Powder Technology 195 (2009) 221–226

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De-agglomeration of hydrophobic and hydrophilic silica nano-powders in a highshear mixer

P. Ding, M.G. Orwa, A.W. Pacek ⁎School of Chemical Engineering, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom

⁎ Corresponding author.E-mail address: A.W.Pacek@bham.ac.uk (A.W. Pacek

0032-5910/$ – see front matter © 2009 Elsevier B.V. Adoi:10.1016/j.powtec.2009.06.003

a b s t r a c t

a r t i c l e i n f o

Article history:Received 12 November 2008Received in revised form 8 June 2009Accepted 12 June 2009Available online 21 June 2009

Keywords:Hydrophobic and hydrophilic silicanano-powdersDe-agglomerationEnergy densitypHRheology

The effect of energy density, pH and solid concentration on kinetics of de-agglomeration of hydrophobicsilica nano-powder in a high shear mixer and on the rheology of resulting suspensions was investigated andcompared with de-agglomeration kinetics and rheology of the suspension of hydrophilic silica nano-powder.In both types of nano-powders large aggregates were broken by fracture and erosion. In hydrophobic nano-powder erosion was more pronounced whilst in hydrophilic nano-powder erosion followed initial fracture oflarge aggregates. At sufficiently high energy input both hydrophobic and hydrophilic aggregates were brokeninto nano-aggregates but, even at the highest energy input, those nano-aggregates could not have beenbroken into single nano-particles. Rheology of the suspensions of hydrophobic nano-aggregates stronglydepends on pH and on solid concentration whilst rheology of suspensions of hydrophilic nano-powder israther weakly dependent on those parameters.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Silica nano-particles manufactured by flame hydrolysis of chloro-silanes are hydrophilic with the size between 7 and 40 nm [1,2] buttheir surface properties are frequently modified by reacting hydroxylgroups with silane coupling agents which leads to hydrophobicsurfaces [2,3]. Both types of nano-particles are commonly used incoating industry to improve physical properties of coatings and tofacilitate production, storage and application of coats [1]. They havestrong effect on rheology of suspensions [4] and are frequently used toenhance dispersion of pigments and to prevent separation inpigments and fillers suspensions [1]. In all those applications silicanano-particles are either added to the pigment suspensions as a drypowder or as a suspension. The quality of paints/fillers depends on theparticle size, size distribution, shape and morphology [5]. Therefore,dry silica nano-powders have to be dispersed in aqueous solutions togive homogenous suspension and it is essential that large aggregatesinherently present in dry nano-powders are broken into primary silicanano-particles or into nano-aggregates.

The mechanism of dispersion of dry, hydrophilic silica nano-powder (AEROSIL®200 V, further refer to as 200 V) in water has beenrecently investigated by Pacek et al. [4]. They observed both ruptureand erosion during de-aggregation and found that sub-micronaggregates (often called primary or hard aggregates) cannot bebroken into single nano-particles even at the highest energy input andat the highest repulsive interparticle forces.

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In this work the kinetics of dispersion of dry hydrophobic silicanano-powder (AEROSIL® R816 further refer to as R816) in water andrheology of resulting suspensions have been investigated andcompared with the kinetics and rheology of the suspensions ofhydrophilic silica 200 V. According to the manufacturer of silica nano-powders (Degussa) the only difference between R816 and 200 V is thecharacter of the surface of nano-particles whilst the nano-particlessize, shape and density are identical. De-agglomeration of bothpowders has been investigated in the same high shear mixer, at thesame energy dissipation rates and energy density, therefore the effectof surface properties on kinetics of de-agglomeration and rheology ofresulting suspensions was identified and it is discussed below.

2. Experimental

Experimental rig, methodology and procedure are discussed indetails elsewhere [4] and they are only briefly summarised below.

2.1. Materials and methods

Hydrophobic silica nano-powder (R816) was produced by treatinghydrophilic silica nano-powder (200 V) with hexadecysilane and itwas supplied as a dry powder. According to manufacturer it hadfollowing properties [1]: particles density of 2200 kg/m3, SiO2 contentN99.8%, specific surface area of 190 m2/g, average size of 12 nm andthe “nature” pH of 4.0–5.5.

