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Pouring of grains onto liquid surfaces: dispersion or lump formation? Xin Yi Ong, Spencer E. Taylor, and Marco Ramaioli Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.9b01277 • Publication Date (Web): 08 Aug 2019 Downloaded from pubs.acs.org on August 9, 2019

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Pouring of grains onto liquid surfaces: dispersion or

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lump formation?

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Xin Yi Onga, Spencer E. Taylorb, Marco Ramaiolia,c,*

4

a Department

5

United Kingdom.

6

b

Department of Chemistry, University of Surrey, GU2 7XH, United Kingdom

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c

UMR 782, GMPA, INRA, 78850 France

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Wetting, capillarity, dispersion, grains, lumps

of Chemical and Process Engineering, University of Surrey, GU2 7XH,

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ABSTRACT: This study considers the consequences of adding grains to an air-liquid

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interface from a funnel. Depending on the grain contact angle and liquid surface tension,

ACS Paragon Plus Environment

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the interface is found to support a single or multiple layers of grains, forming a granular

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stack. By continuing to add grains, the stacks grow until either the lower grains disperse

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in the liquid, or the complete stack breaks free from the surface and sinks as a dry powder

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lump. Herein, the effects of grain contact angle, density and size on these processes are

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studied experimentally and a theoretical analysis given. The maximum number of grains

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contained in a floating stack and its critical depth are observed to increase as the grain

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size decreases. The maximum number of grains scales with the Bond Number (Bo) as

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Bo-1.82 when stack detachment is observed and with a higher exponent > -2.0 when grains

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disperse into the liquid. As a result of these different scaling exponents, a critical Bond

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number above which grains wet and disperse can be identified. Favourable conditions for

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dispersion are achieved with larger grains and, to a lesser extent, by lower surface tension

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and contact angle. The critical Bond number separating grain dispersion from lump

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formation increases with an increasing grain contact angle, thus providing a physical

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justification for increasing grain size with common processes such as granulation or

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agglomeration. Conversely, a quantitative framework to interpret the limitations in

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dispersing small grains is proposed, justifying the need for low contact angle or liquids


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with low surface tensions, both favoured by the use of surfactants. The present findings

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have identified conditions under which lump formation occurs, and hence how these

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undesired phenomena can be avoided in applications requiring the efficient dispersion of

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grains across a liquid interface.

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INTRODUCTION

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The floating and sinking of grains at air-liquid surfaces are of importance in many

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industrial applications. In the food industry, carbohydrates, such as maltodextrins,

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undergo a complex rehydration process 1. Furthermore, the presence of fat can affect

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negatively the wettability and cause the undesirable formation of partially dry powder

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lumps. The reconstitution of food powders can be described by four main stages, namely

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wetting, capillarity, dispersion and dissolution.2 In practice these stages occur

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simultaneously and influence each other, thereby making the analysis of the individual

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processes challenging. In previous work, the influence of contact angle,3,4 grain density,

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size5–9 and mass flow rate of grains added to the interface10–12 have been studied in order to

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improve the reconstitution performance.


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When an object impinges the surface of a liquid, the surface may undergo strong

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deformations and the shape of the meniscus surrounding the object is analytically

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described by the Young-Laplace equation.13 A small object that is denser than the liquid

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can float as a result of the vertical force corresponding to the weight of the liquid displaced

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by the meniscus.14 Hence, the equilibrium position of a floating object is governed by its

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mass, the buoyancy force and the surface tension, under the constraint of the contact

50

angle. Thus, the transition of an object from a floating to a sinking condition may be

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influenced by the effects of contact line.15 For the case of a pile of grains, the situation

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becomes more complicated as the contact line has significant undulations16 and the

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contact angle may vary significantly with the increased curvature.17

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During wettability studies, both Jurin’s law18 and Washburn’s theory19 have been widely

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employed to describe the process of liquid wicking in a porous medium. It is also known

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that water penetration in a porous medium was shown to be dependent on the tortuosity.20

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For example, there is a significant difference when penetration occurs in cylindrical

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capillaries as compared to powder beds consisting of spherical grains, for which the

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different internal structures affect the flow behavior.21 The critical contact angle, 𝜃0∗ , below


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which wicking occurs in monodisperse layers of beads was experimentally observed to

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be 55˚,4 which is considerably lower than the 90˚ limit for cylindrical capillaries.

