ABSTRACT The rheology of concrete is influenced by water content,
the amount, size and size distribution of all the solid components as well as by the dispersion of the finer particles through the addition of superplasticizers. In addition, the rheological behaviour over time evolves as a result of cement hydration. Consequently the a-priori prediction of concrete rheology is a complex task.
In this article, models that have been and are being developed to achieve this task are discussed. The key role of the degree of dispersion will be underlined. A treatment of interparticle forces and a yield stress model integrating these will be presented. Such work is necessary to integrate the dispersion efficiency of superplasticizers based on their dosage and molecular structures into existing models for predicting concrete rheology.
RÉSUMÉ La rhéologie du béton est influencée par la teneur en eau, la
quantité, la taille et la distribution de taille de tous les matériaux granulaires ainsi que du degré de dispersion qui peut être obtenu par l’ajout de superplastifiant. De plus, le comportement rhéologique de ce matériau évolue dans le temps à cause de l’hydratation du ciment. Il en résulte que la prédiction de la rhéologie du béton est particulièrement complexe.
Dans cet article, des modèles existant et en cours de développement dont le but est précisément d’atteindre cet objectif sont discutés. L’importance du degré de dispersion sera soulignée. Un traitement des forces interparticulaires et de leur lien au seuil d’écoulement est présenté. Le but d’un tel travail est de pouvoir en fin de compte inclure l’effet du dosage et de la structure moléculaire de superplastifiants dans les modèles existants pour la prédiction de la rhéologie du béton.
DEDICATION
This review paper is based on the talk given as the award Lecture for the 2003 Robert L’Hermite Medal during the RILEM week that was held in September 2003 in Lisbon. The review part matured from a literature review to which I have been able to devote substantial time to since I joined Sika Technology in May 2002. It has greatly benefited from discussions with colleagues within the European Research Network which is being set-up by Prof. Karen Scrivener (EPFL, Switzerland). In particular on the theme of rheology, I
am grateful to Prof. Henri van Damme (ESPCI, France) and Dr. Christian Vernet (Lafarge LCR, France) for very sharp and enlightening discussions. Sika’s participation in the VCCTL industry consortium run by NIST (USA) has also given me prime access to the excellent modeling work done by Dr. Nick Martys. I am grateful to both him and Dr. Chiara Ferraris for stimulating discussions and collaboration.
The approach to developing a model for yield stress based on interparticle forces and notions of particle packing initiated during my thesis at EPFL. I am tremendously thankful during that period for the continuous encouragement and guidance
R. J. FlattSika Technology A.G, Corporate Research & Analytic, Zurich, Switzerland
Towards a prediction of superplasticized concrete rheology
Editorial Note Dr. Robert J. Flatt presented a lecture of this paper at the 2003 RILEM Annual Meeting in Madrid, as he was awarded the 2003 RobertL'Hermite Medal in recognition of his work on cement and concrete technology. The L’Hermite Award Committee for 2003 decided to recommend the prize to Dr. Flatt since he has demonstrated original approach and independence for fundamental studies of issues which are relevant to cement and concrete technology. In particular he should be noted for his contributions to the basic understanding of the mechanisms of dispersants in concretes. His scientific work added valuable insight into chemical effect dispersants and their impact on physical characteristics of particle interactions and rheology. His studies are highly valuable from a basic point of view and are bound to have practical implication. Dr. Flatt is a RILEM Senior Member.
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that Dr. Paul Bowen gave me. My advisor Prof. Heinrich Hofmann also gave me then his full support allowing me to spend the necessary time on further understanding basic processes dictating interactions of organic admixtures and cements. Dr. Yves Houst who initiated and ran the project on cement admixture interactions gave me his full support and has continued to do so in very productive collaborations since, for which I am very much grateful.
The modeling work on yield stress resumed since I joined Sika in the form of a support to the modeling efforts made in a 5th Framework European project Superplast. I am grateful to all participants in the project for creating a climate of true fruitful scientific collaboration. More specifically I benefited very much from discussions with Prof. Philip F.G. Banfill, Dr. David Swift, Mr. Shawn Monkman (HWU), Prof. Lennart Bergström, Dr. Anika Kauppi (YKI) and Prof. Françoise Lafuma (ESPCI, France). Finally, I would specially like to thank Sika Technology AG for creating outstanding conditions for basic research, as well as my colleagues Dr. Norman Blank and Dr. Urs Mäder for their enthusiastic support in this area. Furthermore, I am particularly grateful to Dr. Irene Schober with whom I have continuously had extremely constructive, challenging and enlightening discussions ever since I joined Sika Technology.
In addition to the people above, I would like to dedicate this paper to the friends and family whose support kept me on what at the time was a difficult and often discouraging road of scientific learning and research, but has since become extremely enjoyable and rewarding.
1. INTRODUCTION
The improvement of superplasticizers has made it possible to produce concrete with reduced amounts of water leading to higher durability without compromising workability. As a result of this new technology, it is now possible to make self compacting concrete (SCC), which does not require the labour intensive vibration usually associated with concrete placing.
Self Compacting Concrete is characterised by substantially lower values of yield stress than ordinary concrete. In a low yield stress concrete, gravity would tend to cause segregation of the coarse aggregates. This can be avoided by proper grading of the constituents, in particular of the finer particles [1] and/or by the addition of water-soluble non adsorbing polymers which increase the viscosity of the continuous phase [2].
Ultimately, it is the combination of yield stress, plastic viscosity and matrix density that contributes to sustaining the particles. Wallevik [3] reports on the most frequently encountered combinations of yield stress and plastic viscosity values found in most SCC mixes (Fig. 1).
These rheological properties characterise the macroscopic response to shear of a material containing particles of which the sizes span many orders of magnitude. Various mechanisms are involved in this macroscopic response, including particle packing [4] physics of granular media [5], hydration chemistry [6, 7], colloidal science [8] and polymer chemistry [9].
Superplasticizers are organic polymers that prevent the finer particles in concrete from agglomerating, greatly improving workability. It is the advent of more effective dispersants that has brought about the most significant progress in recent concrete technology. Despite the important progress in this
field, the effectiveness of superplasticizers does remain susceptible to the nature of the cement. This leads to the so-called cement-superplasticizer incompatibilities. Typically this may mean that the desired workability cannot be achieved or that this requires excessive dosages of superplasticizer.
In this paper, we discuss the mechanisms through which superplastizers affect the rheology of concrete, as well as the role that hydration can play in altering it. We attempt to present this information within the context of concrete rheology.