Zeta potential of nano-particles and nano-aggregates was mea-sured by Zetamaster and size distributions were measured usingparticle size analyzer Mastersizer 2000 (Malvern Instruments). The

Fig. 2. Transient size distributions in 5 wt.% suspension of hydrophobic nano-powder inwater (pH=4) at energy dissipation rate of 89.3 kW/m3: (●) 0 min, (▼) 10 min,(■) 60 min, (♦) 240 min.

222 P. Ding et al. / Powder Technology 195 (2009) 221–226

morphology of aggregates was analysed with an EnvironmentalScanning Electron Microscope (ESEM, Philips XL30) and rheology ofsuspensions was measured using a controlled stress rheometer (TA1000, TA Instruments, UK).

Dry nano-powders were dispersed in a batch rotor/stator mixer(L4R from Silverson) with rotor diameter of 0.028 m, rotor height of0.015 m and a gap between the rotor and the standard disintegratingstator of 0.0005 m.

2.2. Procedure

Silica nano-powder was pre-dispersed in water in a glass stirredvessel and pH was adjusted to the required value. Dispersion wastransferred to the high shear mixer fromwhich the air was completelyexcluded. The rotor speedwas set to the required value (3000, 5000 or8000 rpm) and at each speed the dispersion was sheared for fourhours. Small samples of dispersion were taken at certain times andaggregates size distributions were measured.

3. Results and Discussion

3.1. Zeta potential

Zeta potential of hydrophobic R816 nanoparticles suspended inwater was measured over wide range of pH and it is compared withthe zeta potential of hydrophilic 200 V nano-particles in Fig. 1.

The results show that both R816 and 200 V nano-particles havenegatively charged surfaces and the same iso-electric point at pHbetween 2 and 3. At pH between 3 and 9, the absolute value of zetapotential of R816 was on average 10 mV lower than that of 200 Vwhich is consistent with the data reported by the manufacturer(Degussa). The reduction of zeta potential can be explained by lowerconcentration of SiOH groups at the surface of R816 nano-particles.

3.2. Effect of energy input on kinetics of de-agglomeration

Transient particle size distributions during de-agglomeration ofhydrophobic silica nano-powder (R816) at an average energydissipation rate of 89.3 kWm−3 are shown in Fig. 2 and the imagesof dry powder (before de-agglomeration) and nano-aggregates after 4hours of processing are shown in Fig. 3.

The character of the transient size distributions, morphology of dry,hydrophobic nano-powder and structure of nano-aggregates are verysimilar to size distributions, morphology and structure of hydrophilicnano-powder [4]. However, close comparison of transient distributionsof both nano-powders reveals certain differences between kinetics ofde-agglomeration of both nano-powders. In the case of hydrophobicsilica after 10 min (at energy density of 54 MJ/m3) of processing nano-

Fig. 1. Zeta potential of hydrophobic (R816, solid symbols) and hydrophilic (200 V,empty symbols) silica nano-particles as a function of pH.

aggregates were already present in the suspension as clearly indicatedby the curve at 10 min (Fig. 2), whereas in the case of hydrophilic silicanano-aggregates appeared after more than 30 min of processing (atenergy density of 161 MJ/m3). Comparison of transient size distribu-tions of R816 (Fig. 2) with transient size distribution of 200 V (Fig. 1 in

Fig. 3. Morphology of hydrophobic silica nano-powder: (a) dry nano-powder beforede-agglomeration and (b) nano-aggregates after 4 hours of shearing at 89.3 kW/m3.

Fig. 4. Effect of energy density on: (a) cumulative volume fraction of nano-aggregates and(b) breakage of large aggregates in 5 wt.% suspension ( pH=4) at different energydissipation rates; circles— 4.7 MJ/m3; triangles— 21.7 MJ/m3; squares— 89.3 MJ/m3; solidsymbols—R816, empty symbols— 200 V; lines— best fit to Eq. (1) and Eq. (2) respectively.

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[4]) indicates that during first 20 min large agglomerates of hydrophilicsilica were broken mainly by fracture whilst in the case of hydrophobicsilica fracture and erosion occurred simultaneously.

In both powders the median diameters and the shape of sizedistributions of nano-aggregates (first mode in Fig. 2) practically didnot change with the processing time only the volume of nano-aggregates increased and volume of large aggregates decreased. BothMastersizer (Fig. 2) and ESEM (Fig. 3b) detected nano-aggregatesbetween 50 nm and 1000 nmbut single nano-particles (of the order of12 nm according to manufacturer) were not detected in the suspen-sion. This clearly indicates that sub-micron aggregates (Fig. 3b) couldnot be broken into single nano-particles even at the highest energydensity (1500 MJ/m3) used here. It has been reported that hard(primary) nano-aggregates are frequently formed during manufactur-ing of fumed silica and that strong chemical/sintering bonds aredominant inter-particle forces [5].

