Kendall 2001

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.Powder Technology 121 2001 223229

www.elsevier.comrlocaterpowtec

Adhesion and aggregation of fine particlesq

Kevin Kendall), Carl Stainton

School of Chemical Engineering, Uniersity of Birmingham, Edgbaston, Birmingham B15 2TT, UK

Received 17 October 2000; received in revised form 7 March 2001; accepted 29 March 2001

Abstract

The influence of interparticle adhesion on the formation and properties of aggregates is reviewed. Increasing the adhesive forces

between particles in an aggregate should raise the strength of the aggregate because each particle contact then requires more force for

fracture. However, it is well known experimentally that strongly adhesive fine particles lead to fluffy structures which contain fewer

contacts and which are therefore weak, even though each individual particle contact may be stronger. Thus, adhesion can both increase

and decrease the strength of aggregates, since the process of aggregation is inhibited by adhesion, whereas, the strength of the final

aggregate is proportional to adhesion. This paper reviews the background to this problem and then gives two well-defined examples of

opposing behaviours: one where there is an exact correlation between adhesion and aggregates in the case of red blood cells; the other

where adhesion reduces aggregation in a computer simulation of adhering spheres. q2001 Elsevier Science B.V. All rights reserved.

Keywords: Adhesion; Aggregation; Fine particles

1. Introduction

The idea that interparticle adhesion dominates the for-

mulation of fine particle materials goes back to Isaacw xNewton 1 who wrote in 1704, Ais it not from the mutual

Attraction of the Ingredients that they stick together forcompounding these MineralsB. Although Newton is better

known for his work on mathematics, optics, motion and

gravitation, he carried out most of those studies in a 2-year

period following the great plague. Much more of his time,

around 30 years, was spent mixing particles together and

heating them in furnaces to study their transformations inw xthe alchemical tradition 2 . He wrote about 600,000 words

on this topic, and his comments on adhesion are surely

correct.

By controlling the adhesion between fine particles, it is

possible to transform products into remarkably different

states. A typical example is the addition of water tocolloidal polymer beads. Dry polystyrene particles, 0.2 mm

in diameter, are fluffy in air and do not form a transparent

film at 808C because the particles stick together to inhibit

qPresented at the Conference on Particle Technology, University of

Surrey, July 2000)

Corresponding author. Tel.: q44-121-414-2739; fax: q44-121-414-

5377. .E-mail address: [email protected] K. Kendall .

packing and aggregation. The same particles when mixed

with water can remain in suspension at high volume

fractions and will form a clear film when dried at higher

temperatures. The explanation is that the water has reduced

the adhesion between the particles, allowing them to pack

better in aqueous suspension. Thus, although the finaladhesion between each particle contact remains equally

strong as a result of the heating process, a low adhesion

during the dispersion and drying of the particles can lead

to a more compacted structure and improved final proper-

ties. The conclusion is that adhesion between particles has

opposing effects at different stages of the aggregation

process; low adhesion allows good compaction and struc-

turing in the fluid process; high adhesion then allows high

strength during the coalescence or solidification process

Clearly, it is necessary to understand both these stages of

aggregation to control aggregate strength optimally.

A second example is illustrated in the results of Fig. 1w xwhich shows bend strength data for cement bars 3 . The

left-hand curve shows that cement powder cast with water,

then hardened, gave strengths between 5 and 15 MPa with

a poor reliability reflected in the low slope of the Weibull

line. By contrast, the right-hand curve shows the same

cement powder mixed with water and soluble polymer,

then plastically sheared to break down the aggregates

followed by hardening. This gave much higher bend

strengths, near 150 MPa, with a considerably higher slope

to the reliability curve. The conclusion was that the shear-

0032-5910r01r$ - see front matter q2001 Elsevier Science B.V. All rights reserved.

.P II: S 0 0 3 2 -5 9 1 0 0 1 0 0 3 8 6 -2

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( )K. Kendall, C. Stainton r Powder Technology 121 2001 223229224

ing apart of agglomerates in a cement mix, followed by

low adhesion resulting from polymer stabilisation, fol-

lowed by reaction to produce high final adhesion, could

produce substantial improvements in properties in the final

product by allowing better particle packing before the

aggregation and solidification step.

2. Understanding aggregates

The nature of the forces holding aggregates together canw xbe described in another quote from Newton 4 : AParticles

attract one another by some Force, which in immediate

Contact is exceeding strong . . . and reaches not far from the

particles with any sensible effectB. In effect, the particles

stick where they touch; they are tacky as a result of the

strong short-range attractions.

