Compression of polymer bound alumina agglomerates at the micro deformation scale

www.elsevier.com/locate/powtecPowder Technology 140 (2004) 228–239

Compression of polymer bound alumina agglomerates

at the micro deformation scale

Y. Sheng, B.J. Briscoe*, R. Maung, C. Rovea

Department of Chemical Engineering, Imperial College, Prince Consort Road, London SW7 2AZ, UK

Abstract

In this paper, the mechanical properties of alumina particle based agglomerates, prepared in a tumble mixer by using a poly (vinyl alcohol)

(PVA) binder (ca. 5% by volume) in conjunction with water solvent, are described at a micro deformation level. Uniaxial compressive

deformation profiles of these alumina agglomerates, typically 180–200 Am in diameter, are reported and their diverse behaviour during

compression have been observed, which vary from a clear brittle rupture to progressive ductile deformation. However, certain common

patterns of the agglomerate reaction force response, as a function of the compressive displacement, are identified, such as the similarities in

the reloading response after each discrete fracture event. The Hertz Theory and a slope and peak force analysis are applied to establish the

common patterns and trends, and generalize the intrinsic deformation characteristics of these agglomerates.

D 2004 Elsevier B.V. All rights reserved.

Keywords: Agglomerate; Mechanical behaviour; Compression; Micro deformation; Slope analysis; Peak force analysis

1. Introduction

Agglomeration techniques are adopted widely in areas

such as the pharmaceutics and the ceramic industry where

fine powders are often agglomerated prior to the compaction

and tabletting processes to facilitate the required flow

properties.

Agglomerates often exhibit complex mechanical behav-

iours that vary from brittle, elastic-plastic (for most dry

agglomerates) to elastoviscoplastic and fully plastic (for

wet agglomerates) depending upon the preparation method,

the environment (moisture content/temperature), the micro-

structure, and the loading conditions (mode and loading rate)

[1–15]. It is desirable for a number of reasons, to identify and

quantify the mechanical properties of agglomerates, not least

to ensure that the products or intermediates meet the required

specifications such as their fracture strength and solubility.

Various testingmethods are generally used in industry such as

compression, indentation and tensile testing, but these tests

can be difficult to perform at micro scale due to the difficulties

in manipulating such small entities. In addition, by their

0032-5910/$ - see front matter D 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.powtec.2004.01.016

* Corresponding author. Tel.: +44-171-594-5559; fax: +44-171-594-

5561.

E-mail address: [email protected] (B.J. Briscoe).

nature, normally agglomerates made by nominally the same

method may exhibit a wide spectrum of the mechanical

behaviour. Thus, despite of the potential importance, this

subject has not been studied in the same depth as for larger

monolithic materials. There is much available literature on

the measurement, micro mechanical modelling and numeri-

cal simulations of the mechanical behaviours of agglomerates

in the scale of nanometers to millimetres [1–5,9–16,24].

Most of these focused upon determining the tensile strength,

yield strength, fracture toughness, and attrition properties of

the agglomerate [9,11,13,14,16,24].

In this paper, the agglomerates studied were all made by

the tumbling method. Poly (vinyl alcohol), dispersed in

water (5% by volume), was sprayed into the a-Al2O3

powder as the binder to facilitate the agglomeration process.

Alumina agglomerates made by this way were subsequently

dried and sieved into certain size classifications. Then the

micro scale compression tests of these alumina agglomer-

ates have been carried out in a specially designed micro-

compaction apparatus in order to characterise the various

deformation processes that the chosen agglomerates under-

go when they are compressed between two nominally

smooth and parallel platens. The first order Hertz Theory

[17,20] is applied to describe part of the elastic response of

the agglomerates during their initial compression stage. The

Y. Sheng et al. / Powder Technology 140 (2004) 228–239 229

mechanical response of the agglomerates during the entire

compression process could not be described in terms of a

mean general representative model or associated parameters.

Two different approaches are adopted in this paper. Since

certain similarities in the piecewise reloading curves, after

each fracture events were found, the slope analysis was

carried out to represent the intrinsic evolving stiffness of the

agglomerates. The peak force analysis [22] was introduced

in the studies to describe the overall response of the

agglomerates. A wide diversity of the deformation profiles

and stiffness parameters have been found in the acquired

data, which is most certainly a reflection of the method of

agglomerate manufacture and the inherent heterogeneity of

the resulting agglomerates.

