ABSTRACT
Studies on the mechanical properties of solids and liquids indicate that there are many generalities governing the mechanical behavior of bodies of various natures� One can distinguish several basic types of mechanical behavior, a combination of which yields an approximate description of the more complex mechanical properties of real bodies� The science describing the laws that govern the mechanical behavior of solid-and liquid-like bodies is referred to as rheology (from the Greek words meaning “the study or theory of flow”)�
Modern rheology includes rheology itself as an independent fundamental discipline and rheometry, which addresses rheological measurements� The rheological behavior of various materials and systems is of importance in a great variety of industries, for example, food, oil and gas production, mining, mineral ore processing, and pharmaceutical� There are numerous publications, monographs, and books dedicated to rheological studies, such as the Journal of Rheology, Applied Rheology, and the Rheological Bulletin [1-11]�
Rheology in general addresses the response of materials to stresses applied in various ways� The main principle of rheology is the description of the mechanical properties of systems using simple idealized models containing a relatively small number of parameters� The simplest approach is the so-called “quasi-steady-state regime,” which involves a restriction on uniform shear and low deformation rates�
Inside a physical body, let us consider a cube with an edge of unit length� Let the tangential force, F, (designation used in this chapter) be applied to the opposite faces of the cube to cause a shear stress, τ, numerically equal to the applied force (Figure 3�1)�
The applied shear stress causes the deformation, or strain, of the cube, that is, a shift of its upper face with respect to the lower face by an amount γ� This shift is numerically equal to the tangent of the tilt experienced by the side face, that is, to the relative shear strain, γ� When the strains are low, one can write that tan (γ) ≈ γ� The relationship between the stress τ, the strain γ, and their change as a function of time represents the mechanical behavior, which is the main subject of rheology� Let us start by reviewing three basic models of mechanical behavior: elastic, viscous, and plastic�
3.1.1.1 Elastic Behavior Fully reversible elastic behavior is characterized by a direct proportionality between stresses and deformations and is described by Hook’s law:
t g= G
where the proportionality constant G is referred to as the elastic shear modulus [N/m2] τ is the shear stress [N/m2]
In the rheology of condensed phases, the elastic modulus G is often used as the sole characteristic of elastic behavior� In isotropic media mechanics, it has been established that for solid-like bodies, the modulus G ~ 2/5 of the Young’s modulus [10]�
Graphically, Hook’s law corresponds to a straight line passing through the origin (Figure 3�2), with an inverse slope equal to G� As briefly stated earlier, the characteristic feature of elastic behavior is its complete mechanical and thermodynamic reversibility� This means that the original shape of the body is completely restored upon the removal of the stress, and there is no energy dissipation associated with the application and removal of the stress-causing load� The energy stored by a unit volume of an elastically deformed body is given by the expression
W del = ( ) = =ò 0
2 2
t g g g tG
G (3�1)
A spring with a spring constant k = G (spring constant is the ratio of the force, F, to the elongation, Δl, caused by that force) can be used as a model of elastic behavior (Figure 3�3)�
Elastic behavior is typical of solid bodies� The nature of elasticity is related to the reversibility of small deformations of interatomic bonds� Within the limits of small deformations, the potential energy curve can be approximated with a quadratic parabola, which conforms to a linear relationship between γ and τ� The magnitudes of the elastic modulus G depend on the nature of the interactions in the solid body� For molecular crystals, G is ~ 109 N/m2, while for metals and covalent crystals, the value of G is on the order of ~ 1011 N/m2� The elastic modulus G reveals either a weak dependence from temperature or no temperature dependence at all�
There are also cases in which the elasticity may have a completely different, entropic nature� This is the case in systems consisting of macromolecules or in clay suspensions� In such systems, the applied shear stress results in a change in the chaotic orientation of the segments of macromolecules or of the clay platelet-like particulates, which causes an ordering and hence a decrease in entropy� The return of the system to the original (disordered) state is associated with thermal motion� The modulus associated with entropic elasticity is small and exhibits strong temperature dependence [10]�
As has already been stated, the elastic modulus has the units of pressure, Pa, or N/m2, which is also equivalent to J/m� The latter means that, in agreement with Equation 3�1, it is possible to formally view the elasticity modulus as twice the elastic energy stored by the unit volume subjected to the unit strain� At a given shear stress, in agreement with Equation 3�1, the smaller the modulus G, the higher the elastic energy density stored by the body�
In reality, the elastic deformation of solid bodies is observed only up to a certain limiting value of shear stress τ, above which the body either undergoes destruction (for brittle objects) or shows residual deformation (i�e�, reveals plasticity)�
3.1.1.2 Viscous Flow Viscous flow is characterized by proportionality between the stress and the rate of deformation, that is, between τ and dγ/dt, and is described by Newton’s viscosity law:
t
h g = d
dt
where the coefficient η is dynamic viscosity with the units of Pa s or N s/m2 (1 Pa s = 10 Poise = 1 cP)� Graphically, Newton’s viscosity law can be represented by a straight line passing through the origin in the dγ/dt − τ coordinates (Figure 3�4)� The inverse slope of this line yields the viscosity η�
In contrast to elastic behavior, this idealized viscous behavior is completely irreversible, both mechanically and thermodynamically� Irreversibility implies that the initial shape of the body is not restored after the shear stress has been relieved� Viscous flow is accompanied by the dissipation of energy, that is, by the conversion of all work into heat� The rate of energy dissipation, that is, the power dissipated per unit volume, is given by
d d
d d
d d
dW t t t
= = æ è ç
ö ø ÷
t g h g 2
This quadratic dependence is characteristic of viscous friction� The viscous behavior of a body can be modeled by a dashpot (Figure 3�5)� In this model, it is assumed that the ratio of the acting force F to the velocity of a piston, F/(dl/dt), is equal to the viscosity of fluid�
The nature of viscous flow is related to self-diffusion, that is, to the mass transfer due to atoms sequentially changing their positions in the course of thermal motion�The applied stress lowers the potential barrier to this mass transfer in one direction and increases mass transfer in the other direction, which results in the gradual development of a macroscopic deformation� Consequently, viscous flow is a thermally activated process, and the viscosity η shows a characteristic exponential dependence on temperature� For different materials, the values of viscosity can cover a very broad range� For low-viscous fluids, such as water or metal melts, η ~ 10−3 Pa s, while for high-viscous Newtonian fluids, η can be by higher by three to six orders of magnitude and even higher for structured systems� The probability of thermally activated acts (i�e�, of diffusion) increases with time, even in the case where the height of the potential energy barrier is significant� For this reason, even solid bodies may reveal liquid-like behavior with viscosities around 1015-1020 Pa s or higher� Such flows take place in geological processes�
3.1.1.3 Plastic Flow (Plasticity) In contrast to the two previous cases, plastic flow (plasticity) is characterized by the absence of proportionality between the stress and the strain, that is, plasticity represents a case of nonlinear behavior� For plastic bodies subjected to stresses below the critical value, τ < τ* (the so-called shear yield point), the rate of strain is zero (dγ/dt = 0)� Plastic flow starts at the yield stress, τ = τ*, and does not require further increase in stress (Figure 3�6)� Similar to viscous flow, plastic flow is thermodynamically and mechanically irreversible� However, in contrast to the prior case, the rate of energy dissipation in plastic flow is proportional to the rate of strain:
d d
d d
W t t
d *= t g
This type of dependence is a characteristic of dry friction, that is, corresponds to Coulomb’s dry friction law, Ffr = μFN� Plastic behavior can be modeled by a “friction element,” that is, two flat plates with a friction coefficient, μ, with respect to each other and compressed with a normal force FN in such a way that the applied tangential force Ffr would correspond to the yield stress of a given material (Figure 3�7)�
The nature of plastic flow is a complex of processes involving the rupture and rearrangement of interatomic bonds, which in crystalline solids usually involves the participation of linear defects (dislocations)� The temperature dependence of plastic flow may be considerably different from that of a Newtonian fluid� Under certain conditions, various molecular and ionic crystals (e�g�, naphthalene, AgCl, and NaCl) reveal behavior that is close to plastic� The values of τ* typically fall into the range of 105-109 N/m2� Plastic flow is typical for a variety of disperse structures: powders (including snow and sand) and pastes� In the latter case, the origin of the plastic flow is a sequence of acts involving the rupture and reformation of contacts between the particles of disperse phase� In contrast to a fluid, a plastic body maintains its acquired shape after the stresses have been removed� It is noteworthy that the ancient craft of pottery had originated from the plasticity (Greek-πλαστωσ) of wet clay�
These are the three simplest cases of mechanical behavior and the corresponding rheological models� By combining them as elements, one can obtain more complex models describing the rheological properties of various systems� Every such combination is typically viewed within a framework of a specific deformation regime in which one seeks to reveal the qualitatively new properties of a given model as compared to the properties of its elements�
a� A combination of elastic models and viscous flow models in series yields the Maxwell model (Figure 3�8)� Newton’s third law, applied to such a combination, dictates that equal forces (shear stresses τ) act on each of the constituent elements, while the deformations are added together