A model aggregate can be built up at the macroscopic

scale using these tacky surface adhesive attractions. Take

smooth polymer beads of ethylene vinyl acetate copoly-

mer, about 3-mm diameter, and stir them together in a

beaker. The smooth rubbery surfaces make molecular con-

tact with each other and cause adhesion, eventually form-

ing an aggregate of several hundred beads. Such an aggre-

gate, shown in Fig. 2, displays the porosity, elasticity and

strength that would be anticipated in a micrometre scale

aggregate. The three parameters which are most useful in

describing the aggregate features are packing fraction f .volume of particlesrvolume of aggregate , diameter D of

each sphere and the work of adhesion W, the energy

required to separate 1 m2 of adhering interface.

In typical powder processes, the adhesion W exerts

paradoxical effects. For example, when forming an aggre-

gate by tableting, the dry powder is pressed between metaldies. A high work of adhesion is desirable to hold the dry

particles together in the tablet, but it also causes high

friction and fouling of the die walls, and prevents the

Fig. 1. Weibull plot of cement bend strengths, showing the improvement

produced by shearing the powder in polymer solution to break down

aggregates.

Fig. 2. A model aggregate made from rubbery spheres in adhesive

contact.

particles from getting into a close-packed state. By con-

trast, during wet agglomeration, powders are stirred and

water is added which reduces W, the work of adhesion,

allowing the particles to move into close contact to form

agglomerates which become stronger after drying as a

result of better packing. Such strengths can be measured inw xcrushing tests 5 .

The interparticle adhesion is further reduced in a colloid

suspension, for example, of polymer spheres in latex, by

incorporating surfactant polymer molecules to coat the

particle surfaces. In such latex formulations, the adhesion

between the particles can be close to zero, or even nega-

tive. Yet, aggregation still occurs as indicated by opales-

cent scattering from the structured particles in the suspen-w xsion 6,7 . Thus, it is evident that aggregates can exist over

a wide range of adhesion values. An aggregate may be

identified not only by its adhesive properties, but also by

its light scattering, its differing density, its stagnant Brow-

nian motion and so forth. Indeed, an aggregate may be

difficult to break down even when repulsive forces operate

between the particles, that is, for negative adhesion. This

occurs in sediments of dispersed particles, where gravityhas packed the particles very tightly into a hard bed, which

is difficult to remove. The same is true for filter cakes,

which may be too tightly packed when adhesion is low. It

is better to aggregate the particles with weak adhesive

forces to produce agglomerates, which can give a more

open filter cake structure. It is evident from this discussion

that adhesion is not the only factor which must be consid-

ered in evaluating aggregate properties. External influences

such as gravity, Brownian diffusion, shearing, evaporation

and entropy must also be taken into account.

3. Theory of adhesion

Adhesion is the interparticle force causing aggregation.

It could be a long-range force such as gravitational or

Coulombic, acting from the centre of the particle, but more

usually, it is a short-range force originating from molecular

forces such as van der Waals attractions of the particlew xsurfaces 8 . This adhesion can be defined as the force F

required to pull two particles apart, as shown in Fig. 3a, or

it may be defined more specifically in terms of the shape

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( )K. Kendall, C. Stainton r Powder Technology 121 2001 223229 225

. .Fig. 3. Two definitions of adhesion: a force to separate particles; b

minimum of the energy versus separation curve.

of the energy versus separation curve for unit area of .material Fig. 3b which normally requires two parameters,

a minimum energy and a range of attraction. The work of

adhesionWis the minimum of this energy curve, which in

itself can be sufficient to describe van der Waals forces

because we know the range is normally about 0.1 nm forthat case.

The understanding of the connection between the forcew xand the energy was established by Bradley 9 in 1932. He

created a theory of adhesion for rigid spherical particles by

adding the van der Waals forces for all the molecules in

the two particles and concluded that the adhesion force

should be proportional to particle diameter, and work of

adhesion;

Fs pWDr2 1 .

This theory fitted his experimental results on smooth

silica spheres shown in Fig. 4. Two years later, Derjaguin

w x10 attempted to improve this equation by taking elasticdeformations into account, but did not obtain the correct

answer. The right analysis for adhesion of smooth elasticw xspheres was found in 1971 by Johnson et al. 11 who

Fig. 4. Bradleys results for force required to separate silica spheres.

showed that the work of adhesion of smooth elastic rubber

spheres could be measured from the size of the elastically

deformed contact spot, and that the separation force was:

Fs 3pWDr8 2 .

