Strategy for Selection of Methods forSeparation of Bioparticles From ParticleMixtures

P. van Hee, M.A. Hoeben, R.G.J.M. van der Lans, L.A.M. van der Wielen

Departmentof Biotechnology, Delft University of Technology Julianalaan 67, 2628 BCDelft, The Netherlands; telephone: 0031-15-2792361; fax: 0031-15-2782355;e-mail: L.A.M.vdWielen@tnw.tudelft.nl

Received 2 March 2005; accepted 10 January 2006

DOI: 10.1002/bit.20885

Abstract: The desired product of bioprocesses is oftenproduced in particulate form, either as an inclusionbody (IB) or as a crystal. Particle harvesting is then acrucial and attractive form of product recovery. Becausethe liquid phase often contains other bioparticles, such ascell debris, whole cells, particulate biocatalysts or parti-culate by-products, the recovery of product particles isa complex process. In most cases, the particulate productis purified using selective solubilization or extraction.However, if selective particle recovery is possible, thealready high purity of the particles makes this down-stream process more favorable. This work gives anoverview of typical bioparticle mixtures that are encoun-tered in industrial biotechnology and the various drivingforces that may be used for particle–particle separation,such as the centrifugal force, the magnetic force, theelectric force, and forces related to interfaces. By couplingthese driving forces to the resisting forces, the limitationsof using these driving forces with respect to particle sizeare calculated. It shows that centrifugation is not a generalsolution for particle–particle separation in biotechnologybecause the particle sizes of product and contaminatingparticles are often very small, thus, causing their settlingvelocities to be too low for efficient separation bycentrifugation. Examples of such separation problemsare the recovery of IBs or virus-like particles (VLPs) from(microbial) cell debris. In these cases, separation pro-cesses that use electrical forces or fluid–fluid interfacesshow to have a large potential for particle–particleseparation. Thesemethods are not yet commonly appliedfor large-scale particle–particle separation in biotechnol-ogy and more research is required on the separationtechniques and on particle characterization to facilitatesuccessful application of these methods in industry.� 2006 Wiley Periodicals, Inc.

Keywords: particle-particle separation; bioparticle; inclu-sion body; cell debris; recovery

INTRODUCTION

There is a large and growing market for compounds that

are produced with biotechnological processes. Some of

these compounds obtain a particulate form during their

production. Examples of bulk compounds are b-carotenewith a production of &500 metric tons per year (Chemical

Week, April 19, 1995) and a turnover of&$320 million per

year (www.chemicalmarketreporter.com), vitamin E with a

production of&65,000metric tons per year and a turnover of

approximately $2,000 million per year (Chemical Week,

March 21, 2001 and May 21, 1997), and L-tryptophan with a

production of &750 metric tons per year and a turnover of

approximately $38 million per year (Chemical Week,

November 5, 1997). Alternative processes for the production

of these compounds may avoid the formation of a particulate

product, but are not necessarily more favorable. Dealing with

particulate products thus is an important issue inbiotechnology.

Bioparticles can be formed by aggregation of molecules or

by active accumulation of these molecules in specific

compartments inside cells. Active accumulation involves

active transport of components to specific locations inside the

cells where theymay aggregate or where they are included in

polymers. The latter process does not necessarily yield solid

particles, whereas aggregation does. Aggregation requires

supersaturation of the liquid phase. A particulate product will

therefore only be formed by aggregation when the saturation

concentration is exceeded.

The saturation concentration of a compound is related to

its interactions with all of the surroundingmolecules. If these

interactions are energetically favorable, the compound has a

high saturation concentration and vice versa. Figure 1 depicts

the octanol–water partitioning coefficient (Kow), which is a

measure for polarity, against the molar mass of various

organic compounds. The shaded areas indicate the solubility

of these compounds inwater. The graph shows that a decrease

in solubility corresponds to an increase in molar mass and

Kow. Thus, small polar molecules like many amino acids

remain dissolved in water up to concentrations of 100 g/L or

higher, but with somewhat larger and less polar molecules,

the solubility is already exceeded at lower concentrations.

Most products in biotechnological processes consist of large

complexmoleculeswith highmolarmass. These products are

likely to be produced above their saturation concentration

and will thus form particles during the production process.

Moreover, it has been shown in many enzymatic conversion

processes that precipitation or crystallization of the pro-

duct(s) has a favorable effect on the conversion rate (Ulijn

�2006 Wiley Periodicals, Inc.

Correspondence to: L.A.M. van der Wielen

et al., 2003), thus leading to the design of processes yielding

particulate products.

A particulate product can be easily purified with particle–

liquid separation techniques, such as centrifugation or fil-

tration, if it is the only particle-phase present in the reaction

liquor. Generally, however, other particles are present

that make solid–liquid separation useless. In literature,

many biotechnological processes are described that produce

a particulate product in the presence of other particles.

Examples of such processes are: (1) the production of intra-

cellular particulate microbial products, such as inclusion

bodies (IBs), crystals, cell organelles, viruses, and virus-like

particles (VLPs) by microorganisms, and (2) enzymatic

conversions and biotransformations using (particulate)

biocatalysts that yield a particulate product and/or particulate

by-product. In these processes, the particulate bioproducts

need to be recovered from reaction liquors that contain other

particles. Recovery can be realized using selective solubili-

zation or extraction, but since the particles contain the

product generally at a high purity, selective particle recovery

by particle–particle separation may be more favorable.

In particle–particle separation, particles are physically

separated from one another without affecting their (internal)

chemical composition or their morphology. A method that is

commonly used for particle–particle separation in biotech-

nology is centrifugation. This techniquemakes use of settling

velocity differences that result from size, shape, and/or

density differences of the particles. It has certain limitations

and cannot be used for all particle–particle separation

problems that are encountered in the biotech industry. When

centrifugation fails, other particle–particle separation meth-

ods using other driving forces, such as magnetic separation,

electrophoresis, and air flotation, should be considered.

This study gives an overview of bioparticle mixtures

that are encountered in biotechnological processes (Sections

‘‘Overview of Particle Mixtures in Biotechnological Pro-

cesses’’ and ‘‘Properties of Typical Bioparticles and Their

Suspensions’’). Forces are evaluated that have potential for

physical separation of these mixtures. On the basis of this

evaluation, a strategy for particle–particle separation in

biotechnology is presented using the proposed driving forces.

Application of these forces may be limited because of

specific requirements for the process design. These possible

limitations are taken into consideration in Section ‘‘Selection

of a Particle–Particle Separation Technique,’’ but first the

potential of the forces for particle–particle separation is

determined by direct comparison (Section ‘‘Forces in

Particle–Particle Separation’’). The work is restricted to

the application of single driving forces for separation.

OVERVIEW OF PARTICLE MIXTURES INBIOTECHNOLOGICAL PROCESSES

In this section, an overview is given of reported processes in

which bioparticles are formed in thepresence of other particles.

These bioparticles may be intracellular particulate microbial

products, extracellular particulatemicrobial products,VLPs, or

solid products in biocatalysis. In Tables I and II, references are

0-5 g/l

5-50 g/l

50-250 g/l

Mm (g/mol)

Log Kow

-4

-3

-2

-1

0

1

2

3

4

4003002001000

1

2

3

4

5

6

7

89

10

111312

14 15

16

18

17

19

20

21

22

Figure 1. The relation between solubility, polarity, and molar mass for

various organic compounds. The shaded areas mark the solubility range of

the compounds. Kow is the octanol–water partitioning coefficient, which is a

measure for the polarity of the compounds. Compounds (Weast et al., 1922;

http://www.mdbio.com.tw/AM/am-trimethoprim.htm; http://chemfinder.-

cambridgesoft.com/result. asp; Sangster, 1997): (1) ampicillin, (2) trimetho-

prim, (3) alanine, (4) glycine, (5) methionine, (6) phenylalanine, (7) serine,

(8) tryptophan, (9) DL-valine, (10) codein, (11) phenylbutazone, (12)

quinidine, (13) phenytoin, (14) cimetidine, (15) chlorothiazide, (16)

theophylline, (17) ethacrynic acid, (18) furosemide, (19) phenobarbitol,

(20) caffein, (21) diazepam, (22) meprobamate. [Color figure can be seen in

the online version of this article, available at www.interscience.wiley.com.]

Table I. References on biotechnological processes that yield particle mixtures.

Category References

Intracellular particulate products produced

with microorganisms

Table II

Extracellular particulate products produced

with microorganisms

(Beards et al., 1993; Buque-Taboada et al., 2004; Kometani et al.,

1997; Kumagai, 1999; Leuenberger, 1985; Matsumae et al., 1995;

Michielsen et al., 2000a,b; Miller, 1985; Nakayama, 1985; Spassov

et al., 1996; Takahashi, 2003)

VLPs produced produced with microorganisms (Andrews et al., 1995; Cruz et al., 2000; Kitano et al., 1987; Kuiper et al.,

2002; Meijer et al., 1996; Moran, 1999; Tsoka et al., 2000)

Solid products produced with immobilized

catalysts

(Takahashi, 2003; Davison et al., 1997; Kasche and Galunsky, 1995;

Kuhl et al., 1990; Lee et al., 1999a; Zmijewski et al., 1991)

Production of solid products and solid by-products

with enzymes

(Blacker and Holt, 1997; Youshko et al., 2002)

2 Biotechnology and Bioengineering, Vol. xx, No. x, month xx, xxxx

DOI 10.1002/bit

Tab

leII.

Literature

overview

ofintracellularparticulate

microbialproducts.

