1. Ceramic processing

A ‘process’ is “a systematic series of actions or operations directed to some end”. The ‘end’ in a ceramics context, is an object of pre-defined shape or a part of an object such as a coating. Ceramics processing has, over the last two decades, incorporated the organic and physical chemistry of surfactants, dispersants and polymers in its quest to control particle behavior, i.e. colloidal processing [1]. Colloidal processing provides improved microstructural design and reliability. In colloidal processing a homogeneous and dense green body is formed from a stable suspension. In a colloidal suspension the interparticle forces between the particles can be controlled and altered through careful control of the colloid chemistry [8] [17].

According to J. R. G. Evans manufacturing processes can be divided into five classes (i) casting, (ii) deformation, (iii) material removal, (iv) joining and (v) solid free forming [1]. Casting processing is characterized by state change, reduction of free volume (not always) and the use of a surface (the mould) to define the final object shape. When free volume collapse is non-uniform throughout the body residual stress develop and deformation occurs on sintering [1]. In deformation processing the shape of an object is change by plastic deformation. Material removal processes remove material from a blank by cutting, abrasion, ablation or corrosion to leave a final shape. At a macroscopic level joining is treated as a manufacture process because the final product depends upon the successful assembly of separately manufactured parts. Solid freeforming can be defined as the creation of a shape by point, line or planar addition of material without confining surfaces other than a base [1].

2. Casting processes

State change is the key link in casting processes. State change mechanisms in ceramic suspensions includes: (i) fluid medium removal (e.g. slip casting), (ii) suspension pH sift towards its isoelectric point (IEP), (iii) increase ionic strength (double layer compression), (iv) coagulation using MgO, (v) reaction of organic binder (i.e. gelcasting), (vi) change of dispersant characteristics (e.g. TIF), (vii) freezing the liquid medium… [5]. Casting processes ii – vii where there is no fluid removal are also known as direct consolidation processes.

In direct consolidation, a ceramic suspension consolidates inside non-porous molds (e.g. metal molds) without compaction or removal of water. Direct consolidation can be produced through the formation of a physical gel as a consequence of the creation of a network structure among particles, through the formation of a chemical gel by chemical polymerization of monomers or thermal gelation of polysaccharides [9]. Compared with other methods, direct consolidation can produce complex green bodies with better dimensional accuracy, higher flexural strength and reduced density gradients.

Flocculation and coagulation leds to the formation of a network structure of the particles when electrostatic repulsion is minimised by shifting the pH toward the isoelectric point or by increasing the suspension ionic strength [17]. Another alternative is temperature induced forming (TIF), which ois based on temperature induced modification of polymer configuration to initiate network formation. In TIF, the change of polymer configuration with temperature minimizes steric repulsion inducing particle flocculation under Van der Waals force, forming a physical gel [5].

In gelcasting process, monomer, crosslinker, initiator, catalyst and ceramic powder are mixed in water to form a homogeneous suspension with high solids loading and low viscosity, then monomer is

polymerized to form a polymer– solvent gel, and the macromolecular gel network resulting from the in-situ polymerization of dispersed medium holds the ceramic particles in the shape defined by the mold. After removal from mold, the gelled part contains about one-fourth of its mass as moisture, this moisture is then removed by drying [5] [17].

Thermal gelation of natural binders has been successfully applied in the last decade for manufacturing bulk bodies with near-net-shape [20]. Natural polysaccharides can form a gel either on cooling (as in the case of agar, agarose and carrageenan) or on heating (as for methylcellulose derivatives, dextrines and starches). [9] [17] [18]. However, agar derivades are considered the most effective gelling binders for aqueous media, providing a high gel-strength to the green bodies when added in very low concentrations [17] [20]. In thermal gelation an aqueous suspension that contains a low content of a thermogelling polysaccharide (~ 1 wt.%) is poured at temperatures higher than the gelling temperature (Tg) in a non-porous mould cavity and cooled bellow the Tg, thus leading to a consolidated body with gel consistency [20].

More often than not, technical ceramics materials and ocationally clay based ceramics are of colloidal dimensions (i.e. diameter < 1 micron), there fore, colloidal processing is a key aspect in ceramic science.

3. Colloidal processing

The manufacture of ceramic products needs careful control of the different processing steps in order to prevent defect formation and to develop the microstructure needed to obtain the desired properties. The defects introduced in any one step of the process are retained in the next one and can be present in the final material. Colloidal processing allows the control of the early steps of powder processing and its evolution during forming and consolidation steps. Colloidal processing is based on the study and manipulation of the interparticle forces involved in the suspension in order to obtain dense particle-packing and uniform microstructures [10].