2. MODELLING CONCRETE RHEOLOGY
The rheology of concrete tends to be well described by the Bingham model which states that the relation between shear stress and shear rate is linear, once a yield stress has been passed. The slope is referred to as the plastic viscosity and the positive intercept as the yield stress.
2.1 Homogenisation approach To solve the problem of a broad span of particle sizes, the
so-called homogenisation approach is often presented as a tool for predicting concrete properties, including rheology. The idea is schematically illustrated in Fig. 2 (following the solid line arrows). It advocates that the properties of a matrix of scale n can be predicted by the properties of the matrix of scale n-1 along with those of the particles at the same scale n-1. Incrementally, one then predicts the properties of the matrix at scale n+1, by the matrix properties at scale n and the particles of the same scale.
This approach is followed at NIST, where dissipative particle dynamics is used to predict the plastic viscosity of concrete based on that of the mortar and aggregates, as well as to predict the plastic viscosity of mortar based on the properties of the cement paste and the sand properties [10].
Similarly, Geiker et al. [11] evaluate the yield stress and plastic viscosity of concrete based on the same parameters measured on the mortar and on the properties of aggregates (volume fraction, shape, etc.).
Though very promising, these approaches face the problem of agglomeration. As illustrated schematically in Fig. 2, agglomeration and hydration can lead to the formation of agglomerates which may be of comparable size or even larger
0
40
80
120
160
0 30 60 90 120
Plastic viscosity [Pa s]
Yiel
d va
lue
[Pa
s]
550 mm
600 mm
650 mm700 mm
Slump flow
0
40
80
120
160
0 30 60 90 120
Plastic viscosity [Pa s]
Yiel
d va
lue
[Pa
s]
550 mm
600 mm
650 mm700 mm
Slump flow
Fig. 1 - Proposed area in values of yield stress and plastic viscosity diagram for SCC (adapted from [3]).
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than particle size n-1 or than the gaps in between those particles. Under such conditions, the rheological behaviour of matrix of size n is no more independent of the presence of particles of size n.
Since agglomerates break down under shear, one may expect that for each concrete mix there will be a limiting shear above which the homogenisation approach may be used. This ought to be the case for plastic viscosity. In fact, Ferraris and Martys [12] have recently shown that the plastic viscosity of concrete normalised by the plastic viscosity of its corresponding mortar, tends to follow a master curve which is well matched by their simulation (Fig. 3).
The same curve could account for the relative viscosity of a mortar with respect to cement paste. In that case, sand was replaced by glass spheres. For yield stress, the homogenisation approach will be limited to sizes above those of the cement paste, requiring input of paste yield stress from either experiments or modelling approaches that can deal with networks of agglomerated particles.
2.2 Particle packing approach De Larrard and coworkers from LCPC have developed a
powerful semi-empirical method for selecting the mix design of concrete to achieve specified final properties [4]. Its basis is a sound fundamental treatment of the packing of particles of different classes. The various properties are then expressed in terms of packing densities through functions that contains various fitting parameters. The final result tends to show a broad prediction ability.
For plastic viscosity, general trends as a function of the packing density of the mix are captured, as indicated in Fig. 4 [1]. Nevertheless, there are several significant departures from the model.
For yield stress the approach is based on accounting for the number of friction points among particles of identical class, neglecting friction among different classes. One
important limitation of this concerns agglomerates, once again. The function proposed by de Larrard [4] is:
)])1(910.0224.0[
)]log(216.0736.0[537.2(
'3*
'0
C
iii
KPP
Kd
(1)
where *'
1 i
iiK , i,
*i and di are respectively the
volume fraction, maximum packing fraction and size of particles of class i, subscript C refers to the cement, P and P* refer to the superplasticizer dosage and saturation dosage respectively.
This function is reported as a preliminary step that needs further experimental validation [4]. Use of this model must be done bearing in mind that fitting parameters relate to the choice of materials and rheometer used in the original experiments leading to the above equation [13]. Particular caution must be taken with the treatment of cement yield stress which does not account for the importance of pre-shear, the extend of which has a profound effect on yield
Matrix n-1 Particles class n-1
Matrix n Particles n
Matrix n+1
ShearSuperplasticizers
Agglomerationhydration
Matrix n-1Matrix n-1 Particles class n-1Particles class n-1
Matrix n Particles n
Matrix n+1
ShearSuperplasticizers
Agglomerationhydration
Fig. 2 - Illustration of the principle of homogenisation approach and limitations resulting from agglomeration of finer particles.
1
10
100
0 20 40 60Volume concentration of coarse agg. [%]
Rel
ativ
e vi
scos
ity
Grad. #1Grad. #2Grad. #3BML #1BML #2IBB #1IBB #2IBB #3Beads in paste
1
10
100
0 20 40 60Volume concentration of coarse agg. [%]
Rel
ativ
e vi
scos
ity
Grad. #1Grad. #2Grad. #3BML #1BML #2IBB #1IBB #2IBB #3Beads in paste
Fig. 3 - Relative plastic viscosity of concrete and mortar with various aggregate grading, and results from particle dissipative dynamics. Experimental data are given by squares, triangles and circles, while simulation results are shown by diamonds. The bilinear function is just given as a guide to the eye. Reproduced from [12] with authorisation.
Fig. 4 - Plastic viscosity of various concretes expressed as a function of the packing density, adapted from [1].
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stress of cementitious materials owing to the irreversible breakdown of agglomerates [14].
Nevertheless, this function as well as the general packing approach brings some basic scientific concepts into the often obscure art of concrete mixture proportioning. More importantly, it seems to provide impressive predictive ability. De Larrard and Sedran report on a case where a mix design was proposed on the basis of their model and on the proper material input for the selected raw materials [1]. Table 1 reproduces the predicted mix design as well as the final mix after minor adjustments. The closeness between both is impressive. Nevertheless, within the scope of this article it is important to highlight that the largest mismatches (about 25%) are found for the limestone filler and the superplasticizer, two components for which limitations have been pointed out in the hypothesis underlying the approach.
2.3 Prediction of concrete slump It is generally mentioned that yield stress dominates the
widely used slump test. Recently mathematical approaches have succeeded in describing the evolution of slump value with yield stress [15, 16]. In Fig. 5, we reproduced results for a conical and cylindrical starting geometry. In the second case predictions are less good at low yield stress. However, the ability of these approaches to demonstrate the link between yield stress and slump is convincing. Thus enhanced mix design tools and proper accounting for agglomeration should allow the prediction of the yield stress of concrete and from there the slump test that most norms are based on and which still constitutes a go/no go criterion in field applications.