Again, in this elastic particle case, the adhesion force

increased in proportion to the diameter of the particles,

confirming Bradleys rule. This is paradoxical because our

experience tells us that adhesion force tends to decreasewith increasing particle size, fine particles adhering best

and large particles least. This observation is best explained

by comparing the particle adhesion force with the gravita-w xtional force which tends to pull the particles apart 12 . The

two influences are plotted over a wide range of forces and

particle diameters in Fig. 5.

Taking a large particle, like a smooth racing car tire at

1 m diameter, it is clear that gravity is much larger than

the adhesion force of 0.1 N. If the tire is rough, as

illustrated in the dotted curve, then the short range adhe-

sive forces are reduced further and there is very little

influence of adhesion on the tire behaviour. By contrast, 1

mm diameter particles such as bacteria give a million timesmore adhesion than gravity. Thus, small particles are

dominated by adhesion forces and large particles by grav-

ity.

It is evident from Fig. 5 that adhesion can be controlled

in the mm to mm range by varying the roughness of the

dry particles, where roughness can be measured by atomic

force microscopy. For rough particles to adhere in propor-

tion to particle diameter, it is presumed that roughness

curvature rises in proportion to sphere diameter. Another

way to control adhesion is to wet the particles with con-

taminating molecules. For example, smooth silica particles

with adsorbed water on their surfaces stick 10 times lessw xthan clean silica grains 9 . Thus, the smooth adhesion

curve of Fig. 5 is lowered by an order of magnitude by the

addition of water. Rough particles behave differently be-

cause water can fill the gaps to increase adhesion by liquid

Fig. 5. Bradleys rule for adhesion of smooth and rough particles com-

pared with the gravitational force.

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bridging. Addition of specific surfactant molecules can

decrease adhesion still further, even to zero or negative

values. Consequently, the adhesion of particles can be

varied systematically to test the effects on aggregation

phenomena by experiment. Then, the theoretical argument

can be judged.

4. Strength of aggregates

One obvious influence of adhesion is on the strength of

aggregates. It would be expected that, as adhesion in-

creases, aggregates should become stronger. This is aw x w xtheory which goes back to Rumpf 13 and Zimon 14 in

w xthe 1950s. Rumpf 13 expressed the particle adhesion in

terms of the Hamaker constant and the separation between

surfaces, parameters which are difficult to measure. It is .often better to use the work of adhesion model of Eq. 2

above because it contains only one parameter W, which

can be readily defined and measured. Both theories gave

the same linear dependence for adhesion force on particle

size. This leads to an aggregate strength which decreases

linearly with particle size, as seen from the idealised modelw xof a perfect aggregate shown in Fig. 6a 15 .

If the separation force for each sphere is given by the

JKR equation, Fs 3pWDr8, and if the spheres all comew xapart simultaneously as Rumpf 13 assumed, then the

w xtheoretical tensile strength s of the aggregate is 16

ss 3pWr8D 3 .

This equation shows that the strength of an aggregate

can be surprisingly high if the particle size is small enough

and there are no flaws. For example, a calcium silicate .hydrate cement gel particle with particle size of 10 nm

and a work of adhesion of 0.1 J my2 would have a

theoretical tensile strength of 12 MPa according to this

equation.

However, the problem is more complex because of the

composite structure of the cement product. Also, flaws and

cracks in the aggregate must reduce the strength, while

non-equilibrium cracking tends to increase strength. If

these more complex effects are considered, together with

the widely varying packing fraction fof aggregate struc-

tures, then a more realistic equation for aggregate strengthw xemerges 16

1r24ss 15.6f Rr Dc 4 . .

where packing fraction f, work of fracture R, where R isthe non-equilibrium value of W, and particle size are

important, but flaw length c also plays a part as shown by

the results on cement in Fig. 6b, which indicates a cy1 r2-

dependence. If the aggregate is not defective and, there-

fore, only has flaws which are comparable to the particle

diameter, then c sD and this equation reduces to one .more like Eq. 3

ss 15.6f4RrD 5 .

In conclusion, the strength of aggregates should in-

crease with adhesion, but should also be influenced strongly

by packing, by particle size and by the structure of theaggregate.