4. Interparticle forces

Four main types of interparticle interactions can be distinguished in concentrated suspensions: (i) hard-sphere interactions (Fig. 1a), (ii) electrostatic interaction (Fig. 1b), (iii) steric interaction (Fig. 1c) and (iv) Van der Waals interaction (Fig. 1d) [4]. These four different types of interactions are illustrated in Fig. 1.

Hard-sphere interaction (Fig. 1a): The particles are considered to behave as “hard-spheres” with a radius RH that is slightly larger than the core radius R. When the particles reach a center-to-center distance that is smaller than 2RH the interaction increases very sharply approaching infinity (∞) [4].

Electrostatic (double layer) interaction (Fig. 1b): The particles in this case have a surface charge, either by ionization of surface groups as in the case of oxides or in the presence of adsorbed ionic

surfactants. The surface charge (σ0) is compensated by unequal distribution of counter ions (opposite in charge to the surface) and co-ions (same sign as the surface) which extend to some distance from the surface. The double layer extension depends on electrolyte concentration and valency of the counter ions, in Eq. 1: εr is the permittivity (dielectric constant) for water at 25 °C. ε0 is the permittivity of free space. k is the Boltzmann constant and T is the absolute temperature. n0 is the number of ions per unit volume of each type present in bulk solution (the double layer extension increases with decrease in electrolyte concentration) and Zi is the valency of the ions. e is the electronic charge [4].

𝟏𝜿= 𝜺𝒓𝜺𝟎𝒌𝑻

𝟐𝒏𝟎𝒁𝒊𝟐𝒆𝟐

𝟏𝟐 (1)

When charged colloidal particles in dispersion approach each other such that the double layers begin to overlap (particle separation becomes less than twice the double layer extension), repulsion occurs. The individual double layers can no longer develop unrestrictedly, since the limited space does not allow complete potential decay. For two spherical particles of radius (R) and low surface potential (ψ0) under condition κR<3, the expression for the electrical double layer repulsive interaction energy (GE) is given by Eq. 2 [4],

𝑮𝑬 =𝟒𝝅𝜺𝒓𝜺𝟎𝑹𝟐𝝍𝟎

𝟐𝒆𝒙𝒑 !𝜿𝒉𝟐𝑹!𝒉

(2)

Where, h is the closest distance of separation between the surfaces. The above expression shows the exponential decay of GE with h. The higher the value of κ, the steeper the decay. This means that at any given distance h, the double layer repulsion decreases with increase of electrolyte concentration. The importance of the double layer extension can be illustrated as follows. Let us consider a very small particle with a radius of 10 nm in 10−5 mol/dm3 NaCl. The core radius R=10 nm but the effective radius Reff (the core radius plus the double layer thickness) is now 110 nm. The core volume is (4/3)π (10)3 nm3 but the effective volume is now = (4/3)π(110)3 which is ~1000 times higher than the core volume. The same applies to the volume fraction [4].

Figure 1: Interparticle interactions, (a) hard sphere interaction, (b) electrostatic double layer interaction, (c) steric interaction and (d) Van der Waals interaction.

Steric interaction (Fig. 1c): This occurs when the particles contain adsorbed nonioinic surfactant or polymer layers. One can define an adsorbed layer thickness δ for the surfactant or polymer and hence an effective radius Reff = R+δ. When two particles each with a radius R and containing an adsorbed polymer layer with a thickness δ, approach each other to a surface-surface separation distance h that is smaller than 2δ, the polymer layers interact with each other resulting in two main situations: (i) polymer chains

may overlap with each other and (ii) polymer layer may undergo some compression. In both cases, there will be an increase in the local segment density of the polymer chains in the interaction region. The real situation is perhaps in between the above two cases, i.e. the polymer chains may undergo some interpenetration and some compression [4].

Provided the dangling chains are in a good solvent, this local increase in segment density in the interaction zone will result in strong repulsion as a result of two main effects: increase in the osmotic pressure in the overlap region as a result of the unfavorable mixing of the polymer chains, when these are in good solvent conditions. This is referred to as osmotic repulsion or mixing interaction and it is described by a free energy of interaction Gmix and (ii) reduction of the configurational entropy of the chains in the interaction zone; this entropy reduction results from the decrease in the volume available for the chains when these are either overlapped or compressed. This is referred to as volume restriction interaction, entropic or elastic interaction and it is described by a free energy of interaction Gel. Combination of Gmix and Gel is usually referred to as the steric interaction free energy, Gs [4].