3. FORCES IN THE SYSTEM
From the above, it appears that promising models have been and are being developed for predicting concrete rheology. The major problem they seem to face is a computational one about dealing with the many length scales that agglomerates can span. The same is true for cases with continuous and broad grading of particle sizes. Research in the field of interparticle forces and rheology of cement paste has precisely the objective of elucidating this problem.
Before getting in to a more detailed description of the forces that cause agglomeration and dispersion, we examine overall the ensemble of forces affecting the behaviour of all particles in concrete. These are: gravity, inertia, viscous drag, dispersion (van der Waals), electrostatic and Brownian. To examine their relative importance for each
Fig. 5 - Relation between slump and yield stress in dimensional form. Above: the conical slump used for concrete (Reproduced from [15] with authorisation). Below: a cylinder used for paste tests (reproduced from [16] with authorisation).
Table 1 - Concrete mix design, with requirement theoretically established mixture and adjusted final
mix, adapted from [1]
Specifications Theoretical
mixtureActual
MixtureCoarse aggregate 2/6 (kg/m3)
Dmax 6 mm 60% of total
aggregate volume
935 912
Fine aggregate 0/4 (kg/m3)
623 608
Cement (kg/m3) 406 408 Limestone filler (kg/m3)
101 139
Silica fume (kg/m3)
10% of cement content
40.6 39
Superplasticizer (kg/m3)
4.35 5.62
Water (l/m3) 185 190 Compaction index 7 6.9Slump (mm) = 150 150 160 Viscosity (Pa s) 350 350 120-250a
Compressive strength at 28 days (MPa)
= 80 80 78.1
Total Shrinkage At 50% RH (10-6)
750 750
a After 60 min.
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class of particles, we will use dimensionless numbers in which the normalisation term is gravity. Original force expressions used are listed in Table 2. For simplicity, it is assumed that the particle velocity U can be written as:
MU 0 , where is the applied shear stress and M is
the plastic viscosity of the matrix. Values for these rheological parameters are taken to be the average ones given in Fig. 1, that is about 10 Pa for yield stress and 50 Pa s for plastic viscosity. The applied shear is arbitrarily taken to be double the yield stress.
In Fig. 6, the dimensionless numbers are plotted for coarse aggregate, sand, cement and silica fume. For the sake of simplicity their respective average diameters are taken to be 50mm, 0.5mm, 5 m and 50nm. In Fig. 6a the separation distance is assumed to be 100 times smaller than the particle concerned, while in Fig. 6b a constant separation distance of 1nm is assumed. The double layer thickness -1 is taken as 0.67 nm [17].
In the colloidal range (silica fume), the most important forces are dispersion electrostatic and Brownian, followed by viscous drag. The latter would actually be substantially lower in water. Since we somewhat arbitrarily are using plastic viscosity of concrete for all sizes, the viscous drag of the colloidal range is somewhat over-evaluated. At the other end, gravity, viscous forces and inertia are dominant.
As far as cement is concerned, once again the viscous force is probably overestimated. Gravity becomes a force to take into consideration. Its importance with respect to electrostatic and dispersion forces depends strongly on the separation distance. Thus if one wants to prevent the sedimentation of cement particles, one needs either a high plastic viscosity (dispersed system, larger separation distance) or a sufficient yield stress (agglomerated system, small separation distances).
Because all forces appear to be of comparable magnitude, dealing with cement paste rheology is not
trivial. While a colloidal scientist will approach the problem purely from the dispersion side, a granular state physicist will be tempted to do the same from the point of interparticle friction. To complicate matters, since dispersion plays a crucial role and the system is reactive, a chemist will tend to explain the rheology of cement paste based on bulk chemical analysis of the cement. A mineralogist will do the same with a Bogue or even Rietveld phase analysis of the cement. The truth certainly involves a combination of all of these disciplines, which makes the problem as stimulating as complex. Moving from cement paste to concrete rheology, one adds the complication of having to deal with a broad grading of particles for which the relative importance of the different forces depends on the size of the particle in question.
3.1 Interparticle forces If the net interparticle force is attractive, agglomeration
will occur. For colloidal range particles it may be resisted by Brownian motion if the attraction is not too strong. For larger particles, it is either shear forces or gravity that must be taken into account. In the following section, we examine the nature and quantitative description of these forces in greater detail.
Table 2 - Force expressions Force Force expressions Gravity
ga3
34
Inertia 26 aUM
Viscous aUM6Electrostatic
h
h
r eea
12 2
0
Dispersion (van der Waals) 212 h
aA h
Brownian aTkB
where a is the particle radius, is the density difference between the particle and the matrix, M is the plastic viscosity of the continuous phase, 0 is the permitvity of vacuum, R is the relative permitivity of the continuous phase, is the zeta potential, kB is the Boltzmann constant, T is the absolute temperature, -1 is the double layer thickness, h is the separation distance and A(h) is the Hamaker constant at separation distance h(see below).
Fig. 6 - Schematic presentation of the relative importance of forces in concrete for the different components. a) Separation distance is assumed to be 100 times smaller than the particle size. b) Separation distance assumed to be always 1 nm.
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3.1.1 Individual forces
Shortly after mixing, one of the main forces leading to agglomeration is expected to be the van der Waals force, also called dispersion force, which is a confusing terminology. At separations representative of contact (< 30 nm), the force is well described by:
212~
haAF hvdW (2)
where lk
lk
aaaaa 2 is the harmonic average radius of the
intervening particles of radii ak and al, h is the separation distance and A(h) is the Hamaker constant. Approximations for evaluating the magnitude of A(h) and its dependence on separation distance are given elsewhere [18].
As hydration proceeds, ion correlation forces due to divalent calcium ions will arise between negatively charged surfaces and cause strong linkage at the increasing number of C-S-H contact points [19]. An additional attractive electrostatic force may occur between particles in cementitious systems either because of opposite charge (either of the particles themselves or because of surface charge distributions inhomogeneity). Another situation less often acknowledged is that if two surfaces with identical sign but different magnitude of surface potential will, when they maintain a constant potential upon approach, undergo a charge inversion at close separation which leads to attraction. All these effects are not predicted by the standard application of the DLVO theory.