2. Experimental configurations

2.1. Preparation of the alumina agglomerates

An a-Al2O3 powder (AES-11c, Sumitomo Chemicals,

Japan), with mean particle size 0.5 Am and of high purity

(99.8%) was used in the current investigation for its

commercial importance; i.e. single crystals, tubes for

various uses, cutting tools, substrates for electronics, cat-

alyst supports, and wear resistant components can all be

readily manufactured from Sumitomo AES-11c alumina at

relatively low sintering temperatures; and also, data per-

taining to the processing such as the diffusion coefficient,

the surface energy, the thermal expansion coefficient may

easily be found for the subsequent studies. Another ad-

vantage is that this material is now available in uniform

quality over long periods. It is also a material that has been

extensively used by one of the authors, in conjunction with

the current agglomeration procedure, to study various

facets of the manufacture of monolithic dense alumina

specimens [6–8].

To facilitate the agglomeration, a polymeric binder (poly

(vinyl alcohol), PVA (BDH Chemicals, UK)), dispersed in

Fig. 1. SEM pictures of the alumina agglomerate made by Tumbling method. (A) S

alumina agglomerate (� 50,000). The white areas represent the alumina particles

deionised water, was added to the alumina powder. The

binder serves to combine together the basic alumina par-

ticles to produce free flowing units and also convey strength

or toughness to the agglomerates. In the current investiga-

tions the binder system was produced from a commercial

grade poly (vinyl alcohol), having a molecular weight of ca.

72,000 (BDH Chemicals, UK); degree of hydrolysis, ca.

90%. This binder system has been used externally as an

additive, and with success, in the author’s laboratory to

manufacture alumina greens and sintered dense bodies. A

measure of 2 g of PVA (1% of the alumina weight) was

dissolved with 50 ml of deionised water to thoroughly

break-up large aggregates and then heated at 75–95 jC to

bring about a complete dissolution of the polymer. The

solution was then cooled to ambient. The agglomeration

manufacture was performed in a batch-type tumble mixer:

Pascall, Lab-mixer II (Pascall, UK), with an air-driven

spray-gun (ITW finishing, Bournemouth, UK). The binder

solution was sprayed on to the particles whilst the drum was

rotating for a period of 5–10 min. The compound was then

mixed for an additional 15 min to produce approximately

spherical agglomerates; this process also had the effect of

polishing the agglomerates to remove weak and irregular

surface features. After drying, the agglomerates were sieved

into the size classifications ranged from 180 to 250 Am. A

SEM microscope (SEM; Jeol T-200, with gold coating) was

used to examine the nature of the individual agglomerates.

The clusters of particles are roughly in same range of sizes

and shapes as is shown in Fig. 1(A). The black area in Fig.

1(B) represents the voids within the agglomerates, which

suggests a significant non-homogeneity in the microstruc-

tures of the agglomerates.

2.2. Micro compression apparatus

A specially designed and constructed instrumental

arrangement, schematically illustrated in Fig. 2, was used

to investigate the compressive deformation response of

these single agglomerates between the nominally smooth

EM image of a single alumina agglomerate (� 10,000). (B) SEM image of

and the dark areas represent the pores.

Fig. 2. Schematic figures of the configurations of micro compression apparatus. 1. Antivibrational bench, 2. inverted microscope, 3. particle sample, 4. video

cameras, 5. video switch, 6. video capture computer, 7. video printer, 8. translational stage, 9. force transducer, 10. cantilever, 11. microstep driver, 12.

transducer amplifier, 13. host computer, and 14. position sensor.

Y. Sheng et al. / Powder Technology 140 (2004) 228–239230

flat parallel surfaces [2,4,5]. The instrument system was

based around a Wilovert S inverted optical microscope

(Plund Wetzlar, Germany) (2 in Fig. 2). An inverted

microscope was used so that the deformation elements

generated by the equipment could be attached without

compromising the conventional imaging capabilities of the

microscope in the axis of the compression. A micro-

controlled motion stage (8 in Fig. 2) capable of discrete

Fig. 3. Digital images of the agglomer

micro-steps less than 100 nm was attached to the micro-

scope stage (PTS 1000, Photon Control, UK). Connected

to the motion stage was a small horizontal arm (10 in

Fig. 2) on to which a very sensitive piezoelectric force

transducer (9 in Fig. 2) was mounted. The force trans-

ducer (BG-10, Kulite, USA) was a strain gauged micro-

cantilever beam with a force resolution better than 10� 5

N. A highly polished flat brass platen (2 mm diameter)

ate during the compression test.


was attached to the bottom of the transducer. During the

experiments the platen was raised and lowered using the

motorised control stage with a maximum displacement

capability of 150 mm. To ensure that the displacements

made by the stage were accurate, a linear encoder

operated on the principle of a photoelectric register was

used to verify the magnitude of the imposed displace-

ments (14 in Fig. 2).