Typeofinclusionbody

Size(mm)

Characteristics*

Organism

Function

References

Proteins

0.05–2.25

�,variableshape

Many(m

ainly

recombinant)

microorganisms

Aggregation,incorrectfolding,

lackingtheabilityto

excrete

(Bowden

etal.,1991;Fischer

etal.,1992;

Fischer

etal.,1995;Gram

etal.,1994;

Hellebustetal.,1989;Hoessetal.,

1988;H

ondaetal.,2000;K

olleretal.,1995;

LaV

allieetal.,1993;Mitrakiand

King,1989;Rinas

etal.,1992;Valax

andGeorgiou,1993;Wangsa-

Wiraw

anet

al.,2001b;Wetzelet

al.,1991;

Wongetal.,1997a)

Polyphosphategranules

0.048–1

�,am

orphous,spherical

Manym.o.’s

Phosphatestorageandregulation,

energy

storage,

accumulation

ofunwanted(toxic)metalsor

metalsusedin

themetabolism

(Bodeetal.,1993;LinsandFarina,1999;

Shively,2003)

Starchgranules

1–100

�,variableshape

Plantcells

Energystorage

(Janeetal.,1994)

Cyanophycin

?–>0.5

�,variableshape

Cyanobacteria,recombinant

E.coli

Nitrogen

storage

(Oppermann-Sanio

andSteinbuchel,

2002;Shively,2003)

Glycogen

granules

0.02–0.3

þ/�

variableshape

Manyprokaryotes

Hypothesis:energy/carbonstorage

(Shively,2003)

Polyhydroxyalkanoate

granules

0.1–0.8

þ,spherical

Bacteria,algae,etc.

Energy/carbonreserve

(Shively,2003)

Sulfurglobules

0.1–1

þThiorhodaceaeandother

apochloroticsulfurbacteria

Hypothesis:sulfurstorage

(LinsandFarina,1999;Shively,2003)

Magnetosomes

0.04–0.1

þ,containsFe 3O4orironsulphides

Magnetotacticbacteria

Helpsinsearch

fornutrientsdueto

magnetism

(Dunin-Borkowskietal.,1998;L

insandFarina,

1999;Prokschetal.,1995)

Carboxysomes

0.09–0.5

þBlue–green

algae,m

anynitrifying

bacteriaandthiobacilli

Hypothesis:storageof

ribulose-1,5-diphosphate

carboxylase

(Shively,2003)

Other

crystals

1–15

�,variableshape

Variousm.o.’s

Dueto

increasedproduction

(CN1294191;Eonseonetal.,2003;

Jeaongetal.,1999)

*,þ

indicates

thattheinclusionbodyissurrounded

byamem

braneand�

indicates

thatthereisnomem

brane.

Van Hee et al.: Separation of Bioparticles from Particle Mixtures 3

Biotechnology and Bioengineering. DOI 10.1002/bit

listed of the various categories of particle mixtures that are

described in this section.

Intracellular Particulate Microbial Products

Non-excreted microbial products accumulate inside the

microorganism and often cause the formation of crystals or

IBs that have size in the range of 0.05–1.2 mm. Crystals are

formed when the product is produced above its saturation

concentration, such as in the case ofb-carotene production byBlakeslea trispora (Jeaong et al., 1999) and the production of

xanthophylls by microorganisms or microalgae (Eonseon

et al., 2003). IBs are formedwhen the product is accumulated

in a specific compartment inside the cells, such as with

polyhydroxyalkanoates (Lee et al., 1999b), or when it

aggregates inside the microorganism, which often happens

with peptides and proteins that are folded incorrectly due to

chaperones that are lacking in the organism (Buchner et al.,

1992; Honda et al., 2000; Wong et al., 1997a). Besides

protein IBs, crystals, and PHA granules, many other forms of

IBs can be found in nature, such as phosphate granules, starch

granules, sulfur globules, carboxysomes, magnetosomes,

glycogen inclusions, and cyanophycin inclusions. In Table II,

an overview is given of the characteristics of these IBs. It is

important to note that not all of these IBs are (yet) of

economical interest, but the overview gives a good

impression of what microorganisms are capable of.

The structure and characteristics of IBs may depend on the

cell compartment where the compound is accumulated. For

instance, b-lactamase IBs in E. coli were found to be

amorphous when produced in the periplasmic space and

highly regular-shaped when produced in the cytoplasm

(Bowden et al., 1991). It is difficult to predict the IB

properties and its location inside the cell just by looking at its

chemical composition. In addition, the surface properties of

the IBs may be influenced by intracellular dissolved

compounds that adsorb onto the IBs. This phenomenon

makes prediction of the IB properties even more difficult.

The separation of particulate intracellular microbial

products from biomass is generally performed by product

release through homogenization in combination with cen-

trifugation. The IB enriched fraction is subsequently

dissolved and extracted with chemicals and, if required, the

product is refolded in a proper refolding buffer (Fischer et al.,

1992; Gram et al., 1994; Hellebust et al., 1989; Honda et al.,

2000). Since most of the IBs contain the product at high

purity, dissolution and extraction does not seem to be the

most efficient separation operation and direct recovery by

particle–particle separation may increase the separation

efficiency and reduce the number of process steps.

Extracellular Particulate Microbial Products

A reasonable number of cases have been reported where an

excretedmicrobial product forms crystals (5–100 mm) due to

supersaturation of the liquid phase (Beards et al., 1993;

Buque-Taboada et al., 2004; Michielsen et al., 2000a,b;

Spassov et al., 1996). For instance, both the substrate and

product are crystals in the conversion of Ca-maleate to Ca-D-

malate by permeabilized Pseudomonas pseudoalcaligenes

(Michielsen et al., 2000a,b). These processes yield mixtures

of microbial cells and particulate product(s), again mak-

ing particle–particle separation a key step in downstream

processing.

Currently, many of these products are separated from other

particles by dissolution in a second (organic) liquid phase

followed by crystallization (Buque-Taboada et al., 2004).

Particle–particle separation is only considered in those cases

where the density and/or size differences between the

product particles and the other particles are large enough

for separation by centrifugation.

Virus-Like Particles (VLPs)

Viruses and VLPs are particles with sizes between 20 and

200 nm that are produced intracellularly. The particles often

consist of an inactivated virus or a surface antigen of a virus

that is produced by a genetically modified microorganism.

For recovery of these intracellular VLPs, the cells have to be

disrupted, thus creating a mixture of cell debris and VLPs.

When the virus is still active, however, the cells may be lysed

spontaneously. Again it is clear that the production of these

bioparticles gives rise to a particle–particle separation

problem. Purification of VLPs is currently performed with

a wide variety of separation techniques, such as centrifuga-

tion (Cruz et al., 2000; Tsoka et al., 2000), filtration (Cruz

et al., 2000;Kuiper et al., 2002; Tsoka et al., 2000), extraction

without dissolution (Andrews et al., 1995; Kitano et al.,

1987), and chromatography (Cruz et al., 2000; Kuiper et al.,

2002; Tsoka et al., 2000) that are used in a large train of

separation operations. Direct VLP isolation by particle–

particle separation would reduce the number of process steps

and might thus lower the downstream processing costs.

Particles in Enzyme Catalysis andBiotransformation

Solid-to-solid conversion in enzymatic synthesis is thermo-

dynamically possible if the apparent equilibrium constant is

larger than the solubility ratio of the product(s) and

substrate(s) (Diender et al., 1998). Besides the advantages

of conventional enzymatic synthesis, such as high regio- and

stereoselectivity, absence of racemization and reduced need

for side-chain protection, solid-to-solid conversion has

additional advantages. The reaction yields may increase

by product precipitation in water (Ulijn et al., 2003), which

eliminates the use of organic solvents to shift the thermo-

dynamic equilibrium toward synthesis. Using water instead

of solvents is of course favorable for environmental reasons,

but more importantly, it may also avoid inactivation of the

biocatalyst by the solvent (Eichhorn et al., 1997).

Enzymatic reactions yielding solid products that have a

typical size of 5–100 mm require particle–particle separation

in subsequent process steps when there are other particles

4 Biotechnology and Bioengineering, Vol. xx, No. x, month xx, xxxx

DOI 10.1002/bit

present. These other particles may be particulate biocata-

lysts, particulate by-products or crystals of unconverted

enantiomer in case of conversion of a mixture of racemic

crystals into chiral crystals (Straathof et al., 1998). When an

immobilized catalyst is the only other particle present in the

mixture, separation may be very easy by careful selection of

the catalyst carrier material with respect to size, density, and

structure, for example catalytic membranes (Kasche and

Galunsky, 1995) and catalytic expanded bed media (Van der

Wielen et al., 1990, 1996, 1997).With other particlemixtures

in enzymatic conversion processes particle–particle separa-

tion may be a key step in the purification process. Currently,

product purification is performed in most cases with

(selective) extraction and (selective) crystallization of the

product (Kometani et al., 1997; Kuhl et al., 1990; Matsumae

et al., 1995; Miller, 1985; Yan et al., 1999).

PROPERTIES OF TYPICAL BIOPARTICLES ANDTHEIR SUSPENSIONS

In order to determinewhich separation process is suitable for

the recovery of a particulate bioproduct from a particle

mixture, the particle properties as well as the liquid phase

properties should be known. In this section, properties of

bioparticles and their suspensions are reviewed.