For the more simple case of particles with identical surface potential, the following expression may be used to calculate the magnitude of their repulsion [17]:
h
h
ES eeaF
12 2
0 (3)
with is the relative dielectric constant for water, 0 the permittivity of vacuum, and -1 the Debye length, given by:
bB
nzeTk
2201
2 (4)
where e is the electronic charge, kB is the Boltzmann constant, T is the absolute temperature, bn and z+ are respectively the bulk concentration and charge of an equivalent symmetric electrolyte which best describes the behavior of the cement aqueous phase [17].
Because of the high concentration of electrolytes, electrostatic forces are generally insufficient alone to prevent agglomeration and superplacticizers are needed. These are typically polyanionic organic polymers [20, 21] that induce negative electrostatic potential to the particles onto which they adsorb. Many studies have focussed on this aspect concluding that electrostatic repulsion was the means by which dispersion was produced. However, Banfill [22] had raised doubts about this interpretation stating in a discussion paper of Daimon and Roy [23]: “Adsorption of anionic resins onto cement particles clearly changes the zeta potential and the rheological behaviour, but the effect of zeta potential on rheology is not proved and steric stabilisation may be equally
important.” In a later paper Gartner et al also conclude to the predominance of steric hindrance [24]. It is only more recently with the development of new generation superplasticizers which contain side chains of polyalkylene oxide, that the possibility of steric stabilisation has become a widely acknowledged mechanism [25, 26]. These polymers are also referred to as comb-type copolymers, referring to the adsorption conformation that they are supposed to adopt at the surface of cement particles. Various expressions exist to describe steric repulsion. They all arise from considerations on the unfavorable entropy of mixing resulting from the overlap of adsorbed layers as particle separation distance decreases below twice the adsorbed layer thickness. These expressions depend on the morphology that the polymer adopts at the surface. In the case of the so-called mushroom type adsorption morphology (where the density of polymer chains is highest not at the surface but some distance from the surface) the following expressions can be used [27]:
125
3 35
2 hL
sTkaF B
Ste (5)
where s is the distance between the centers of two neighboring mushrooms. L is the maximum length extending into the solvent.
On particle approach the polymer layers are compressed and the above relation is said to hold until the polymer layers reach their maximum degree of compaction. At that point the compressed layers behave as hard walls and no further approach is possible, the force becomes infinite.
It should be stressed that steric repulsion is based on the assumption that there is no type of attraction among the polymers. This is however not evident. It is conceivable that calcium ions might in cementitious systems lead to some degree of attraction by acting as a bridging complexation agent or through ion correlation forces in a similar way as for C-S-H cohesion [28]. Furthermore, it has been demonstrated by Total Internal Reflectance Microscopy (TIRM) for layers of pluronic triblock polymer (PEO-PPO-PEO) that there is an attractive dispersion force among the polymer layers that is not negligible [29]. Nevertheless, the magnitude of this additional attractive force remains small with respect to the entropic term. The magnitude of these attractive forces will depend on the dielectric properties of the adsorbed layers. The closer they are to those of water, the smaller the interlayer attraction will be. Conversely more dissimilar properties could lead to partial loss of the expected steric repulsion. The less dense an adsorbed layer is at equal layer thickness, the less probable it is to suffer from this problem. Thus, the comb-type copolymers may have the additional benefit of reduced interlayer dispersion attraction.
3.1.2 Net interparticle force
It may be assumed following the classic DLVO approach that the various interparticle forces are additive. Because all forces listed above depend linearly on the a , we define G,the interparticle force parameter, which is the sum of these forces divided by a . This makes G independent of particle size, for sizes and separation in the range of interest for cementitious systems [18]. Negative values of G indicate
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attraction while positive values indicate repulsion. In Fig. 7, G is plotted as a function of separation distance for two values of adsorbed layer thickness. In each case three potentials are considered: 0, 10 and 15 mV. In each case the maximum compaction of the adsorbed layer is arbitrarily taken to be 0.7 nm, while the spacing of the adsorbed molecules is 10 nm. The main observation from Fig. 7 is the difference in scale between the two graphics, which reveals the predominant effect of adsorption layer thickness. The second observation is that in all cases the layer thickness of 3 nm is sufficient to prevent the polymer layer from reaching its maximum compaction. For 1 nm layer thickness, the combined effect of steric hindrance and electrostatic repulsion (with 10 mV) is insufficient to prevent the dispersion forces from completely compressing the adsorbed layers and stopping the particle form falling into the attractive primary minimum. The depth of the
minimum is limited by the maximum degree of compression of the adsorbed layers in those cases.
This suggests that there is a threshold thickness necessary to provide sufficient stabilisation. These calculations would suggest that this threshold lies between 1 and 3 nm. Owing to the approximation underlying several of these calculations, we will not attempt a more precise identification of this threshold.
Furthermore, in all cases considered in Fig. 7, electrostatic repulsion decreases the magnitude of the maximum attractive force. This suggests that this mechanism of repulsion although secondary must not be completely overlooked. It is important to highlight however that this conclusion is limited by the validity of the assumption that the plane of origin of the electrostatic potential is located at the outer side of the adsorbed layer. In the case where this plane is located at the cement surface, the high ionic strength lead to negligible electrostatic interaction for separation distances on the order of 2 adsorption layers of 1 nm each.
4. ADSORPTION
4.1 Adsorption versus and consumption Because polymer adsorption plays such a crucial role in
dispersing agglomerates, many studies have focussed on determining to what degree added superplasticizer adsorbs onto cement particles. These studies are typically done by solution depletion. In this method a given amount of polymer is added to a suspension. After a chosen mixing time, the suspension is centrifuged. The amount of superplsticizer remaining in the aqueous phase is then measured. By difference with the added amount one then obtains the adsorbed amount.
This holds provided that polymer disappears from the aqueous phase only through adsorption, excluding products of intercalation as well as possibly micellization or co-precipitation products. However, there is clear indication, at least in the case of sulphonated naphtalene formaldehyde condensate (SNFC) that intercalation does take place into a layered calcium alumino hydrate denoted Afm [6, 7]. Because of these reactions, the amount of polymer usually termed adsorbed should rather be referred to as consumed.
-600
0
600
1 2 3 4 5 6 7 8Separation [nm]
G
[pN
/µm
]
0 mV 5 mV 15 mV
-40
-20
0
20
40
4 6 8 10 12 14Separation [nm]
G
[pN
/µm
]
0 mV 5 mV 15 mV
b)
a)
Fig. 7 - Interparticle force parameter given as a function of separation distance. Negative values indicate attraction, positive values indicate repulsion. Three different surface potentials 0, 5, 15 mV are considered. In both cases the maximum compaction is taken as 0.7 nm giving a complete repulsion at 1.4 nm if not earlier. a) layer thickness assumed to be 1 nm thick, b) layer thickness assumed to be 3 nm thick.
