initially

smaller than ϕ= 0.19 but then increases and becomes larger at

the end. These results suggest again that an eﬀective viscosity

based on the initial composition of the suspension provides a

good ﬁrst estimate of the resulting coating ﬁlm but cannot ex-

plain the systematic deviation obtained during one experiment.

We shall come back on this point later.

Figure 6: Rescaled thickness of the coating layer h/R as a

function of the eﬀective capillary number Caϕfor varying initial

volume fraction in the suspension plug, from ϕ= 0.05 to ϕ=

0.25. The experiments are performed with a suspension of 80

µmparticles dispersed in silicone oil and the capillary tube has

a radius R= 750 µm. The solid line is given by Eq. (6) and the

viscosity used to calculate Caϕis based on the Maron-Pierce

correlation with ϕcorresponding to the initial volume fraction

of the suspension in the syringe and indicated in the legend.

4.2 Influence of the volume fraction of

the suspension

To provide a more exhaustive characterization, we now consider

diﬀerent initial volume fractions ϕ= 0.05,0.1,0.15,0.2, and

0.25 for 80 µmparticles. The results are reported in ﬁgure

6, which shows the evolution of the rescaled coating thickness

h/R as a function of the eﬀective capillary number Caϕand the

comparison with the prediction given by Eq. (6). Here again,

the value of η(ϕ)is calculated with the initial volume fraction in

the syringe and the Maron-Pierce correlation. Apart from the

very small values of h/R < 0.1, where the coating ﬁlm is thinner

than the particle diameter, the experimental data matches well

the prediction given by Eq. (6). Again, some deviations around

the average values are observed, and the experimental points

seem more scattered at larger volume fractions.

4.3 Influence of the particle diameter

To investigate if the scattering of the data observed can be

due to a peculiar coating regime, we consider the inﬂuence of

the diameter of the particles on the coating thickness in ﬁgures

7(a)-(b). Similar to previously, we report in ﬁgure 7(a) the

evolution of h/R as a function of Caϕ(and as a function of Ca

in the inset of ﬁgure 7(a)). Again, for small enough particles,

the prediction given by Eq. (6) using η(ϕ)calculated using

Eq. (5), provides a fair estimate for the coating thickness for

small and moderate sizes of particles. However, the dispersion

of the points around the expected value seems to increase when

increasing the particle diameter. In particular, for d= 140 µm

particles the collapse is poor.

We rescale in ﬁgure 7(b) the coating thickness hby the par-

ticle diameter d. Past experiments in the dip coating conﬁg-

uration have reported that the thick ﬁlm regime is observed

for eﬀective capillary numbers such that h≥d.26,27 A similar

approach in the present situation using Eq. (6) together with

the condition h/d ≥1leads to the threshold eﬀective capillary

number to be in the thick regime:

Caϕ

∗=d

1.34 −3.35 d/R 3/2

.(9)

We report the capillary threshold in ﬁgure 7(b) for the dif-

ferent particle diameters. We observe that for Caϕ<Caϕ

∗the

thickness indeed seems to saturate around the particle diame-

ter, h≃d. This observation is reminiscent of what was reported

in the dip coating of a plate and corresponds to the heteroge-

neous regime with a monolayer of particles.26 The larger the

particle, the larger the capillary number needs to be to recover

the eﬀective viscosity regime. These results show that our ex-

perimental results quantitatively match the prediction of the

threshold value Caϕ

∗. Beyond Caϕ

∗the thickness increases fol-

lowing Eq. (6) but with some scattering of the data. This result

and the threshold capillary number echo the observations made

in the dip coating conﬁguration.26,27

In summary, the diﬀerent coating regimes, their domains of

existence, and the estimated coating thicknesses are compara-

ble between the present case of coating of the inner wall of a

capillary tube and the dip coating process. 26 At ﬁrst order,

this result can be expected since the two conﬁgurations share

very similar ﬂuid mechanics features. The topology of the ﬂow

in the two situations are comparable and, in particular, they

both present a stagnation point followed by a coating ﬁlm of

Figure 7: (a) Rescaled thickness of the coating ﬁlm h/R as

a function of the eﬀective capillary number Caϕfor varying

particle diameters: 20 µm(yellow circles), 40 µm(green circles),

80 µm(light blue circles) and 140 µm(dark blue circles). The

volume fraction of particles in the initial suspension is ϕ=

0.19, the viscosity used to calculate Caϕis calculated using the

Maron-Pierce correlation, and the inside radius of the tube is

R= 750 µm. Inset: Evolution of h/R as a function of the

capillary number based on the viscosity of the interstitial ﬂuid

Ca. The solid line is the coating thickness given by equation

(3). (b) Evolution of h/d when varying the eﬀective capillary

number Caϕand for diﬀerent size of particles. The horizontal

dashed line indicates h=d, the vertical dotted lines are the

threshold eﬀective capillary number Ca∗

ϕgiven by Eq. (9) at

which the coating ﬁlm starts to be in the thick ﬁlm regime.