The use of two high-resolution video cameras allowed

visual observations of agglomerate deformation to be made

during the experiments. One camera was placed on the

microscope viewing system and another was placed orthog-

onal to the axis of the compression equipment. Digital

images of the deforming agglomerates during the compres-

sion tests could be captured and the various deformation

stages could also be monitored during both loading and

unloading, as is shown in Fig. 3. The configuration of the

system not only simplified the particle selection process but

also assisted in interpreting the results obtained.

In order to provide a value for the actual, as opposed, the

imposed displacement, a compliance characterisation was

performed to remove the deformation of the force transducer

from the recorded displacements [3]. The stiffness of the

cantilever beam in force transducer was found to be

2� 10� 3 N/m in the compliance test, so the deformation

of the cantilever beam could be calculated by dividing the

corresponding force by the stiffness of the beam.

Before any mechanical investigations were conducted,

the size of the agglomerates was measured across the two

principle axes (in the plane of view) using a graticule and

the simple arithmetic average was used to describe the

effective agglomerate diameter.

Fig. 4. Typical force–displacement curves with two general categories of deforma

with a smoother curve. Particle size: 200 Am, Compression rate: 1 Am/s.

The chosen agglomerates were then placed on the lower

platen (a glass platen) to ensure they were placed directly

underneath the centre of the platen. The upper platen was

lowered very slowly ( < 2 Am/s) until it just touched the

sample as judged by the sensed transducer signal. Once

contact with the sample had been identified the computer-

driven deformation program was initiated. Experiments

were conducted at three different velocities, 1, 5 and 10

Am/s, and were performed in an ambient environment

within a temperature of 20–30 jC and 25–40% RH.

3. Force–displacement curves

Typical force–displacement curves obtained from the

micro scale compression experiments are shown in Fig. 4,

from which the sequence of discrete rupture events can be

identified through the fluctuations of the reaction force.

Also illustrated in Fig. 4 is the result of the correction for the

compliance of the force transducer; the transducer deflection

is found to become a significant fraction of the imposed, as

opposed to the actual, displacement for large imposed loads.

Despite showing the definite diversity, the complete force

against displacement curves are seen to fall into two general

categories, as demonstrated in Fig. 4.

The first is a group of ‘‘jagged’’ data curves with lower

overall rupture forces. Agglomerates that primarily fracture

are usually characterised by a sudden decrease in the

reaction force with the progressive increase of the imposed

displacement. When the force drops to zero or a finite value

before increasing again, partial fracture takes place within

the agglomerate, which is accompanied by a spatial re-

tion response: a brittle response with a jagged curve and a ductile response


orientation of the fractured primary particles. The partial

fracture phenomenon can also be observed in the digital

pictures taken during the micro compression tests, as is

shown in Fig. 3. It will be realized that the recorded force–

displacement response observed (see Fig. 4 for example) is a

function of the interactions between the transducer system

and the deformation properties of the agglomerates. The

stiffness of the mechanical beam and the agglomerates do

not differ greatly; by a factor of five to ten. The beam was

not damped either mechanically or electronically (by filter-

ing). Hence appreciable discontinuous motion is to be

anticipated where the sample facilitates this process. The

phenomenon has many things in common with stick–slip

motion [4,5,23].

The second generic type of response is a set of smoother

continuous loading curves with less abrupt and discrete

force discontinuities, which indicate that the agglomerates

do not appear to experience frequent and significant fracture

events while being compressed, and exhibit a comparatively

more ductile behaviour. In this case, the extent of force

‘‘drops’’ are comparably small when compared with the

general trend of the curve, and this is represented by a

horizontal plane interrupting the general increasing trend, as

can be seen in Fig. 4. It is believed that these events are not

a ‘large fracture’ but a sliding or geometric re-orientation

response of the stressed material. It will be clearly evident

that this class of response also provides a significantly larger

reaction force than the other class.