Initial surfaceAdsorption extent and
adsorbed propertiescondition initial rheology
Newly formed surfaceConsume excess polymerReduce surface coverage
AdsorbedInvolved in dispersion
mechanisms
ReactedPrecipitated, intercalated,
micellized
ExcessRemaining in solution
Added polymer
ConsumedAssociated with solid phases
time
Initial surfaceAdsorption extent and
adsorbed propertiescondition initial rheology
Newly formed surfaceConsume excess polymerReduce surface coverage
AdsorbedInvolved in dispersion
mechanisms
ReactedPrecipitated, intercalated,
micellized
ExcessRemaining in solution
ExcessRemaining in solution
Added polymer
ConsumedAssociated with solid phases
ConsumedAssociated with solid phases
ConsumedAssociated with solid phases
time
Fig. 8 - Schematic presentation differentiating the polymer that is usually identified as adsorbed (consumed) and the polymer that is actually effective in the dispersion mechanisms.
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Schematically, we may divide the polymer that disappears from the solution into an adsorbed and a reacted part (Fig. 8). Only the first truly contributes to dispersion. The other may be considered as lost.
The reacted fraction mainly concerns interaction with the aluminate phases to degrees that depend on the nature of the polymer as well as the aluminate phase, the calcium sulphate phases and the soluble alkali. While these reactions dominate the effect of SNFCs, there are indications that such reactions are less problematic for the comb-type copolymers. The side chains of PEO in these polymers probably both inhibit intercalation and/or leave protruding chains capable of providing steric stabilisation even if the backbone is already incorporated into hydration products [7].
4.2 Competitive adsorption We have mentioned that comb-type copolymers could be
less extensively affected by reactions with hydrating aluminate phases. However, these polymers still exhibit the so-called cement-superplasticizer incompatibilities, which remain unpredictable and mysterious. It is quite possible that the underlying mechanism leading to such incompatibilities depends on the nature of the superplasticizer.
In the case of SNFCs, they are probably strongly linked to reactions with aluminate phases. In the case of comb-type copolymers it has been suggested that the concentration of dissolved sulphates could be crucial [30]. In this study the authors showed that adsorption and workability decreased when sulphates were added. On the other hand the opposite was observed after calcium addition. This could be repeated in a cyclic way. The interpretation was that sulphates compete with the polymer for adsorption sites which decreases the dispersion efficiency in presence of these ions. However calcium addition leads to gypsum precipitation which consumes sulphates and allows increased polymer adsorption and thus dispersion.
The possibility of ion competition for adsorption sites of PEO containing polymers had already been pointed out in a study of the temperature dependent adsorption of some superplasticizers containing PEO and others that did not [31]. The model system studied consisted of suspensions of MgO at pH 12. Under those conditions, it was found that adsorption was insensitive to temperature for the polymers without PEO (high density of adsorbing groups), while polymers with PEO (fewer ionic groups) showed an increased adsorption with temperature. This was attributed to the lower adsorption energy of these polymers making them subject to adsorption competition by hydroxyl ions.
Thus it appears, that while inclusion of PEO side chains enhances steric stabilisation and renders polymers less sensitive to the reactions associated with the hydration of aluminate phases, the decrease in ionic concentration in the polymer backbone may make efficiency of adsorption the determining step. In fact with a given family of comb-type copolymers, Schober and Mäder showed that there is a linear relation between adsorption and flow, (the spread diameter from a sample initially placed in a vertical cylinder) [9]. This result may be rationalised in two quite simple ways.
Following the derivation of slump by Christensen and Schowalter [15], it appears that yield stress should scale with the inverse of the base surface of the spread sample, or with the inverse second power of the diameter. The yield stress
should also scale with the interparticle force. From Fig. 7 it is apparent that if the layer thickness is beyond a minimum threshold, the steric force soon resists the dispersion force. Thus a simplified force expression would be the dispersion force at a distance given by the adsorption layer thickness. A simple way of dealing with incomplete surface coverage would be to take an average thickness which depends linearly on surface coverage. Following Equation (2), this means that yield stress should scale with the inverse of the second power of surface coverage. Thus we find that the flow diameter should scale with adsorption as found by Schober and Mäder [9]. This relation is not expected to hold at too low dosages or for polymers that undergo substantial intercalation.
5. YIELD STRESS OF SUSPENSIONS
Much work has been put into the prediction of yield stress of particulate suspensions. Important results have been obtained with alumina suspensions of sub-micrometer particles with various levels of electrostatic stabilisation [32]. That data clearly shows that yield stress scales with the square of the zeta potential, indicating that yield stress is indeed proportional to the interparticle force.
Yield stress modelling also involves a microstructural aspect. That is, one needs to know how many contact points exist and how many of those need to be broken in order for the suspension to begin flowing. In the work by Zhou et al [32], it is suggested that there are two areas of different power law dependence on volume fraction and a coordination number depending on the inverse particle size. The later point is rather unsatisfactory and the authors clearly state that more work is needed to understand the origin of this unexpected functionality.
In fact, work under way [33] indicates that the size dependent co-ordination number is probably an artefact, resulting from an inappropriate scaling with volume fraction. Following a former model [8] and a packing model by Suzuki and Oshima [34], analytical expressions for the co-ordination number have been introduced into a yield stress model [33] that includes interparticle force, particle packing considerations, particle size and distribution. This leads us to the following expression for yield stress 0 .
mmm
2
10 (6)
where is the volume fraction, m is the maximum packing volume fraction, and m1 is a prefactor that is a function of the volume mean particle size Dv, the particle size distribution and the interparticle force parameter G:
PSDv
FDGm 41
8.1 (7)
The particle size distribution function FPSD is given in Annex A.
This result may be extended to included a volume fraction percolation threshold 0 into Equation (6) in the following way:
mmm
20
10 (8)
Materials and Structures / Matériaux et Constructions, Vol. 37, June 2004
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This addition of a percolation factor stems from the idea that until percolation is achieved there should be no yield stress. Thus volume fractions below this percolation limit does not contribute to the measured yield stress. The expression in Equation (8) fits the volume fraction dependence of many independent sets of yield stress data for undispersed cement paste extremely well as indicated in Fig. 9. In both Figs. 9 (a) and (b), the maximum packing fraction of 65% was taken from a filter pressing result on a well dispersed OPC [35]. Work in progress has shown that this parameter gives maximum packing fraction in accord with those obtained by fitting rheological data with 1-2% [35]. In both figures, the fit is extremely good despite both
sets of data showing differences in yield stress of a factor close to 10. Such differences between these data sets are probably not only linked to the nature of the cement but also to extent of pre-shear suffered by the suspension prior to their measurement [14]. We note that while values of m1 are relatively similar, there is an important difference between the fitted values of the percolation thresholds (m3= 0 ).