Particle Properties

Morphology, Shape, and Composition

Bioparticles have a large variation in shape, composition, and

morphology.Most extracellular particulatemicrobial products

and bioparticles that are formed in biocatalysis are produced as

highly pure crystalswith varying shape. IBs, on the other hand,

maybecrystallineor amorphous evenwhen they are composed

of the samematerial but produced indifferent compartments of

a microbial cell (Bowden et al., 1991). In addition, IBs

composed of the same material may have a surrounding

membrane in one organism, while in another organism it lacks

this membrane (Shively, 2003). It is, therefore, impossible to

give a general impression of the morphology, shape, and

composition of intracellular particulate microbial products.

Particle Size Distribution

The particle size distribution is a key parameter for almost all

particle–particle separation processes and it will be treated in

more detail in Sections ‘‘Forces in Particle–Particle Separa-

tion’’ and ‘‘Optimization and Control of Driving Forces.’’

The overview of bioparticle properties presented in Table III

shows that the particle size ranges from 0.05 to 100 mm and

that there is a large probability for the particle size

distributions of two types of bioparticles to overlap.

Particle Charge

Bioparticles have a surface charge density that is dependent

on their surface chemistry and the composition of the fluid

phase. In many cases, the surfaces of bioparticles contain

carbonyl, amino, and hydroxide groups. All of these groups

can exchange OH� or Hþ with the surrounding medium

causing a change in surface charge. When a charged particle

is in solution, it attracts ions with opposite charge and repels

ionswith like charge. This phenomenon, in combinationwith

the thermal motion of the ions surrounding the particle,

causes the formation of an electrical double layer around

the particle. The thickness of this electrical double layer

is dependent on the ionic concentration of the liquid,

which is typically in the range of 0.01 mol/L to 1 mol/L in

biological systems, the valency of the ions and their thermal

motion.

A measure for the electrokinetic behavior of a particle is

given by its electrokinetic potential or zeta-potential, which

is the potential at the surface of shear between the charged

surface and the electrolyte solution. The relation between pH,

ionic strength, and the zeta-potential of bioparticles, such as

microbial cells and IBs, has been studied to a reasonable

extent (Egorova, 1994; Van der Wal et al., 1997; Wangsa-

Wirawan et al., 2001a,b; Yan et al., 1992). The typical zeta-

potential for bioparticles lies in the range of �100 mV to

30 mV (Van der Wal et al., 1997; Yan et al., 1992).

Suspension Characteristics

Density of suspension. The density of a suspension is

dependent on the density and concentration of the particles

and the density of the liquid. Table III shows that the

bioparticle density is in the range of 900 to 1,540 kg/m3.

Since most biotechnological processes are carried out

in aqueous systems with densities ranging from 1,000 to

1,050 kg/m3, the density of the suspension will be in between

900 and 1,540 kg/m3. In thiswork, the density ofwaterwill be

used for all particle suspensions because most products are

produced in water.

Viscosity of suspension. In most biotechnological pro-

cesses, water is the main liquid phase. The viscosity of the

liquid phasemay, however, bemuch higher than the viscosity

of water due to dissolved compounds. For example, in

fermentative processes that produce an intracellular product,

cells have to be disrupted in order to release the product. As

cells are ruptured, DNA and other molecules are released

causing a viscosity increase. This viscosity increase can be

limited by making use of cell disintegration methods that

hydrolyze these molecules. Nevertheless, the liquid phase in

fermentative processes is very likely to show non-Newtonian

behavior with viscosities exceeding the viscosity of water.

Besides dissolved molecules, the particle volume fraction

and the particle shape influence on the viscosity. Einstein

(Investigations on the theory of Brownian movement, 1926)

obtained the following theoretical relation for the viscosity of

identical non-interacting rigid spherical particle suspensions

at low particle volume fractions:

h ¼ h0 � ð1þ 2:5cÞ ð1Þ

Van Hee et al.: Separation of Bioparticles from Particle Mixtures 5

Biotechnology and Bioengineering. DOI 10.1002/bit

Tab

leIII.

Anoverview

ofbioparticlecharacteristics.

Particle

Diameter

(mm)

Density

(kg/m

3)

Iso-electricpoint

Wholecellsbacterial

0.5–5.0

(AgerkvistandEnfors,1990;Bowden,

1985;Harrison,1991;Hayashietal.,2001b;

Kulaetal.,1990)

1,080–1,120(Erbeldinger

etal.,1998;Wongetal.,1997b)

3.0–5.0

(Hayashietal.,2001a,b)*

Yeast

2.0–10(Bowden,1985;Harrison,1991;Kulaetal.,

1990;Siddiqietal.,1996)

�1,040(Lipschutzetal.,2000;NikolaiandHu,1992)

—

Fungiandalgae

40–70(H

arrison,1991)

——

Mammalian

5–40(Bowden,1985;Lipschutzetal.,2000;Nikolai

andHu,1992)

——

Plant

50–100(Bowden,1985)

——

Celldebrisbacterial

0.05–3.0

(AgerkvistandEnfors,1990;Bowden,1985;

Kulaetal.,1990;Wongetal.,1997a;

Wongetal.,1997b)

1,061–1,090(W

ongetal.,1997b)

�2.8(W

angsa-W

iraw

anetal.,2001b)*

Yeast

0.05–8.0

(Kulaetal.,1990;Siddiqietal.,1996)

——

Inclusionbodies

0.05-1.2(Tayloretal.,1986;Walker

andLyddiatt,

1999;Wangsa-W

iraw

anetal.,2001b;

Wongetal.,1997b)

900-1,260(Preustingetal.,1993;Tayloretal.,1986;Wong

etal.,1997a,b)

�5.0(W

angsa-W

iraw

anetal.,2001b)*

Crystals

1.0–100(Belletal.,1982,1983;Jauregietal.,2001;

Spassovetal.,1996;Wolffetal.,1997)

1,000-1,540(A

bsolom

etal.,1986;Belletal.,1983;Jauregietal.,

2001;Leungetal.,1999;Matthew

s,1974)

3–11(W

eastetal.,1964;Chang,1981)**

VLPs

0.02–0.20(A

ndrewsetal.,1995;Cruzetal.,2000;

Kitanoetal.,1987;Tsokaetal.,2000)

1,140-1,190(Cruzetal.,2000)

—

*Theseiso-electricpointsareavailablein

literature

butforother

bioparticles

differentvalues

may

beobtained.

**Iftheiso-electricpointofcrystalisequalto

themoleculesofwhichitiscomposed.

6 Biotechnology and Bioengineering, Vol. xx, No. x, month xx, xxxx

DOI 10.1002/bit

where c is the particle volume fraction, h the apparent

viscosity, and h0 the viscosity of the suspending medium.

This equation is valid for c� 0.1. In the concentration

range, where this equation is applicable, the influence of

particle concentration is small. At higher particle concen-

trations, the influence can be much larger (Bird et al., 2002).

In general, it can be said that the viscosity of bioparticle

suspensions may vary from the viscosity of water to very

high values (Agerkvist and Enfors, 1990; Mosqueira et al.,

1981; Wong et al., 1997a). In this work, the viscosity of

water will be used, as it can be regarded as the common

lower limit in bioprocesses, which can be approached by

dilution and chemical or enzymatic treatment.

FORCES IN PARTICLE–PARTICLE SEPARATION

The forces acting on a particle moving in a fluid are depen-

dent on the presence of force fields, the particle properties,

the properties of the fluid phase(s), the properties of other

particles in the suspension and themutual movement of each.

The resultant of these forces may be the driving force for

particle–particle separation. In this section, various forces

are estimated using the particle (and liquid phase) proper-

ties that were discussed in ‘‘Properties of Typical Bioparti-

cles and Their Suspensions.’’ The particle shape is assumed

to be spherical to avoid complex calculations, even though

shape differences may be the basis for particle–particle

separation.

External Field Forces

Centrifugal Force

The centrifugal force, which is often applied in biotechnol-

ogy, can be used for particle–particle separation if particles

in amixture show different settling velocities under influence

of this external field. In the equation below, the net resultant

of the centrifugal force on a particle in a fluid phase is

shown.

FC ¼ a � Dr � V ð2Þ

where a is the centrifugal acceleration (m/s2), Dr the density

difference between the particle and the suspension (kg/m3),

and V the particle volume (m3). The difference in FC for two

types of particles depends on their differences in density

and volume. An increase in centrifugal acceleration may

cause an increase in the absolute force difference between

the particles. The typical upper limit of the centrifugal

acceleration that can be achieved with industrial scale

centrifuges is 20,000g. Therefore, the maximum FC that can

be achieved in bioparticle mixtures is approximately

5.5� 10�5 N when the particles have a maximum size and

density of 100 mm and 1,540 kg/m3 respectively. The force

decreases rapidly with a decreasing particle diameter due to

its dependence on the particle volume and thus on the

cubical particle diameter.

Electric Force

The electric force is proportional to the electric field strength

and the particle charge. The effective charge of a particle can

be described as a function of its zeta-potential and the Debye

decay length (Van de Ven, 1989) that is a measure for the

thickness of its electrical double layer around the particle.

The reciprocal of theDebye decay length, 1k, can be calculated

with the following equation (Schulze, 1984):

k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2 � Nav

e � k � T �X

ðCi � z2i Þr

ð3Þ

where 1/k is the Debye decay length (m), e the elementary

charge (C), Nav the Avogadro number, E the dielectric

permittivity of the fluid (F/m), k theBoltzmann constant (J/

K), T the temperature (K), Ci the ionic concentration of the

solution (mol/m3), and zi the ion valency (�). In the equation

below, the influence of the Debye decay length on the electric

force exerted onto a particle is shown (Schulze, 1984).