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Several authors22–24 studied the behavior of self-assembled monodisperse layers of

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grains, also called rafts, at an oil-water interface, to predict the shape and the size of

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𝜌𝑆 ― 𝜌𝑊 𝑅 these assemblies before sinking. The dimensionless number 𝐷 = ( 𝜌𝑊 ― 𝜌𝑂)𝑎, is

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defined based on the grain density (𝜌𝑆), the densities of two fluids (𝜌𝑊, 𝜌𝑂), the radius (R)

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of the grains and the capillary length, 𝑎 = (𝛾/(𝜌𝑊 ― 𝜌𝑂)𝑔) 1/2. The monolayer grain rafts

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were found not to sink when 𝐷 ≤ 1. 22–24. . The dimensionless number D compares the

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weight of a grain to the maximum buoyancy force originated when the meniscus reaches

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its maximum depth, i.e the capillary length.

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Jones et al.24 discussed how the addition of the grains to the surface affects the

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resulting geometry (i.e., “rafts” versus “stacks”) and the size limit of the assemblies before

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they sink through the interface. These workers also found that grain rafts sink with a

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higher number of grains than stacks. In these studies,22–24 the grains are assumed to be

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fully wetted by the oil.


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Raux et al.4 studied the creation of a stack using grains having a contact angle θ > 𝜃0∗

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and showed that wicking occurs when the stack depth, h, exceeds a critical stack depth,

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h*. For grains that are small compared to the capillary length and poured very gently onto

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the interface, the critical stack depth was found to depend only weakly on grain size.

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Consistent with most of the studies cited above, the present work used insoluble grains

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to focus on the conditions leading to an effective wetting and dispersion, without the

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influence of dissolution. The influence of the grain size, density, contact angle and surface

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tension on stack formation was considered. We report experimental results and use

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dimensional analysis to interpret the conditions leading to grain dispersion and stack

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detachment, to identify conditions for the effective dispersion of grains in liquids which

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avoid the formation of dry lumps.

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EXPERIMENTAL

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Materials

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The impact of grain size and density was studied using three types of spherical grains:

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glass beads (from Sigma-Aldrich, UK), poly(methyl methacrylate) beads (PMMA, from


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Sigma-Aldrich, UK) and yttrium-stabilized zirconium oxide beads (ZY-S, from Sigmund

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Lindner,

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dichlorodimethylsilane solution in heptane, and toluene) were purchased from Sigma-

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Aldrich UK.

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Methods

UK).

All

other

reagents

(absolute

ethanol

≥

99.8%,

glycerol,

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The grain size distribution of the powders was determined using a QICPIC image

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analysis system with a gravity dispenser. Table 1 summarizes the mean grain diameter

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d50 , the span s = (d90 – d10) / d50, the aspect ratio (sphericity), the grain densities,

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measured by gas pynometry (AccuPycnometer 1330, Micrometrics Instrument Corp.,

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Norcross, GA, USA) and the bulk densities, measured by weighing powder in a FT4

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powder rheometer (25 mm diameter and 25 mL volume) with split vessels. As shown in

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Table 1 (and in Figure S1 in the Supporting Information), all the grains have a high degree

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of sphericity (aspect ratio ≈ 0.93). The glassy grain surfaces (glass and PMMA) are

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smooth on a micron scale, and ZY-S exhibits some surface structure, as observed using

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digital optical microscopy (DSX 500, Olympus IMS).


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Table 1. Characteristics of the grains used in this study.