Typical views of the rupture patterns of the agglomerates,

observed at the bottom contact in the direction of the

loading axis, for these two generic types of force–displace-

ment responses at the end of the deformation, are shown in

Fig. 5. For the typical brittle case, the agglomerate is usually

crushed and fragmented at a lower applied force, whilst the

agglomerates with ductile response can sustain a higher

compression force up to the pre-fixed deformation and the

unloading phase commences with few fractures. These

observations are consistent with the recorded force–dis-

placement data, shown in Fig. 4, in which the ductile

agglomerates always have higher reaction forces, with the

Fig. 5. Digital images of the different rupture patterns: brittle (left), and ductile (r

using a � 10 lens.

same imposed displacement, compared with the compara-

tively brittle cases.

Although the irregular, often jagged shape and irrepro-

ducibility of the curves makes a rigorous interpretation of

the obtained data a difficult task, certain trends can still be

abstracted from the data by means of contact mechanical

theories and some data treatment methods specially devel-

oped for such fluctuating data. As there are definite simi-

larities between the slopes of ‘‘reloading’’ curves after each

fracture events amongst all of the tests conducted, the

average value of those slopes will be a useful parameter

to characterize the intrinsic stiffness of these agglomerates.

The general increasing trend and similar shapes of both the

‘‘jagged’’ and ‘‘smooth’’ force–displacement curves can

also provide an overall description of the deformation

profiles and mechanical responses of agglomerates under

compression.

First, the elastic response of agglomerates, at the initial

stage of the micro compression, is analysed by applying the

Hertz Theory.

4. Hertz analysis

A number of theories have been proposed to describe the

deformation of solid and liquid filled spheres which may

provide an insight into describing the deformation of these

alumina agglomerates at small compressive loads prior to

their fracture [2,4,5,11,17–21]. The Hertz theory [17]

describes the small strain contact deformation of spheres

deforming elastically. The JKR theory [18] modifies the

Hertz theory by including an auto adhesive effect, which

may play an important role in the very small deformation

range. The Tatara theory [19] invokes non-linear elasticity

and a large deformation formulation to predict the compres-

sive behaviour of elastomeric spheres at large deformations.

The nearly spherical agglomerates studied in this paper

often showed a high degree of elasticity, as judged by the

clear recovery of the imposed strain, up to an overall

imposed deformation of ca. 5%; this is the contact com-

ight), corresponding to the tests shown in Fig. 4, viewed from the bottom,

Table 1

Result of Hertz analysis

Deformation

velocity

Hertz Index Mean R2 Mean reduced

modulus (MPa)

Mean R2

1 Am/s 1.52 0.907 385 0.909

5 Am/s 1.58 0.937 388 0.913

10 Am/s 1.44 0.948 323 0.926

Mean 1.51 0.931 365 0.916


pression for two contacts. The data manipulation process

suggested by Johnson [20] was conducted to investigate

whether the Hertz analysis was applicable. The mechanical

response of a spherical body compressed between two flat,

smooth rigid surfaces may be approximated as [21]:

F ¼ 4

3� R1

2 � E

1� v2

� �ðh1=2;HÞ

32 ð1Þ

where F is the applied force, R is the initial radius of the

sphere, E is the Young’s modulus of the material, and m is

the Poisson’s ratio of the sphere and h1/2,H is the Hertzian

approach distance. Assuming the system is of perfect

symmetry, the above equation can be rewritten with respect

to the total imposed displacement h = 2h1/2,H as:

F ¼ 4

3� R

12

232

� E

1� v2

" #ðhÞ

32 ð2Þ

Thus, if the initial force–displacement curves in a log–log

plot are linear with a slope of 1.5, then Hertz analysis may

be considered to be applicable and can provide a physically

realistic description of the compressive behaviour of the

agglomerates at the initial stage of the compression. It is

extremely difficult to precisely define a force/displacement

Fig. 6. Results of Hertz Factor f

range for the Hertz analysis, or indeed to accurately estimate

a mean or effective radius of the agglomerate. In this paper,

an imposed displacement up to 5% was selected for the

analysis and the range selected was also terminated a few

microns before the first fracture. Results of average slopes

data of the log F against log h (named Hertz Factor)

obtained from such an analysis are summarised in Table

1, and illustrated in Fig. 6 for agglomerates of nominally the

same composition deformed at three different velocities.

The assumptions upon which the Hertz Model is based on

are not completely satisfied in the practical tests. The

geometry of the agglomerates is not perfectly spherical and

it is very difficult to ensure that the contact between the platen

and the agglomerate is absolutely normal and frictionless.