Fe ¼ q � E ¼ 2 � p � dp � ð1þ1

2� k � dpÞ � e � zp � E ð4Þ

where q is the particle charge (C), E the electrical field

strength (V/m), dp the particle diameter (m), and zp the zeta-potential of the particle (V). Equation 4 shows that the

difference inFe for two types of particles is governed by their

radius and zeta-potential differences. The absolute force

difference increases with an increase of the electrical field

strength.

Bioparticles have sizes up to 100 mmand zeta-potentials as

low as �100 mV. They are generally suspended in water

(E¼ 7.12� 10�10 F/m) with ionic concentrations ranging

from0.01mol/L to 1mol/L. This causes themaximumFe that

can be reached in biological systemswith a typical maximum

electrical field of 6� 106 V/m (Perry et al., 1998), to be in the

order of 3.8� 10�7 N. To our knowledge, the electric force

has not yet been applied for large-scale particle–particle

separation in biotechnology, though it is used in the mineral

industry for the recovery of minerals. In biotechnology, the

electric force has been mainly used for particle surface

charge determination as well as the recovery/purification of

bioparticles and biomolecules on analytical scale by electro-

phoresis using membranes, gels, and capillaries (Henskens

and Dieijen-Visser, 2000; Jones et al., 1983; Micale et al.,

1980; Netz, 2003; Preece and Luckman, 1981; Young, 1976),

and by dielectrophoresis (WO200196857; Washizu et al.,

2000). Besides these small-scale applications, there are

indications that the force has potential for large-scale

applications as well (Douglas et al., 1995; Ivory, 1993;

Morgan et al., 1997).

Magnetic Force

Some bioparticles contain compounds that are (para)mag-

netic. Examples of these bioparticles are magnetotactic

Van Hee et al.: Separation of Bioparticles from Particle Mixtures 7

Biotechnology and Bioengineering. DOI 10.1002/bit

bacteria and red blood cells. Magnetotactic bacteria use

magnetosomes, which are intracellular magnetic particles,

to find the optimal oxygen concentration and/or redox

potential in water and at sediment–water interfaces

(Dunin- Borkowski et al., 1998). Magnetosomes consist

of iron oxide magnetite crystals or iron sulphate crystals,

which are enveloped by amembrane (Lins and Farina, 1999).

Red blood cells contain hemoglobin, which gives them

paramagnetic properties (Melville et al., 1975). Besides

bioparticles with natural magnetic properties, non-magnetic

bioparticles can be magnetized. Microorganisms and micro-

bial products can be highly efficient bioaccumulators of

soluble and particulate forms of metals (Lins and Farina,

1999). Because the metals are (para)magnetic, the bioparti-

cles will obtain magnetic properties as well. The extent to

which the bioparticles become magnetic depends on the

absorption time and metal concentration (Bahaj et al., 1989).

Adsorption of magnetically labeled antibodies onto cells or

other bioparticles can have a similar effect (McCloskey et al.,

2003). This process is very selective because of the specific

interactions between the antibodies and their binding sites.

Magnetized bioparticles can move in a magnetic field,

where the magnetic force exerted onto the particle can be

described with the following equation:

FM ¼ w � H � V � dBdx

ð5Þ

where w is the magnetic susceptibility of the particle (�), H

the magnetic field intensity (A/m), dB/dx the magnetic field

gradient (T/m). Equation 5 shows that the magnetic force

difference is proportional to the differences in particle

volume and particle magnetization, which is the product of

susceptibility and magnetic field intensity.

The magnetic susceptibility is the intensity of magnetiza-

tion of a particlewhen it is placed in a uniformmagnetic field.

For microorganisms with absorbed metal ions, this suscept-

ibility is in the range of 0–28.7� 10�5 (SI units) (Bahaj et al.,

1989), but other conditions may yield higher susceptibilities.

The magnetization of magnetosomes in magnetotactic

bacteria has been reported to be close to that of bulk

magnetite (Dunin-Borkowski et al., 1998; Proksch et al.,

1995), which has a saturation magnetization of 4.8� 105 A/

m with a susceptibility of about 3.1 (SI units) (Heider et al.,

1996). These high values will, however, only hold for

particles containing a very large fraction of magnetite.

Therefore, the data for magnetized microorganisms, as

presented in Bahaj et al. (1989), will be used for the

estimation of the maximal magnetic force in this work.

The magnetic field gradient, which has a typical maximum

value of 2.5� 105 T/m in an industrial-scale superconducting

electromagnet (Perry et al., 1998), influences the absolute force

difference between the particles. Themaximummagnetic field

intensity that can be obtained in such a device is 3.98� 106 A/

m.Using this value for themagnetic field intensitymay give an

overestimation of the magnetic force since the saturation

magnetization of the bioparticles is probably lower than this

magnetic field intensity. Nevertheless, if the saturation

magnetization is neglected and the maximum magnetic field

gradient can be applied, themaximalFM that can be reached in

biological systems is in the order of 1.5� 10�4 N for 100 mmparticles. Even though this force can possibly exceed the

centrifugal force, it has been applied in biotechnology mostly

on a small scale for the concentration and separation of cells

(Comella et al., 2001; Cullison and Jaykus, 2002; McCloskey

et al., 2003;Melville et al., 1975; Owen, 1978; Radbruch et al.,

1994; Zborowski et al., 2003) and the purification ofmolecules

by their adsorption onto magnetic adsorbents followed by

recovery of the magnetic adsorbents (Heeboll-Nielsen et al.,

2003; Hubbuch and Thomas, 2003).

Interaction Forces

Besides the external field forces, there are forces acting between

particles and objects, such as other particles, part of the

separation device or fluid–fluid interfaces. The magnitude of

these forcesdependson theparticle properties and theproperties

of the object/interface that the particle is interacting with. Here,

only spherical and planar geometries are considered.

Van der Waals Interaction

The Van der Waals force results from interactions between

atoms that are dipoles and/or induced dipoles. In most cases,

this force is attractive since the free energy of a system is

lower when the dipoles and/or induced dipoles are aligned

and thus attract one another. In Equation 6, an approximation

of the interaction force between two particles is given

(Schulze, 1984).

FvdW ¼ � A

12 � h2 �dp � ddp þ d

� �if

dp � ddp þ d

� �>> h

ð6Þ

where A is the Hamaker constant (J), h the distances

between the surfaces of the two particles, and d the diameter

of the other sphere (m).

For the interaction between a flat surface and a spherical

particle at close distances, the diameter of the other sphere, d,

can be chosen much larger than the particle diameter dp,

which results in the following equation (Schulze, 1984):

FvdW ¼ � A � dp12 � h2 for h << dp << d ð7Þ

The parameters determining the selectivity of particle–

particle separation with the Van der Waals force are the

particle diameter and the Hamaker constant. The latter

typically lies between 10�21 and 10�19 J (Fielden, 1996;

Han, 2002; Hayashi et al., 2001b; Koliadima, 1999; Okada

et al., 1990b; Shaw, 1966).

Electrostatic Interaction

For two spherical particles with zeta-potentials between

�60 mV and 60 mV (Hogg et al., 1965) the electrostatic

8 Biotechnology and Bioengineering, Vol. xx, No. x, month xx, xxxx

DOI 10.1002/bit

interaction force can be approximated with the following

equation (Hogg et al., 1965; Okada et al., 1990a):

FE ¼ 2 � p � e � dp � ddp þ d

� k � zp � z�

e�k�h

1þ e�k�h �ðzp � zÞ2 � e�2�k�h

2 � zp � z � ð1� e�2�k�hÞ

" #

fordp � ddp þ d

� �>> h and k � dp >> 1

ð8Þ

where z is the zeta-potential of the surface interacting

with the particle (V). In order to calculate the interaction

force between a surface and a spherical particle at close

distances the diameter of one of the two spheres can again

be chosen much larger than the other. This results in the

following approximation (Hogg et al., 1965; Okada et al.,

1990a):

FE ¼ 2 � p � e � dp � k � zp � z �e�k�h

1þ e�k�h

� �

for h << dp << d; k � dp >> 1 andðzp � zÞ2

zp � z<< 1

ð9ÞThe selectivity of the electrostatic interaction force for

particle–particle separation is determined by the differences

in radius and zeta-potential of the particles that need to be

separated. The absolute force difference can be increased by

an increase of the zeta-potential of the surface the particles

are interacting with.

The Van der Waals force and the electrostatic interaction

force are both dependent on the distance between the particle

and the interacting surface. The resultant of these two forces

is called the DLVO force. In Figure 2, a typical force–

distance curve is presented for the DLVO force, the Van der

Waals force, and a repulsive electrostatic interaction force.

Forces larger than zero represent repulsive forces and

negative forces are attractive. When two types of particles

interact with the same object or fluid–fluid interface, their

DLVO force–distance curves will be different. Particles with

a repulsiveDLVO forcewill move away from the surface and

particles with an attractive DLVO force will move towards

the surface. This phenomenon has been applied for separa-

tion in potential barrier field-flow fractionation (Hansen and

Giddings, 1989;Koliadima andKaraiskakis, 1990), amethod

in which the particle suspension flows parallel to a (charged)

surface. Another example of the use of the DLVO force for

bioparticle separation was given by Hayashi et al. (2001a) in

a study that showed that various microorganisms having

different zeta-potentials can be separated on the basis of these

differences by adsorption onto a charged surface.