The sign of Gmix depends on the solvency of the medium for the chains and Gel is always positive and hence in some cases one can produce stable dispersions in a relatively poor solvent [4]. The repulsive energy can be calculated by considering the free energy of mixing of two polymer solutions, as for example treated by Flory and Krigbaum. The free energy of mixing is given by two terms: (i) an entropy term that depends on the volume fraction of polymer and solvent and (ii) an energy term that is determined by the Flory–Huggins interaction parameter (χ). Using the above theory one can derive an expression for the free energy of mixing of two polymer layers (assuming a uniform segment density distribution in each layer) surrounding two spherical particles as a function of the separation distance h between the particles [4]. The expression for Gmix is given by Eq. 3,

𝑮𝒎𝒊𝒙𝒌𝑻

= 𝟐𝑽𝟐𝟐

𝑽𝟏𝝂𝟐𝟐

𝟏𝟐− 𝝌 𝜹 − 𝒉

𝟐𝟑𝑹 + 𝟐𝜹 + 𝒉

𝟐 (3)

Where, k is the Boltzmann constant, T is the absolute temperature, V2 is the molar volume of polymer, V1 is the molar volume of solvent and ν2 is the number of polymer chains per unit area. The sign of Gmix depends on the value of the Flory–Huggins interaction parameter χ: if χ < 0.5, Gmix is positive and the interaction is repulsive; if χ > 0.5, Gmix is negative and the interaction is attractive; if χ = 0.5, Gmix = 0 and this defines the θ-condition [4].

The elastic interaction arises from the loss in configurational entropy of the chains on the approach of a second particle. As a result of this approach, the volume available for the chains becomes restricted, resulting in loss of the number of configurations. Gel is given by Eq. 4 [4],

𝑮𝒆𝒍𝒌𝑻= 𝟐𝝂𝟐𝑹𝒆𝒍 𝒉 (4)

Where, Rel(h) is a geometric function whose form depends on the segment density distribution. It should be stressed that Gel is always positive and can play a major role in steric stabilization. It becomes very strong when the separation distance between the particles becomes comparable to the adsorbed layer thickness δ. One can also define an effective volume fraction φeff that is determined by the ratio of the adsorbed layer thickness δ to the core radius R: If (δ/R) is small (say for a particle with radius 1000 nm and δ of 10 nm) φeff ~ φ and the dispersion behaves as near “hardsphere”. In this case one can reach high φ values before the system becomes “concentrated”. If (δ/R) is appreciable, say > 0.2 (e.g. for particles with a radius of 100 nm and δ of 20 nm) φeff > φ and the system shows strong interaction at relatively low φ values [4].

Van der Waals interaction (Fig. 1d): Is well known that atoms or molecules always attract each other at short distances of separation. The attractive forces are of three different types: Dipole–dipole interaction (Keesom), dipole induced dipole interaction (Debye) and London dispersion force. The London dispersion force is the most important, since it occurs for polar and non-polar molecules. It arises from fluctuations in the electron density distribution [4]. At small distances of separation (r) in vacuum, the attractive energy between two atoms or molecules is given by Eq. 5 [4],

𝑮𝒂𝒂 =!𝜷𝟏𝟏𝒓𝟔

(5)

where β11 is the London dispersion constant. For colloidal particles which are made of atom or molecular assemblies, the attractive energies may be added and this results in Eq. 6 for two spheres (at small h) [4],

𝑮𝑨 =!𝑨𝟏𝟏(𝟐)𝑹

𝟏𝟐𝒉 (6)

Where, A11(2) is the effective Hamaker constant of two identical particles with Hamaker constant A11 in a medium with Hamaker constant A22. When the particles are dispersed in a liquid medium, the van der Waals attraction has to be modified to take into account the medium effect. When two particles are brought from infinite distance to h in a medium, an equivalent amount of medium has to be transported the other way round. Hamaker forces in a medium are excess forces. The effective Hamaker constant for two identical particles in a medium (A11(2)) is given by Eq. 7 [4],

𝑨𝟏𝟏(𝟐) = 𝑨𝟏𝟏 + 𝑨𝟐𝟐 − 𝟐𝑨𝟏𝟐 (7)

Eq. 7 shows that two particles of the same material attract each other unless the Hamaker constant of the particles exactly matches that of the medium. The Hamaker constant of any material is given by Eq. 8 [4],

𝑨 = 𝝅𝟐𝒒𝟐𝜷𝒊𝒊 (8)

Where, q is number of atoms or molecules per unit volume. In most cases the Hamaker constant of the particles is higher than that of the medium. As shown in Fig. 1d the attraction potential (VA) increases very sharply with h at small h values. A capture distance can be defined at which all the particles become strongly attracted to each other (coagulation) [4].