As the transient size distributions are bi-modal, the kinetics of de-agglomeration cannot be analysed in terms of an average aggregatesize and it requires a full solution of population balancemodel recentlypublished for a similar problem [6]. However, as the aim of this work isto quantify differences between two powders simplified descriptionhas been employed here. The kinetic of de-agglomeration wasanalyzed in terms of cumulative volume fraction of nano-aggregates(Eq. (1)) and transient median diameter of aggregates larger than1 µm (Eq. (2)) [7,8].

yðEtÞ = 1− exp½−A1ðE−EdÞ� ð1Þ

d50ðEÞ = α⋅E−β ð2Þ

In Fig. 4 transient cumulative volume fractions of nano-aggregatesin the 5 wt.% suspensions of hydrophobic (R816, solid symbols) andhydrophilic (200 V, empty symbols) silica nano powders as well astransient median diameters of large agglomerates are compared.

Fig. 4a indicates that de-agglomeration depends on both theenergy density and on the average energy dissipation rate. There is acertain critical energy dissipation rate below which large agglomer-ates cannot be broken into nano-aggregates even if the energy densityis relatively high. Whilst the hydrophobic (R816) nano-powder wasfractured at energy dissipation rate of 4.7 kW/m3 (Fig. 4b) the furthererosion/fracture into nano-aggregates was not observed at thisenergy dissipation rate even at the energy density of the order of100 MJ/m3 (Fig. 4a). At the same energy density but at energydissipation rate of 21.7 kW/m3 nearly 50 V% of R816 powder wasdispersed into nano-aggregates. However, further increase of energydissipation rate to 89.3 kW/m3 has negligible effect on the rate oferosion and coefficients in Eq. (1) calculated from experimental data:Ed=0 and A1=0.0055 m3/MJ were the same at both levels of energydissipation rate. De-agglomeration of hydrophilic (200 V) nano-powder into nano-aggregate required much higher energy dissipationrate. Nano-aggregates were not observed at 4.7 and 21.7 kW/m3 evenat energy density as high as 500 MJ/m3. At the same energy densitybut at energy dissipation rate of 89.3 kW/m3 more than 80 V% ofhydrophilic nano-powder was dispersed in nano-aggregates. De-agglomeration of hydrophilic (200 V) nano-powder also requiredmore energy density as indicated by the values of constants in Eq. (1)Ed=158 MJ/m3 and A1=0.0051 m3/MJ. 90 V% hydrophobic (R816)powder was broken into nano-aggregates at energy density of450 MJ/m3 whereas de-agglomeration of the same mass of hydro-philic powder required 700 MJ/m3. It is worth to notice that whilst incase of hydrophobic nano-powder erosion was observed from verybeginning of the process (delay time equal zero), in hydrophilic nano-powder erosion started after nearly 30 min of processing at thehighest energy dissipation rate.

The transient median diameters of large hydrophobic and hydro-philic aggregates are compared in Fig. 4b. In both cases the size-

energy model (Eq. (2)) [7,8] seems to correlate the median diameterwith energy density rather well. The reduction rate of mediandiameter of hydrophobic aggregates is much lower than the reductionrate of median diameter of hydrophilic aggregates and constants β inEq. (2) are equal to 0.18 and 0.27 respectively. All the above resultsindicate that de-agglomeration of hydrophobic aggregates is con-trolled by erosion and fracture whereas the initial stage of de-agglomeration of hydrophilic aggregates is controlled by fracture.

3.3. Effect of solid concentration on de-agglomeration kinetics and rheologyof suspensions

Transient cumulative volume fractions of hydrophobic and hydro-philic nano-aggregates at different solid loads are compared in Fig. 5.