The blue region denotes the regime of thick ﬁlm, the yellow

regime is the monolayer regime, and the white region is when

only liquid is observed.

constant thickness.34,38 Previous studies in the dip coating con-

ﬁguration and in the present capillary tubes conﬁguration but

with isolated spherical particles have reported that the thresh-

old for single particle entrainment are similar.36,39 . In the case

of non-dilute suspensions, the coating of the capillary tube can

Figure 8: Examples of evolution of (a) the dimensionless coating thickness h/R as a function Caϕ, (b) the calculated corresponding

eﬀective viscosity ηeﬀ and (c) the calculated corresponding solid fraction ϕeﬀ based on the viscosity calculated as a function of

the coating thickness. The experiments are performed with a suspension of 80 µmparticles and the capillary tube has a radius

R= 750 µm. The dotted line in (a) corresponds to Eq. (6) where η(ϕ)is based on the viscosity of the suspension in the syringe.

Each series of colors indicates one experiment going from the light to the dark colors (the direction of time is indicated by the

arrow).

be predicted at ﬁrst order using the average volume fraction,

and the corresponding eﬀective viscosity. However, as reported

above, a signiﬁcant diﬀerence is observed in the capillary tube

conﬁguration: during one experiment, a systematic deviation

from the mean value of h/R is observed. It leads to much more

scattering of the data than what was observed for the dip coat-

ing. We consider in the next section this point in more detail,

and in particular, we highlight the diﬀerences between the dip

coating and the intermittent ﬂow in a tube.

4.4 Evolution of the coating thickness

and shear-induced migration

At the beginning of an experiment, we introduce a plug of

length Lwith a controlled volume fraction ϕ. Then, once the

suspension plug translates, it deposits a suspension ﬁlm on the

wall of the capillary tube, making the plug shrink in volume

and thus in length. We show in ﬁgure 8(a) three examples

of experiments at volume fractions ϕ= 0.1,0.15 and 0.2where

we report the evolution of the dimensionless ﬁlm thickness from

the beginning of the experiment when the plug is long (light

colors) to the end of the experiment (dark colors) when the

plug is signiﬁcantly shorter. In all cases, we observe the same

evolution of h/R. In average, the coating thickness has a mean

value centered around the prediction given by Eq. (6) based on

the viscosity η(ϕ)calculated with the initial volume fraction.

However, h/R is smaller than the mean value at the beginning

of an experiment and larger at the end. This eﬀect is peculiar

to the present conﬁguration in a tube and was not observed

for dip coating.26 To shed light on this peculiar behavior, we

extract the local viscosity ηeﬀ when the ﬁlm is deposited in ﬁg-

ure 8(b) and the corresponding volume fraction ϕeﬀ in ﬁgure

8(c). Assuming that the suspension behaves as a homogeneous

mixture, we estimate the eﬀective viscosity of the suspension

ηeﬀ leading to a coating thickness husing Eq. (7).34 Then,

we estimate the volume fraction of particles ϕeﬀ using Eq. (8).

It appears that the volume fraction of particles in the coating

ﬁlm increases during an experiment. It is initially smaller than

the volume fraction in the syringe and becomes larger at the

end of the experiment when the suspension plug has shrunk

signiﬁcantly.

The ﬂow of a suspension plug in a capillary tube presents a

key diﬀerence with the dip coating: the shear-induced migra-

tion of the particles,48–50 and a "fountain-ﬂow" where particles

are more concentrated at the front meniscus.41,51–54 Indeed,

the ﬂow of a plug in a cylindrical channel is a shear ﬂow where

the particles migrate from high shear stress to low shear stress

region.48 As a result the volume fraction of particles is not

constant along a cross-section of the tube, but instead larger

at the center and smaller near the outer wall. As a result, the

measured local viscosity of the suspension also changed spa-

tially. The migration of the particles is the result of combined

eﬀects of hydrodynamic forces between the particles and non-

hydrodynamic interactions such as surface friction or repulsive

forces between particles. Diﬀerent models have been devel-

oped to predict the distribution of particles, such as the dif-

fusion model, 55 , and the suspension balance model. 56,57 How-

ever, the present conﬁguration exhibits an additional challenge,

the plug is limited in length compared to an inﬁnite Poiseuille

ﬂow. Therefore, a "fountain ﬂow", with particles at the cen-

ter going faster than the one near the boundary, leading to an

internal recirculation is also observed.51 In particular, in ad-

dition to being non-uniform along a cross-section, the volume

fraction is also not uniform along the length of the suspension

plug.53 A theoretical description of the distribution of particles

at every location in the suspension plug is beyond the scope

of the present study given the high complexity of the problem.

Nevertheless, we can provide a qualitative explanation for the

evolution of the ﬁlm thickness during an experiment.