Thus there is an inevitable distributions of the fitted slopes of

initial linear part of force–displacement curves which can be

noted in Fig. 6. Nevertheless, the average values of the Hertz

Index for the experiments at all three deformation rates are

fairly closed to 1.5, which indicates the applicability of the

Hertz Analysis approximations in the early stages of the

micro compression of the agglomerates. There is no signif-

icant effect of the small variations of loading velocities upon

the values of average slopes; similarly, no time dependent

modulus is identified; however, the approach velocity, and

the nominal rate of strain, was only varied by a factor of 10.

The mean radius of agglomerates was estimated by

adopting an arithmetic mean; the value of reduced agglom-

erate modulus was then calculated using Eq. (2). The

computed mean value of Er (the reduced modulus) for the

different deformation velocities are listed in Table 1. The

results for the studies at the lower deformation velocities of

1 and 5 Am/s are fairly similar. Modulus compacted at the

deformation velocity of 10 Am/s is significantly lower.

or different loading rates.


Due to the porous and variable nature of the agglomer-

ates, it is very difficult to define the precise value for

Poisson’s Ratio experimentally. However, a fair estimation

can be made by assuming the Poisson’s Ratio of alumina

agglomerate to be in the range of 0 to 0.3, and then the

Young’s Modulus, E, is estimated as between 400 and 500

MPa:

Er ¼E

1� v2ð3Þ

The reported tensile modulus for crystallised alumina is

between 200 and 500 GPa [7], whilst the tensile modulus for

anhydrous PVA is about 10 GPa at an ambient temperature

of ca 25 jC and modest strain values [10]. Thus, the

estimated value of the agglomerate modulus is ca. three

orders of magnitude less than that of the constituent alumi-

na, and about a tenth or one twentieth or so of that of the

binder. One may reasonably assume that in such a porous

structure, the rigid particulate will have little direct influence

as the strain will be accompanied by the deformation of the

binder and the associated geometric deformation including

the closure of the adjacent voids. Voids closure, and the

anticipated strain hardening, is not an apparent major feature

of these data for either types of the generic response. The

force/displacement curves do not resemble the typical con-

strained uniaxial compaction data for a collection of

agglomerates [7,10,16]. We must assume therefore that a

significant fraction of the imposed displacement is simply

accommodated by the radial expansion of the deformed

agglomerates. The application of a simple scaling law would

suggest that the porosity of the binder matrix would be

about 90% to 95%; this is quite close, maybe fortunately so,

to the nominal value.

5. General stiffness analysis

When examining the events that occur during the entire

micro compression process, despite the fact that the exper-

iment data are distributed over a broad spectrum of values,

there are, however, similarities when considering the general

trends and the slopes of the reloading after the partial

fracturing events. To clarify the trends observed from these

diverse data, which exhibit a similar response, they are

grouped into two general categories as was mentioned

above; that is the primarily ‘‘Brittle’’ and ‘‘Ductile’’

responses. The data analyses for the two cases were carried

out and are reported separately. For the group of the more

ductile agglomerates, the force–displacement curves are

‘‘smoother’’ and monotonic, increasing during the loading

process with very few identifiable fractures, as a loss of load

bearing capacity event. Thus it is easier to fit these curves

into a general equation, such as a power law model.

However, for the brittle agglomerates with the ‘‘jagged’’

force–displacement curves, the fracture events appear to

include a random process, and it is thus potentially difficult

to identify a general trend in the behaviour of these

agglomerates. Therefore, different approaches are required

in order to estimate the stiffness of these agglomerates from

the micro-compression data. A slope analysis is used to take

account of the intrinsic mechanical non-homogeneity of the

agglomerates and the consequent fracture events in terms of

the average of slopes of the ‘‘piecewise’’ reloading curves

after each fracture event. The overall stiffness of the

compression curve has been defined in a more general

way as a parameter to provide an indication of the general

force resistance response of the material to the stresses

applied. This can be generated after numerical treatments

in order to smooth the severely fluctuated data. A Peak

Force Analysis [22] is introduced in this paper to treat the

‘‘jagged’’ force–strain curve cases in order to produce a

generalized smoothed loading curve, and develop a more

generic description of the overall response of agglomerates.

5.1. Slope analysis

Due to the somewhat apparent random ‘‘oscillating’’

force–displacement data behaviour of the agglomerates

under compression, in the context of their reaction force-

imposed displacement response, it is very difficult to deter-

mine properties which are directly connected with the

fracture mechanics, such as the ‘‘strength’’ of a single

agglomerate. The value of the force causing the first evident

fracture of the agglomerate is found to vary, over a very

wide distribution in the current experiments.