Cumulative volume fraction of hydrophobic (R816, Fig. 5a) nano-aggregates at 1 wt.% and 5 wt.% practically overlap and at 500 MJm−3

approximately 90 V% of nano-powders was dispersed into nano-aggregates. At these solid concentrations the critical energy densityEd=0and the erosion constantA1=0.0051[m3/MJ]. However, at 10 wt.%of solid, hydrophobic nano-powder could not be dispersed into nano-aggregates even at the highest energy density (1300 MJ/m3). In thecase of hydrophilic nano-powder (200 V, Fig. 5b) the cumulativevolume fractions of nano-aggregates at 1 and 5 wt.% practically overlap(critical energy density Ed=138 MJ/m3 and the erosion constantA1=0.0045 m3/MJ) and 90% of powder was dispersed into nano-aggregates at energy density of approximately 700 MJm−3. The increaseof solid load to 10 wt.% led to the increase of critical energy densityEd=170 MJ/m3 and to reduction of erosion constant (A1=0.0024 m3/MJ).Overall efficiencyof de-agglomerationwasalso reducedand70 V%of

Fig. 5. Transient cumulative volume fraction at different concentration of (a) hydrophobicand (b) hydrophilic nano-aggregates suspended in water; circles — 1 wt.%; triangles —

5 wt.%; squares— 10 wt.%; pH=4, lines: best fit to Eq. (1).

Fig. 6. Effect of solid concentration on rheology of suspensions of (a) hydrophobic and(b) hydrophilic nano-powders after complete de-agglomeration at nature pH; circles—1 wt.%; triangles — 5 wt.%; squares — 10 wt.%.

224 P. Ding et al. / Powder Technology 195 (2009) 221–226

nano-powder was dispersed into nano-aggregates at 700 MJ/m3

whereas at the same energy density 90 V% of nano-powder wasdispersed at two lower solid concentrations.

There are two factors contributing to the reduction of de-agglomeration rate with the increase of solid concentration. Firstly,as the mass of aggregates increases more energy is needed to breakthem into nano-aggregates and secondly the drastic change ofrheology of suspensions induced by the increase of solid concentra-tion reduce the efficiency of de-agglomeration as illustrated in Fig. 6.

1 wt.% suspension of hydrophobic nano-powder (R816, Fig. 6a)was Newtonian with viscosity slightly higher than viscosity of water(1.26mPas). As the solid concentration was increased to 5 wt.% thesuspension became shear thinning with the viscosity at the highestshear rate one order of magnitude higher than viscosity of 1 wt.%suspension. On the one hand, the higher the viscosity the higher thelaminar shear stress on the surface of agglomerates the higher theerosion rate. On the other hand, the increase of viscosity leads to thereduction of the flow intensity through the mixing head therefore tothe reduction of number of aggregates exposed to high shear rate andto the reduction of erosion rate. It appears that at 5 wt.% solid botheffects cancels out and there is practically no change in erosion rate asshown in Fig. 5a. Further increase of solid content to 10 wt.% leads toextremely shear thinning suspension with viscosity at the highestshear rate three orders of magnitude higher than viscosity of 1 wt.%suspension at the same shear rate. Laminar shear rate also increasedbut, as the pumping capacity of the high shear mixers is very low it ispossible that at such high viscosity the suspension was not pumpedthrough mixing head.

The effect of concentration of hydrophilic nano-powder (R200,Fig. 6b) on the rheology of suspension is much weaker. The differencebetween viscosity at 1 wt.% and 5 wt.% is marginal what explains why

the increase of solid load to 5 wt.% does not affect erosion rate(Fig. 5b). At 10 wt.% the suspension is slightly shear thinning withviscosity around eight times higher than 1 wt.%. This increase shouldnot affect the de-agglomeration rate (see discussion above and Figs. 5aand 6a). As the solid concentration increases to 10 wt.%more energy isneeded to break all the aggregates therefore at constant energydensity de-agglomeration rate is lower (Fig. 5b).

3.4. Effect of pH on kinetics of de-agglomeration and rheology of suspensions

It has been reported [9] that pH strongly affects both kinetics ofwet de-agglomeration of nano-powders and rheology of resultingsuspensions by affecting electrostatic charges on the agglomeratesurfaces what in turns affects the balance between attractive (van derWaals) and repulsive (electrostatic) interparticle forces. The van derWaals forces are practically independent onpHwhilst the electrostaticrepulsive forces strongly depend on zeta potential, e.g. on pH.

Transient cumulative volume fractions of hydrophobic and hydro-philic nano-aggregates in 5 wt.% suspensions at different pH com-pared in Fig. 7 indicate that the effect of pH also depends on thecharacter of the particle surface and that it is different for hydrophobicand for hydrophilic nano-particles.