Material

Grain

size Span, s

diameter, d50

Aspect

Density,

Bulk

ratio

𝜌𝑆

density,

(kg/m3)

𝜌𝐵

(mm)

𝜃0∗ (°)

(kg/m3) PMMA

0.907

0.22

0.926

1200

750

70 ± 1.7

PMMA

0.497

0.41

0.928

1170

695

74 ± 1.5

Glass

1.158

0.21

0.944

2495

1570

71 ± 0.8

Glass

0.606

0.42

0.945

2497

1595

68 ± 1.5

Glass

0.266

0.44

0.941

2497

1550

70 ± 0.8

Glass

0.082

0.58

0.931

2490

1530

72 ± 1.5

ZY-S

1.152

0.20

0.931

6022

2200

77 ± 0.8

ZY-S

0.738

0.42

0.930

6020

2160

69 ± 1.5

ZY-S

0.305

0.29

0.933

6010

2120

74 ± 1.3

106 107

The as-received grain samples were initially hydrophilic. The grains were cleaned by

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contacting with 1 mol/L hydrochloric acid solution for 1 hour, followed by thoroughly

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rinsing with deionized water and drying for 4 hours at 60˚C. The grains were then silanized

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to induce hydrophobicity, following the silanization protocol by Hamlett et al.3, using a 5%

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dichlorodimethysilane solution in heptane at room temperature. This was followed by

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rinsing with acetone (except PMMA), allowing to air-dry for 2 hours and oven-drying for 8

113

hours at 60˚C.


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Deionized water-ethanol mixtures of varying composition (expressed as mass fraction

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of ethanol, Mf) were used as the liquid phases, enabling the liquid density and the wetting

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properties of the grains to be systematically altered.4 Contact angles of single grains that

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had been carefully positioned at the air-liquid interface of the different ethanol/water

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mixtures contained in a 1 cm  1 cm quartz cuvette, were measured, as shown in the

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inset to Figure 1. Each measurement was repeated on ten different beads to provide

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mean values. The DropSnake plugin of ImageJ25 was used to compute the contact angles

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from images obtained using an FTA100 Drop Shape Analyzer (First Ten Angstroms,

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Portsmouth, VA, USA). The surface tensions of the ethanol/water mixtures were

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determined at room temperature by the Wilhelmy plate technique using a Krüss K10

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tensiometer.

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The effect of increasing viscosity on wetting properties of the grains was also verified in

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the case of ternary mixtures of glycerol, deionized water and ethanol. The viscosity of the

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liquids at 22℃ were measured by a Paar Physica UDS 200 controlled stress rheometer

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(Anton Paar GmbH, Graz, Austria).


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In order to study powder dispersion, a glass cell (15.5 cm  11.0 cm  8.0 cm) containing

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the test liquid was used. Initially, a uniform single layer (raft) of dry grains was deposited

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on the liquid surface. Further grains were then poured on the surface through a paper

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cone with a known diameter aperture positioned at a fixed position and distance from the

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surface. Depending on their contact angle and density, the grains either pass straight

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through the surface and disperse in the liquid, or are retained by the surface as a

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multilayer stack. The creation of a raft initially avoids a lateral movement of the

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subsequent stack.

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The process was monitored in terms of growth and evolution of the stack using a Basler

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camera with a resolution of 2.3 Mpixels (acA1929-155 μm). The mass flow-rate of the

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grains was recorded continuously using a Sartorius 2250 balance connected to a

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computer. Funnels with different orifice diameters, dO = 2, 2.5 and 3 mm, were used to

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maintain a constant mass flow rate, while changing the grain size and density. The

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average mass flow rate was (3.8 ± 1.1)  10-4 kg/s. The height of the orifice above the

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undisturbed liquid surface, dH, was 30 mm. The maximum depth of the stacks, h*, and the

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maximum stack mass, m*, were recorded at the point of either wicking or stack


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detachment being observed. All experiments were carried out at ambient temperature at

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30-40% relative humidity.

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RESULTS

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Effect of liquid composition on surface tension and grain contact angle

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Figure 1 shows the effects of the ethanol mass fraction, Mf, on the liquid-air surface

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tension and the contact angle of the different types of hydrophobized grains. In general,

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both the contact angle and surface tension data are seen to decrease monotonically with

152

increasing Mf. The contact angle varied in the range 100 ± 1.4˚ to 48 ± 1.1˚ and depended

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on the material and grain size.