A common feature of all the ‘‘jagged’’ force–displace-

ment charts is the readily identifiable repetitive alternate

increasing–decreasing trends in the reaction force. The

amplitude of the force oscillations is not regarded to be a

significant parameter, at this stage, since it varies signifi-

cantly in the cases examined. However, the slopes of the

quasi-linear sections after each fracture show a certain

similarity both in each experiment and amongst all of

the experiments conducted. The average slope figure for

these slopes reflects the intrinsic elastic characteristics of

the agglomerate and its subsequent deformation induced

debris during the progressive compression and fracturing.

As is shown in Fig. 7, the values of the measured slopes

are relatively consistent in their magnitude for a given

deformation velocity, which indicates that, after each

fracture event, there must be a subsequent rearrangement

and re-orientation of the primary particles within the

ruptured agglomerate to form a new and mechanically

stable structure in order to transmit the compression force

until the next rupture event occurs. Reloading the agglom-

erate with the instantaneous stable structure after fracturing

is found to follow the same slopes. Therefore, the average

of these reloading slopes can be regarded as a representa-

tive parameter for the intrinsic evaluating stiffness of the

progressively disrupting agglomerates. As may be ex-

pected, a degree of variation in the results is evident,

Fig. 7. Slope analysis of the reloading curves after partial fractures.


and this is believed to be due to the non-homogeneity of

the agglomerates; actually these data suggest that there are

no very major internal spatial varieties in the properties of

these agglomerates. A detectable decreasing trend of aver-

age slopes with the deformation rates can be identified in

the results, which is probably not significant. It is worth

noting that the binder material is intrinsically viscoelastic

and hence an increasing trend may have been anticipated.

One of the explanations for the decreasing trend could be

the corresponding and associate creep effect. At the lower

velocities there is more time available for a rearrangement

Fig. 8. Peak force analysis and power law fitting for a typical jag

process to take place within the agglomerate to enhance

the overall stiffness.

5.2. Peak force analysis

In order to identify and define an overall stiffness

parameter, directly, taking the data recorded would produce

a very wide spectrum of results, due to the wide variable

degree of the ‘‘fluctuations’’. An averaging scheme may

reduce the force oscillations to some extent, but may

underestimate the mean effective stiffness of the specimens

ged force–nominal strain curve (loading velocity: 5 Am/s).

Fig. 9. Result of K for brittle agglomerates.


because it considers in an equal way for both the positive

and the negative force deviations. Additionally, the inciden-

tal force minima, caused by fracturing, will considerably

affect the magnitude of the deduced values. Since we are

currently more interested in the rupture phenomena (com-

plete failures, partial failures, sliding events), then the

stiffness parameter required would be more realistic if it

were to be identified with the peak force values, rather than

with a mean value set composing of the maxima and

minima, of these force fluctuations.

An algorithm, to identify the local maximum values in a

typical ‘‘jagged’’ force–deformation chart, has been devel-

oped in order to provide a monotonic increasing envelope

Fig. 10. Result of n for b

for the fluctuated loading curves [22]. Each point of the

original curve is evaluated and is removed if its value is

lower than or equal to the previous higher force or to the

next value. In an algebraic form, each specific point, i, is

removed if one of the following conditions is not satisfied:

PðiÞ > max ½Pðj < IÞ� ð4Þ

and

PðiÞ > Pðiþ 1Þ ð5Þ

A major advantage of this approach is that all the data

files produced have a considerably smaller number of data

rittle agglomerates.

Fig. 11. Result of K for ductile agglomerates.


points, and hence the informed fitting of the data sets

becomes a much simpler task; especially because the new

shape of the resulting force–deformation curve is more

regular and smoother than the original. A power law

model can then be reasonably fitted to the new data file,

where;

P ¼ Ken ð6Þ

where the K and n values are numerical constants to be

determined and a is the nominal strain.

For the convenience of comparing the results from these

data that have different agglomerate sizes, a nominal strain

Fig. 12. Result of n for du

is introduced as the deformation of the agglomerate

divided by the diameter of the tested agglomerate. The

force–displacement curves are translated into the force–

nominal strain curves for the subsequent peak force

analysis.

The peak force analysis and the curve fitting procedures

are demonstrated in Fig. 8, in which square symbols are the

local maxima and are identified by the above algorithm. A

power law model was fitted to these points in order to

provide optimised values of the constants such as K = 0.016,

and n = 0.39. The result of the direct fitting to the jagged

curve, which equally considers the maxima and minima

deviation, is also presented in Fig. 8. Compared to the peak

ctile agglomerates.