5. Aggregates which grow stronger with adhesion

In order to demonstrate these concepts, the simplest

possible aggregates of human red blood cells were studiedw x17 . Such erythrocytes display very low adhesion under

normal circumstances so that only small aggregates, typi-

cally doublets, are seen under ordinary dilute conditions,

as shown in Fig. 7a. These doublets can be broken by

Brownian collisions which cause the adhering cells to

move apart. But new doublets are also forming by Brown-

ian diffusion, so a dynamic equilibrium is attained eventu-

ally. Stronger adhesion should displace this equilibrium to

higher concentrations of doublets. Therefore, the measure-

ment of doublet numbers should lead to a direct measure

of interparticle adhesion. Zero adhesion, shown by the

. .Fig. 6. a Cubic structure of spheres adhering in a perfect aggregate; b experimental strength results for cement aggregates with flaws.

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. . .Fig. 7. a Doublet formed by adhering red cells; b hard wall potential between cells; c square-well potential between cells.

hard wall potential in Fig. 7b, should give no doublets, but

as an energy well is introduced, as in Fig. 7c, doublets start

to form.

The most interesting thing about this situation is that thew xproblem can be solved analytically 17 if a theoretical

square-well interaction potential is assumed as shown in

Fig. 7c. In a square-well potential, the particles have hard

walls at a radius a, but also have an attractive zone of

attractive energy at la. Beyond that range, the spheres

do not interact but travel in straight lines at constant speed.

This hard sphere square well, which was first used byw xAlder and Wainwright 18 , has been solved to predict the

number of doublets in a suspension.

The mathematical result is that the ratio of doublets to

singlets N rN is proportional to the volume fraction fof2 1the cells and depends on the range l and the energy of

the well according to the equation below.

NN rN2 s 4 f l3 y 1 exp rkT fN rN 6 . . .2 1 2 1

The conclusion of this argument is that a plot of doublet

to singlet ratio versus particle volume fraction should yield

a straight line passing through the origin. The gradient of

the line is a measure of the adhesion which depends on

range land energy of the interactions. Therefore, a high

gradient signifies high adhesion and a low gradient, low

adhesion as shown below in Fig. 8. Thus, an adhesion

number can be defined as the gradient of this plot, to give

a measure of the bonding of the cells. This idea has been

proved for polymer latex, for colloidal silica spheres andw xfor emulsion particles 19 . Experimentally, the objective

was to define this non-dimensional adhesion number for

three different species of red cells, horse, rat and human.

Red blood cells, erythrocytes, were used because ofw xtheir low and reversible adhesion 20 . Cells were prepared

from three species: human blood from North Staffordshire

Hospital, fresh horse blood in EDTA, and fresh rat blood

from Central Animal Pathology. Each blood sample was

washed six to seven times in phosphate buffered saline to

remove the non-red cell components before suspending in

physiological saline solution, then examined by optical

microscope to measure the number of doublets. Each

species of cell was treated in two ways to judge the effect

of surface adhesion molecules: by adding glutaraldehyde,

which is used to prevent cell adhesion; or fibronectin

which is known to coagulate red cells. The results forhuman red cells gave an adhesion number of 420. For

comparison, the results for rat and horse erythrocytes are

shown in Table 1 below.

These results show conclusively that rat cells are almost

twice as sticky as human red cells, while horse erythro-

cytes are almost twice as adhesive as rat cells, causing

Fig. 8. Results for human erythrocyte doublets, showing the effect of glutaraldehyde which reduced adhesion and fibronectin which increased it.

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Table 1

Comparison between adhesion of various red cells

Animal Adhesion number N rN f2 1

Horse 1488"200

Rat 750"4

Human 420"5

larger aggregates. Whether this can be explained in terms .of the higher energy of the bonds, as defined by Eq. 1 , or

in larger range of bonds remains to be determined.

6. Aggregates reduced by adhesion

A particular example which has demonstrated a de-

crease in aggregates as the adhesion was increased hasw xbeen defined recently 21 . Uniform spherical particles

were modeled in a molecular dynamics computer simula-w xtion using a code developed in a PhD study 22 , following

w xthe method of Alder and Wainwright 18 . Several benefits

arise from the use of the computer model: firstly, the

particles can be followed individually in the computer,

whereas, it would be very difficult to see inside a high

volume fraction dispersion of zirconium oxide, for exam-

ple; secondly, the particles can be made equally sized and

identically adherent in the model, a tough task experimen-

tally; thirdly, problems such as gravitation, which strongly

affect experimental results can be eliminated in the model;

finally, the computer can provide a unique answer to the

simple question, does increasing adhesion change the

aggregation behaviour?.