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Figure 1. Effect of the ethanol mass fractions in the liquid (Mf) on the contact angle of

156

silanized grains of different sizes (d50 indicated in the legend) and on the air-liquid surface

157

tension (black squares). The inset shows images of 1.158 mm glass beads at the air

158

interface of different solutions.

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Critical contact angle for stack formation

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Figure 2 shows schematically the conditions leading to stack formation when pouring

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grains continuously. When the contact angle of the grains is lower than a critical contact


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angle (θ ≤ 𝜃0∗ ), no granular stack is formed, the interface only being able to support a

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granular raft, as previously reported by Raux et al.4

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Table 1 summarizes the critical contact angles (𝜃0∗ ) measured experimentally for

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different grain densities and sizes. As expected, such critical angles are significantly lower

166

than 90°. However, they are higher than the critical contact angle 𝜃0∗ ≈ 51° 26,27 predicted

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theoretically for stacks composed of monodisperse grains that are smaller than the

168

capillary length and placed gently onto the interface.4 Although polydispersity and packing

169

defects could contribute to this difference, pouring grains continuously from a funnel is

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more likely to be the dominant factor. As a result, the kinetic energy of the grains can

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provide perturbations that are able to overcome the shallow energy barriers4 preventing

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wicking into the pores within grains having a contact angle higher than but close to 𝜃0∗ .

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The maximum size of granular stacks, limited by wicking or by lump formation

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When 𝜃 > 𝜃0∗ , two different scenarios occurred when grains were fed continuously from

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a funnel. It was observed that the stacks grew until either: (i) The size became limited by


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the progressive dispersion of wetted grains into the liquid, or (ii) a sudden detachment of

177

the stack led to the formation of a dry lump (as schematically illustrated in Figure 2).

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Wicking is detected by the detachment of individual grains from the stack, as liquid

179

invades the pores of the stack. Conversely, the stack is said to detach when it sinks as a

180

whole, air being maintained in the pores, with individual grains not dispersing into the

181

liquid. The corresponding maximum number of grains (𝑁𝑊 or 𝑁 ∗ , respectively) are

182

computed using the formula 𝑁𝑊 or 𝑁 ∗ = 𝑚 ∗ /3𝜋( 2 ) 𝜌𝑆. Each experiment was repeated

183

at least five times.

4

𝑑50 3


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Figure 2. Experiments investigating the dispersion of grains poured onto a static air-liquid

186

interface from a funnel. When the contact angle θ ≤ 𝜃0∗ , the grains cross the interface

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and disperse individually. When the contact angle θ > 𝜃0∗ , the grains form a stack. Wicking

188

of the liquid in the pores can lead to the progressive dispersion of the grains if N = NW
Bo*. Conversely, if N = N* < NW, i.e. when Bo < Bo*, the granular stacks

190

detach and sink forming dry powder lumps.


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Indeed, either the liquid was progressively able to wick into the pores, with some grains

192

able to disperse into the water (as illustrated in Figure 3 a and b), or the equilibrium of the

193

whole stack was suddenly compromised and it sank while most grains remained dry (as

194

illustrated in Figure 3 c and d).

195 196

Figure 3. Different mechanisms limit the size of the granular stacks obtained by pouring

197

grains continuously onto an air-liquid interface. The sequences (a) and (b) show the

198

sedimentation of grains occurring when the liquid is able to wick into the stacks (a) for d50

199

= 0.907 mm PMMA grains with θ = 88 ± 3.8˚ and (b) d50 = 0.606 mm glass grains with θ

200

= 77 ± 1.6˚. The sequence (c) shows wicking followed by the detachment of the stack for

201

d50 = 0.606 mm glass grains with θ = 90 ± 1.7˚. In (d) the detachment of and stack forming


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a dry lump can be observed for d50 = 0.266 mm glass grains with θ = 90 ± 1.4˚. The

203

average mass flow rate was 0.00038 ± 1.07 x 10-4 kg/s and the funnel was located 30

204

mm above the liquid surface.