Energy-distance curves

Combination of these interaction energies results in three main energy-distance curves [4]: (i) electrostatic energy-distance curves, (ii) steric energy-distance curves and (iii) electrosteric energy-distance curves

Electrostatic energy-distance curves: Combination of double layer repulsion (GE) and Van der Waals attraction (GA) results in the well known theory of stability of colloids (DLVO Theory). A plot of the total energy (GT=GE+GA) versus h is shown in Fig. 2, which represents the case at low electrolyte concentrations, i.e. strong electrostatic repulsion between the particles. GE decays exponentially with h, i.e. GE → 0 as h becomes large [4].

At long distances of separation, GA > GE resulting in a shallow minimum (secondary minimum-floculation). At very short distances, GA >> GE resulting in a deep primary minimum (coagulation) [4]. At intermediate distances, GE > GA resulting in energy maximum, Gmax, whose height depends on surface potential (ψ0) and the electrolyte concentration and valency. At low electrolyte concentrations (<10−2 mol/dm3 for a 1:1 electrolyte), Gmax is high (>25 kT) and this prevents particle aggregation into the primary minimum.

The higher the electrolyte concentration (and the higher the valency of the ions), the lower the energy maximum. Under some conditions (depending on electrolyte concentration and particle size), flocculation into the secondary minimum may occur. This flocculation is weak and reversible. By increasing the electrolyte concentration, Gmax decreases till at a given concentration it vanishes and particle coagulation occurs [4].

Figure 2: Electrostatic energy-distance curve

Figure 3: Steric energy-distance curve

Steric energy-distance curves: Combination of streric interaction free energy (Gmix + Gel) with the Van der Waals attractive energy (GA) gives the total free energy of interaction GT (assuming there is no contribution from any residual electrostatic interaction), see Eq. 9 [4],

𝑮𝑻 = 𝑮𝒎𝒊𝒙 + 𝑮𝒆𝒍 + 𝑮𝑨 (9)

A schematic representation of the variation of Gmix, Gel, GA and GT with surface - surface separation distance h is shown in Fig. 3. Gmix increases very sharply with decrease of h, when h < 2δ. Gel increases very sharply with decrease of h, when h < δ. GT versus h shows a minimum, Gmin, at separation distances comparable to 2δ [4]. When h < 2δ, GT shows a rapid increase with decrease in h [4]. The depth of the minimum depends on the Hamaker constant (A), the particle radius (R) and adsorbed layer thickness (δ). Gmin increases with increase of A and R. At a given A and R, Gmin increases with decrease in δ (i.e. with decrease of the molecular weight, Mw, of the stabilizer). The larger the value of δ/R, the smaller the value of Gmin will be. In this case the system may approach thermodynamic stability, as is the case of nanoparticle dispersions [4].

Electrosteric energy-distance curves: Combination of electrostatic repulsion, steric repulsion and van der Waals attraction is referred to as electrosteric stabilization. This is the case when using a mixture of ionic and nonionic stabilizer or when using polyelectrolytes. In this case the energy-distance curve has two minima, one shallow maximum (corresponding to the DLVO type) and a rapid increase at small h corresponding to steric repulsion. This is illustrated in Fig. 4.

Figure 4: Energy-distance curve for electrosteric stabilization.

5. Colliodal suspension stabilization

The properties of colidal systems are determined by the nature of the interface which separates the internal phase (e.g. particles) from the medium in which it is dispersed. Clearly with colloidal systems the

interfacial region represents a significant proportion of the whole system. The structure of the interfacial region determines, in particular the tendency of the particles to form aggregate units or remain as individual particles under the influence of Van der Waals attraction [4]. For all disperse systems, the van der Waals attraction energy (GA) is inversely proportional to the particle–particle distance of separation h and it depends on the effective Hamaker constant A of the suspension. There are three ways to overcome the van der Waals attraction (i) electrostatic stabilization, (ii) steric stabilization and (iii) electrosteric stabilization.