The magnitude of the maximum DLVO force difference

for two particles can be approximated by calculating the

force difference at the Debye decay length for a particle with

the largest Hamaker constant and maximal electrostatic

attraction and another particle with the lowest Hamaker

constant and maximal electrostatic repulsion. As mentioned

before, theHamaker constant for bioparticles lies in the range

of 10�21–10�19 J and their zeta-potentials are in the range of

�100 mV to 30 mV. Thus, the maximal driving force for

particle–particle separation of 100 mm particles using the

DLVO force is approximately 3.8� 10�7 N. In this esti-

mation, the zeta-potential limits for Equations 8 and 9 are

ignored. The magnitude of the DLVO is thus reasonable

compared to the centrifugal force but it has not yet been

applied for large-scale particle–particle separation in bio-

technology.

There is another interaction force that depends on the

distance between the interacting objects, that is, the hydro-

phobic interaction force. This force will not be discussed

because there is a lack of literature data for bioparticles,

which makes estimation of the magnitude of this force

impossible. It is worth mentioning, however, that an

expression similar to that for the Van der Waals force has

been proposed to quantify the hydrophobic force with a

constant that is different from the Hamaker constant

(Rabinovich and Yoon, 1994).

Interfacial Tension Force

The interfacial tension between two fluid phases can be

lowered by adsorption of particles at the fluid–fluid interface.

The energy reduction that is related to this process can be

calculated using (Binks, 2002):

EI ¼ p=4 � d2p � g � ð1� cosyÞ2 ð10Þ

where g is the interfacial tension (N/m) and y is the

equilibrium contact angle between the particle–fluid inter-

face and the fluid–fluid interface measured through the fluid

Figure 2. A typical force distance curve for the DLVO force between two

particles with charges of equal sign. [Color figure can be seen in the online

version of this article, available at www.interscience.wiley.com.]

Van Hee et al.: Separation of Bioparticles from Particle Mixtures 9

Biotechnology and Bioengineering. DOI 10.1002/bit

where the particle is pulled into. The increase in energy of

the system that is caused by removal of the particle from the

interface gives rise to a force that pulls the particle back into

the interface. This force is related to the static and dynamic

fluid–fluid interfacial tension, the particle diameter, and the

velocity of the particle during its removal from the interface.

Exact calculation of the interfacial tension force would

require a detailed description of the particle removal process

(Nguyen, 2003), but it is approximated with the derivative of

Equation 10 (Clarke and Wilson, 1983) that is shown in

Equation 11. This equation only takes into account the

energy difference between the initial state when the particle

is at its equilibrium position in the interface and the final

state when the particle is fully immerged into one of the

fluid phases and neglects the phenomena at intermediate

particle positions.

FI ffi@ðp=4 � d2p � g � ð1� cosyÞ2Þ

@dp2� ð1� cosyÞ

� � ¼ p � dp � g � ð1� cosyÞ

ð11Þ

where ½ � dp � (1� cosy) is the immersion depth of the

particle into the fluid where it will be pulled out of, which is

equal to the distance the particle has to be transported for its

removal from the interface. Equation 11 shows that the

selectivity in particle–particle separation using the inter-

facial tension force is dependent on differences in contact

angle, particle diameter, and the fluid–fluid interfacial

tension.

Contact angles of bioparticles depend on the properties of

the two fluid phases at the interface. Theoretically, this

contact angle can be between 08 and 1808 but in practice, lessextreme values are encountered, such as the air–water

contact angle for bacterial cells that ranges from 58 to 1078(Absolom et al., 1986; Busscher et al., 2000; Chattopadhyay

et al., 1995). The large variation in these contact angles is

caused by variation of the surface properties of the bacterial

cells, as well as the variation in composition of the aqueous

phase. For polyhydroxybutyrate, air–water contact angles

have been measured with a capillary rise technique that vary

from 218 to 908 (Marchessault et al., 2001). The large

variation in these values is due to the use of various film

preparation techniques. When these contact angles are used,

the maximum interfacial tension force that can be encoun-

tered for bioparticles of 100 mm diameter is in the order of

5.8� 10�5 N when adsorbed to an air–water interface

(g¼ 0.072 N/m). The force thus has a large potential for

particle–particle separation in biotechnology.

Particle–particle separation using the interfacial tension

force has only been studied to a small extent (Boucher, 1989;

Jauregi et al., 2001; Winitzer, 1973a,b). In biotechnology,

interfaces have been used for the recovery and/or separation

of bioparticles and solutes from mixtures. In these applica-

tions, aqueous two-phase systems (Andrews et al., 1995;

Asenjo et al., 1991; Guereca et al., 1994; Heywood-

Waddington et al., 1986; Kula, 1986, 1993; Walker and

Lyddiatt, 1999) and other liquid–liquid systems (Borbas

et al., 2001; Dennison and Lovrien, 1997; Hoeben et al.,

2004; Jauregi et al., 2001, 2002; Kiss et al., 1998; Pike and

Denisson, 1989; Tan and Lovrien, 1972) were used to capture

the desired products in the interface between the two liquids.

In addition, air flotation whichmakes use of the adsorption of

particles and/ormolecules to air bubbles that rise in the liquid

phase due to buoyancy, has been applied for the recovery of

whole cells (Bahr and Schugerl, 1992; De Dousa et al., 2003;

Gahr and Schugerl, 1992; Gaudin, 1975; Grieves and Wang,

1966, 1967; Kalyuzhnyi et al., 1965; Palmieri et al., 1996;

Sadowski and Golab, 1991; Tybussek et al., 1994; Vlaski

et al., 1996; Wang et al., 1994). In all of these examples,

interfaces are used to capture the product, but this does not

necessarily mean that separation is accomplished solely with

the interfacial tension force.

The use of the interfacial tension force for complete

particle–particle separation in a single stage requires one par-

ticle to adsorb into the interface while the other particle

remains in one of the fluid phases. When this is accom-

plished, the adsorbed particle can be selectively removed

from the system by separating the interface from the fluid

phase that contains the other particle. There are two methods

that can be applied for obtaining a fluid–fluid interface that is

enriched with one of the two particles that are present in the

system. One method is to create conditions that allow both

particles to adsorb but forces one of the two particles to

detach from the interface. In this case, detachment can be

accomplished by using another force such as the drag force

and the centrifugal force, or by competition between the two

particles for adsorption to the interface as was reported by

Jauregi et al. (2001, 2002). Another method is to create

conditions that prevent one of the types of particles from

adsorbing into the interface. This can be accomplished by

using fluid phases that make it energetically unfavorable for

one of the particles to adsorb at the interface or by creating an

energy barrier for one of the particles that prevents the

particle from approaching the interface close enough for

adsorption to take place. This energy barrier could be related

to the DLVO force or the hydrodynamic conditions in the

system. It is clear that both methods require careful selection

of the fluid phases and the hydrodynamic conditions in the

separation device. There are many possible fluid phase

combinations that have a wide variety of interfacial tensions.

For instance, the interfacial tension for aqueous two-phase

systems is in the order of 0.3� 10�3 N/m, while for an air–

water interface, it is in the order of 0.072 N/m. This indicates

that the range of possibilities for particle–particle separation

using the interfacial tension force is comprehensive.

Dynamic Interactions Between Particle and Fluid

Forces resulting from the dynamic interactions between

particles and fluids depend on particle movement and the

change in particle movement relative to the fluid phase. This

movement is governed by all forces that act on the particle.

The forces discussed in this section are therefore in many

10 Biotechnology and Bioengineering, Vol. xx, No. x, month xx, xxxx

DOI 10.1002/bit

cases dependent on other forces and can only be calculated

when these other forces are known. Moreover, the dynamic

interactions between particles and fluids are often only

applicable in combination with other forces. For this reason,

they end up in the force balance on the opposite side of the

other forces, which from now on will be regarded as driving

forces, as is shown in Equation 12.

Xi

Fi ¼�Xj

Fj ð12Þ

where Fi are the driving forces and Fj are the dynamic

particle–fluid interaction forces. Furthermore the dynamic

particle–fluid interaction forces will be regarded as inevi-

table forces that influence separation using the driving

forces even though they can be useful for separation. Their

applicability for particle separation will not be discussed in

this work.

There are three dynamic particle–fluid interaction forces

that are frequently mentioned in literature. These are the drag

force, the inertia force and the force resulting from the virtual

mass of a particle. Besides these forces, many other forces

exist such as the Magnus force that results from a spinning

motion of a moving particle and the Kutta-Joukowski lift

force that results from the fluid-flowprofiles around a particle.

These forces are not often applied for particle–particle

separation on a large scale but they have been shown to be

applicable on a small-scale through for instance field-flow

fractionation (Giddings et al., 1976). In field-flow fractiona-

tion (FFF), a force field acts perpendicular to a flowing liquid

that has a certain flow profile. This force field can be the

gravitational force or one of the other field forces that were

discussed in Section ‘‘External Field Forces.’’ The applied

force field causes the particles to move through the flow

profile in the flowing liquid. This causes a differentiation of

the particle velocities in the separation device that is the basis

for their separation. This process is influenced by the dynamic

particle–fluid interaction forces and Brownian motion (see

Section ‘‘Other Phenomena: Brownian Motion’’). In flow

FFF, a cross-flow liquid stream is applied perpendicular to a

flowing suspension (Giddings et al., 1976; Wahlund and

Litzen, 1989). In this process, the particles are distributed on

the basis of the dynamic particle–fluid interaction forces, the

gravitational force and Brownian motion. Separation on the

basis of the lift force is feasible in FFFwhen it is large enough

in comparison to the applied force field (Giddings, 2000).

Below the drag force, the inertia force and virtual mass are

discussed.

Drag Force

The drag force results from friction between the fluid and the

particle (friction drag) and the displacement of the fluid phase

by the particle (form drag). In the equation below, the general

form of the drag force is given.