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Figure 4 illustrates quantitatively how the mechanisms (i) and (ii) limit the size of

206

granular stacks. All results were carried out with average mass flow rate of (3.8  1.1) 

207

10-4 kg/s and the funnel was located 30 mm above the undisturbed liquid surface. The

208

maximum depth of the stacks, h*, increased with the contact angle for all the grain sizes

209

considered. Furthermore, h* increased significantly when the glass grain size decreased

210

from 1.158 to 0.082 mm.

211

The stacks built with smaller grains and with higher contact angle detach from the

212

interface without grain dispersion, forming dry lumps. Conditions leading to stack

213

detachment are represented by blue symbols in Figure 4. The results obtained for the

214

0.082 mm grains are comparable with the experimental results for grains of radius R = 52

215

μm by Raux et al.4 who focused on smaller contact angles and only observed wicking, as

216

shown in the red dashed line in Figure 4.


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Conversely, the depth of the stacks formed from larger grains with low contact angle is

218

limited by the liquid wicking into the pores, leading to the sedimentation of individual

219

grains and clusters of grains. These conditions are represented using red symbols in

220

Figure 4. For intermediate size and wettability conditions, both liquid wicking and stack

221

detachment can be observed. It is important to notice that stack detachment does not

222

only occur for completely hydrophobic grains, as grains with a contact angle as low as

223

75° have also been observed to form stacks that detach from the interface and sink as

224

dry lumps.

225

The effect of a ten-fold increase in viscosity on stack stability was also verified in the

226

case of ternary mixtures of glycerol, deionized water and ethanol. The maximum depth of

227

the stacks obtained with these solutions was found to depend on the grain size and

228

contact angle, but not on the liquid viscosity, suggesting that the dynamics of liquid

229

wicking into the pores is not a controlling mechanism.

230

Similar results have also been obtained for the other materials. An approximately linear

231

relationship exists between the contact angle θ and the maximum stack depth, h* as

232

shown in the Supporting Information (Figure S2), but no clear interpretation can be


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obtained for these raw results. A dimensional analysis has been applied to highlight the

234

phenomena governing the critical stack size, as described below. In general, the

235

maximum depth of the stacks, h*, is found to increase with increasing contact angle, and

236

decreases with increasing grain size.

237

238 239

Figure 4. Effect of the contact angle θ on the maximum depth of a granular stack, h* for

240

glass grains of different sizes. Blue symbols indicate stacks that detach leading to lump


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241

formation, red symbols indicate conditions leading to grain dispersion. The average mass

242

flow rate was (3.8 ± 1.1)  10-4 kg/s. The funnel was located 30 mm above the liquid

243

surface and using water/ethanol solutions except where stated otherwise. The black

244

dashed line was obtained from Raux et al.4

245

By knowing both the maximum depth of the stacks, h*, and the maximum stack mass,

246

m*, at the point of either wicking or stack detachment being observed, a linear relationship

247

exists between them for grains of different densities and sizes, as shown in the Supporting

248

Information Figure S3. Also shown in Figure S3 (inset), it is seen that the stacks formed

249

from different density grains have different shapes. In fact, apart from the maximum stack

250

depth, the present study does not consider the detailed shapes of the stacks and their

251

height above the liquid surface. This helps to explain why, for a given h*, the m* was found

252

to decrease as the density of grain increases. Also, at a given h*, the stacks formed with

253

different density grain have different contact angles. Although the density of ZY-S is six

254

times higher than the PMMA, when the contact angle of ZY-S is above 90 it is still

255

possible to form the stack and achieve comparable h* to the stack formed with PMMA (θ


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= 77). However, the perimeter of the ZY-S stacks are much smaller than found for

257

PMMA.

258

DISCUSSION

259

The above results indicate that the stability of an assembly of granular stack on a liquid

260

surface is governed by a number of critical factors, and the fate of the grains can be

261

dispersion or formation of a dry lump. The important factors identified herein include the

262

wettability of the grains and grain size.