Fig. 7a shows that pH practically has no affect on the kinetics of de-agglomeration of hydrophobic nano-powder (R816). Transient cumu-lative size distributions of hydrophobic nano-aggregates at differentpH (zeta potential between 0 and −40 mV) practically overlap. Thefines generation constant A1 slightly increased from 0.0068 at pH=4to 0.0077 at pH=9 and at all values of pH critical energy density

Fig. 7. Effect of pH on transient cumulative volume fraction of nano-aggregates in 5 wt.%suspensions of (a) hydrophobic and (b) hydrophobic nano-powders; circles — pH=4;triangles — pH=7; squares — pH=9.

Fig. 8. Effect of pH on rheology of 5 wt.% suspensions of hydrophobic nano-aggregatesafter de-agglomeration, (●) pH=3; (▼) pH=4; (■) pH=7; (♦) pH=9.

Table 1Parameters in the power-law model of 5wt% suspension of hydrophobic R816 nano-powder.

pH 3 4 7 9k [Pasn] 8.871 5.8790 0.0110 4.24×10- 3

n 0.0946 0.1007 0.8316 0.9168r2 0.9999 0.9996 0.9459 1.000

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Ed=0 and all the experimental data in Fig. 7a can be approximated byEq. (1) with A1=0.0074 and Ec=0.

Theeffectof pHonde-agglomeration rate of hydrophilic nano-powder(200 V) is much more pronounced as shown in Fig. 7b. In this case anincreases of pH from 4 to 9 drastically speeds up de-agglomerationprocess by reducing the critical energy density (Ed) from 158MJ/m3 atpH=4 to 55 MJ/m3 at pH=9 and increasing fines generation constant(A1) from0.0053 m3/MJ at pH=4 to0.012 m3/MJat pH=9. The increaseof de-agglomeration rate of hydrophilic nano-powderwith the increaseofpHcanbeexplainedby the increaseof thenumberof silanol groupson thesurface of silica nano-aggregates [10]. As pH was increased from 4 to 9,more silanol groupswere formedand ionisationof these groups increasednegative charge of the surface as indicated by zeta potential (Fig. 1)leading to the increase of electrostatic repulsive force and consequently tothe increase of de-agglomeration rate.

The lack of the effect of pH on de-agglomeration of hydrophobicnano-powder can be explained by the modification of surfaceproperties during the reaction of hydroxyl groups with silane [2], [3]but, as the details of these reactions are commercially sensitive andwere not disclosed by manufacturer detail analysis of the effect of pHon surface properties is not possible. It is also possible that long rangeattractive hydrophobic forces make kinetics of de-agglomeration lesssensitive to the changes of electrostatic forces.

Whilst pH drastically affected de-agglomeration rate of hydrophilicnano-powder (200 V) the viscosity of the suspensions were practi-cally independent of pH and at pH between 4 and 9 the 5 wt.%suspensions were water-like (viscosity of 2.5mPas). It is possible thathydration/structural inter-particle forces induced by the surfacesilanol groups [4,10] dominate over electrostatic, pH dependentrepulsive forces when the concentration of these groups at the surfaceof hydrophilic silica nano-particles is high.

In case of R816 situation was opposite with pH having relativelysmall effect on de-agglomeration rate but very strong effect on therheology of the suspension as shown in Fig. 8. It appears that thestrong effect of pH can be explained within the framework DLVOtheory developed for lyophobic systems [11,12]. At pH=3 (absolutevalue of the zeta potential b10 mV), electrostatic repulsive forces aresmall and the rheology is controlled by attractive van der Waals forcesleading to very viscous and shear thinning suspension. At pH=7,absolute value of zeta potential increases to 30 mV therefore inter-particles repulsive force also increase that leads to a drastic reductionof apparent viscosity of suspension. Further increase of pH to 9 leadsto much smaller increase of absolute value of zeta potential to 37 mVtherefore the reduction of viscosity is also rather small.

The changes of rheology of the suspension of hydrophobic nano-powder were quantified using power law model:

η = k⋅γ̇n−1 ð3Þ

and the consistency constants (k) and power law index (n) calculatedfrom flow curves shown Fig. 8 are summarised in Table 1.