Elecstrotatic stabilization: Electrostatic stabilization occurs when a double layer develops as a result of the presence of a surface charge around amphoteric particles immersed in a polar liquid which is compensated near the interface by unequal distribution of counter ions and co-ions [4] [10]. In systems containing double layers, interaction leads to repulsion as soon as the double layers begin to overlap. This repulsion which is determined by the extent of the double layer (which is determined by the electrolyte concentration and valency), may overcome the van der Waals attraction, leading to a colloidally stable system. On the other hand if the repulsive force is not sufficiently large (e.g. when the

double layers are compressed), the van der Waals attraction dominates the interaction and aggregated systems result [4].

Steric stabilization: Steric stabilization arises when nonionic surfactants and/or macromolecules are adsorbed at the particle surface. Repulsion occurs when such layers overlap: this repulsion occurs as a result of the unfavorable mixing of the stabilizing chains when these are in good solvent conditions and the loss of configurational entropy on significant overlap. Steric repulsion will overcome the van der Waals attraction showing a rapid increase at distances smaller than twice the adsorbed layer thickness [4] [10]. Floculation occurs when the stabilizing chains reach poor solvency ( i.e. χ > 0.5) [4].

Electrosteric stabilization: Electrosteric stabilization arises from the combination of the other two, due to the action of charged macromolecules (i.e. polyelectrolytes) which combine the steric barrier provided by the polymer at short separation distances with the electrostatic repulsion related to the double layer overlap [10]. The effects of polyelectrolytes are well known, anionic polyelectrolites shifts the isoelectric point towards lower pH, and on the other hand cationic polyelectrolytes have the opposite effect [28].

6. Colliodal suspension characterization

Rheology of suspensions

The rheology of suspensions depends on the balance between three main forces: Brownian Diffusion; Hydrodynamic Interaction; and Interparticle Forces [2] [4]. These forces are determined by four main parameters: (i) the solids volume fraction φ (total (effective) volume of the particles divided by the volume of the dispersion). (ii) the particle size distribution and shape. (iii) the net energy of interaction GT, and (iv) the balance between repulsive and attractive forces [4].

Macroscopic point of view of rheology

From the macroscopic point of view suspensions can be considered as continuous media. The constitutive equation of so-called generalized Newtonian fluids (non compressible and isotropic) for three-dimensional flow geometry is [3],

𝝉 = 𝟐𝜼𝑫 (10)

𝜼 ≡ 𝝉𝜸 𝜸!𝑭𝑰𝑿

(11)

In Eq. 10, D is the deformation-rate tensor, τ is shear stress tensor and η es shear viscosity. For Newtonian liquids the shear viscosity η is a constant, while for non-Newtonian liquids the shear viscosity is a function of the shear rate. In the case of non-Newtonian liquids, the apparent (shear) viscosity is defined as the ratio of shear stress and shear rate at a certain shear rate [3]. Although not explicitly written in Eq. 11 the apparent shear viscosity (η) is a function of temperature [3].

Microscopic point of view of rheology

The continnum description for suspensions rheology is a valid approach. As long as suspensions are considered from a macroscopic length scale, i.e. in processes (e.g. viscometric flow experiments) in which the external characteristic length (e.g. the viscometer gap) is much larger than the intrinsic characteristic length of the microscopic heterogeneities (e.g. the size of suspended particles) [3]. In micromechanics, suspensions are considered as multiphase mixtures and effective material properties

are defined for the mixture as a whole. The minimum amount of microstructural information necessary for a suspension is the solid phase volume fraction (φ). Thus, from the viewpoint of micromechanics, the effective viscosity of a suspension is assumed to be a function of the solids volume fraction [3]. As before, it is understood, although not explicitly written in Eq. 12, that the effective viscosity is also a function of temperature [3].

𝜼 = 𝜼 𝝓 (12)

Distinction between diluted, solid and concentrated suspensions based on micromechanics

A possible distinction between dilute, solid and concentrated suspensions may be made if one considers the balance between the thermal (or Brownian) motion, hydrodynamic interaction and interparticle interaction [4].

If the Brownian diffusion predominates over the effect of hydrodynamic interaction, the suspension may be considered dilute. In this case the distance between the particle surfaces is large compared to the range of the interaction forces (hydrodynamic or surface). For dilute dispersions, the particle interactions can be represented by two-body collisions. Provided the gravitational force can be neglected (no settling occurs), the properties of dilute suspensions are essentially time-independent. Any time-average quantity, such as light scattering, osmotic pressure and viscosity, may be extrapolated to infinite dilution to obtain fundamental properties of the system (e.g. hydrodynamic radius) [4].