Table 2

Result of the peak forces analysis, and the power law fitting; Eq. (6)

Deformation

velocity

Brittle Group

(jagged curves)

Ductile Group

(smoother curves)

K n Mean R2 K n Mean R2

1 Am/s 0.032 0.473 0.916 0.068 0.638 0.955

5 Am/s 0.024 0.437 0.912 0.068 0.626 0.968

10 Am/s 0.031 0.479 0.928 0.064 0.626 0.962

Mean 0.029 0.463 0.918 0.067 0.630 0.961


force results, the power index of average fitting is much

lower, which indicates a general underestimation of the

stiffness of agglomerates.

The average values, and their corresponding numerical

deviations, for the power law fitting results for the different

deformation rates are illustrated in Figs. 9–12, and the

parameters are listed in Table 2. The average numerical

results of the power law constants are relatively consistent

for all of the loading conditions with a significant amount of

deviations presumably caused by the inherent non-homoge-

neity of the samples.

There is also a significant decrease in the numerical

values of k and n as the loading rate increases from 1 to 5

Am/s for the more brittle cases. The average result of 10 Am/

s tests increase to near the ones of deformed at 1 Am/s. This

may be reasonably attributed to the intrinsic non-homoge-

neity of the samples studied. As is shown in Figs. 11 and 12,

the average figures of the power law constants K, and n for

the ductile agglomerates are found to be greater in their

convergence than the corresponding values for the brittle

cases. There are smaller differences between the results

from different loading rates. These features may be obvi-

ously identified when comparing Figs. 9 and 10 with Figs.

11 and 12; the power law constants of the ductile agglom-

erates are significantly larger than those of the brittle cases,

which indicates that agglomerates exhibiting a ductile

behaviour in the compression test actually have a higher

overall stiffness by a factor of ca. 2, in K (also see Table 2).

This phenomenon can be reasonably explained as arising

from the results of the nature of uneven distributions of the

binder content within the agglomerates that the tumbling

method will naturally produce. According to the current data

and the associated analysis, the power law model is seen to

be a useful and economic method to describe the overall

mechanical response of the agglomerates under the uniaxial

compression.

6. Conclusions

Based upon the results obtained from micro compression

experiments, the deformation profiles and the mechanical

response of the alumina agglomerate are grouped into two

general categories: a ‘‘brittle’’ response with an associated

‘‘jagged’’ force–displacement curve and a ‘‘ductile’’ re-

sponse with an associated ‘‘smooth’’ force–displacement

curve.

The Hertz theory has been applied to describe that

part of the apparently reversible elastic behaviour of these

agglomerates during the early stages of the micro-com-

pression ca. 5% of the imposed displacement. The

numerical value of the reduced modulus and Young’s

Modulus are then obtained from the application of the

Hertz theory, and both parameters, the Hertz Index and

Reduced Modulus, for both types of generic response, are

seen to have sensible numerical values. The typical value

of the estimated Young’s modulus is about 5% of that of

the polymeric binder.

The slope and peak force analyses, adopted in this

paper, have revealed some intrinsic mechanical character-

istics associated with the deformational response of the

agglomerates. The average values of the slopes of the

‘‘piecewise reloading’’ curve, after each fracture event,

are found to be very consistent, and thus are regarded as

an effective and useful measurement of the intrinsic or

effective essentially unconstrained stiffness of the ag-

glomerates. A detectable, but probably not significant

rate-dependent behaviour of these slopes has also been

observed. The peak force algorithm is introduced in order

to ‘‘smoothen’’ the fluctuated force–displacement curves,

and a power law model has been proposed to describe

the overall ‘‘peak’’ stiffness of these agglomerates during

the entire compression process. The agglomerates exhib-

iting the more ductile behaviour are found to have a

higher overall stiffness than the relatively more brittle

examples, as is expressed in the K and n values in power

law Eq. (6).

It is believed that the observed spectra of the

behaviour noted in the results is a natural consequence

of the intrinsic compositional non-homogeneity and the

associated imperfect geometry of the agglomerates ex-

amined. This may be due to several causes, i.e. each

agglomerate could potentially have different binder con-

tents since the tumbling technique is probably not a

very effective means to ensure that a uniform distribu-

tion of the binder is provided throughout the agglomer-

ates; and the localized distribution of defects (such as

porosity, small hard inclusions, very small fractures),

within each of the agglomerates; also the shape and

size of the agglomerates studied were often neither

uniform nor well defined.