Ff ¼ CD � A? � 1=2 � rl � v2 ð13Þ

where CD (�) is the drag coefficient, A\ is the cross-

sectional area of the particle (m2) and v (m/s) is the particle

velocity relative to the fluid phase. The drag coefficient is

dependent on the shape of the particle, on its orientation in

the fluid and on its relative velocity. The drag coefficient for

spherical particles at Reynolds numbers below 0.1 is given

by Stokes’ law:

CD ¼ 24

Refor Re < 0:1:

This relation can also be used for Reynolds numbers

between 0.1 and 1 but then the drag force is underestimated

with about 10%. For Reynolds numbers between 0.5 and

6� 103, the drag coefficient can be calculated with the

following expression (Bird et al., 2002):

CD ¼ffiffiffiffiffiffi24

Re

rþ 0:5407

!2

for 0:5 < Re < 6� 103:

At Reynolds numbers between 5� 102 and 105 the drag

coefficient obtains a constant value of about 0.44 (Bird et al.,

2002).

Inertia Force

An expression for the inertial force is given below:

FIE ¼ V � rp �dv

dtð14Þ

where rp is the particle density (kg/m3). If the inertia force is

included in the force balance, particle acceleration becomes

a function of particle density, particle size, and the other

forces. This causes a differentiation in acceleration of parti-

cles with different driving forces, sizes, and/or densities,

which could be the basis for particle–particle separation.

Application of this principle on an industrial scale will be

very difficult. In addition, including the inertia force in

the force balance requires time iterations for calculation of

the relative particle velocities and their derivative to time,

making simple estimation of the particle–particle separa-

tion efficiency impossible. Therefore, the inertia force will

not be considered in any force calculations in this work.

For small particles, the inertia force generally is very

small because these particles tend to follow the liquid

streamlines (Caulet et al., 1996), which makes this a proper

assumption. With larger particles, the inertia force may

reduce or even prevent particle acceleration. This can

prevent the particles from reaching their terminal velocity

that was calculated on the basis of the drag force and the

driving forces.

Virtual Mass

The virtual mass of a particle accounts for the resistance of

the fluid against changes in particle velocity relative to that of

the fluid. For spherical particles in a constant fluid flow

Van Hee et al.: Separation of Bioparticles from Particle Mixtures 11

Biotechnology and Bioengineering. DOI 10.1002/bit

profile, the virtual mass is equal to half of the particle volume

(Zhang et al., 2000), so that

FVM ¼ 1

2� V � rl �

dv

dtð15Þ

where rl is the liquid density (kg/m3) and dv/dt is the

acceleration (m/s2). The same argument holds for this

force as for the inertia force with respect to its influence

on the relative particle velocity and its applicability for

particle–particle separation. Therefore this force will be

excluded from the force balance as well.

Other Phenomena: Brownian Motion

The time-average kinetic energy of a particle, regardless of

its size, is equal to the average thermal energy of the

surrounding molecules, which is 3/2 kT (1/2 kT in one

dimension). The kinetic energy of a particle may fluctuate in

time and may thus differ from the time-average value. This

makes estimation of the microscopic effects of the Brownian

motion very difficult. The time-average value will, therefore,

be used for estimation of the macroscopic effect of Brownian

motion. Equation 16 shows the relation between the one-

dimensional concentration gradient and the Brownian force

(Van de Ven, 1989).

FB ¼ � k � Tn

� dndx

ð16Þ

where dn/dx is the particle concentration gradient (mol/m4).

Particles can thus move from a high particle concentration

to a low concentration as a result of the Brownian force.

In addition, particle motion occurs in homogeneous

suspensions where dn/dx is zero, but will be random. This

random motion does not cause a net particle transport and

is therefore not useful for particle–particle separation by

itself. However, it can act as a dominant transport

mechanism and facilitate other processes such as particle

adsorption at interfaces. Brownian motion has been used in

biotechnology mainly for the separation of molecules by

making use of the differences in their transport through

membranes and in matrices. Transport in these systems

depends on the thermal motion of the compounds, their

partitioning in the media and the hindrance they experience

within the matrix or membrane. In addition, Brownian

motion can play an important role in particle separation

using field-flow fractionation (Giddings, 2000).

For estimation of the maximum Brownian force in

bioparticle mixtures, the maximum obtainable particle

concentration gradient is required. If it is assumed that

the highest particle concentration is reached at 60 %v/v

particles (dense packing of spheres) and decreases to 0 %v/v

over a distance of 10�3 m, then 1/n � dn/dx is approximately

2� 103/m at an average value of n resulting in a Brownian

force of 8.1� 10�18 N at a temperature of 298 K. Although

this estimationmay be off by a couple of orders ofmagnitude,

we can readily see that the Brownian force acting on 100 mm

particles is very small compared to the other forces that were

discussed in this section. For very small particles the

Brownian force becomes more important because it is

independent of particle size while the other forces decrease

with particle size.

Overview of Driving Force DifferencesBetween Particles

The upper limit of the driving force difference for two

bioparticles with similar sizes and different properties is

estimated for bioparticles and liquid phases with properties

that are presented in the Section ‘‘Properties of Typical

Bioparticles and Their Suspensions’’ and Section ‘‘Forces in

Particle–Particle Separation.’’ The result of this estimation is

depicted in Figure 3 for the whole range of particle sizes that

may be encountered in biotechnology. The graph shows that

for particle diameters between 20 nm and 100 mm, the inter-

facial tension force has the largest potential for separation of

particles with the same size. For particle diameters below

0.5 mm, the electrical force also has large potential. The

centrifugal force has a much lower potential than these two

forces and the graph confirms that itmay evenbe inapplicable

for the separation of small particles. Whether or not the

interfacial tension force and the electric force are the largest

driving forces for particle–particle separation in practice

depends on the liquid phase properties and the particle

properties for each specific separation problem. These

properties can be manipulated to some extent to increase

the selectivity for many of the driving forces (Section

‘‘Optimization and Control of Driving Forces’’).

Direct comparison of the driving forces (Fig. 3) can help to

determine the forces and processes that should be investi-

gated in more detail. After this initial screening, the

limitation to the applicability of the driving forces has to be

100 101 102 103 104 105

particle diameter (nm)

10-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

force (N)

centrifugal force

magnetic force

Brownian forceDLVO force

electrical force

interfacial tension force

Figure 3. An overview of the maximal force difference that can be

obtained with the driving forces when they act on two bioparticles with

similar sizes and different properties.

12 Biotechnology and Bioengineering, Vol. xx, No. x, month xx, xxxx

DOI 10.1002/bit

taken into consideration. This is done in Section ‘‘Selection

of a Particle–Particle Separation Technique.’’

Before discussing the optimization of driving forces, a

small comment has to be made with respect to combining

driving forces in particle–particle separation. In many cases,

we are forced to deal with combinations of forces because

some forces cannot be excluded from the process. Examples

of these forces are the Brownian force, the gravitational force

and many of the forces that are related to the dynamic

interactions between particles and fluids. It is therefore

essential to take into account the influences of these forces

when designing a process. Furthermore, combinations of

driving forces may give additional options in particle–

particle separation. Force estimations like those presented in

Figure 3 can aid the selection process. Evaluation of these

combinations is only valuable when a specific separation

problem is regarded because of the larger number of possible

combinations and the difficulty of combining some of the

forces. Therefore this topic will not be dealt with in this work.

In literature, this strategy is discussed to some extent. For

instance, the thesis of Tils (Tils and Tels, 1992) deals with

particle separation byflotation in a centrifugal force field, and

in field-flow fractionation dynamic particle-fluid interaction

forces are often combined with field forces and Brownian

motion (see section ‘‘Dynamic Interactions Between Particle

and Fluid’’).

OPTIMIZATION AND CONTROL OF DRIVINGFORCES

Particle properties can be manipulated by changing their

surface chemistry, which influences the electrical force, the

magnetic force, theDLVO force, the interfacial tension force,

and possibly the aggregate size.

The surface properties of particles can be controlled by

adsorption of surfactants. This technique is commonly

applied in flotation and may change the zeta-potential of

particles and their contact anglewith the fluid–fluid interface

(Fraunholcz and Dalmijn, 1998; Hanumantha Rao and

Forssberg, 1997). Polymer or polyelectrolyte adsorption

may have a similar effect as the surfactants that are used in

flotation.

The particle size distribution can be manipulated using

aggregation. This process only has a positive effect on

particle–particle separation if it is selective for one kind of

particle or if the particles aggregate specifically with their

own kind resulting in aggregate sizes that yield a larger force

difference compared to the force difference for the primary

particles. It is therefore very important that heteroaggrega-

tion and entrapment of particles in aggregates of other

particles is prevented. The aggregation rate of particles in a

suspension may be increased by reducing the electrostatic

repulsion between the particles, by adding polymers or

polyelectrolytes, with shear, by changing the temperature,

with high pressure treatment or by enzymatic action

(renneting) (Dickinson, 2003). The first two methods will

be briefly discussed.

The electrostatic repulsion between particles can be

reduced by changing the nature and concentration of the

ions in solution (Eq. 3), by reducing their zeta-potential

through pH change (Eq. 8) or by specific adsorption of

surfactants or ions on their surface (Chang and Hshieh,

1991). The latter strategy has been shown to work

successfully for air–water interfaces with aluminium (Li

and Somasundaran, 1992) and magnesium ions (Li and

Somasundaran, 1991) as well as for polystyrene particles

with surfactants (Okada et al., 1990b). Addition of polymers

or polyelectrolytes can also lower the electrostatic repulsion,

but more importantly these compounds may form bridges

between the particles by adsorbing onto the particle surface.