Due to the shear-induced migration and the fountain ﬂow,

the volume fraction of particles near the outer wall at the rear

of the plug is initially smaller than the average volume fraction

ϕprepared in the plug. Since the suspension ﬁlm initially de-

posited on the wall comes from the streamlines near the outer

wall (see e.g., Ref.39 for a description of the ﬂow), the initial

coating ﬁlm also has a smaller volume fraction, as observed in

ﬁgure 8(c). Therefore, the volume fraction in the remaining

suspension plug increases in time. As a result, even if the vol-

ume fraction of the suspension near the wall remains smaller

than near the centerline the total volume fraction in the sus-

pension plug increases signiﬁcantly during an experiment. Con-

sequently, later in one experiment the volume fraction ϕeﬀ in

the region of thickness hnear the wall increases in time, and

eventually becomes larger than the initial volume fraction, as

can be seen in the late stage of all examples shown in ﬁgures

8(a)-(c).

One important consequence of this eﬀect is that, even if in

average it is possible to deposit a ﬁlm of thickness and composi-

tion of the same order of magnitude as the initial composition

of the suspension, the actual thickness and composition will

both increase along the tube during the translation of the plug.

Therefore, although this method can be interesting to deposit

particles on the inner wall of a capillary tube,40 it may be

challenging to control very well the resulting coating and prop-

erties. We should emphasize that such an eﬀect would also be

observed at a constant ﬂow rate, where urwould be constant,

but where ϕwould follow a similar evolution.

5 Conclusion

In summary, despite the shear-migration of particles leading

to a non-homogeneous particle distribution in a cross-section

of the tubing, our results suggest that for signiﬁcant coating

thickness, larger than the diameter of the particles (h > d), a

good estimate of the ﬁlm thickness is given by equation (6).

In this expression, the usual classical capillary number has to

be replaced by a capillary number based on the eﬀective vis-

cosity of the suspension [Eq. (4)]. This observation is reminis-

cent of the observation obtained with the dip coating conﬁgura-

tion.26,27 More speciﬁcally, the same three coating regimes are

recovered: (i) thin-ﬁlm of pure liquid only, (ii) a heterogeneous

regime where the thickness is almost constant, h∼dover a

range of capillary number, and (iii) a thick-ﬁlm regime consist-

ing of multiple layers of particles. Yet, in this third regime,

the presence of local variation of volume fraction leads to an

evolution of the coating thickness and composition of the liq-

uid ﬁlm during the translation of the plug. Nevertheless, the

mechanisms of capillary ﬁltration and sorting proposed in the

dip coating case should still mostly hold in the present conﬁg-

uration.34,36,37

However, a main diﬀerence highlighted by our experiments

is a systematic evolution of the coating thickness, and the lo-

cal volume fraction, during the translation of the plug. This

systematic evolution is associated with the complex ﬂow in the

plug: a combination of shear-induced migration and fountain

ﬂow. As a result, even if at ﬁrst order the average volume frac-

tion in the coating is the one of the plug, it is initially smaller

and at later stages larger. It leads to two main consequences;

First, the composition of the coating ﬁlm and the thickness

changes along the capillary tube. This result would still be ob-

served for a constant translation velocity. Second, contrarily to

a Newtonian homogeneous ﬂuid, there is a strong inﬂuence of

the history of the system. Indeed, the evolution of the volume

fraction is going to depend on the volume of suspension already

left on the wall of the tube.

A more reﬁned theoretical model that captures the observed

discrepancies due to the inhomogeneity of particles in the plug

needs to be developed. Such a model could contribute to ratio-

nalizing the experimental results at the next order and account

for the local volume fraction around the air-liquid meniscus.

This prediction would also allow a uniform coating on the wall

of capillary tubes by varying the velocity of the plug, for in-

stance, in a ﬂow-rate driven system, to account for the variation

of thickness and volume fraction. Note that the coating ﬁlms

observed in the present conﬁguration typically contain only a

few layers of particles so that conﬁnement could introduce addi-

tional eﬀects. For instance, since the interface exerts a capillary

force on the particles, force chains could be generated between

particles leading to second-order eﬀects. In addition, the pres-

ence of the interface and the local curvature could modify the

local volume fraction of particles, as considered recently during

the motion of a contact line.25,58

We should also emphasize that the variation in volume frac-

tion during the translation of a suspension plug in a porous me-

dia could also favor clogging by bridging at constrictions. 6,59,60

The present studies illustrate the heterogeneities and complex-

ity brought by the presence of particles in capillary systems

where the size of the particle is comparable to the capillary

object.

Acknowledgements

This material is based upon work supported by the National

Science Foundation under NSF Faculty Early Career Devel-

opment (CAREER) Program Award CBET No. 1944844 and

by the American Chemical Society Petroleum Research Fund

through the ACS-PRF 60108-DNI9 grant.

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