Carl Stainton set up 12,000 spheres in his desktop

computer, started them moving randomly around in athree-dimensional box with periodic boundaries, and then

compacted them to 0.576 packing fraction, where structur-

ing was then observed. The structures were picked out on

the computer by their particular signatures, and then fol-

lowed as they grew with time. Obviously, the growth of

structure is random and is different in each simulation,

depending on the position of all the spheres at the starting

point. In other words, structuring is a random process.

However, the general pattern of the structural growth could

be observed to form some overall average conclusions

about the ultimate geometry. .

Although the face-centred cubic fcc crystal is the onethat is most stable, having the lowest energy, it was noted

that this phase was not the first to be nucleated. In fact the

body-centred cubic, i.e. bcc lattice was the one that tended

to grow first in the simulation, closely followed by fcc and .hexagonal close-packed hcp . This suggests that Ostwalds

step rule is being followed. Ostwald found that the stable

crystal form does not emerge first in crystallisation experi-

ments. The phase closest in energy to the random phase

appears first and then mutates with time into the stable

crystal structure. This happens because the random phase

more quickly adapts to the bcc rather than fcc even though

the bcc is less stable over long times.

As the model was observed, small regions of bcc

structure were seen appearing. These crystals were unsta-

ble and readily transformed into a faulted mixture of fcc

and hcp structures by a slip of the lattice. Thus, fcc regions

closely followed the appearance of bcc material, with hcp

lagging behind. Slow conversion and growth of the nuclei

occurred, but with an unpredictable path giving differingamounts of fcc and hcp. There seemed to be no one single

pathway through to the final stable fcc structure. As the

crystal regions began to dominate, with the random phase

disappearing to almost zero, the development of fcc struc-

ture became spasmodic as sudden conversions of material

took place.

The final structure of the particles after a long period of

computing is shown in Fig. 9. It was evident that the

structure was dominated by fcc shown as dark spheres

while most of the random material shown as light spheres

had disappeared. But there were still significant regions of

bcc and hcp shown as intermediate greys. The four pic-

tures in Fig. 9 show the randomness of the final structures,

although they each contained roughly the same volumes of

each phase.

In the above case of zero adhesion, the spheres re-

mained in a fluid state, allowing aggregated structures to

form very easily. As adhesion was introduced, using the

square-well model shown in Fig. 7b, then this adhesion

must pull the spheres together in addition to the com-

paction pressure, eventually making the material turn solid.

This is the question now addressed: how does the attrac-

tive potential of the particles influence the structuring of

Fig. 9. Final structures of computer models showing the dominance of fcc . . .dark over bcc and hcp grey and random light , for four different

starting conditions.

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Fig. 10. Computer results showing 12,000 spheres after aggregating at 2

kT.

w xthe spheres? Alder and Wainwright 18 had attempted to

answer this question in the late 1950s but computingpower was insufficient and he could not resolve it. Now, a

desktop computer can provide some answers. The obvious

answer is that high adhesion causes the particles to gel,

preventing them from easily structuring in the equilibrium

positions.

The two cases of zero adhesion or finite adhesion were

simulated on the computer one after the other, with identi-

cal starting configurations. The adhesion was first inserted

at a low level, less than kT, which had no influence on

aggregation, and was then gradually increased to 2 kT to

show its effect. One resulting structure is illustrated in Fig.

10. It is evident that most of the particles remain in the .fluid disorganised state light particles under the adhesion

influence, and do not aggregate as before.

7. Conclusions

Adhesion between particles has been studied in relation

to its effect on aggregation phenomena. It has been demon-

strated that adhesion has the direct effect of increasing

aggregation, if the structure remains constant. Experiments

with blood cells show this effect clearly. Strength of

aggregates is directly proportional to the adhesion energy

in this case. However, it has also been shown that structure

is also influenced by adhesion in more subtle ways, as

indicated by a molecular dynamics computer model. Adhe-

sion between particles causes solidification of aggregates,

and this can interfere with the structure, making it looser

by reducing packing fraction, or slowing the structure

formation for ideal monosize spheres at high packing

fraction. Since the equation for strength of aggregatesdepends on packing fraction to the power four, this effect

of reducing aggregates can be the dominant one.

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Kendall 2001

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