At this point, we need to examine whether values for m1 are well predicted by Equation (7). For this we resort to published data on the yield stress of alumina suspensions [32]. The authors of this study have used four different alumina powders with mean particle diameters between 0.18 and 0.7 m. They have examined yield stress over a wide range of pH, thereby varying the degree of electrostatic stabilisation. Here we only examine the data obtained at the isoelectric point (IEP), so that only the dispersion force acts
0
200
400
600
800
1000
0.36 0.38 0.4 0.42 0.44 0.46
Yie
ld s
tress
[Pa]
Volume fraction [-]
y = m1*(m0-m3)^ 2/(0.65*(0.65-m0))ErrorValue650.16091.7m1
0.006380.318m3 NA2690ChisqNA0.997R
0
200
400
600
800
1000
0.36 0.38 0.4 0.42 0.44 0.46
Yie
ld s
tress
[Pa]
Volume fraction [-]
y = m1*(m0-m3)^ 2/(0.65*(0.65-m0))ErrorValue650.16091.7m1
0.006380.318m3 NA2690ChisqNA0.997R
(a)
0
20
40
60
80
100
120
0.3 0.35 0.4 0.45 0.5 0.55
PFB average
Yie
ld s
tress
[Pa]
Volume fraction [-]
y = m1*(m0-m3)^ 2/(065*(0.65-m0))ErrorValue
2222.68925.9m10.0380.174m3
NA175.5ChisqNA0.990R
0
20
40
60
80
100
120
0.3 0.35 0.4 0.45 0.5 0.55
PFB average
Yie
ld s
tress
[Pa]
Volume fraction [-]
y = m1*(m0-m3)^ 2/(065*(0.65-m0))ErrorValue
2222.68925.9m10.0380.174m3
NA175.5ChisqNA0.990R
(b)
Fig. 9 - Yield stress of cement suspensions, fit by Equation (8). (a) Data from Wallevik [3], (b) data obtained from Banfill [36].
0
1000
2000
3000
4000
5000
6000
7000
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
AKP-50AKP-30AKP-20AKP-10
Yiel
d S
tress
[Pa]
Volume fraction [-]
0
1000
2000
3000
4000
5000
6000
7000
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
AKP-50AKP-30AKP-20AKP-10
Yiel
d S
tress
[Pa]
Volume fraction [-]
0
1000
2000
3000
4000
5000
6000
7000
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
AKP-50AKP-30AKP-20AKP-10
Yiel
d S
tress
[Pa]
Volume fraction [-]
0
1000
2000
3000
4000
5000
6000
7000
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
AKP-50AKP-30AKP-20AKP-10
Yiel
d S
tress
[Pa]
Volume fraction [-]
(a)
(b)
Fig. 10 - Volume fraction dependence of suspensions at their isoelectric point. The data is taken from Zhou et al [32], the fit is done from Equation (6). (a) Data fitted for m1 and 0. (b) As in (a) except that 0 for AKP-10 is taken as 19.6% according to procedure described below.
Flatt
298
between the particles. The published values of yield stress versus volume fraction are reported in Fig. 10. The maximum packing for all was assumed to be 64%, the expected theoretical random close packing figure for such narrow size distributions [37]. This was experimentally confirmed for the AKP-50 powder from the slip casting of stable suspensions [38].
The percolation threshold ought to vary with the size distribution of the powders. Bowen has found that the size distribution of these powders could generally be well fitted to a log normal distribution [38]. Values of Dv and g are given in Table 3.
When plotting 0 versus g AKP-10 clearly stands out from the trend exhibited by the other powders (Fig. 11). It is believed that this is somewhat of an artefact from the fitting procedure and it was decided to modify the value of
0 for AKP-10 to make it compatible with the linear trend
of the other powders shown in Fig. 11. The corresponding fit is shown in Fig. 10(b).
After this modification of the fitting routine for AKP-10, we examine whether the values of m1 are compatible with Equation (7). As indicated in that equation, m1depends on the particle size and distribution as well as on interparticle forces. In Fig. 12, the ratio of m1 to the particle size distribution function, FPSD, is plotted versus 1/Dv. Values of FPSD calculated using values of Dv and g
in Table 3 are shown in the same table. The linear relation obtained in Fig. 12 indicates that the
above model correctly captures the scaling of yield stress with volume fraction, particle size and particle size distribution. From the slope in the figure, one can calculate that the particle-particle separation distance is about 0.28 nm. It is realistic that particles are in very close contact since the data considered is at the IEP. However, it is not possible to judge the absolute value since at such close separations the force changes very sharply with separation distance. Thus, we may say that this looks plausible but that more data for better dispersed systems would be needed to examine whether this is or not a credible result in terms of scaling with interparticle force.
In summary, at this stage the above model can be considered to capture the functionality of yield stress with respect to volume fraction, particle size and particle size distribution. This result is based on the derivation of a first principle based a calculation of yield stress which predicts a linear relation with the interparticle force parameter. The examination of separation distance obtained for the alumina system is realistic, but further work is needed to confirm the proper scaling of interparticle forces. An important question to be examined is whether percolation threshold 0 depends on the interparticle forces. If such is the case, an additional interparticle force dependence would be introduced, which is currently not predicted by the above model. The same is true about the maximum packing.
In fact more recent results indicate that an interparticle force dependent maximum packing implemented in a way that is compatible with centrifugal consolidation results greatly improves the fitting of yield stress versus volume fraction. The conclusions concerning the dependence on the prefactor m1 on interparticle force are unchanged. This additional encouraging development is beyond the scope of this paper and will be reported elsewhere. On another note, attention must be drawn to the risk of misinterpretation experimental data, for example because of gravity (sedimentation of load on agglomerated network). This delicate aspect must be given further consideration.