263

Grain dispersion versus stack formation

264

As shown in Figure 4, the maximum stack depth h* was observed to increase more or

265

less linearly with θ, for high contact angles. As discussed above, in the present study the

266

kinetic energy of the grains impacting the liquid surface triggered wicking for higher

267

contact angles than the critical contact angle (𝜃0∗ ) observed by Raux et al.4

268

The results obtained using larger grains indicated that only smaller stacks were formed.

269

This observation is also consistent with the theory for wicking in a powder,4 because


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Page 22 of 37

270

grains larger than the capillary length of the system deform the interface. This increases

271

the meniscus curvature, facilitating wicking and limiting the stack depth.

272

In the literature, the ratio 𝐷 = 𝜌𝑆/𝜌𝐿 has been used to characterize the behavior of rafts

273

and stacks at a liquid interface23,24,28,29. It is interesting to note that the shapes of stacks

274

constructed from PMMA grains (Figure 3a) always involves a large pile above the liquid

275

surface. As a result of a lower grain density than the other types of grain studied, stacks

276

formed with PMMA grains only experience wicking, even though the PMMA grain density

277

is greater than the liquid density (𝜌𝑆 > 𝜌𝐿). It is therefore suggested by the present results

278

that the relevant density to be compared to the liquid density to predict whether a stack

279

sinks or not is the bulk density, rather than the grain density. Indeed, the PMMA grains

280

used have a bulk density lower than the liquid density (𝜌𝐵 < 𝜌𝐿), which prevents stacks

281

from detaching and forming lumps similar to glass grains (Figure 3 c and d).

282

For grains of comparable size, PMMA formed stacks that are larger than for glass, in

283

absolute terms. By normalizing the maximum depth of the stacks by the capillary length

284

and by 𝜌𝐵/𝜌𝐿 , PMMA and glass results obtained with large grains become similar, as

285

shown in Supporting Information, Figure S4. These results therefore suggest that for large


22

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286

grains the maximum depth of the stacks scales approximately as 𝜌𝑠―1/2, i.e., a four-fold

287

increase in grain density is expected to halve the maximum stack depth. However, this

288

scaling does not work for small grains and a better approach is presented below. Figure

289

S4 shows also that the normalized depth of stack increases with increasing contact angle

290

until it reaches 90. When the contact angle is higher than 90°, the normalized maximum

291

depth of the stacks does not depend on the contact angle. Instead, the change in the

292

maximum depth of stacks for contact angles higher than 90 observed in Figure 4 is

293

caused by the changes in liquid/air surface tension and capillary length that result from

294

using different water/ethanol mixtures.

295

Maximum number of grains for detachment or dispersion

296

The results presented in Figure 4 can be interpreted by normalizing the mass of grains

297

poured to create the stacks by the mass of a single grain, obtaining the maximum number

298

of grains present just before detachment or wicking, leading to grain dispersion. The

299

relative importance of buoyancy and surface tension acting on the grains can be captured

300

by the Bond number, 𝐵𝑜 = (𝜌𝐿)𝑔𝑅2/.


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Page 24 of 37

301

Figure 5 shows (in blue) the maximum number of grains 𝑁 ∗ in stacks undergoing

302

detachment and forming dry lumps, plotted against Bo. The stacks were prepared from

303

glass and ZY-S with different grain sizes. Surface tension and liquid density were varied

304

by changing the ethanol mass fraction.

305 306

Figure 5. The dependence of the maximum number of grains N in an island on the Bond

307

number Bo. The blue symbols represent the maximum number of grains before the

308

detachment of a stack occurs. The red symbols represent the maximum number of grains


24

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309

before wicking is observed, leading to the dispersion of grains into the liquid. The average

310

mass flow rate was 0.00038 ± 1.07 x 10-4 kg/s and the funnel was located 30 mm above

311

the liquid surface. The Bo-1.5 scaling proposed by Jones et al,24 is shown for reference.

312

It can be seen that the maximum number of glass grains, 𝑁 ∗ , decreases strongly when

313

the grain size increases from d50 = 0.082 mm up to d50 = 1.158 mm. When the glass

314

granular stacks sink, forming lumps, 𝑁 ∗ scales roughly as 𝑁 ∗ = 𝑘 𝐵𝑜 ―1.82, with 𝑘𝑔𝑙𝑎𝑠𝑠

315

≈ 5.34, as the solid black fitting line demonstrates. Also, the ZY-S granular stacks sink

316

forming lumps, with 𝑁 ∗ scaling approximately in the same way, with 𝑘𝑍𝑌 ― 𝑆 ≈ 0.17, shown

317

by the dashed black fitting line.