g g g t t hh
= + = + æ
è ç
ö
ø ÷òG G d
t
where γ is the net strain� Consequently, the rates of strain are additive too:
d d
d d
d d
g g gh t t t
= +G (3�2)
A characteristic regime that manifests the specifics of the mechanical behavior of this model is the regime in which the strain first instantaneously reaches the value of γ0 and then stays at that level, that is, γ = γ0 = const� At the initial moment, t = 0, the strain in the viscous element is zero, so the entire deformation (and the entire work) is concentrated within the elastic element� Consequently, the initial stress is τ0 = Gγ0� This stress further causes the deformation of the viscous element� Since the total deformation is constant, the deformation of the elastic element decreases, resulting in a decrease in the stress�Under conditions of constant net strain, γ = const, Equation 3�2 can be written as
1 0 G
+ æ è ç
ö ø ÷ =
d d t t
ht
Integrating this expression, using the initial condition τ(t = 0) = τ0 = Gγ0, yields
t t= -
æ
è ç
ö
ø ÷0 exp
t tr
The value tr = η/G has the units of time and is referred to as the relaxation period. This value graphically corresponds to the point at which the line tangent to the τ(t) curve at point t = 0 intersects with the x axis (Figure 3�9)� This gradual decrease of the stress (stress relaxation) is typical in viscoelastic systems� The energy that was previously stored in the elastic element dissipates in the viscous element� As a result, the behavior of such a system is mechanically and thermodynamically irreversible�
The relaxation period defines the behavior of the system, in accordance with the Maxwell model with respect to the timescale of the applied stress�If the time t during which stress is applied is greater than the relaxation period, that is, t > tr, the system has properties similar to those of a viscous liquid, while at t ≪ tr, the system behaves like an elastic solid� The flow of glaciers and other processes of strain development in mountains and cliffs are representative examples of such behavior� In rheology, the ratio of a material’s characteristic relaxation time to the characteristic flow time is referred to as the Deborah number� This parameter plays an important role in describing the response of various materials to different stresses�
b� Connecting the elastic and viscous elements to each other in parallel yields the Kelvin model, schematically shown in Figure 3�10� In this case, the strain in both elements is the same, while the net stress is the sum of the individual stresses, τ = τG + τη� A strain regime corresponding to a constant shear stress, τ = τ0 = const, is of the most interest here� In contrast to the Maxwell model, the presence of the viscous element in the Kelvin model does not allow for an immediate deformation of the elastic element� As a result, the deformation gradually develops over time at a rate given by
d d g t
h t t
h t g
h h
t = =
-( ) =
-( )0 0G G
The integration of the aforementioned expression yields deformation as a function of time, namely,
g t= - -éë ùû
( ) exp( )
1 /G
/ rt t
This type of dependence of strain on time is referred to as the elastic aftereffect� It is schematically shown in Figure 3�11� The deformation increases at a declining rate to the limit of γmax = τ0/G, which is determined by the elasticity modulus of Hook’s element�
The elastic aftereffect is encountered in solid-like systems with an elastic behavior� The elastic behavior is reversible: when the stress is removed, the strain drops gradually to zero, that is, the initial shape of the body is restored, using the energy stored by the elastic element� However, in contrast to true elastic behavior, the elastic aftereffect is thermodynamically irreversible: the dissipation of energy takes place in the viscous element� The damping of mechanical oscillations in rubber, caused by harmonic stresses, is the example of a process conforming to the Kelvin model�
c� A rheological model that describes the internal stresses can be obtained by connecting the elastic element and the nonlinear dry friction element in parallel, as shown in Figure 3�12�
Under conditions when the applied stress τ exceeds the yield stress (τ > τ*), the strain γ = (τ − τ*)/G appears in the body� This deformation results in the accumulation of energy in the elastic element� If at the same time, τ < 2τ*, then the “frozen” residual stresses remain in the body after the stress has been removed due to the action of the dry friction element� This “frozen” stress equals τ0 − τ* and has the sign opposite to that of the initial stress and can’t exceed τ*�
d� The Bingham model represents the situation when the viscous Newtonian element and Coulomb’s dry friction element are connected in parallel (Figure 3�13)� This model is commonly used for the description of colloidal structures, such as aqueous suspensions of claylike minerals� Since the elements in the Bingham model are connected in parallel, their deformations are the same, while the stresses are additive� The stress at the Coulomb element can’t exceed the yield stress, τ*� Consequently, the rate of deformation in the viscous element should be proportional to the difference between the acting stress and the yield stress, namely,
d d g t t
ht =
-( )* B
No flow takes place at τ < τ* (Figure 3�14)� Since the ηB parameter defines the derivative, ηB = dτ/(dγ/dt), this constant parameter is referred to as the differential viscosity, which differs from a variable effective viscosity, τ/(dγ/dt) = ηeff (dγ/dt)� In rheology literature, dγ/dt is typically denoted as g.�
e� Complex combination models are often used to describe the behavior of real systems, especially in cases with broadly varying conditions (time, stress, etc�) In these complex models, the described simpler models are present as individual elements� For example, the system may reveal more than one characteristic relaxation time (or a spectrum of relaxation times), as shown in Figure 3�15� As the resulting rheological models become more complex, the mathematical description also becomes more complex�
Employing the so-called electromechanical analogies can allow one to substantially simplify the use of rheological models� This method is based on modeling rheological properties using electric circuits� This is possible because the laws describing electric circuits are mathematically identical to the laws describing the deformation of solid and liquid bodies� To illustrate this, let us turn to Figure 3�16� The energy stored by the spring, Gγ2/2, has exactly the same mathematical form as the expression for the energy of a charged capacitor, q2/2C, where q is the electric charge and C is the capacitance� The energy dissipation in the viscous element, η(dγ/dt)2, is equivalent to the Joule’s heat dissipated by the
resistance R through which the current I flows, that is, to RI2� For instance, using these formally identical relationships allows one to model the relaxation of the mechanical stresses in the Maxwell model with a voltage drop occurring during the discharge of a capacitor connected to a resistance in the circuit with the time constant tr = RC = η/G� However, at the same time, it is not always possible to describe the behavior of real systems, even with complex models containing only the parameters that do not change in the process of strain: G, η, τ*� In such cases, one must switch to models that contain variable parameters, for example, the elements of nonlinear elasticity, G = G(γ), nonlinear viscosity η = η(dγ/dt), and variable yield stress (hardening) τ* = τ* (γ)�
For a large number of disperse systems with a globular structure mechanical characteristics, such as strength, Pc (N/m2), and the ability to resist the action of external stresses in general are manifested by the cohesive forces between the particles at points of their contact� Experiments have confirmed [11,12] that the strength Pc can be estimated using the additivity approximation, Pc = χp1, where p is the strength (the force measured in N) of individual contacts between particles, and χ (m−2) is the number of such contacts per unit area of the “fracture surface�” Estimates for both p and χ can be obtained on the basis of both theory and experimental data�
The value of χ is determined by the geometry of the system, primarily by the particle size (radius, r, for spherical particles) and by the packing density of particles described by porosity, Π� The porosity is a dimensionless characteristic defined as the ratio of the volume of pores, Vp, to the total volume of the porous structure, V, that is, Π = Vp/V� The χ = χ(r, Π) dependence can be estimated from data on the degree of dispersion of the particles and the porosity of samples by employing the specific models for disperse structures� For example, in the case of loose monodisperse structures with spherical particles connected into crossing chains with n particles per chain between the nodes (Figure 3�17), the χ function for the case when the porosity Π does not exceed 48% can be described as
c
p
=
= -
1 2
1 6 3 2
[( )/ ]
/[ / ( )] ( )
r n
n n
where na is the average number of particles between the nodes in the chains (see Section 7�3 for more details)�
For moderately dense structures (such as those described by a primitive cubic lattice with a coordination number of 6, employing a first and most crude approximation yields an estimate of χ ≈ 1/(2r)2� This provides one with the means to estimate the order of magnitude of χ in real systems, namely, for particles having a diameter 2r ~ 100 μm, we get a value of χ ~ 103-104 contacts per cm2;
for particles with 2r ~1 μm, χ ~ 107-108 contacts per cm2; while for particles with 2r ~10 nm, χ ~ 1011-1012 contacts per cm2� These ranges of values will be different for systems with polydisperse or anisometric particles�
While in the previous chapters we have already addressed particle-particle interactions in some detail, it would be worthwhile to recall the principal concepts here� In coagulation contacts, the particle-particle interactions are restricted to contact between the particles via either the remaining equilibrium narrow gaps filled with dispersion medium or directly with each other� Contacts of this type form either in the absence of the Derjaguin-Landau-Verwey-Overbeek potential energy barrier or when the interaction energy is sufficient to overcome the potential energy barrier� Coagulation contacts correspond to particle interactions at the primary potential energy minimum� The adhesive energy between the particles in coagulation contact is determined by the depth of the potential energy minimum, namely,
u
A r hc
» * , 12 0
where A* is the complex Hamaker constant h0 is the equilibrium particle-particle distance r is the curvature radius of particle surface at the point of contact