Bridging causes an increase in particle aggregation rate due

to more effective particle collisions and a decrease in

aggregate break-up rate. Examples of chemicals that can be

used for particle size manipulation are borax, which is an

effective flocculation agent for cell debris, as reported by

Tsoka et al. (2000) and calcium ions that can be used to

flocculate microorganisms by forming metal-polymer com-

plexes with the polymers on the surface of the microorgan-

ism, as reported by Sanin and Vesilind (1996).

SELECTION OF A PARTICLE–PARTICLESEPARATION TECHNIQUE

In the following sections, a method for creating a window

of operation for the driving forces is presented and the

selection procedure for suitable separation methods is

discussed for typical particle mixtures that are encountered

in biotechnology.

Window of Operation for the Driving Forces inParticle–Particle Separation

The window of operation for the field forces depends on the

particle properties and the minimum velocity difference that

is required for their separation. The velocity of a particle can

be calculated by solving the force balance between the

driving forces and the dynamic particle–fluid interaction

forces (Eq. 12). From the dynamic particle–fluid interaction

forces only the drag force will be taken into account for

reasons that were discussed in Section ‘‘Overview of Driving

ForceDifferences Between Particles.’’ The driving forces for

separation of bioparticles will often be small near their

separation limits causing particle velocities to be small as

well. Stokes’ law can therefore be applied for the drag force

in most cases. As an example, in Equation 17 Stokes’ drag is

included in the force balance of Equation 12 in case the

centrifugal force is the only driving force.

FC ¼ Ff ! a � Dr � p6� d3p

¼ 3 � p � � � dp � v ! dp;lim

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi18 � � � vmin

a � Dr

s ð17Þ

Van Hee et al.: Separation of Bioparticles from Particle Mixtures 13

Biotechnology and Bioengineering. DOI 10.1002/bit

The limiting particle diameter that can be separated with the

driving force is a function of the particle properties, the fluid

properties, the centrifugal acceleration, and the minimum

required terminal particle velocity. By subtracting the mini-

mum required terminal particle velocities of all particles in a

mixture, the minimum particle diameters that are required

for the separation of the mixture can be calculated. The

relation between the minimum particle diameters and

the minimum required velocity difference in a binary parti-

cle mixture is shown below for the field forces in case of

Stokes’ drag. A similar approach can be taken when other

hydrodynamic conditions apply by including the appro-

priate expression for the drag coefficient (see section ‘‘Drag

Force’’).

Centrifugal Force : Dvmin ¼ v1 � v2j j ¼ a

18 � �� Dr1 � d2p lim;1 � Dr2 � d2p lim;2

��� ���ð18Þ

Electric Force : Dvmin ¼2 � e � E3 � � � jð1þ 0:5 � k � dp lim;1Þ

� z1 � ð1þ 0:5 � k � dp lim;2Þ � z2jð19Þ

Magnetic Force : Dvmin ¼H

18 � � �dB

dx�

w1 � d2p lim;1 � w2 � d2p lim;2

��� ��� ð20Þ

These equations show that the field forces can be applied for

all particle sizes that cause the particle velocity difference to

exceed the minimum required difference. For the DLVO

force, a similar expression can be obtained (Eq. 21), which

shows that the window of operation of the DLVO force is

independent on the particle diameter. It can thus be applied to

all particle sizes as long as the actual velocity difference

exceeds theminimum required difference. Since this velocity

difference is a function of the distance between the particles

and the interacting surface, it will be estimated by using the

Debye decay length.

DLVO Force : Dvmin ¼2 � e � k � z

3 � � � e�k�h

1þ e�k�h � ðz1 � z2Þ����þ A2 � A1

36 � p � � � h2���:

ð21Þ

The window of operation for the interfacial tension force

cannot be constructed in a similar manner as for the field

forces and the DLVO force because it completely depends

on the design of the separation device, as was discussed

in Section ‘‘Dynamic Interactions Between Particle and

Fluid.’’ In Section ‘‘Selection of a Driving Force for Part-

icle–Particle Separation,’’ this design will be treated in

more detail to facilitate construction of the operation

window for the interfacial tension force. The operation

window of the Brownian force will not be discussed because

estimation of the particle concentration gradient is arbitrary

and will therefore not give an accurate selection criterion.

Nevertheless, the Brownian force may be applicable if the

differences in particle transport rates through membranes, in

matrices, and in field-flow fractionation are large enough.

Estimation of the minimum required velocity difference

requires knowledge of the feed flux, particle concentration in

the feed, and the area available for separation. All of these

parameters depend on the design of the separation device. To

make comparison of the driving forces possible, wewill limit

ourselves to using the minimum particle velocity that is

required for centrifugal separation. This velocity can be

calculated with the Sigma theory (Ambler, 1961), which

states that separation of a particle from the liquid phase with

an efficiency of 100% requires a minimum particle velocity

perpendicular to the liquid flow equal to the ratio of the

volumetric liquid flux and the available settling area

perpendicular to the liquid flow. In order to make this theory

applicable for particle–particle separation instead of parti-

cle–liquid separation, we have to allow for different

apparatus designs dependent on the direction and magnitude

of the particle velocities. In Figure 4, two possible designs are

presented (Giddings, 1991). These designs show that for

complete separation in devices with similar dimensions that

are used for processing equal volume fluxes, the minimum

required velocity difference between the particles should be

in the same order as the terminal velocity of a single particle

that is required for particle–liquid separation. With a feed

feed suspension

parallel

liquid flow

particle 2 particle 1

particle 2

particle 1

feed suspension

perpendicular

v1

v2

v2v1

vf

Figure 4. Examples of the design of separation devices for cases where

particle movement is perpendicular and parallel to the liquid flow. In case of

particlemovement parallel to the liquid flow, the velocity difference between

the two types of particles should be such that the net velocity of Particle 2 is in

the direction of the liquid flow (v2� vf < 0), while the net velocity of

Particle 1 is in opposite direction of the liquid flow (v1� vf > 0). Feed

distribution will become an issue in this design. The vectors indicate the

direction of particle movement relative to the liquid phase.

14 Biotechnology and Bioengineering, Vol. xx, No. x, month xx, xxxx

DOI 10.1002/bit

flux of 1 m3/h that is reasonable for large-scale processes and

an equivalent settling area of 2� 104 m2 (Perry et al., 1998),

which is approximately the upper limit for industrial centri-

fugation at high gravitational acceleration (11,500g), the

minimum required particle velocity difference is approxi-

mately (1� 11,500)/(3,600� 2� 104)¼ 1.6� 10�4 m/s.

For the magnetic and electric force, this minimum required

velocity difference is in the same order when industrial-scale

electric and magnetic separators having a separation section

with an approximate length, width, and height of 3, 1, and

0.05m (dimensions are estimated from handbook data (Perry

et al., 1998)) are considered for treating the same feed.

Therefore, the same velocity difference will be applied in all

calculations

In the following section, a window of operation is created

for each of the driving forces when applied to typical particle

mixtures in biotechnology.

Selection of a Driving Force forParticle–Particle Separation

The overview of particle mixtures that was presented in

section ‘‘Overview of Particle Mixtures in Biotechnological

Processes’’ shows that there are different classes of separa-

tion problems that may require different purification strate-

gies. In most of these separation problems product

purification requires removal of whole cells or cell debris.

Construction of the operation window for the centrifugal

force requires information on particle size and density, which

is available in literature for most particle mixtures. Estima-

tion of the electric, and magnetic force, on the other hand,

requires the magnetic susceptibility and particle charge,

which are scarcely reported in literature. These particle

properties are therefore estimated on the basis of the data that

were presented in Sections ‘‘Electric Force’’ and ‘‘Magnetic

Force.’’ In Figure 5, the relation between the limiting particle

diameters for the centrifugal, electric and magnetic force are

shown for extreme cases of IB purification (Fig. 5A and B),

VLP purification (Fig. 5C), and extracellular particulate

microbial product purification (Fig. 5D and E) that may be

encountered in industry. In the calculation of these limits,

the liquid phase density is assumed to be 1,000 kg/m3, since

most biotechnological processes are carried out in water.

Higher liquid densities may give larger velocity differences

in centrifugation and may thus lower separation limits,

but possibilities for density manipulation depend on each

specific separation problem and will not be considered here.

Furthermore, the density of whole cells and cell debris is

assumed to be 1,085 kg/m3 (Wong et al., 1997b).

In Figure 5, the area marked with horizontal dotted lines

represents all particle diameter combinations that cannot be

separated with the magnetic force and the area marked with

vertical dotted lines represents the combinations that cannot

be separated with the centrifugal force. The electric force

only fails to give separation for particle diameter combina-

tions exactly on the boundary line that is depicted in Figure 5.

If larger velocity differences are required for the electric

force the two particle diameter limits will move apart in a

similar manner as with the magnetic force. Furthermore, the

DLVO force may be applied to all particle diameter

combinations since the minimum particle velocity difference

is exceeded in all cases. However, applicability of this force

will depend very much on process design. The limits of

the interfacial tension force are not depicted for reasons that

were discussed in Section ‘‘Window of Operation for the

Driving Forces in Particle–Particle Separation.’’