318

To predict when the whole stack should detach from the interface, Jones et al.24

319

considered the balance of vertical forces acting on the floating stack, including the weight

320

of the liquid displaced by the stack and the weight of the liquid displaced by the meniscus.

321

Considering the stack characteristic length scale, this leads to a scaling 𝑁 ∗ = 𝑘 𝐵𝑜 ―1.50,

322

with 𝑘𝐽𝑜𝑛𝑒𝑠 ≈ 3.1.


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Page 26 of 37

323

In our study, considering different grain densities and sizes allowed to identify a scaling

324

on a much wider range of Bo, however the exponent is lower than that proposed by Jones

325

et al.24. This could be due to the shape of the stack, which is flatter than a sphere in our

326

experiments, or to grain polydispersity that might affect the packing fraction. The higher

327

density of ZY-S reduces 𝑁 ∗ strongly, as indicated in the ratio 𝑘𝑔𝑙𝑎𝑠𝑠/𝑘𝑍𝑌 ― 𝑆 ≈ 31.

328

Conversely, the effect of contact angle on 𝑁 ∗ is very minor and is not discussed here for

329

brevity.

330

Figure 5 also shows (in red) the maximum number of grains 𝑁𝑊 in stacks undergoing

331

wicking, leading to wetted grains dispersing in the liquid. For a given Bo, when the liquid

332

wicks into the stack pores, allowing the dispersion of grains, the number of grains in the

333

stacks (𝑁𝑊) is lower than the maximum number that would induce the detachment of the

334

stack (𝑁 ∗ ). As shown in Figure 5, the curves for wicking were observed to move towards

335

lower 𝑁𝑊, when the density of the grains is higher.

336

For glass grains with d50 = 0.606 mm, wicking occurs when θ < 83. This corresponds

337

to reaching the condition 𝑁 = 𝑁𝑊 < 𝑁 ∗ . When θ = 73, the stacks built from glass grains

338

of sizes d50 = 0.606 (Bo = 2.22  10-2) and 0.266 mm (Bo = 4.41  10-3) both experience


26

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339

wicking. However, for d50 = 0.082 mm (Bo = 4.19 x 10-4), 𝑁 ∗ < 𝑁𝑊 and a stack reaches

340

the limit for lump formation before the limit for wicking.

341

For a given grain size, using a liquid with a higher surface tension decreases Bo and

342

leads to an increase in 𝑁 ∗ and 𝑁𝑊. It is important to observe that the contact angle and

343

surface tension are both changed simultaneously in our system. The dashed red lines in

344

Figure 5 connect values of 𝑁𝑊 obtained with different grain sizes and similar contact

345

angles and show that 𝑁𝑊 increases with increasing contact angle. This dependence is

346

stronger when the contact angle is lower than 90° and weaker when above 90° (blue

347

points).

348

The dashed red lines show that 𝑁𝑊 scales with Bo roughly with the exponent -2.0 for

349

both glass and ZY-S grains and -3.0 for PMMA grains. Such a dependence is stronger

350

than the dependence of 𝑁 ∗ (exponent -1.82) and this leads to two regimes. For any

351

contact angle above 𝜃0∗ , there exists a critical Bond number 𝐵𝑜 ∗ such that when 𝐵𝑜 >

352

𝐵𝑜 ∗ , wicking and grain dispersion occur for a smaller number of grains than the

353

detachment of the granular stack (𝑁𝑊 < 𝑁 ∗ ). Therefore, by progressively pouring grains


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Page 28 of 37

354

onto a liquid surface, the number of grains leading to wicking (𝑁𝑊) will be reached and

355

grains will be wetted and dispersed without forming lumps.