Acknowledgements

The research work in this paper is supported by the

Engineering and Physical Sciences Research Council

(EPSRC) in UK, through grant number GR/N13050. The

authors would like to thank Dr. D.R. Williams for his

contributions in design and construction of the micro

compression apparatus used in this study.


References

[1] M.J. Adams, R. McKeown, Micromechanical analysis of the pres-

sure–volume relationship for powders under confined uniaxial com-

pression, Powder Technology 88 (1996) 155.

[2] B.J. Briscoe, D.C. Andrei, P.F. Luckham, D.R. Williams, The defor-

mation of microscopic gel particles, Journal de Chimie Physique et de

Physico-Chimie Biologique 93 (1996) 960.

[3] B.J. Briscoe, K.S. Sebastian, S.K. Sinha, Application of the compli-

ance method to microhardness measurements of organic polymers,

Philosophical Magazine. A 74 (1996) 1159.

[4] B.J. Briscoe, K.K. Liu, D.R. Williams, The compressive deforma-

tion of a single microcapsule, Physical Review. E 54 (6) (1996)

6673–6680.

[5] K.K. Liu, D.R. Williams, B.J. Briscoe, Adhesive contact deformation

of a single micro-elastomeric sphere, Journal of Colloid and Interface

Science 200 (2) (1998) 256–264.

[6] N. Ozkan, B.J. Briscoe, The surface topography of compacted

agglomerates: a means to optimise compaction conditions, Powder

Technology 86 (2) (1996) 201–207.

[7] N. Ozkan, B.J. Briscoe, Compaction behaviour of agglomerated alu-

mina powders, Powder Technology 90 (3) (1997) 195–204.

[8] N. Ozkan, B.J. Briscoe, Overall shape of sintered alumina compacts,

Ceramics International 23 (1997) 521–536.

[9] J.R. Cory, M.L. Aguiar, Rupture of dry agglomerates, Powder Tech-

nology 85 (1995) 37.

[10] P. Coupelle, S. Baklouti, T. Chartier, J.F. Baumard, Compaction be-

haviour of alumina powders spray-dried with organic binders, Journal

De Physique. III 10 (1996) 1283.

[11] D.G. Bika, M. Gentzler, J.N. Michaels, Mechanical properties of

agglomerates, Powder Technology 117 (2001) 98–112.

[12] W. Pietsch, Size Enlargement by Agglomeration, Wiley, UK, 1991.

[13] M.E. Fayed, L. Oten (Eds.), Handbook of Powder Science and Tech-

nology, Chapman & Hall, New York, 1997.

[14] K. Gotoh, H. Masuda, K. Higashitani, Powder Technology Hand-

book, 2nd ed., Marcel Dekker, New York, 1997.

[15] R.C. Rowe, R.J. Roberts, Advances in Pharmaceutical Sciences, Ac-

ademic Press, London, 1995.

[16] M.J. Adams, M.A. Mullier, J.P.K. Seville, Agglomerate strength mea-

surement using a uniaxial confined compression test, Powder Tech-

nology 78 (1994) 5–13.

[17] H. Hertz, Uber die Beruhrung fester elastischer Korper, Zeitschrift fur

Reine Angewandte Mathematik 92 (1882) 156.

[18] K.L. Johnson, K. Kendall, A.D. Roberts, Surface energy and the

contact of elastic solids, Proceedings of the Royal Society of London.

A 334 (1973) 95.

[19] Y. Tatara, Extensive theory of force–approach relations of elastic

spheres in compression and in compact, Journal of Engineering Mate-

rials and Technology, ASME 111 (1989) 163.

[20] K.L. Johnson, One hundred years of Hertz Contact, Process Institute

of Mechanical Engineers 196 (1982) 363.

[21] T.J. Landner, R.R. Archer, Mechanics of Solids. An introduction,

McGraw-Hill, New York, 1994.

[22] M. Pleg, M.D. Normand, Stiffness assessment from jagged

force–deformation relationships, Journal of Texture Studies 26

(1995) 353.

[23] J.H. Smith, J. Woodhouse, The tribology of rosin, Journal of the

Mechanics and Physics of Solids 48 (2000) 1633–1681.

[24] W.J. Beekman, et al, Failure mechanism determination for industrial

granules using a repeated compression test, Powder Technology 130

(2003) 367–376.

Compression of polymer bound alumina agglomerates at the micro deformation scale

Documents

Transcript of Compression of polymer bound alumina agglomerates at the micro deformation scale