As the number of particles in suspension is increased, the volume of space occupied by the particles relative to the total volume (i.e. the volume fraction φ) increases as does the probability of particle–particle interaction (hydrodynamic and surface). Eventually a situation is reached whereby the interparticle distances become relatively small compared with the particle radius. Under this condition any particle in the system interacts with many neighbors and the repulsive interactions produce specific order between the particles. This system is refered to as a solid suspension it shows an elastic response and no time dependence of its properties [4].

In between the above two extremes of dilute and solid suspensions, one may define concentrated suspensions. In this case, the volume fraction ϕ is sufficiently high for many-body interactions to occur and both hydrodynamic and surface interaction play a major role in determining the properties of thesystem. However, the interparticle distances are comparable to the (effective) particle size and this

allows the particles to diffuse (albeit slowly) and the properties of the system show time dependence [4].

Rheology of concentrated suspensions

In concentrated suspensions ηr is a time-dependent complex function of φ. Concentrated suspensions show non-Newtonian flow ranging from viscous, to elastic response depending on the Deborah number (De). Three responses can be considered: (i) Viscous response: De < 1, (ii) elastic response: De > 1 and (iii) Viscoelastic response: De ~ 1. Clearly the above responses for any suspension depend on the time or frequency of the applied stress or strain. Four different types of systems (with increasing complexity) can be considered as described below (i) hard sphere suspensions, (ii) electrostatic double layer, (iii) steric and (iv) floculated [4].

Rheology of systems with hard-brownian-sphere interactions

Hard-sphere interaction takes pace when both the repulsive and attractive forces are screened. In this case the rheology of the suspension is determined by the balance between Brownian diffusion and hydrodynamic interaction. This simplifies the analysis and different theories are available to describe the

variation of relative viscosity ηr with the volume fraction ϕ of the dispersion [4]. One can define a maximum hard-sphere volume fraction (φc) above which the flow behavior suddenly changes from fluid-like to solid-like (viscous to elastic response) [4]. For random packing of equal sized spheres φC=0.64 [3] [4].

The earliest theory for prediction of the relationship between the relative viscosity (ηr) and solids volume fraction (φ) was described by Einstein. Einstein assumed that the particles behave as hard-spheres with no net interaction [3] [4]. At the dilute limit (φ ≤ 0.01) the disturbance around one particle does not interact with the disturbance around another and ηr is related to φ by Eq. 13 [4], In Eq. 13, η denotes the effective suspension viscosity and η0 the viscosity of the suspending medium. For the above hard-sphere dilute suspension, the flow is Newtonian [4].

𝜼𝒓 =𝜼𝜼𝟎= 𝟏 + 𝟐.𝟓𝝓 (13)

Eq. 14 gives the reduced shear rate (shear rate x time for a Brownian diffusion tr) [4]. In Eq. 14, R is the particle radius, η0 is the viscosity of the medium, k is the Boltzmann constant and T is the absolute temperature.

𝜸𝒓𝒆𝒅 = 𝜸𝒕𝒓 =𝟔𝝅𝜸𝜼𝟎𝑹𝟑

𝒌𝑻 (14)

A plot of relative viscosity Vs. (η0R3/kT) is shown in Fig. 5 at ϕ=0.4 for particles with different sizes. The curves are shifted to higher relative viscosity values for larger ϕ and to lower values for smaller ϕ [4].

The curve in Fig. 5 shows two limiting (Newtonian) viscosities at low and high shear rates that are separated by a shear thinning region. In the low shear rate regime, the Brownian diffusion predominates over hydrodynamic interaction and the system shows a “disordered” three-dimensional structure with high relative viscosity. As the shear rate is increased these disordered structure starts to form layers coincident with the plane of shear and this results in the shear thinning region. In the high shear rate regime, the layers can “slide” freely and hence a Newtonian region with lower viscosity is obtained. In this region the hydrodynamic interaction predominates over the Brownian diffusion [4].

Figure 5: Relative viscosity Vs. (η0R3/kT)

Figure 6: Relative viscosity Vs. Volume fraction.