356

𝐵𝑜 ∗ identifies a critical grain size above which grain disperse. From Figure 5, it is

357

possible to obtain the dependence of 𝐵𝑜 ∗ on 𝜃 for glass grains when the dashed red lines

358

at specific contact angle intersects with the solid black fitting line. When the contact angle

359

𝜃 increases from 70 to 83, the critical Bond number, 𝐵𝑜 ∗ , increases from 3  10-4 to 4

360

 10-3.

361

Given the quadratic dependence of Bo on the grain size, increasing the grain size is an

362

effective way to exceed 𝐵𝑜 ∗ and promote grain dispersion and avoid lumps. More

363

specifically, increasing the grain contact angle from 70° to 83° requires increasing the

364

grain radius R by approximately 3.3 to obtain dispersion in a given liquid.

365

Another dimensionless number, 𝐷 = 𝜌𝐵/𝜌𝐿 comparing the bulk density of the grains and

366

the liquid density is also important in determining whether dispersion or lump formation

367

occurs. 𝐷 was varied by considering grains of different densities. For a given Bo, when

368

the bulk density and 𝐷 increase, the maximum number of grains sustainable before

369

wicking or stack detachment, 𝑁𝑊 or 𝑁 ∗ , they both decrease. Only wicking occurs when


28

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Langmuir

370

𝐷 < 1, as for PMMA grains, leading to grain dispersion. The definition of 𝐷 proposed in

371

this study is different from the definition commonly proposed in the literature, as the latter

372

could not explain the PMMA grain behavior.

373

More favorable conditions for dispersion are therefore achieved for larger grains and

374

lower density. Lump formation is favored for smaller, higher density and more

375

hydrophobic grains, for which the impact of grain size, density and contact angle have

376

been quantified. Overall, both Bo and D have been shown to govern whether wicking or

377

stack detachment occur.

378 379

380

Interesting directions for future investigation include studying systematically the effect of grain flow rate and the effect of grain cohesion.

CONCLUSIONS

381

The behavior of grains poured continuously from a funnel onto a static air-liquid interface

382

was studied experimentally. Depending on grain size, density, contact angle θ and liquid

383

surface tension, granular stacks were observed to experience wicking, resulting in grain

384

dispersion or to create dry lumps, detaching from the interface.


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385

When the contact angle is lower than the critical contact angle 𝜃0∗ , the interface is only

386

able to support a granular raft. When θ > 𝜃0∗ , grains are poured continuously, until either

387

individual grains disperse into the liquid or the whole stack sinks. The critical contact

388

angle, 𝜃0∗ , found in this study is higher than the value of 51 4, owing to the kinetic energy

389

of the grains promoting wicking into the pores within grains. When the contact angle is

390

greater than 𝜃0∗ , the maximum size of the granular stacks formed on the interface

391

increases with increasing contact angle and decreases with increasing grain size.

392

The occurrence of the wicking and stack detachment regimes can be interpreted based

393

on three dimensionless numbers: the contact angle, Bond number and the density ratio

394

D. The experimental results show that wicking occurs when the critical number of grains

395

leading to wicking is lower than the critical number of grains leading to stack detachment

396

(𝑁𝑊 < 𝑁 ∗ ), the opposite leads to lump formation. The experimental scaling with the Bond

397

numbers is discussed in the light of theoretical analysis. A critical Bond number exists,

398

above which wicking causes grain dispersion. This critical Bond number increases with

399

the contact angle and identifies the grain size above which grain will disperse.


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400

These results provide a quantitative understanding of the behavior of grains poured

401

onto a liquid surface, paving the way for improving powder dispersion in many industrial

402

applications.

403

404

ASSOCIATED CONTENT

405

The Supporting Information is available free of charge on the ACS Publications website

406

at DOI:

407

Wicking or stack detachment for grains of different densities and sizes (PDF)

408

409

AUTHOR INFORMATION

410

*Corresponding Author

411

E-mail: [email protected].

412

Notes

413

The authors declare no competing financial interest.


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414

ACKNOWLEDGMENT

415

This work was supported by funds from the Chemical and Process Engineering

416

Department of the University of Surrey.

417

REFERENCES

418

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