6. CONCLUSIONS
From the above review it appears that the action of superplasticizers is to limit or suppress agglomeration among the finer particles in concrete, that is cement, slag, fly-ash, fillers and silica-fume. This microstructural change originates at the molecular level, while its effect is observed on the macroscopic scale by rheology measurements. Such measurements show the effect of dispersion and are essential for comparing the efficiency of superplasticizer according to dosage or chemical structure. However, in absolute terms,
Table 3 - Characteristic of alumina powders for fitted log normal distributions
AKP-50 AKP-30 AKP-20 AKP-10 Dv 0.147 0.17 0.33 0.55
g 0.275 0.400 0.600 0.550 FPSD 0.87 0.79 0.65 0.69
y = 0.1577x + 0.1093
0.10
0.15
0.20
0.25
0.30
0.35
0.2 0.3 0.4 0.5 0.6 0.7Standard deviation
Per
cola
tion
thre
shol
d [-]
Fig. 11 - Variation of percolation threshold of powders from fit shown in Fig. 10(a). The outliner is AKP-10. Arrow shows choice to readjust the percolation threshold to AKP-10 to make it compatible with values of other powders.
R2 = 0.9622
0
3000
6000
9000
12000
0 2 4 6 81 / Dv
m1
/ FP
SD
Fig. 12 - Prefactor m1 from Equation (6), showing inverse dependence on volume average particle size.
Materials and Structures / Matériaux et Constructions, Vol. 37, June 2004
299
rheological parameters depend on the grading of particle sizes involved. Thus interpretation of absolute changes must be carried out with care.
Examination of the interparticle forces, reveals that steric stabilization should be the most efficient way of preventing agglomeration. Thus comb-type copolymers with non adsorbing side chains should provide good stabilisation.
Hydration can interfere with dispersion and this aspect must not be overlooked. For earlier generations of superplasticizers, interaction with the hydrating aluminate phases appears to lead to the consumption of polymer through intercalation, thus decreasing the amount available for dispersion. Comb-type copolymers may be less sensitive to this effect because of the PEO side chains and reduced number of ionic groups. On the other hand their lower adsorption energy might render their adsorption more sensitive to the ionic composition of the solution.
State of the art polymer synthesis offers many possibilities of varying the structure/function properties of superplasticizers [9]. Improved understanding of the action of superplasticizers on dispersion of the finer particles in concrete, and increased capacity in interpreting macroscopic measurements of rheology will lead to superplasticizers with increased performance and robustness.
Superplasticizers play a major role in bringing concrete into the workability range of Fig. 1 defining combinations of yield stress and plastic viscosity allowing the production of self-compacting concrete. They are, however, not the exclusive ingredient and careful mix design must be done in particular to limit segregation. The a priori definition of concrete mix design for defined requirements appears to be within reach. Rheological models are quite successful in predicting the effect of the aggregates, but deeper understanding is required at the interparticle force range for the prediction of yield stress.
A model has been presented which pursues this objective. At this point, it appears to correctly capture the volume fraction, particle size and size distribution scaling of yield stress of cement paste. However, there is still need to identify independent measurements that may supply the percolation term it includes. If this succeeds we will have the ability to predict on a fundamental basis the yield stress of cement paste. From there, composite theory approaches should lead to the prediction of mortar and concrete yield stress.
There is much work to be done before the a priori mix design, choices may be made to target specific rheological behaviour of concrete (for example SCC compatible with Fig. 1). However, important progress is being made, and this task does appear, more than ever before, a possible one.
ACKNOWLEDGEMENTS
The author would like to acknowledge constructive comments on this paper by Dr. Paul Bowen (EPFL), Dr. Yves Houst (EPFL), Prof. George W. Scherer (Princeton University) and Dr. Irene Schober (Sika Technology AG). Prof. Banfill (HWU) is thanked for helpful discussions and compiled data on the yield stress of cement suspensions.
REFERENCES
[1] De Larrard, F. and Sedran, T., ‘Mixture-proportioning of high-performance concrete’, Cem. Concr. Res. 32 (2002) 1699-1704.
[2] Ambroise, J., Chabannet, M. and Péra, J., ‘Basic properties and effects of starch on self-levelling concrete’, Proceedings of the RILEM international symposium on the role of admixtures in high performance concrete (Ed. J.G. Cabrera and R. Rivera Villarreal), RILEM Publ., France, 357- 376.
[3] Wallevik, O., ‘Rheology of cement suspensions’ (The Icelandic Building Research Institute, 112 Keldnaholt, Iceland, 2002).
[4] De Larrard, F., ‘Concrete mixture proportioning’ (E & FN Spon, London, 1999)
[5] Van Damme, H. Mansoutre, S., Colombet, P., Lesaffre, C. and Picart, D., ‘Pastese: Lubricated and cohesive granular media’, C.R. Physique 3 (2002) 229-238.
[6] Fernon, V., Vichot, A., Le Goanvic, N., Colombet, P., Corazza, F. and Costa, U., ‘Interaction between portland cement hydrates and polynaphtalene sulfonates’ in Proc. 5th Canmet/ACI Int. Conf. on Superplasticizers and Other Chemical Admixtures in Concrete (editor: Malhotra V.M.), (American Concrete Institute, Framington Hills, SP-173, 1997) 225-248.
[7] Flatt, R.J. and Houst, Y.F., ‘A simplified view on chemical effects perturbing the action of superplasticizers’, Cem. Concr. Res. 31 (8) (2001) 1169-1176.
[9] Schober, I. and Maeder, U., ‘Compatibility of polycarboxylate superplasticizers with cement and cementitious blends’, in Proc. 7th Canmet/ACI Int. Conf. on Superplasticizers and Other Chemical Admixtures in Concrete, (editor: Malhotra, V.M.), (American Concrete Institute, Framington Hills, SP-217, 2003) 453-48.
[10] Martys, N.S. and Mountain, R.D., ‘Velocity Verlet algorithm for dissipative-particle-dynamics-based models of suspensions’, Phys. Rev. E. 59 (1999) 3733-3736.
[11] Geiker, M.R., Brandl, M., Thrane, L.N. and Nielsen, L.F., ‘On the effect of coarse aggregate fraction and shape on the rheological properties of self-compacting concrete’, Cement and Concrete Aggregates 24 (1) 3-6.
[12] Ferraris, C.F. and Martys, N.S., ‘Relating fresh concrete viscosity measurements from different rheometers’, J. Res. Natl. Inst. Stand. Technol. 108 (2003).
[13] Ferraris, C. and de Larrard, F., ‘Testing and modelling of fresh concrete rheology’, NISTIR 6094, National Institute of Standards and Technology, February 1998.