In each of the graphs depicted in Figure 5, the area that is

marked with dashed lines confines the range of particle sizes

that are encountered in biotechnology. These areas show that

separation of extracellular particulate microbial products

from whole cells can be performed with centrifugation in

most cases regardless of the density of the product. Puri-

fication of IBs andVLPs by centrifugation, on the other hand,

will be difficult or even impossible in many cases because

many particle size combinations lie outside of the working

area for centrifugation. In addition, cell debris may have a

broad particle size distribution causing part of the cell debris

particle size to be below the separation limit for centrifugal

separation inmany cases. This inevitably leads to incomplete

separation. When the centrifugal force fails, one of the other

driving forces must be selected.

The magnetic force, the interfacial tension force (in

combination with the DLVO force), and the electric force are

used in the paper and/or mineral industry, which may make

application of these forces in biotechnology possiblewithin a

reasonable time span. The electric force and the interfacial

tension force are most favorable because in theory there are

very few separation problems that cannot be resolved with

these forces. Even though the actual zeta-potentials may be

different from those that were assumed for constructing

Figure 5, a small difference in zeta-potential is sufficient

for particle separation with the electric force (Dvmin still

exceeded). Since the zeta-potential can be easilymanipulated

in biological systems by varying the pH, nearly all particles

with a different relation between pH and zeta-potential can be

separated with the electric force. It is much more difficult to

indicate what particle mixtures can be separated with the

interfacial tension force (in combination with the DLVO

force) because the mechanisms of flotation and interfacial

partitioning are not fully understood yet. In addition, the

mixtures that need to be treated in biotechnology are often

very complex due to the large number of components that are

present, whichmakes prediction of the particle behavior very

difficult. Nevertheless, an attempt will bemade to construct a

window of operation for the interfacial tension force. As

indicated in Section ‘‘Dynamic InteractionsBetween Particle

and Fluid,’’ there is a number of strategies for the use of the

interfacial tension force for particle–particle separation.

These strategies have in common that one type of particles

needs to remain adsorbed at the interface while the other

particles remain in the continuous phase, for instance, due to

an energy barrier that prevents adsorption of these particles at

the interface or due to competition between the particles

for adsorption at the interface. In most biotechnological

Van Hee et al.: Separation of Bioparticles from Particle Mixtures 15

Biotechnology and Bioengineering. DOI 10.1002/bit

10-8 10-7 10-6 10-5 10-4

10-8

10-7

10-6

10-5

10-4

diameter particle 2 (m): IBs

diameter particle 1 (m): cell debris

magnetic force

electric force

centrifugal force

A

10-8 10-7 10-6 10-5 10-4

10-8

10-7

10-6

10-5

10-4

magnetic force

electric force

diameter particle 2 (m): IBs

diameter particle 1 (m): cell debris

centrifugal force

B

10-8 10-7 10-6 10-5 10-4

10-8

10-7

10-6

10-5

10-4

magnetic force

electric force

diameter particle 2 (m): VLPs

diameter particle 1 (m): cell debris

centrifugal force

C

10-8 10-7 10-6 10-5 10-4

10-8

10-7

10-6

10-5

10-4

magnetic force

electric force

diameter particle 2 (m): product

diameter particle 1 (m): whole cells

centrifugal force

D

Figure 5. An overview of the particle–particle separation limits for the centrifugal force, the electric force, and the magnetic force for extreme particle–

particle separation problems that are encountered in biotechnology. Particle 1 represents cell material with a magnetic susceptibility of 2.5� 10�4 and a zeta-

potential of�0.030 V. Particle 2 represents the particulate product with a magnetic susceptibility of 1.0� 10�4 and a zeta-potential of�0.010 V. The vertical

lines indicate the conditions where centrifugation is not applicable for separation and the horizontal lines indicate the area where the magnetic force is not

applicable for separation. The electric force can be applied for all particle size combinations that are not on the line that indicates the separation limit. The area

confined by the dashed lines indicates all particle size combinations that are encountered in biotechnology for each class of separation problems. A: Separation

of IBswith a high density (1,260 kg/m3) frommicrobial cell debris.B: Separation of IBswith a lowdensity (900 kg/m3) frommicrobial cell debris. C: Separation

ofVLPswith a density of 1,190 kg/m3 frommicrobial cell debris.D: Separation of extracellular particulatemicrobial productswith a high density (1,540 kg/m3)

from whole cells. E: Separation of extracellular particulate microbial products with a low density (1,000 kg/m3) from whole cells.

16 Biotechnology and Bioengineering, Vol. xx, No. x, month xx, xxxx

DOI 10.1002/bit

processes, the density of the continuous (often aqueous)

phase will be higher than the density of the dispersed phase.

This density difference can be utilized in the process by

making the dispersed phase rise through the continuous phase

under influence of the gravitational force or centrifugal force.

The interfacial tension force acting on the particle that is

adsorbed at the interface should therefore exceed the

gravitational (and shear) force(s) that acts on the particle to

prevent detachment from the interface. In Figure 6, the relation

is given between contact angle, interfacial tension, and the

maximumparticle diameter that still allows particles to remain

at the interface under gravity. For construction of this graph,

the particle density and continuous phase density are assumed

to be 1,540 and 1,000 kg/m3, respectively and the contact

angles are assumed to be in the range that was reported in

section ‘‘Forces in Particle–Particle Separation.’’ The opera-

tion window shows that the interfacial tension force can be

applied for separation of particle sizes ranging frommolecular

scale up to approximately 0.01 m regardless of the differences

in particle densities as long as particle adsorption at the

interface is selective for one of the particles in themixture. The

process of creating selectivity canbevery complex, however. It

depends on the hydrodynamics near the interface, the

interaction forces (Section ‘‘Interaction Forces’’) and compe-

tition effects. Thorough discussions on the application of

interfaces for particle separation lie outside of the scope of this

work and can be found elsewhere (Clarke and Wilson, 1983;

Finch and Dobby, 1990; Gaudin, 1975; Schulze, 1984).

CONCLUSION

Selection of a particle–particle separation technique requires

knowledge of the particle properties, such as density,

magnetic susceptibility, zeta-potential, and hydrophobicity.

When these properties are known, the optimal particle–

particle separation technique can be chosen by solving the

force balance between the drag force and the driving forces.

Force estimations reported in this work show that the

interfacial tension force (in combination with the DLVO

force) and the electrical force, in whatever way applicable,

have large potential for particle–particle separation in

biotechnology. The electrical force requires very small

differences in zeta-potential to make complete separation

possible and the window of operation of the interfacial

tension force indicates its applicability to a very broad range

of particle sizes (up to 0.01m). The use of the electrical force

and the interfacial tension force becomes desirable when

centrifugation cannot be applied for particle–particle

separation. This is generally the case when particle sizes

are small (�1 mm) and/or when the density differences

between the particles are small. Examples of such separation

problems are the separation of IBs andVLPs from cell debris.

Currently, only limited use is made of these alternative

particle–particle separation techniques due to a lack of

knowledge. Therefore R&D efforts should increase in this

area to extend the downstream processing toolbox that is

currently available to the biotech industry.

NOMENCLATURE

a centrifugal acceleration (m/s2)

A Hamaker constant (J)

A\ cross-sectional area of particle (m2)

B magnetic field (T)

c particle volume fraction (-)

CD drag coefficient (-)

Ci ionic concentration (mol/m3)

10-8 10-7 10-6 10-5 10-4

10-8

10-7

10-6

10-5

10-4

magnetic force

electric force

diameter particle 2 (m): product

diameter particle 1 (m): whole cells

centrifugal force

E

Figure 5. (Continued )

10-4 10-3 10-2 10-1

10-5

10-4

10-3

10-2

10-1

θ = 107 o

θ = 5 o

interfacial tension (N/m)

maximum particle diameter (m)

ATPSorganic phase/ aqueous phase gas/liquid

Figure 6. Window of operation for the interfacial tension force showing

the relation between contact angle, interfacial tension and the maximum

particle size that remains adsorbed at the interface under gravity. The particle

density and continuous fluid phase density are assumed to be 1,540 and

1,000 kg/m3, respectively. In the graph approximate ranges of the interfacial

tension of different types of fluid–fluid systems are given.

Van Hee et al.: Separation of Bioparticles from Particle Mixtures 17

Biotechnology and Bioengineering. DOI 10.1002/bit

d diameter of sphere (m)

dp particle diameter (m)

dp,lim limiting particle diameter (m)

e elementary charge (C)

E electric field (V/m)

EI energy required to pull a particle out of a fluid-fluid interface (J)

FB Brownian force (N)

Fe electric force (N)

FE electrostatic interaction force (N)

FF drag force (N)

FC centrifugal force (N)

Fi general expression for the driving forces (N)

FI interfacial tension force (N)

FIE inertia force (N)

Fj general expression for the dynamic particle–fluid interaction

forces (N)

FM magnetic force (N)

FvdW Van der Waals force (N)

FVM force caused by virtual mass of particle (N)

h distance between the surfaces of a particle and an object (m)

H magnetic field intensity (A/m)

k Boltzmann constant (J/K)

n particle concentration (mol/m3)

Nav Avogadro number (#/mol)

t time (s)

T temperature (K)

q particle charge (C)

Re Reynolds number (�)

v relative velocity between particle and fluid (m/s)

vmin minimum required particle velocity (m/s)

V particle volume (m3)

x distance (m)

zi ion valency (�)

w magnetic susceptibility (�)

E dielectric permittivity of the fluid (F/m)

g interfacial tension (N/m)

h apparent viscosity (Pa s)

h0 liquid viscosity (Pa s)

k�1 Debye decay length (m)

y contact angle between particle and fluid-fluid interface (8)rl liquid phase density (kg/m3)

rp particle density (kg/m3)

z zeta-potential of a surface (V)

zp particle zeta-potential (V)

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