As pH increases from 3 to 9 the rheology of suspension graduallychanges from strongly non-Newtonian and shear thinning with a veryhigh viscosity at pH =3 to Newtonian water like at pH=9. Thereduction of pH from 9 to 3 leads to an increase of viscosity and toreduction of power law index. Similar effect of pH on the rheology ofother types of nano-particles has been recently reported by theauthors [9].

4. Conclusions

De-agglomeration of both hydrophobic (R816) and hydrophilic(200 V) silica nano-powders follows similar general pattern. Fractureand erosion leading to bi-modal transient size distributions ofaggregates was observed in both powders. The qualitative analysisand comparison of the kinetics of de-agglomeration carried out interms of median diameters of large aggregates and cumulative volumefractions of nano-aggregates revealed that initially erosion dominated

226 P. Ding et al. / Powder Technology 195 (2009) 221–226

in hydrophobic nano-powder and fracture in hydrophilic nano-powder. In both nano-powders nano-aggregates could not be brokeninto single nano-particles even at the highest energy density and thehighest repulsive interparticle forces. Experimental results clearlyshow that the erosion of hydrophobic nano-aggregates occurs atmuchlower energy dissipation rates and energy density than the erosion ofhydrophilic aggregates. In other words, less energy is required todisperse a unit mass of hydrophobic nano-powder than to disperse aunit mass of hydrophilic nano-powder. Very similar results werereported by manufacturer [1]. Those results are rather unexpectedbecause in hydrophilic nano-powder van der Walls force is the onlyone attractive inter-particle force whereas in hydrophobic nano-powder there is also strong hydrophobic long range attractive force[13–15] The presence of extra attractive force should made de-agglomeration of hydrophobic silica nano-powder more energydemanding than de-aggregation of hydrophilic silica which clearly isnot a case. It is very difficult to explain this disagreement betweentheory and experiments and one possible explanation is that theinteraction between silica nano-particles cannot be described usingstandard DVLO related models as discussed by authors in previouswork [4]. It has also been suggested in literature [16], [17] thatcoagulation of silica nano-particles is strongly dependent on alkalinecations and pH and simultaneous effect of those on aggregation/de-aggregation kinetics is well outside standard DVLO model.

The solid concentration has opposite effect on erosion in bothpowders. The increase of the concentration of hydrophilic nano-powder (200 V) in the suspension from 1 wt.% to 10 wt.% leads torelatively small reduction of the erosion efficiency and nano-powdercan be dispersed into nano-aggregates. At 10w/w% hydrophobicnano-powder (R816) could not be dispersed into nano-aggregates asthe erosion efficiency dropped to zero.

There is very strong effect of pH on erosion efficiency inhydrophilic nano-powder (200 V) where the increase of pH from 4to 9 caused a drastic increase of erosion efficiency, whilst similarincrease of pH in the suspension of hydrophobic nano-powder haspractically no effect on the erosion of nano-aggregates.

The rheology of hydrophobic nano-aggregates strongly depends onpH and on solid concentration whilst rheology of suspension ofhydrophilic nano-powder is rather weakly dependent on thoseparameters.

Acknowledgement

This work is a part of PROFORM (“Transforming Nano-particles intoSustainable Consumer Products Through Advanced Product and ProcessFormulation” EC Reference NMP4-CT-2004-505645) project which ispartially funded by the 6th Framework Programme of EC. The contentsof this paper reflect only the authors’ view. The authors gratefullyacknowledge the useful discussions held with other partners of theConsortium: Bayer Technology Services GmbH; BHR Group Limited;Centre for Computational Continuum Mechanics (C3M); KarlsruheUniversity, Inst. of Food Process Eng; Loughborough University,

Department of Chemical Eng; Poznan University of Technology, Inst.of Chemical Technology and Eng; Rockfield Software Limited; UnileverUK Port Sunlight, Warsaw University of Technology, Department ofChemical and Process Eng.

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Glossary

A1 Constant defined in Eq. (1) [m3/MJ]d0.5 Median diameter [µm]E Energy density [MJ/m3]Ed Critical energy density defined in Eq. (1) [MJ/m3]k Consistency constant [Pa⋅sn]n Power law index [–]r2 Coefficient of determination [–]y(t) Cumulative volume fraction of fines [–]α Constant defined in eq. (2) [(MJ)βm1−3β]β Constant defined in Eq. (2) [−]γ̇ Shear rate [s−1]η Viscosity [Pa⋅s]