For typical lyophobic colloidal systems with a complex Hamaker constant ~ 10−19 J and a particleparticle separation of 0�2-1 nm, the adhesive energy in the contact is significantly larger than kT, which indicates that thermodynamics favors the formation of coagulation contacts� The primary potential energy minimum is even deeper in systems composed of coarser particles� At the same time, for the case of low values of the complex Hamaker constant, 10−21-10−22 J, the adhesion between the particles in systems that are not too coarse (particles with a diameter up to a micron) is overcome by the Brownian motion, and the formation of structures with coagulation contacts is impossible�
The strength of a coagulation contact, that is, the adhesive force between particles, is determined primarily by the molecular forces� For spherical particles, this force is given by
p u
h A r
h1 0 0212 » »c *
The corresponding estimates of adhesive force, p1, with a Hamaker constant A* value on the order of ~ 10−19 J and h0 ~ 0�2-1 nm (i�e�, A*/12h02 ~ 10−1-10−2 J/m2) for particles with radii of 10 nm, 1 μm, and 1 mm are, respectively, 10−9-10−10, 10−7-10−8, and 10−4-10−5 N� Forces that are on the order of 10−9 N or higher can be measured experimentally�
In cases when a complete displacement of the dispersion medium from the gap between particles takes place, that is, upon the rupture of the adsorption-solvation layer (or in a vacuum) a direct point-like contact between the particles may be established� Such contact may be formed by one or several atomic cells� In this case, along with the van der Waals forces, the short-range (valent) forces acting over the area of contact may also play some role in particle adhesion� The contribution of such forces to the strength of the contact can be estimated as p1 ≈ N e2/(b24πε0), where N is the number of valent bonds, e is the elementary charge, ε0 is the electric constant, and b is the typical characteristic interatomic distance� In this case where there is a minimum number of valent bonds, N ~ 1-10, one finds that p1 ~ 10−8 N� Under these conditions, one does not yet reach the conditions necessary for establishing the bridge-like phase contacts, which will be discussed in detail in Chapter 6� Consequently, in lyophobic systems for particles with r ~ 1 μm, the contribution of shortrange forces to the strength of the contact may be of the same order (or less) as the contribution of the van der Waals forces�
Structures with coagulation contacts typically show low strength and mechanical reversibility, that is, these structures are capable of spontaneous restoration after mechanical destruction (i�e�, thixotropy)� As an example, let us estimate the macroscopic mechanical strength, Pc, of a coagulation structure formed by densely packed particles having radius r ~ 1 μm� For a lyophobic system with A* ~ 10−19 J, one gets Pc ≈ χp1 ≈ (1/2r)2 A*/12 02h ~ 104 N/m2� For a powder or suspension, this value of Pc should be interpreted as the yield stress, τ*� In systems with coarse particles, for example, those having r ~100 μm, the estimate for the Pc value is only ~ 102 N/m2� This value is characteristic of systems with high mobility, such as sand in an hourglass� In contrast to these cases, a system with a high degree of dispersion, for example, with particles having r ~ 10 nm, is characterized by a mechanical strength, Pc, on the order of ~102 N/m2or higher, which indicates that the system is capable of displaying a substantial resistance to deformation�
It is worth recalling here that a dispersion medium akin to the particles, as well as surfactant adsorption, can lower both the interfacial energy, σ, and the complex Hamaker constant, A*, by two to three orders of magnitude� In such a lyophilized system, the adhesive energy and force are also lowered by several orders of magnitude� In a concentrated disperse system in which the dispersed particles are mechanically forced to come into contact with each other, the lyophilization manifests itself as a decrease in the resistance to strain τ*� This means that in concentrated colloidal systems, plasticizing takes place, while in systems with a low concentration of dispersed particles, the lyophilization results in enhanced colloid stability of the free-disperse system (see Chapter 4)�
In generally encountered disperse systems, the particle sizes and characteristics of the particleparticle interaction can cover a very broad spectrum of values, which explains the large variation in the rheological properties of different colloidal systems utilized in different areas of technology� At the same time, disperse systems are the main carriers of mechanical properties in both inanimate and live nature [10-17]�
Large variety of rheological properties of disperse systems is manifested via a broad range of values of the elasticity modulus, G; Young’s modulus, E; viscosity, η; and the yield stress τ* (the yield point, the ultimate strength)�
In continuous systems with solid phases (mineral rocks, construction materials), the parameter G is the elasticity modulus of a solid body, that is, it falls in the range between 109 and 1011 N/m2� It is noteworthy that the elasticity modulus of common liquids under uniform compression conditions is of the same order of magnitude� However, due to low viscosity, the elasticity in liquids can be experimentally determined only by very fast measurements in which the impact time is very close to the relaxation time� For this reason, under common conditions, liquids with low values of η behave like viscous media�
In systems with solid and liquid phases, the elasticity modulus is determined by the interactions between the particles of the dispersed phase� For porous disperse structures of globular type with phase contacts between the particles, the elasticity modulus of the system is determined by the elasticity modulus of the substance making up the solid phase and by the number and area of the
contacts between the particles� This is typically so, regardless of whether the other phase is liquid or gaseous� For example, the elasticity modulus of porous crystallization structures may fall in the range between 108 and 1010 N/m2� These types of structures are brittle and reveal a tendency toward irreversible erosion without any noticeable preceding residual deformation� The disintegration occurs at a yield stress value that is below the stress level at which plastic flow can occur�
The elasticity modulus of coagulation colloidal structures with solid and liquid phases can have a different nature if the formation of such structures takes place under conditions of a relatively low volume fraction of the dispersed phase, when the degree of dispersion is high or in the case when the particles are anisometric� Examples of such systems include hydrogels of vanadium pentoxide and structured colloidal suspensions of bentonite clay in water� As will be discussed in detail in the following text, the shear elasticity in such systems may be stipulated by a higher or lower degree of coorientation between the particles in the course of the strain� Such coorientation increases the ordering in the system and hence decreases entropy� When the load is removed, the Brownian (rotational) motion of the particles restores their chaotic orientation and thus restores the original shape of the body due to the change in the configuration entropy [18]� Shear elasticity has, therefore, an entropic origin, similar to the entropic origin of the elasticity of gas pressure or osmotic pressure� The elasticity modulus, Gel, is on the order of nkT, where n is the number of particles per unit volume that participate in the Brownian motion, that is, the number of kinetically independent units� For instance, in a dilute suspension of finely disperse clay particles having n ~ (3-5) ×1023 particles per m3, the value Gel ~ 103 N/m2�
Elastic deformation can also be observed in foams and concentrated emulsions� In such cases, the yield stress is determined by the increase in the interfacial area upon the deformation of the particles� The mechanical properties of solidified foams and other solid-like materials with a cellular structure are defined by the degree of their dispersion, their backbone structure, and the combination of the mechanical properties of dispersion medium and dispersed phase�
The viscosity of dilute free-disperse systems is mainly determined by the viscosity of the dispersion medium, which, in principle, may vary over many orders of magnitude� Gases, for instance, have a viscosity on the order of 10−5 Pa s, while liquid-like materials have viscosities ranging from 10−2 to 1010 Pa s and for glasses and solids viscosity values fall in the range between 1015 and 1020 Pa s or higher�
Einstein showed that the viscosity of dilute suspensions in the absence of interactions between the particles is proportional to the volume fraction of the dispersed phase, ϕ, that is, the addition of particles to the dispersion medium results in energy dissipation due to the rotation of the particles in the shear force field:
h h h
f -( )
k
where η0 is the viscosity of dispersion medium� For spherical particles, k = 2�5� Thus, in the absence of interactions between the particles, the system behaves as a Newtonian fluid but with a viscosity slightly higher than that of a dispersion medium�
In a suspension containing anisometric particles (ellipsoids, platelets, rods) or “soft” deformable particles (droplets or macromolecules), various tendencies may be revealed� Shear stresses accompanied by particle rotation tend to deform the particles and orient them in the flow in a particular way (Figure 3�18)� The rotational diffusion of the particles in this case tends to oppose the orienting action� As a result, the extent of the particle orientation strongly depends on the rate of the strain� That is, at a low flow rate, the particles may be completely disoriented, while at a high flow rate, they may exhibit a high degree of orientation, which can be registered by optical methods� This in turn leads to the dependence of the viscosity on the flow rate (or the shear stress)� In this case, a single variable of the Newtonian viscosity, η = dτ/d(dγ/dt), is no longer sufficient for describing the system and the shear-rate-dependent effective viscosity, ηeff = τ/(dγ/dt)� The effective viscosity is maximal at a low strain rate, and then it gradually increases with the increase in the shear rate to
some minimum value corresponding to the state in which the particles are completely oriented in the flow (Figure 3�19)� Once that minimum value has been reached, the effective viscosity does not undergo any further change�
When certain hydrophilic polymers are dissolved in water (e�g�, polyacrylamide), the flow rate may increase significantly due to the suppression of the turbulence (the Toms effect of friction reduction) resulting in an increase in the efficiency of pumping� This phenomenon is commonly used in the petroleum industry to fracture subterranean formations and in firefighting to increase the power of pressure pumping� Oppositely, under certain conditions, some systems may undergo an increase in the effective viscosity as the flow rate increases� Such systems are referred to as dilatant or shear thickening� This may be caused, for instance, by a significant deformation of the macromolecules in the flow due to conformation changes� In a concentrated suspension, a significant increase in viscosity with increases in flow rate is referred to as rheopecty (or rheopexy)� This phenomenon can be observed during the initial stages of the starch dissolution process when solid starch is suspended in water�
These types of phenomena can’t be described in terms of simple rheological models with constant parameters� Systems that reveal the dependence of the viscosity on the flow rate are referred to as anomalous or non-Newtonian� In dilute suspensions, changes in the viscosity associated with the orientation and deformation of the particles in the absence of particle-particle interactions are typically not too large�
The viscosity in connected-disperse systems with coagulation structures changes more abruptly than the viscosity in free-disperse systems� In this case, one can encounter an entire spectrum of states between two limiting cases: that of a completely intact structure and one corresponding to the
fully destroyed state� Depending on the magnitude of the applied stress (flow rate), the rheological properties of structured systems can vary over a broad range: from those typical of solid-like materials to those characteristic of Newtonian fluids� This rheological behavior diversity of a real disperse system with coagulation structure is described by a complete rheological curve� An example of such a curve in the form of dγ/dt = f(τ) for aqueous suspension of finely dispersed bentonite is shown in Figure 3�20� A complete rheological curve may also be presented as a dependence of the effective viscosity, ηeff = τ/(dγ/dt), on the shear stress, τ (Figure 3�21)�
As seen in Figure 3�21, a complete rheological curve contains four characteristic regions� Region I corresponds to low stresses under which the system may demonstrate a solid-like behavior with high viscosity (Kelvin model)� This case is characteristic of the already mentioned bentonite clays� The studies of relaxation structures in moderately concentrated suspensions of bentonite clays indicated the appearance of elastic aftereffect at low shear stresses� This effect has an entropic nature, as it is associated with the
mutual orientation of the anisometric particles that are capable of participating in the thermal motion� High values of viscosity are associated with the flow of the dispersion medium from the cells that are shrinking in size to the neighboring ones and by the occurring sliding of particles against each other�
When the shear stress reaches a particular value, tSchw* , the system reveals a viscoplastic flow with an essentially preserved structure and enters the so-called Schwedow creep region (region II in Figures 3�20 and 3�21)� In this region, shear is caused by the fluctuation process of the destruction and subsequent restoration of the coagulation contacts� Due to external stresses, this process becomes directional�This mechanism of creep may be considered by the analogy with the mechanism of flow of liquids developed by Frenkel [19]�
As a result of Brownian motion, the particles agglomerated into a single coagulation structure undergo oscillation around their position in the contacts� Because of thermal fluctuations, some contacts are ruptured, but new contacts between particles in other locations are formed� On average, the number of contacts in the formed structure does not change over time and remains close to the maximum possible number of contacts� In the absence of shear stresses, the process of the rupture and reformation of the contacts in any section occurs with equal probability in all directions� Upon the application of an external force field, these processes become directional, and slow macroscopic shear motion (creep) is observed� Creep is possible over some particular range of τ values, at which a balance between the small number of rupturing and reforming contacts is reached� Creep region II, as well as the subsequent region III (Figure 3�21), can be described in terms of a viscoplastic flow model with a low yield stress, tSchw* , and a very high differential viscosity, ηSchw:
t - h gtSchw Schw* =
d dt
where ηSchw is given by the inverse slope ϕII of the curve in region II, that is, tan−1 ϕII = ηSchw� Consequently, the effective (variable) viscosity in this region is also high:
h t g
h t
Schw/ /* =
( ) =
-d dt [ ( )]1 t
Overall, strains γ on the order of a few percent are typical for low shear stress regions I and II in Figure 3�21� Upon the action of long-lasting shear stresses, large deformation can develop� This is the case in geological processes, such as during the movement of glaciers�
Once a certain shear stress, tB*, is reached, the equilibrium between the formation and rupture of the contacts is shifted toward rupturing� The higher the shear stress τ, the larger the shift� This flow regime, in which the structure undergoes intensive destruction, corresponds to the viscoplastic flow region III in Figure 3�20� This region can be described in terms of Bingham’s model with a relatively large yield stress tB* and a low differential Bingham viscosity, ηB:
t h g-=tB B*
d dt
Bingham’s yield stress, tB*, corresponding to the onset of an intensive structure degradation, can be viewed as a system’s strength characteristic�
The shift in the equilibrium toward the rupture of the contacts results in a drop of the effective viscosity, sometimes by several orders of magnitude:
h t g
h
t eff
B / /*
= ( )
= -é
ëê ù ûú
d dt 1 ( )t
For suspensions of bentonite clays, the values of the effective viscosity may vary over several orders of magnitude, that is, from 106 to ~10−2 Pa s�
After the structure has been completely destroyed, under the conditions of a laminar flow, the disperse system behaves as a Newtonian fluid (region IV in Figure 3�20) with a constant and lowest viscosity, ηmin (Figure 3�21)� The viscosity ηmin of such a system is higher than that of the dispersion medium to a degree that is greater than predicted by Einstein’s viscosity law� The reason for such a larger increase in viscosity is the interactions between particles at the sufficiently high concentration of a suspension� Further increases in shear stress result in a deviation from the Newtonian behavior due to the transition into a turbulent flow regime� Region IV (Figure 3�21) may sometimes not be observed due to the early appearance of turbulent flow�
The rheological properties of structured disperse systems may undergo a substantial variation under the action of vibration� Vibration favors the rupture of the contacts between particles and results in a liquefying of the system at lower shear stresses� As a result, the dγ/dτ − τ curve can shift to the left, as illustrated in Figure 3�22� Vibration is commonly used in various applications for controlling the rheological properties of various disperse systems, such as concentrated suspensions, powders, and pastes�
When the contacts between the particles and the entire structure of the solid phase dispersion are completely destroyed, the rheological behavior of the system is similar to that of a liquid medium� In the opposite case, there is a critical shear stress that is the source of the internal friction, preventing compact packing� Liquids, including those with high viscosity, form a horizontal surface upon spilling� At the same time, dry powders, when poured, form a cone-shaped “mountain” with an angle ϕ (Figure 3�23)�
A single particle stays on a ramp if the component of its weight, mg sin ϕ, is balanced by the friction force mgμ cos ϕ, where μ is the friction coefficient� This establishes a critical tilt angle of the ramp, that is, mg sin ϕ = mgμ cos ϕ or μ = tan ϕ� For many systems, this tilt angle is around 30°� The surface of a powder that has been loosely poured into a vessel may also form a conical tilt� In contrast to a liquid medium, the pressure on the vessel walls may also have a tangential (vertical) component in addition to the normal component� This tangential component corresponds to the Coulomb friction acting along the surface of the vessel (Figure 3�24)� This friction prevents a complete filling of the vessel with a powder�
An important subject that one needs to address here is the role of moisture� Typically, in the case of a complete flooding of powder with a wetting liquid, the internal friction and critical tilt angle are significantly lower than in a dry system� However, at a certain “average” content of the liquid phase, the cohesion between particles and the resistance to shear significantly increase due to the appearance of capillary attractive force (see Chapter 1)�
The specifics of the rheology of powders are completely revealed in the process of their compaction, that is, pressing in a mold in a cylindrical matrix� The axial pressure, P0, acting in a given compact structure results in the appearance of the tangential components of a stressed state, which manifest shear deformation and compaction� At the same time, the compression stresses are acting in directions normal to the possible directions of sliding� According to Coulomb’s law, these stresses oppose shear deformations and compaction (Figure 3�24)� When there are no factors limiting the expansion of a body in the direction perpendicular to the applied uniaxial compression, the tangential stresses are maximal in planes oriented at a 45° angle to the axis� The values of these stresses reach ½P0� Similar normal compressive stresses also act along these planes� These are the conditions that one encounters in the course of pressing a disc or a coin� If there is a resistance from the wall of cylindrical matrix, the stress state also includes uniform compression, and the shear deformations become “jammed�”
In the case of plastic particles, the residual deformations in the structure may occur due to the deformation of the particles themselves and due to the growth in the contact area between the particles� For solid particles, the compaction of powders becomes possible only by the comminution of particles and requires very high stresses� The latter constitutes one of the major problems of powder technology, in particular, that of carbide materials [20]� The problem can be solved by applying vibration�
Let us now provide a quantitative description of the compaction of a powdered material on the basis of the work by Spasskiy et al� [21-25]� These studies deal with tungsten powders compacted in a cylindrical “floating” matrix with two mobile punches with diameters of 2 cm and a cross-sectional area, S = 3�14 cm2� Punch (1) with a mass m1 ~ 400 g is compressed against a powder column with force, F, from 3 to 30 kg by means of a spring (Figure 3�25)� The compaction is determined by the displacement of the punch from the value h0 based on the initial height of the powder column to the current one, h(t),