[14] Tattersall, G.H. and Banfill, P.F., ‘The rheology of fresh concrete’ (Pitman advanced publishing program, London, 1983).
[15] Schowalter, W.R. and Christensen, G., ‘Toward a rationalization of the slump test for fresh concrete: Comparisons of calculations and experiments’, J. Rheol. 42(4) (1998) 865-870.
[16] Davidson, M.R., Khan, N.H. and Yeow, Y.L., ‘Collapse of a cylinder of Bingham fluid’, ANZIAM J. 42 (E) pp. C499-C517, 2000, in Proceedings of the 1999 International Conference on Computational Techniques and Applications (CTAC-99).
[17] Flatt, R.J. and Bowen, P., ‘Electrostatic, repulsion between particles in cement suspensions: domain of validity of linearized Poisson-Boltzmann equation for non-ideal electrolytes’, Cem. Concr. Res. 2253 (2002) 1- 11.
[19] Nachbaur, L., Mutin, J.C., Nonat, A. and Choplin, L., ‘Dynamic mode rheology of cement and tricalcium silicate pastes from mixing to setting’, Cem. Concr. Res. 31 (2001) 183-192.
Flatt
300
[20] Ramachandran, V.S., Malhotra, V.M., Jolicoeur, C. and Spiratos, N., ‘Superplasticizers: properties and applications in concrete’, (CANMET Publication MTL 97-14 (TR), Ottawa, Canada, 1998)
[21] Flatt, R.J., ‘Polymeric Dispersants in Concrete’, in ‘Polymers in Particulate Systems: Properties and Applications’, Eds: Hackley V.A., Somasundaran P., Lewis J.A., (Marcel Dekker, New York, 2001) 247-294.
[22] Banfill, P.F.G, ‘A discussion of the paper Rheological properties of cement mixes’ by M. Daimon and D.M. Roy, Cem. Concr. Res. 9 (1979) 795-798.
[23] Daimon, M. and Roy, D.M, ‘Rheological properties of cement mixes I’, Cem. Concr. Res. 8 (1978) 753-764.
[24] Gartner, E.M., Koyata, H. and Scheiner, P., ‘Influence of Aqueous Phase Composition on the Zeta Potential of Cement in the presence of Water- Reducing Admixtures’, Ceramic Transactions (American Ceramic Society), 40(1994) 131-140.
[25] Sakai, E. and Daimon, M., ‘Mechanisms of superplastification’ in ‘Materials science of concrete IV’, Ed.: Skalny, J.P. and Mindess, S. (The American Ceramic Society, Westerville, OH, USA, 1995) 91-111.
[26] Yoshioka, K., Sakai, E. and Daimon, M., ‘Role of steric hindrance in the performance of superplasticizers in concrete’, J. Am. Ceram. Soc. (1997) 2667-2671.
[27] Pedersen, H.G. and Bergström, L., ‘Forces between zirconia surfaces in poly(acrylic acid) solutions’, J. Am. Ceram. Soc.82 (5) (1999) 137-1145.
[28] Van Damme, H., ‘Colloidal chemo-mechanics of cement hydrates and smectite clays: cohesion vs. Swelling’ in ‘Encyclopedia of surface and Colloidal Science’, (Marcel Dekker Inc, New York, 2002) 1087-1103.
[29] Prieve, D.C. and Bevan, M.A., ‘Effect of physisorbed polymers on the interaction on latex particles and their dispersion stability’, in ‘Polymers in Particulate Systems: Properties and Applications’, Eds: Hackley, V.A., Somasundaran, P. and Lewis, J.A. (Marcel Dekker, New York, 2001) 1-26.
[30] Yamada, K. and Hanehara, S., ‘Interaction mechanism of cement and superplasticizers – The roles of polymer adsorption and ionic conditions of aqueous phase’, Concrete Science and Engineering 3 (2001) 135-145.
[31] Flatt, R.J., Houst, Y.F., Bowen, P., Hofmann, H., Widmer, J., Sulser, U., Maeder, U. and Bürge, T.A., ‘Interaction of superplasticizers with model powders in a highly alkaline medium’ in Proc. 5th Canmet/ACI Int. Conf. on Superplasticizers and Other Chemical Admixtures in Concrete, Ed: Malhotra V.M., (American Concrete Institute, Farmington Hills, Mi, USA, 1997, SP-173) 743-762.
[32] Zhou, Z., Boger, D.V., Scales, P.J. and Healey, T.W., ‘Shear and compressional rheology principles in ceramic processing’ in ‘Polymers in Particulate Systems: Properties and Applications’, Eds: Hackley, V.A., Somasundaran, P. and Lewis, J.A., (Marcel Dekker, New York, 2001) 157-195.
[33] Flatt, R.J. and Bowen, P., Work in progress (2003). [34] Suzuki M. and Oshima T, ‘Estimation of the coordination
number in a Multi-Component Mixture of Spheres’, Powder Technology 35 (1983) 159-166.
[35] Flatt, R.J., Ferraris, C., Martys, N. and Banfill, P., Work in progress (2003).
[36] Banfill, P.F.G., personal communication (2003). [37] Nolan, G.T. and Kavanagh, P.E., ‘The size distribution of
interstices in random packings of spheres’, PowderTechnology 78 (1994) 231-238.
[38] Bowen, P., personal communication (2003).
Annex A
The function of the particle size distribution is as follows:
lkk
lkm
k
m
kl C
Skak g
aV
AAS ,3
.
1
', (9)
where m is the number of particle classes considered, k is the volume fraction of the kth class of particle radius, surface contact ratio between a particle of radius a k and one of radius al [34]:
21
12
k
l
k
l
k
l
k
l
C
S
aa
aa
aa
aa
AA (10)
S a,k is the surface fraction of particles of class k in the powder, given by:
m
iii
kkka
a
aS
1
,
/
/ (11)
And dimensionless term that has to do with the relative sizes of the intervening particles. It may be expressed as a function of the harmonic average radius, a , and arithmetic average radius, a , of the intervening classes as well as the overall volume average particle diameter of the powder Dv.
aaaDag v
lk ˆ2ˆ, (12)
The term lkV . is a key term in the model. It is the so-called additional volume. It is linked to the hypothesis that any undispersed bond (particle-particle contact in an agglomerate) increases the effective volume of solids in the suspension by that amount. For a truncated cone approximation, this is given by:
aaaV lk ˆ43
2 2, (13)
Paper received: August 20, 2003; Paper accepted: October 29, 2003
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