If the relative viscosity in the first or second Newtonian region is plotted versus the volume fraction one obtains the curve shown in Fig. 6. The curve in Fig. 6 has two asymptotes: The slope of the linear portion at low φ values (the Einstein region) that gives the intrinsic viscosity [η] that is equal to 2.5. The asymptote that occurs at a critical volume fraction φC at which the viscosity shows a sharp increase with increase in φ [4]. The best analysis of the ηr Vs. φ curve is due to Dougherty and Krieger. They arrived at the

following simple semi-empirical equation (Eq. 15) that could fit the viscosity data over the whole volume fraction range [4]:

𝜼𝒓 = 𝟏 − 𝝓𝝓𝒄

! 𝜼 𝝓𝒄 (15)

Suspensions of non-Brownian spheres

When the particle dimensions are larger than a few micrometers, hydrodynamic effects should dominate the rheological behaviour. In addition the lack of Brownian (thermal) motion makes it impossible to achieve a truly “random” distribution of particles; hence rheological test results depend on initial conditions and shear history [2].

Suspensions of non-spherical particles

Real systems may contain polydisperse non spherical particles (e.g. ellipsoids, platelets and other irregular shapes). Hydrodynamic interactions and other interparticle forces become more complex in such cases. The maximum random close packing for non-spherical particles has not been studied in detail. It decreases with increasing aspect ratio for rods and fibres but it can be larger than that for monodisperse

spheres [2]. In the case of ordered stacks of monodispersed disk-shaped particles φc can reach values close to 0.91. Also, with increasing polydispersity the maximum packing fraction is always higher than for monodisperse systems [3].

I. Santamaría-Holek and Carlos I. Mendoza (2010) presented a model based on an effective-medium theory for the calculation of the viscosity of suspensions of arbitrarily-shaped particles as a function of particle concentration. The model considers excluded volume interactions between the particles through an effective filling fraction φeff. The effective filling fraction introduces a universal scaling that may be used to reduce both experimental and theoretical results to a master curve (η(φ)/η0 Vs. φ) which is independent of the experimental details (shear history) or particle shape [5].

𝜼 𝝓𝜼𝟎

= 𝟏 + 𝜼 𝝓𝟏!𝒄𝝓

(16)

𝒄 = 𝟏!𝝓𝒄𝝓𝒄

(17)

When applied to a suspension of spherical particles, Eq. 16 improves considerably the predictions obtained using the well known Krieger and Dougherty model (Eq. 15) and other models tested in the whole concentration range. Authors have employed the model to predict the viscosity of elliptical, cylindrical, dumbbell, sponge, square ring, jack-like particles. Furthermore, authors stated that in all cases where numerical or experimental data are available, the agreement with the proposed model was good [5].

Rheology of systems with electrostatic interaction

These are systems containing electrical double layers with long-range repulsion [4]. In this case the rheology is determined by the double layer repulsion particularly with small particles and extended double layers. In the low shear rate regime the viscosity is determined by the Brownian diffusion and the particles approach each other to a distance of the order of ~4.5 κ−1 (where κ−1 is the “double layer thickness”). This means that the effective radius (Reff) of the particles is much higher than the core radius R. The effective volume fraction (φeff) is also much higher than the core volume fraction. This results in rapid increase in

the viscosity at low core volume fraction. In the high shear rate regime, the increase in relative viscosity (ηr) occurs at much higher φ values [4].

Rheology of systems with steric interaction

The rheology is determined by the steric repulsion produced by adsorbed nonionic surfactant or polymer layers — The interaction can be “hard” or “soft” depending on the ratio of adsorbed layer thickness to particle radius (δ/R) [4].

Reology of flocculated sistems

The rheology of unstable systems poses problems both from the experimental and theoretical points of view. This is due to the non-equilibrium nature of the structure. For this reason, advances on the rheology of suspensions, where the net energy is attractive, have been slow and only of qualitative nature. On the practical side, control of the rheology of flocculated and coagulated suspensions is difficult, since the rheology depends not only on the magnitude of the attractive energies but also on how one arrives at the flocculated or coagulated structures in question [2] [4].

Suspension microstructure probing from rheological measurements

Rheological measurements are useful tools for probing the microstructure of suspensions. This is particularly the case if measurements are carried out at low stresses or strains. In this case the special arrangement of particles is only slightly perturbed by the measurement. In other words the convective motion due to the applied deformation is less than the Brownian diffusion. The ratio of the stress applied (σ) to the “thermal stress” (that is equal to kT/6πR3, where k is the Boltzmann constant, T is the absolute temperature and R is the particle radius) is defined in terms of a dimensionless Peclet number (Pe – Eq. 18) [4]. For a colloidal particle with radius of 100 nm, σ should be less than 0.2 Pa to ensure that themicrostructure is relatively undisturbed ( Pe < 1). In order to remain in the linear viscoelastic region,

the structural relaxation by diffusion must occur on a time scale comparable to the experimental time (Deborah numner De ~ 1) [4].

𝑷𝒆 =𝟔𝝅𝑹𝟑𝝈𝒌𝑻

(18)

Suspension stability in time

The prediction of stability through zeta potential and/or rheological measurements is common practice; the studies dealing with the stability of ceramic suspensions against time are much less frequent. The predictive methods are used to select the conditions for the preparation of a stable suspension but are not able to predict howmuch time these suspensions maintain stable [10]. Settling tests give useful information about durability and evolution of the suspension, but may lead to contradictory results because rheological measurements are dynamic tests in which the sample is submitted to shear forces whereas in tests intended to probe suspension stability in time the sample is in equilibrium at rest. Moreover, rheological techniques provide a useful picture of the overall stability at a macroscopic scale but do not allow determining the mechanism through which destabilisation occurs on ageing. This requires the use of a specific technique able to determine the agglomeration kinetics of particles stored in

a cell at rest such as multiple light scattering [10]. Multiple light scattering technique analyses the effect of a light source through the sample, and the transmitted and the backscattered light allows us to obtain information about the agglomeration processes that take place during ageing, such as sedimentation and particle aggregation (coalescence, flocculation) [10]. A major advantage of MLS as compared to other optical techniques such as microscopy, laser diffraction or dynamic light scattering, is that the former is non-destructive as no sample dilution is needed. The instrument is able to detect particle size variation or particle migration in concentrated and optical thick media [10]. According to obtained results authors concluded that MLS technique is a promising tool giving useful information about the rheological behaviour of concentrated suspensions complementary to that obtained through rheological measurements, especially regarding the ageing evolution at rest [10].

7. References

[1] J.R.G. Evans. Seventy ways to make ceramics. Journal of the European Ceramic Society 28 (2008) 1421–1432.

[2] Jan Mewis, Norman J.Wagner. Current trends in suspension rheology. J. Non-Newtonian Fluid Mech. 157 (2009) 147–150.

[3] WILLI PABST. FUNDAMENTAL CONSIDERATIONS ON SUSPENSION RHEOLOGY. Ceramics − Silikáty 48 (1) 6-13 (2004).

[4] Tharwat Tadros. Interparticle interactions in concentrated suspensions and their bulk (Rheological) properties. Advances in Colloid and Interface Science 168 (2011) 263–277

[5] Santamaría-Holek, Carlos I. Mendoza. The rheology of concentrated suspensions of arbitrarily-shaped particles. Journal of Colloid and Interface Science 346 (2010) 118–126.

[8] R. Benavente, M.D. Salvador, M.C. Alcázar, R. Moreno. Dense nanostructured zirconia compacts obtained by colloidal filtration of binary mixtures. Ceramics International xxx (2011) xxx–xxx.

[9] M.H. Talou, M.A. Villar, M.A. Camerucci, R. Moreno. Rheology of aqueous mullite–starch suspensions. Journal of the European Ceramic Society 31 (2011) 1563–1571.

[10] Olga Burgos-Montes, Rodrigo Moreno. Stability of concentrated suspensions of Al2O3–SiO2 measured by multiple light scattering. Journal of the European Ceramic Society 29 (2009) 603–610.

[17] Arnaldo J. Millán1, Rodrigo Moreno*, María Isabel Nieto. Thermogelling polysaccharides for aqueous gelcasting—part I: a comparative study of gelling additives. Journal of the European Ceramic Society 22 (2002) 2209–2215.

[18] Arnaldo J. Milla´n1,Marı ´a Isabel Nieto,Carmen Baudı´n,Rodrigo Moreno. Thermogelling polysaccharides for aqueous gelcasting—part II: influence of gelling additives on rheological properties and gelcasting of alumina. Journal of the European Ceramic Society 22 (2002) 2217–2222.

[20]Isabel Santacruz, Begoña Ferrari, M. Isabel Nieto, Rodrigo Moreno. Ceramic films produced by a gel-dipping process.

[28] João B. Rodrigues Neto a, Rodrigo Moreno. Rheological behaviour of kaolin/talc/alumina suspensions for manufacturing cordierite foams. Applied Clay Science 37 (2007) 157–166.