and further to the final value of h4� The compaction is then calculated as γ = (m/S h∞ρ), where m is the powder mass and ρ is the powder density (19�3 g/cm3 for tungsten)� The second punch (2) is connected to a vibrator� The experiments were conducted with vibration frequencies, f, in the range between 100 and 1000 Hz� The acceleration, a, of punch (2) was measured with a special sensor� The amplitude of the acceleration was chosen between 10 and 100 m/s2� The tungsten powders contained particles between 3 and 10 μm� The mass of the specimen was only 10 g, that is, the tungsten columns were essentially 0�5 cm tall tablets� This allowed one to neglect matrix wall effects�
The parameters of interest were the static compression force F, applied by a soft spring to punch (1), vibration frequency f, the acceleration a, the power of vibrations W, and the compaction achieved γ, as a function of these parameters� The combined data are shown in Figure 3�26�
The experiments conducted allowed one to draw the following conclusions, which define one’s possibility to approach optimal compaction conditions�
First, there is a clear and reproducible nonmonotonous dependence between the compaction achieved, γ, and the vibration frequency, f� It is a priori clear that the optimum vibration frequency, the frequency at which maximum energy absorption takes place, corresponds to a certain resonance frequency in the system, fres� Intuitively, one could assume that this is related to the mass of the punch (1) (400 g) and the stiffness of the compressing spring, ks (reasonably low, around a few kg/mm)� However, direct observations of the oscillations of a weight of a given mass on this spring point to a frequency f ~ 1 Hz, which is far from the realistically optimal frequencies of 100-1000 Hz� Our experiments have shown that, with a high probability, the resonance frequency is determined by the mass m1 and the effective elasticity of the powder structure, k f k mp res p: ( / )( / ) ./= 1 2 1 1 2p Using the solution of the Hertz problem and employing dimensional analysis, one finds that the effective elasticity modulus of the pack of spheres with Young’s modulus E under these special conditions (i�e�, when the area of the contacts is a nonlinear function of the load) is E E Peff µ ( ) ./2 0 1 3 For tungsten, using E = 4 × 1012 dyn/cm2 and for P0 = 5�8 × 106 dyn/cm2, one finds that Eeff = 4�5 × 1010 dyn/cm2, which is much smaller than E� For a given volume of powder in a matrix with the area S = 3�14 cm2 and a column height, h = 0�5 cm, one gets kp = bEeffS/h ≈ b ⋅ 30 × 1010 dyn/cm� Depending on the structure model utilized, the dimensionless coefficient b, describing the degree of structure damageability, may be much smaller than 1� Assuming that b is on the order of 1%, one can estimate the elasticity of the powder structure, kp ~ 3 × 109 dyn/cm� In this case, for P0 = 5�8 × 106 dyn/cm2, one finds fres ~ 430 Hz, which is close to the experimental observations� One can thereby see significant differences in the application of the elasticity of a continuous medium to a partially damageable structure� This difference manifests itself in the parameter b�
Second, the main stage of the compaction in the h(t) curve can be approximated using an exponent (Figure 3�25)� On the one hand, at the extreme compression of the punch (1), the powder is jammed and does not undergo compaction� On the other hand, when F is too small, the powder “swells�”
Comparing the optimal values of the axial compression stresses, P0, with the corresponding resonance frequencies, fres, at a punch acceleration, a = 40 m/s2, one gets a peculiar linear dependence between P0 and fres with a slope of 1�6 × 104 dyn/cm s (Figure 3�27)� The values of P0 in the vibration compaction regime are on the order of a few kg/cm2 instead of hundreds of kilograms or tons per cm2 for static powder pressing�
Third, let γmax be the compaction corresponding to the resonance frequency, fres� With respect to achieving maximum compaction, the dependence of γmax on the amplitude of the acceleration, a, is important� The amplitude of acceleration is indicative of the intensity of the vibration action at a given P0 and corresponding fres� The variation of the punch acceleration at optimal values, P0 = F/S, corresponds to the monotonous increase of γmax (Figure 3�28)� The dynamic component of the pressure can be estimated as Pd ~ am1/S ~ (10-100) × 102 cm/s2 ~ (0�1-1) × 106 g/cm s2 ~ 0�1-1 kg/cm2� This value is by an order of magnitude lower than P0, which also reflects the specifics of the rheology of the partially damaged structure�
Fourth, it is also of interest to obtain estimates of the power used, W� To obtain this estimate, one can turn to the role of the contact interactions and specifically to the dependence of the frequency of the contact rupture on the vibration frequency� For particles with a diameter of 10 μm, one gets the number concentration of about 109 particles per cm2� The number of contacts ruptured per cycle is of the same order of magnitude� Since the work of rupture of an individual contact in a given system is on the average estimated by u1, that is, it is approximately 10−8 erg (see Sections 1�3 and 4�4), for 103 cycles the power is no more than W ~ 103 s−1 × 109 cm−3 × 10−8 erg × 4 × 3 cm3 ~ 105 erg/s ~ 10−2 W� In reality, the power consumed under the optimal conditions is around 0�1 W�
A discrepancy of one or two orders of magnitude would be indicative of the ineffectiveness of the method, that is, of the dissipation of 99% of the power into heat inside the machine and especially into the friction of the particles, that is, dissipation due to the entire spectrum of contact interactions
in the structure� However, in actual applications of vibration technology (e�g�, in filling hoppers, transport in pipes, liquifaction of cement slurries or pulp), even these excessive power amount constitute a rather small contribution to the total balance of power consumption; hence, vibration technology can be regarded as a rather economical one [26,27]�
The crude estimate provided here indicates that the problem of reaching optimal parameters in vibration technology is many sided and needs to be taken into account in solving various engineering problems�
The rheological behavior of a structured thixotropic disperse system depends to a great extent on to which side the equilibrium between the formation and the rupture of contacts is shifted� Since the
rate of contact restoration, which is associated with the Brownian motion of the particles, is finite, a certain time is required for the equilibrium to establish� For this reason, a spontaneous thixotropic restoration of the structure takes place in real time� Due to a complete destruction of the structure, which takes place in region IV, the strength of the system, that is, the yield stress τ*, sharply drops (all the way to zero in the limiting case), and the system exhibits a liquid-like behavior�
When at rest, the system gradually restores its strength, that is, again acquires solid-like behavior� The strength (i�e�, the yield stress, τ*) of a fully restored structure does not depend on the number of structure destruction cycles (Figure 3�29)� The time needed for a complete thixotropic restoration of the structure is referred to as the period of thixotropy, tT� If the rheological measurements are conducted before the dynamic equilibrium between the rupture and the formation of the contacts has been reached, the measured mechanical properties of the system will strongly depend on the structure of the system at the time of measurement, that is, on the extent of the structure destruction at the time of measurement�
In practice, when working with thixotropic systems, one often uses some apparent values of the rheological characteristics, rather than the equilibrium (“true”) ones� The apparent characteristics can be evaluated at a particular time, for instance, after a complete degradation of the thixotropicreversible structure� In some instances, a prolonged constant-rate deformation of the system is required for the equilibrium between the contact rupture and restoration to be established� Such conditions may not be always achievable in the laboratory�
Let us list some examples illustrating the role that the thixotropy of disperse systems plays in nature and technology� The thixotropic properties of bentonite clays are the main reason for the use of bentonite clay suspensions in drilling muds in oil industry� These suspensions behave as ordinary fluids under the normal operation of a bore� One of the functions of the drilling mud is to carry the drilled-out rock to the surface� When there is a need to stop the bore, as in the case when the casing needs to be extended, there is a danger that coarse rock particles will sediment and cause jamming of the bore� The thixotropic properties of the fine disperse clay suspension will result in the formation of a coagulation structure network that will trap the suspending particles and prevent sedimentation� Once the operation of the bore is restored, the coagulation structure formed is rapidly degraded, and the system again acquires liquid-like properties�
The thixotropic properties of flooded clay-based grounds also need to be taken into account in civil construction projects, such as in planning the construction of buildings, bridges, and roads�
The thixotropic properties of pigment structures in oil-based paints provide the paint with the necessary rheological properties� Mixing results in the destruction of the coagulation structure and allows one to apply the paint as a thin layer on the surface� Quick restoration of the coagulation structure prevents the paint from gravity-caused draining downward�
Another area in which the structuring of colloidal systems needs to be taken into account is in the introduction of fillers into various types of polymeric materials� If the task is to obtain a material with high strength and hardness (at the expense of elasticity), it is beneficial to reach a more complete particle packing and use as much filler as possible� To achieve this, one has to prevent the formation of a loose spatial network of particles, that is, it is necessary to weaken interparticle cohesion while maintaining a good cohesion between the particles and the matrix� Since fillers typically consist of polar particles, while the matrix is typically nonpolar, good cohesion is achieved by using chemisorbing surfactants that make the surface of the filler particles hydrophobic� In the case of aluminosilicates, this can be achieved by using sufficient amounts of cationic surfactants�
If the task is to preserve a high degree of elasticity, then one must cut back on the use of filler, so that a loose network of filler particles can be created� This task also requires fine-tuning of the cohesive forces between the filler particles� Indeed, on the one hand, if the degree of lyophilization of the system is too high (weak cohesion between the particles), the filler particles will sediment and a nonuniform material will be obtained as a result� On the other hand, excessive lyophobization (too strong cohesion) will also result in a nonuniform material because of filler caking due to aggregation� It has been demonstrated by Taubman and Nikitina [28] that the ideal structuring of the system takes place under the conditions when the adsorption layer is approximately half of the monolayer� This illustrates the universal role of surfactants in fine-tuning the cohesion between the particles of the disperse phase, as a result of the structural and rheological (mechanical) properties of disperse systems and materials�
The analysis of a full rheological curve indicates how the complex mechanical behavior of the system can be decomposed into a combination of a number of regions, where each region can be described in terms of a simple model utilizing one or two constant parameters, as shown in Figure 3�30� This allows one to use the same model with very different parameters to describe such complex and molecularly different phenomena as Schwedow’s creep and Bingham’s viscoplastic flow� The decomposition of a complex behavior into a limited series of regions described by simpler models is the main technique used in macrorheology for solving complex engineering problems�
At the same time, the understanding of the mechanism of each of these elementary behaviors requires the application of the framework of molecular-kinetic theory and can be characterized as the microrheological approach� To further illustrate the microrheological approach, let us address in detail two particular studies that reveal the role of contact interactions in physical-chemical mechanics�
One such study is the analysis of the nature of elastic deformation, that is, the mechanically reversible behavior of dilute clay suspensions, which find broad use in drilling muds� We will address this subject by presenting a theoretical description of the microrheology of the behavior of a coagulation structure with anisometric particles resulting from the action of small stresses� Under these conditions, the system reveals elastic aftereffect without contact rupture [18]� This phenomenon underlies the ability of clay suspensions to prevent the sedimentation of mineral particles in drilling operation when the bore is stopped�
The second study focuses on the summation of the contact interactions using the estimation of the rheological parameters of a coagulation structure formed with substantially anisometric particles (fibers) [29]�
The rheology of bentonite suspensions has been studied by numerous authors� A peculiar high elasticity (the so-called elastic aftereffect) in the coagulation thixotropic structures that develop in aqueous bentonite suspensions was demonstrated in early studies by Serb-Serbina et al� [30-33]� Under constant shear stress, the strain develops in bentonite suspensions� The strain gradually increases with time and gradually decays, once the shear stress has been removed� Earlier in this chapter,
we explained the entropic nature of these phenomena [18]� The original work by Fedotova et al� [34], focusing on the rheology of dilute (<3% v/v) aqueous colloidal dispersions of bentonite, has shown that the coagulation structures that develop in such suspensions can be spontaneously restored in an isothermal process after degradation, that is, these suspensions reveal thixotropy�
The dependence of the strain rate on the applied uniform shear stress in a steady-state flow reveals several characteristic regions that correspond to different physical phenomena�
In the stress interval between 50 and 500 dyn/cm2,thereis a slow creep regime with a constant Schwedow’s viscosity, η0 ~ 109 poise� This corresponds to a solid-like structure in which the fraction of the ruptured contacts is not yet very large� Under these conditions, the structure is restored in the flow by thixotropy� At higher stresses, a viscoplastic regime with significant rates of deformation is observed (Bingham’s region)� With the increase in shear stress, the effective viscosity decreases to the minimal constant value, ηmin ~ 0�1 poise, which is a characteristic of the viscous flow of a system with a totally destroyed structure� At the same time, at small shear stresses, lower than about 40-50 dyn/cm2, there is practically no creep, and the deformations observed in this regime are completely reversible (by magnitude)� These deformations develop until they reach an equilibrium value corresponding to the magnitude of the applied stress, that is, these deformations are the elastic deformations of elastic aftereffect� Figure 3�31 shows shear deformation, ε, as a function of time for shear stress levels slightly lower than 50 dyn/cm2� The deformation characteristic, ε, corresponds to the “technical deformation,” γ, defined in continuum mechanics, that is, it corresponds to twice the pure
shear value (see Section 5�1 for more details)� This region corresponds to the onset of a rheological curve and is characteristic of a system without any residual strain�
The dependence of the shear strain on time, shown in Figure 3�31, clearly reveals two characteristic stages of the elastic strain: “fast” and “slow�” Both of these strain regimes can be approximated using exponential functions with the corresponding time constants� These exponents show an increase in the strain with time during the action of shear and a time-dependent decay in the strain upon removing the load� During the first “fast” stage the time constant, tf, is on the order of ~ 10−2 s, while during the second, “slow” stage the time constant, ts, is on the order of ~ 102 s� The corresponding limiting equilibrium deformations at a shear stress ~50 dyn/cm2 are εfm ~ 0�1% and εsm ~ 0�3%, respectively� The elasticity modulus value is thus ~104-105 dyn/cm2�
A qualitative explanation of this peculiar behavior of bentonite suspensions observed even at very low volume fractions of the disperse phase is as follows� The reversibility of the strain under conditions when there is none of the “normal” elasticity typical for a solid (elasticity modulus ~ 1010-1012 dyn/cm2) can be explained by the changes in the configuration entropy of the system due to changes in the mutual orientation of the particles� This description of the elasticity is similar to the description of the elasticity in polymer systems, in which it is explained by the statistics of the polymer chain configuration [35]� The fast stage of the aftereffect may be due to the particles turning and rolling without any translational motion of the particles with respect to each other� The slow stage may be related to the further coorientation of particles with some sliding motion, resulting in a translational motion of the points of the coagulation contacts along the particle surface�
The processes involved in the elastic aftereffect can be described by a simple rheological model consisting of two Kelvin’s elements connected in series, as shown in Figure 3�32a� It is worth emphasizing here that this model is applicable only in the region of low shear stresses, below the onset of Schwedow’s creep� From this model, one gets the following values for the slow and fast elastic strain moduli:
G
G
dyn cm
. dyn cm
= = = ´
= = ´
= ´
t e
t e
50 10
5 10
50 3 10
1 7 10
/
/
The superposition of Kelvin models is suitable here only for the analysis of the equilibrium states and for the estimation of the aftereffect time constants, while the kinetics of the development of the strain and the kinetics of the relaxation are considerably more complex and can’t be described in terms of this model�
Both “fast” and “slow” strains have the same entropic nature, for which reason it is possible to introduce a single “common” elasticity modulus, Gel, corresponding to the net equilibrium deformation (Figure 3�32b):
e e em fm sm= + » + »0 1 0 3 0 4. . . %
mainly
Gs
. dyn cm= » ´
» ´- t
e 50
4 10 1 25 103
The deformabilities at these two stages, which corresponding to the different viscous elements, ηf and ηs, are different and amount to ~¼ and ~¾, respectively (Figure 3�32b)� Since the time constant of the elastic aftereffect is defined as the ratio of the viscosity to the corresponding elasticity modulus, the two corresponding viscosities are
h h
poise= ´ » ´ ´ » ´ = ´ » ´ ´ » ´
-G G
t
t
5 10 10 5 10 1 7 10 10 1 7 10
4 2 6 . . poise
Consequently, at the stage characterized by “fast” elasticity, the effective viscosity of the structure formed exceeds the viscosity of the dispersion medium (water) by a factor of 104-105� Where the elastic deformation is “slow,” the ratio of the structured system viscosity to the viscosity of the dispersion medium is ~108�
The estimated microrheological parameters corresponding to the behavior of disperse systems subjected to small shear stresses may be described quantitatively utilizing the concepts of the microprocesses taking place in the thixotropic coagulation structures� In the section that follows, we will provide such a quantitative analysis�
3.2.1.1Elasticity Modulus Let us examine a structured system under the condition of uniform shear at constant temperature, T, and constant shear stress, τ, which, in this case, acts as pressure, p�The behavior of such a system can be described using the thermodynamic potential (per unit volume), Φ(T, τ) = U − ST − τε, where the deformation ε = ε(T, τ), and the τε term is positive when the work performed over the system is positive� Neglecting the true elastic deformations, which are very small at these low stresses, and focusing only on the elastic deformations associated with the change in particle configuration, we may state that U = const, S = S1 = S2(ε), where S1 is independent of the deformation ε, and S2 = S2(ε) is the “configuration component” of the entropy� Furthermore, ε = 0 and S2 = 0 at τ = 0�
At constant values of T and τ, the equilibrium in a given system corresponds to the minimum of the potential Φ� The strain ε is then the only variable parameter, that is,
¶ ¶
æ è ç
ö ø ÷ = -
¶ ¶
æ è ç
ö ø ÷ - =
F e e
t tT
T S ,
2 0
Since ε and τ are of the same sign, the entropy should decrease with an increase in the strain� The required condition that an extremum is a minimum is that
¶ ¶
æ
è ç
ö
ø ÷ = -
¶ ¶
æ
è ç
ö
ø ÷ >
2 0 F e e
T S
which indicates that the entropy decrease with increases in strain must be at least as steep as –ε2�
Let us model a coagulation structure by using a system of identical anisometric particles with a linear size δ� For strongly anisometric particles (platelets or rods), let δ = l, where l is the largest size� For particles that are not strongly anisometric (ellipsoids), let δ = l − d, where l and d are the largest and the smallest linear dimensions, respectively�
The equilibrium state of the system in the absence of shear stresses is characterized by a chaotic arrangement of the particles, and the mean projection of the particle dimension, δ, on any direction has the same value:
z
N ziav =
æ è ç
ö ø ÷ =
1 3 2 d
where N is the number of disperse phase particles in a unit volume of the system� Indeed, in the θ − ϕ coordinate system, the projection of particle dimension onto the polar axis
is z = δ sin θ, as evident from Figure 3�33� Within the element of the solid angle dω = cosθ dθ dϕ, the number of particles, dN, equals (N/2π) cosθ dθ dϕ. We restrict this consideration to a single hemisphere, that is, 0 < ϕ < 2π and 0 < θ < π/2�
The mean projection of the sizes of all the particles on the selected axis is
z
N N
N N
av sin sin= æ è ç
ö ø ÷ =
æ è ç
ö ø ÷ æ è ç
ö ø ÷ =ò òò1 1 2 2d q
d p
q q qd d dcos f d
The fraction of the particles in the interval dz = δ cos θ dθ, that is, within the “belt” dω’ = 2π cos θ dθ, is
dN N
f z z f z= æ è ç
ö ø ÷ = ( ) = ( )
1 2
2 p
p q q d q qcos cosd d d
Consequently, the distribution function can be written as f(z) = const = 1/δ, and the dispersion of z can be written as
s d
d d d
z z z f z z z z2 2 0
2 1
12 = - ( ) = æ
è ç
ö ø ÷
-æ è ç
ö ø ÷ =
æ è ç
ö ø ÷ò ò( )av d d
Let us now assume that, due to the stress applied along the selected polar axis, a partial coorientation of the particles has taken place (i�e�, the system acquired “texture”)�As a result, the value of the particle size projections mean on that axis has increased� To estimate the probability of this state, we will use the approach developed by Yushenko on the basis of suggestions offered by Kolmogorov� Since the projection z of each particle is a random number, the Lyapunov central limit theorem can be applied to the system� Then the distribution around the mean value of the projection, zav, is distributed normally with a mean, μ = δ/2, and a dispersion, s s d2 2 2 12= =z N N/ / . The probability of a deviation of a given mean of a projection on a given axis from the most probable value of the mean, δ/2, in the absence of coorientation is given by the function
f z z Nav av/ / / /( ) = ( ) -ùû éë -( ) =ùû éë[ exp ( ) ]
/ / /1 2 2 61 2 2 2 1 2 1 2p s m s p d exp {( ) [ ( / )] }- -( )3 2 2 2 2/ / /avN z d d
At the same time, this deviation has the physical meaning of a quantity close to the value of the relative strain in the system:
( ) ( )
zav / /
- »d d
e2 2
The case of a strong anisometry of the particles is assumed in this case� If the anisometry is weak, it is possible to assume that the strain, ε, is a function of the characteristic describing the degree of anisometry, which in the simplest case is a liner function:ε~(δ/l)(zav − δ/2)/(δ/2), where l is the maximum linear dimension�
The probability of deformation, ε, due to the coorientation of the particles is given by the distribution function [( ) ]exp {( ) }/ /6 3 21 2 1 2 2/ / /p d eN N-, and the change in the natural log of the probability upon the transition from the state corresponding to ε = 0 to a state with some particular value of ε is given by ΔlnW = −(3/2)Nε2� This corresponds to a change in the system entropy, ΔS2 = −(3/2)Nkε2� Along the equilibrium curve ε = ε(τ) corresponding to the minimum of the thermodynamic potential Φ, one has τ = −T(∂S2/∂ε) = 3NkTε, which yields the equilibrium elasticity modulus as
Gel k= =
t e
3N T
The particles that make the colloidal suspension of bentonite are small platelets with a maximum linear dimension, l ~ 100-200 μm, and having thickness, d ~ 10 μm� The number of particles per unit volume in a 3% dispersion is
N C
l d » » -( )2
17 310 cm
The estimate for the equilibrium elasticity modulus at room temperature is then Gel ≈ 3 × 1017⋅4 × 10−14 = 1�2 × 104 dyn/cm2, which agrees well with the order of magnitude of the experimentally determined value of 1�25 × 104 dyn/cm2� This agreement suggests that the proposed views on the mechanism of the reversible deformations are valid�
3.2.1.2Viscosity of the Elastic Aftereffect The analysis of the elastic aftereffect also requires that a particular model be introduced�Let us first examine the first stage of the elastic aftereffect: the “fast” elastic strain related to the appearance of the viscosity, ηf ~ (104-105)ηw, where ηw is the viscosity of water�
Figure 3�34a and b shows the scheme of deformation of coagulation structure cells� This strain is the result of mutual turning of particles (platelets) without the sliding motion of points of contact� It is assumed here that the contacts can be viewed as “ideal joints” and do not require that any work be expended in turning the particles� The mutual turning of the particles results in their coorientation, that is, in an increase in the mean value of the projection on a given axis� It is clear that within the framework of this idealistic model, there is no reason for a substantial increase in the effective viscosity of the system relative to that of the immobilized water forming the dispersion medium� Indeed, the deformation of each cell is caused by the same uniform shear as that which causes the strain of the whole system�
Here, we are not going to consider the possibility of a viscosity increase in small volumes of liquid resulting from the contact of the liquid with the particle surface� While this factor requires a separate discussion and analysis, it still can’t explain the high effective viscosity of the system, which exceeds that of water by several orders of magnitude� This becomes clear if one examines the thixotropic reversibility of the coagulation structures� Indeed, a complete destruction of the structures results in a substantial drop in the viscosity� The resulting viscosity is not too different from that of a dispersion medium (e�g�, within one order of magnitude of the latter) and agrees with Einstein’s viscosity law� The mean size of the dispersion medium “cells” remains approximately the same, that is, on the order of 100 Å� This means that the total volume of the dispersion medium is in fact a thin boundary layer with a thickness of several tens of Å� If the main source of the viscosity increase had been the interactions between the dispersion medium layers and the neighboring solid phase, the destruction of a structure would not have been expected to cause a strong decrease in the viscosity�
If the shear ε occurs over a time t with the average rate of dε/dt = ε/t, the estimate for the work of the viscous forces per unit volume is η dε/dt ε= ηε2/t, and for a cell with a volume on the order of about l3, it is
W l
t ~
The natural resolution to an effort to understand the issue of increased viscosity is the consideration of a real chaotic structure of particles in which the deformation of the cells is strongly influenced by the flow of the dispersion medium from one cell to another via relatively thin bridges (Figure 3�34c and d)� By utilizing this approach, we can state that the shear, ε, causes a change in the cell volume approximately by l3ε and consequently results in the flow of water, amounting in volume to V = βl3ε, between the neighboring cells� Here, β is the coefficient on the order of several units, which takes into account the microscopic process of the flow of multiple elementary volumes of water between the neighboring cells in a chaotic, “nonideal” particle structure� If the flow takes place in a channel having thickness a, width b, and length c (Figure 3�35), the average flow rate is wav = βl3ε/abt� The distribution of the flow rates is then given by a parabolic law
w x p
c
a x
a ( ) = æ
è ç
ö
ø ÷ æ è ç
ö ø ÷ -
æ è ç
ö ø ÷
é
ë ê ê
ù
û ú ú
1 2 2
1 1 2
2 2 D hW /
where Δp is the pressure drop along the channel length, c� It can be shown that
Dp c
a w
c
a l abt= =é
ë ê
ù
û ú
é
ë ê
ù
û ú3 1 2
3 1 22 2
3h h b eW av W/ / /( ) ( )
and the dissipation of the energy in the channel is given by
W p l c ba l tf W= =D b h b e 3 3 3 212 1e ( / )( ) /
Since we have shown that Wf ≈ ηfε2l3/t, the effective viscosity of the “fast” elastic deformation is
h h
ba læ
è ç
ö ø ÷
Assuming that the thickness and length of the channels are approximately the same as the thickness of the particles (platelets), and that the width of channels approximately equals the maximum particle diameter, a = c ≈ d, b ≈ l, one finds that
h h
= æ è ç
ö ø ÷12
d
If the coefficient β is on the order of a few units and the ratio l/d ~ 10 (or several tens), the aforementioned expression indicates that the viscosity of the fast elastic deformation should exceed the viscosity of water by a factor of 104-105, which agrees well with experimental observations�
It is necessary to emphasize here that the estimate for ηf is still rather crude� The aforementioned relationship does not include in any direct form the concentration of the dispersed phase� However, this concentration is dictated by the same model and amounts to several particles (n) per volume βl3, where β ≈ n� Consequently, the concentration of the dispersed phase can be assumed to be equal to nl2d/βl3 ≈ d/l, which is on the order of 10−1 or somewhat less, which again agrees well with the experimental conditions�
The described method of the viscosity estimation for the “fast” stage of the elastic aftereffect can be further extended to obtain an estimate for the much higher viscosity of the second stage of the elastic aftereffect� In this case, it is assumed that the particle orientation changes owing to a small-scale sliding of the points of contacts along the particle surface without the destruction of the majority of the contacts� To a degree, this sliding process can also be described using the flow of viscous fluid in a gap, but the thickness of the gap needs to be much smaller than that in the case described earlier�While in the previous case, we assumed the gap to be on the order of the thickness of the platelets, d ~ a ~ 10Å (or slightly larger), in the present case, the gap thickness is on the order of several angstroms, that is, a = θd, where the coefficient θ is around 0�1� Substituting this value into the equation for the effective viscosity, one obtains the following result for the viscosity of the “slow” elastic aftereffect regime:
h h
b q
2 æ è ç
ö ø ÷ æ è ç
ö ø ÷
l d
which is three orders of magnitude higher than the viscosity of the “fast” stage and is close to the experimentally determined viscosity of ~106 poise� However, the latter estimate requires further accounting for the changes of the fluid properties in the gap, since the gap thickness is comparable to several molecular dimensions� Furthermore, additional analysis is also needed to understand the ratio of the strains at the “fast” and “slow” stages, εfm:εsm ∝ 1:3�
In the next section, we will describe the approximate microrheological estimation of the effective viscosity of a fibrous suspension in a steady-state thixotropic regime carried out by Shchukin [29]�
The system of interest here is that of a slurry containing fibrous matter with randomly crossing fibers� A common and well-studied example of such a system is aqueous slurry of cellulosic fibers, that is, papermaking pulp� We will present a model that allows one to estimate the number of contacts between the fibers and to provide an estimate for the mechanical properties of the structure� We will also illustrate the impact of common papermaking coagulants (e�g�, polyethyleneimine [PEI] and quaternary alkyl ammonium salts) on the macroscopic rheological properties of pulp suspensions� These studies have an important practical application, as they provide an insight into the control of the rheology of cellulosic slurries at different stages of papermaking� The rheological
behavior of papermaking furnish undergoes drastic changes throughout the papermaking process� The slurry must behave as a liquid at the point of entry from the paper machine head box to the wire and must display a fast increase in the effective viscosity and the critical yield stress upon dewatering and forming a sheet�
In the description of the simplest models of globular disperse structures, that is, suspensions, porous media and powders consisting of isometric (spherical) particles, one needs in reality only two parameters to describe the system: the average particle radius, r, the dimensionless porosity, Π (cm3 of pores/cm3 of porous body) or conversely the continuity of the solid phase, Vs = 1 − Π, cm3� Alternatively, one can also operate with the mean coordination number in a chaotic structure or the mean knot-to-knot number of particles in a model of crossing chains [36]� The two parameters allow one to determine the number of contacts between the particles per unit cross-sectional area, χ� Multiplying the parameter χ by the mean contact strength pav allows one to obtain an estimate for the strength (i�e�, the critical shear stress), Pc = χpav� Numerous experimental studies frequently referenced throughout this book have proven that such a summation is indeed a valid approach�
One can utilize various models for modeling fibrous structures� In the case of a system composed of long elastic entangled threads and fibers (fabrics and felts), the own strength of fibers becomes a determining factor in the strength of the entire structure� For hard fibers that are not too anisometric (linear dimensions ratio up to 100), the model based on the summation of the contact interactions may still be valid� However, in describing the structure and calculating the number of contacts, the use of the two-parameter model is no longer possible, and one needs at least three parameters� These parameters may be the volume fraction of the solid phase, Vs, the fiber length, l, and the fiber diameter, d = 2r� In the following text, we will present the model for calculating the number of contacts between the fibers and compare those results with the results of experimental studies on the cohesion of cellulosic fibers and the rheology of pulp suspension [37-44]�
Let us consider the uniform deformation of a unit volume of a disperse system under the condition of a steady-state viscoplastic flow or the “slow” stage of the elastic aftereffect under the action of a shear stress τ� The system consists of fibers of length l and diameter d occupying the volume fraction Vs� The contact between the crossed fibers can be characterized by the average tangential friction force (which in turn characterizes the mean force of resistance to the shear motion in the contact), ptg� The number of particles, ν, per unit volume of the disperse system is
n
p = 4 2
V d l
The mean distance between the fibers (distance between their centers) in a given direction (x, y, or z) can be estimated as
b b l= -n 1 3/ ,
In a cylindrical volume with radius b and length l, the approximate number of neighboring fibers next to a given fiber, νc, is given by
n p n pn pc » = =b l l
l b
If the fraction α of these neighboring fibers is in contact with a given reference fiber, one can write for the number of contacts along the reference fiber, nc,
n lc =
1 2
where the numerical coefficient ½ takes into account the fact that the contact belongs to both fibers in contact� The number of contacts per unit volume of the system, n, can be estimated as
n n l= =cn apn
1 2
The most complex step here is the estimation of the contact cross section layer thickness, that is, the thickness of the layer in which the destruction of contacts take place upon the destruction of the structure or conversely of the layer in which the contacts together form the basis of the resistance to shear, that is, the net friction force acting under conditions of steady-state flow� In the two limiting cases, this thickness is either l or b, while in the general case, it is a combination of l and b with the units of length, that is, lkb1−k, and 0 ≤ k ≤ 1� It is then possible to calculate the number of contacts per unit area in the contact section as
2 /
Assuming that k = 1, one can write
c apnl nl l= =
1 2
for k = 0, one has
c apnb nb l= =
1 2
The ratio of χ values corresponding to these two limiting cases is l/b = lν1/3 times� If direct measurements of the molecular component of the cohesive force between crossed fibers
yield an average value of pn and independent measurements yield the value of μ for the friction coefficient, the contact friction force is
p pntg = m
And the macroscopic resistance to shear is given as
1 2
The term “molecular component” implies that the normal component of the load (i�e�, the compression in the contact) is absent� It may appear strange that in the final expression for χ, there are only two and not three components� The reason for this is that within the framework of the model of rigid thin fibers, the smaller dimension, d ≪ l, was neglected in the process of counting the number of contacts� This smaller dimension has been used only in the transition from the concentration of the disperse phase, Vs, to the number of particles per unit volume, ν�
Let us further utilize the literature values of Vs, l, d and the contact cohesive forces [39-43]� The length l of softwood and hardwood fibers used in the experimental studies was 1-2 mm and 20 μm, respectively�The consistency of the pulp suspensions was 1% and 2%�The experimental rheological studies conducted in the elastic aftereffect regime were performed with the constant shear stress ranging between 10 and 90 Pa� From the experimental data for 2% softwood pulp (l = 2 mm), the shear rate
in the uniform shear, dγ/dt ≈ 10−4 cm−1 s−1 at τ = 20 Pa, which yields a macroscopic viscosity, η ~ 2 × 105 Pa s� Assuming that the thickness of the contact layer equals l, for α = 1, one finds the following values: ν ~ 3 × 104 cm−3, b ~ 0�03 cm, and l/b ~ 7, nc ~ 10 and n ~ 3 × 105 cm−3, χl ~ 6 × 104 cm−2�
According to the published data [42], the molecular component of the friction force in the individual contact between two fibers, F = ptg = μpn in the presence of PEI coagulant is on the order of 10−3 dyn� For the shear stress, τ, this yields τ = χl ptg~6 × 104 cm−2 × 3 × 10−3 dyn = 180 dyn/ cm2 = 18 Pa
This value agrees reasonable well with the experimentally determined value of 20 Pa, despite some apparent approximations, such as with regard to the homogeneous flow of the pulp suspension�
Earlier experimental studies [37-39] indicate clear trends in the influence of the surface active agents (PEI, quaternary alkyl ammonium salts, etc�) on the rheological properties of pulp and on the strength of the individual contacts between fibers� This has already been discussed in detail in Chapter 2� In both cases, the dependence of these parameters on the additive concentration passes through maxima, the location and depth of which depend on the type of the additive and the type of the fiber� It has been demonstrated that such dependence is related to the effect of surface active additives on the surface charge of the fibers and the possibility of generating patches of opposite charges on the fiber surface� It has also been suggested [29] that the fiber surface can be made both hydrophilic and hydrophobic, depending on how much of an additive has been adsorbed� The reader is referred to Section 2�3�1 for a detailed discussion on the effect of polyelectrolytes and surfactants on the cohesive forces between the fibers�
A significant increase in the pulp mobility (liquification) and homogenization can be achieved by vibration [27,44]� According to the published data [44,45], in a rotational setup operating at a rate of 3000 rev/min, the viscosity of a 15% pulp slurry was about 10 Pa s with the power consumption on the order of a few watts per cm3� It is of interest to estimate the fraction of consumed power that has been used in overcoming the friction forces in the contacts between fibers�
To estimate the number of particles and the number of contacts per unit volume, ν, let us employ the approach described earlier� Even if we assume that all the contacts are ruptured with every half-period of vibration (i�e�, 100 times/s), we may underestimate the frequency of the fiber collisions� The rate s of the relative fiber motion reported in [44,45] was on the order of 10 cm/c� With the average distance between the particles being b = ν−13 (in the present case b ≈ 0�016 cm), this would correspond to the frequency of the formation and rupture of each contact equal to ω = s/b ~ 10 cm/s 0�016 cm ~ 103 s−1�
This estimate yields the maximum number of the collisions between the fibers� Then the number of fiber collisions and the number of ruptured contacts per unit volume of the disperse system is nω (cm3/s)�
Let us assume that the work, u, needed to rupture the contact is predominantly the work that needs to be spent to overcome friction upon moving the contact point along the fiber by a distance l/2 on average� This elementary work is then u ≈ ptgl/2� The power U dissipated by all contacts in a unit volume can be estimated as U ~nωu~n(s/b)ptgl/2 ~ 106 erg/cm3/s� This value represents the highest possible estimate, and it is an order of magnitude lower than the actual consumed power, N. This large discrepancy between the estimated and the real power consumption can be explained by the cyclic processes of flow (turbulent) of the dispersion medium between the neighboring small volumes (“flocs”) of the pulp slurry� This flow of the dispersion medium is characterized by a high effective viscosity, ηeff ~ 104ηw, as is schematically illustrated in Figure 3�36� The flocs with a linear size on the order of 10l were indeed observed in the experiments� Contact interactions are responsible for the formation of flocs upon the action of vibration� In order to explain the observed high value of the effective viscosity, we can refer here to the model of the rheological behavior of clay suspensions (see Section 3�2�1): the flow of the dispersion medium between neighboring microscopic volumes yields a very similar estimate for ηeff�
