Rheological Aspects

Although there is a close relation between the deformation of solid materials and the flow of fluid materials, there are several rheological aspects that deserve separate attention. In this chapter we first indicate where rheology is of importance and thereafter discuss a few of these aspects, starting with a recapitulation of the basics of rheology. Thereafter we deal with the influence of particles, thickeners, and curing on the flow behavior.

As during application coatings are for a large part fluid, it may be useful to point out the influence of the rheological behavior for coatings in general. This relates to transport, application methods, film formation, and final appearance. Let us address these aspects briefly in turn.

Although not of major concern for the coating technologist or scientist applying a coating, the first aspect is that paints, after being *formulated*, have to be produced and transported. This requires not too high viscosity levels in order to be able to pump the paint from one vessel, for example, the supply vessel at the producer's plant, to another, say, the tin to be used by the professional or do‐it‐yourself painter.

Once the tins have arrived at the painter's place, the paint has to be applied. Each application method is linked with a certain shear rate, requiring an appropriate viscosity (behavior) to be able to realize a proper application, and therefore the required viscosity behavior is strongly related to the choice of the application method.

Once applied, the coating material should be able to form a decent film on the substrate, while the drying (reaction) time should not be excessively long. The leveling of the wet paint under the influence of the surface tension, but counteracted by the (changing) viscosity, should lead to a smooth film, preferably of uniform thickness and without defects. The rheology is also important for effect coatings containing flakes, for example, in automotive coatings. During the (high shear rate) spraying, the viscosity should be low so that the flakes can orient, but as soon as the sprayed droplets reach the substrate (thereafter experiencing a low shear rate), the viscosity should be sufficiently high so that the flakes cannot disorientate anymore.

Finally, the appearance of the dry film, that is, their color and the variation in gloss, is strongly related to the quality of application method and the rheological behavior of the wet paint. This relates not only to defects deteriorating the function of the film but also to defects mainly affecting the appearance. In particular, surface irregularities on a larger and smaller length scale are easily detected by the naked eye and may lead to rejection of the coated product. These defects are discussed in Section 11.1.

In rheology one measures the mechanical response of a liquid sample as a function of shear rate and possibly of temperature *T*, that is, the *complex viscosity**η** = *η*′ − i*η*″ with as real part *η*′ and as imaginary part *η*″ (i = √−1). One also uses the *shear viscosity**η* defined by and the *kinematic viscosity**ν* = *η*/*ρ*, where *ρ* is the mass density. The units involved are Pa s for *η* (or the still often used unit centipoise, 1 cP = 10^{−3} Pa s) and m^{2} s^{−1} for *ν* (or the older unit centistokes, 1 cSt = 10^{−6} m^{2} s^{−1}). For many fluids is almost identical to |*η**| ≡ (*η*′^{2} + *η*″^{2})^{1/2}, an empirical finding known as the *Cox–Merz rule*, but the rule is usually not reliable for complex structured fluids. Furthermore we will need the *differential viscosity* d*τ*/d, the “*zero*” *shear rate viscosity**η*_{0} and the “*infinite*” *shear rate viscosity**η*_{∞}. As indicated in Section 9.2, the formal theory for *η** is similar to that for the complex modulus *G**, and Section 9.2 should be read in conjunction with the present. In particular we recall that *η** ≡ *η*′ − i*η*″ = *G**/i*ω* ≡ (*G*″ + i*G*′)/*ω*. The various application processes of coatings have rather different typical shear rates, as shown in Table 10.1, rendering basic knowledge of rheology a must. A classic on paint rheology and dispersion is the book by Patton [2], brief reviews of rheology for coatings are given by Berker [3] and Eley [4], while Macosko and Tanner provide detailed treatises [5, 6]. We also particularly recommend the book by Larson [7]. Finally, we note that the relation between structure and rheology is often not very clear (sometimes vividly formulated as: morphology by rheology is theology), and this has led to the frequent use of empirical constitutive equations.

Table 10.1 Typical maximum shear rates for different coating processes.

Source: Bieleman 2000 [1]. Reproduced with permission of John Wiley & Sons.

Process |
(s^{−1}) |
Process |
(s^{−1}) |

Sedimentation | 10^{−6} to 10^{−4} |
Brushing | 10^{2} to 10^{4} |

Leveling due to surface tension | 10^{−2} to 10^{−1} |
Spraying | 10^{3} to 10^{6} |

Sagging due to gravity | 10^{−2} to 10^{1} |
Pigment dispersing | 10^{3} to 10^{5} |

Dipping | 10^{0} to 10^{2} |
Transfer of inks by rolling | 10^{4} to 10^{6} |

When *η* is a constant, independent of , the fluid is called a *Newtonian fluid*, and we have (Figure 10.1a). The simplest extension to non‐Newtonian behavior is a description where *η* depends only on the current value of , often labeled as *generalized Newtonian* (GN) behavior. For the *power law model*, the viscosity *η* is given by

10.1

with *m* and *n* positive constants, labeled as *consistency index* and *power law* (or *flow*) *index*, respectively. For *n* < 1 we have *shear thinning*, and the fluid is frequently called *pseudoplastic*, while for *n* > 1 we have *shear thickening*,^{1} and the fluid is often denoted as *dilatant* (Figure 10.1b,c). Strictly speaking, the latter designation indicates that the material shows an increase in volume when shear is applied. Most, but not all, shear thickening systems are dilatant. In a ln *η* versus ln plot, power law behavior yields a straight line with slope *n* − 1, which is indeed typically observed at high shear rate. The power law model is often inadequate at low shear rate where shear thinning (or thickening) is limited and, moreover, has no intrinsic time constant. If we use the *truncated power law model*

10.2

the parameter serves as a value above which significant shear thinning sets in and acts as an intrinsic time constant. If we describe the fluid by the *Ellis model* or the more general *Meter model*, that is, by

10.3

there is no need to introduce a somewhat artificial cutoff value like *η*_{0} for the truncated power law model, but the shear rate dependence becomes implicit via value of *τ*. The parameter *τ*_{1/2} is the value of the shear stress for which *η* = *η*_{0}/2, while the intrinsic time constant for this model is given by *η*_{0}/*τ*_{1/2}. Another rather general model is the *Yasadu–Carreau model*:

10.4

with all constants positive. The parameter *λ* is a time constant, while the parameter *b* characterizes the slope of the flow curve, similarly as *n* in the power law model. The width of the transition from Newtonian to power law behavior is determined by the parameter *a*. There is frequently no need to introduce the parameter *η*_{∞}, except for dilute solutions. Setting *b* = −1, the *Cross model* is obtained that still can cover both Newtonian and power law behavior.

Further complexity results if we add a yield strength *τ*_{0}, and such a material shows *plastic behavior* that can be described by, for example, a *Bingham model* (; Figure 10.1d) or a *Herschel–Bulkley model* (). In the latter case, *n* = ½ is frequently taken. For this type of behavior, it is sometimes argued that the required measuring time is merely too long for low shear stresses (or that instrument limitations do not permit such measurements). Hence one often extrapolates the linear part of the viscosity curve to the intercept with the stress axis and calls the intercept the yield value. In this connection a frequently used description is the *Casson equation* reading

10.5

where *η*_{∞} is again the viscosity at infinite shear rate and *τ*_{0} acts as yield value. The parameter *n* characterizes the sharpness of the transition between both regimes for which *n* ≅ ½ is often found. Figure 10.1 depicts the various types of behavior. Still further complexity results if *viscoelastic behavior*, as discussed in Section 9.2, is included. In this case the behavior becomes time dependent. If for a shear thinning system *η* with decreasing is higher than *η* with increasing , the system is (positive) *thixotropic* [9] and *τ* as a function of shows a hysteresis loop (Figure 10.2a). Negative thixotropy is much less common. Note that not all shear thinning systems are thixotropic. For large values of , *η* decreases with time, while for small values of , *η* increases with time (Figure 10.2b).^{2} A characteristic for thixotropy is the area of the hysteresis loop or the change in slope in a Casson plot (Figure 10.2c). Both are dependent on the prior shear history, rate of shear applied, and the time between measurements. The phenomenon may be reversible, partly reversible, or irreversible (common in food products).

Viscosity measurements rely on a variety of driving forces. For low viscosity fluids one uses gravity, as cleverly applied in a *capillary* (or *Ostwald*) *viscometer*, and measures the time required for a known amount of liquid to flow through a capillary tube. The technique permits measuring kinematic viscosities in the range of 10^{−7} to 10^{−1} m^{2} s^{−1}, but only for Newtonian fluids. However, the accuracy obtained is high, in particular when the instrument is calibrated with liquids of known viscosity. For high viscosity fluids one uses an external pressure to obtain a similar result. For non‐Newtonian liquids viscosity is usually measured using an externally driven rotation as implemented in a *rheometer*, in which a measuring head rotates (oscillates) at a certain distance with respect to another, fixed head with the fluid between the heads (Figure 10.3a). These heads can be both plates, a cone and a plate, or two concentric cylinders with a small gap between them, labeled as plate–plate, cone–plate, or Couette viscometer, respectively. Plate–plate viscometers measure the average flow behavior covering a range of shear stresses and need calibration. For a cone–plate viscometer, the angle of the cone takes care that the shear stress is independent of the distance to the center of the cone, and, hence, the absolute value of *η* is determined directly. One either varies the strain rate and measures the resulting torque or varies the torque while measuring the strain. The latter mode is generally advantageous for very low shear rates. A cone–plate viscometer is the instrument of choice for research on the flow behavior of coatings in view of its versatility (wide range of measurable viscosity, absolute measurement of viscosity, relatively easy temperature, and atmosphere control). Couette viscometers are mainly used for low viscosity liquids. Another viscometer using mechanical rotation is the *rotating disk rheometer* (in the jargon often called a Brookfield, after one of the producers of these instruments) in which a disk connected to a dynamometer is rotating through the fluid. Measurements should be made in a container of the same dimensions as that in which the calibration is done with the same depth of immersion. As calibration is usually done with a Newtonian fluid, the results are nonreliable for complex fluids. Related is the *mixing rheometer*, essentially a mixer in which the fluid is subjected to intense mixing by dual rotors in the form of sigma‐shaped blades in a relatively small space. Heat generation can be considerable for high viscosity fluids. Relatively simple is the *bubble viscometer*, measuring the rate of rise of an air bubble through a liquid in a tube. As various tube diameters can be used, the range of application is about 10^{−5} to 10^{−1} m^{2} s^{−1}. Finally, we mention *efflux cup viscometers* in which the time required for a known amount of liquid streaming through an orifice is measured. Both techniques are only useful for Newtonian liquids. Although simple to use, results of efflux cup viscometers are nonreliable as the result is determined by the overall behavior of the fluid, and consequently, their use is not recommended.

The dynamic oscillatory test method is in principle very suitable for characterizing the drying process of paint. However, the geometry of a setup such as plate–plate or cone–plate hinders the evaporation of solvent for solventborne coatings, which reduces the drying rate, and access of water and/or oxygen, which frequently accelerates the crosslinking rate. To describe the drying process, suppose that we apply on a surface a layer of a traditional paint, containing approximately 50% volatile solvent, 50% *binder*, and a small amount of crosslinking catalyst, in this connection often called a *dryer*. The binder itself, a low molar mass polymer, is typically a highly viscous liquid at room temperature. Any solid character developed during drying is due to crosslinks between the polymeric molecules. Schematically the development of *consistency* can be represented as shown in Figure 10.3b. The first interval after application indicates that the solvent is evaporating (*physical drying*). At the end of this interval, we have the undiluted binder, that is, a highly viscous liquid with virtually no elasticity. After some time the crosslink reaction(s) starts to generate large polymeric molecules, and eventually a space‐filling network results (*chemical drying*). This second stage is characterized by a steep rise of the *overall* consistency but, more importantly, also a change from liquid to solid nature. This transition also encloses the rubber state; no plateau is visible as the crosslink density continuously rises until eventually the glassy state is reached and further crosslinking becomes virtually impossible due to a lack in mobility. In reality physical and chemical drying are not that well separated in time. Whether in reality the final *T*_{g} is above or below room temperature depends on the details.

To obtain usable coatings, good leveling after application is essential, and for achieving optimal rheological characteristics, *rheology modifiers* are added. In the following two sections, the basic effects of the addition of dispersed particles and completely dissolvable polymers on the viscosity of a simple liquid are dealt with.

Adding solid particles (fillers) to a fluid, we obtain a suspension. The processability of a suspension is dependent on the shear viscosity *η* as a function of shear rate and temperature *T*. The viscosity of suspensions depends also on the volume fraction of dispersed material *φ* as well as the size (distribution) of the particles.^{3} For suspensions one frequently uses the *relative viscosity**η*_{r} *= η*/*η*_{s}, where *η*_{s} is the viscosity of the pure liquid. Also used is the *intrinsic viscosity* defined by [*η*] ≡ (*η*_{r} − 1)/*φ* for *φ* → 0.

When particles are added to a liquid, the flow of the liquid in the suspension is disturbed by the presence of the particles. This leads to an additional dissipation of energy or, equivalently, to an increased shear stress at a fixed shear rate. For a suspension with a low volume fraction of hard spherical particles *φ*, typically below *φ* ≅ 3%, Einstein [10] derived

10.6

For this low value of *φ*, the number of encounters between individual particles is negligible, and the increase of the viscosity is exclusively due to the particle–liquid interactions (Figure 10.4a). For rods with radius *r* and length *l*(*l*/*R* ≫ 1), the factor *K*_{1} becomes *K*_{1} = *l*^{2}/π*r*^{2} [11]. Hence, for large *l*/*r*, the viscosity increases much more rapidly as for spheres. This is attributed to rotation of the rods during shear.

For *soft* particles, like droplets in an emulsion, *K*_{1} depends on the ratio *η*_{d} = *η*_{drop}/*η*_{s}, as the droplets deform for small deformation to spheroids during shear. With the deformation process, a relaxation time *τ* = *aη*_{s}/*γ* (*a* droplet radius, *γ* interfacial tension between droplet and matrix) is associated, which is generally rather short, say, 10^{−8} to 10^{−7} s. The measure of deformation *d* = *a* − *b*/(*a* + *b*), that is, the relative difference between the long axis^{4}*a* and the short axis *b*, is given by *d* ≅ 2*τ*(1 + 19*η*_{d}/16)/(1 + *η*_{d}) and thus is relatively small, even at high shear rate. For such droplets the viscosity in steady simple shear stress is *η*_{r} = 1 + [(1 + 5*η*_{d}/2)/(1 + *η*_{d})]*φ*, reducing for large value of *η*_{d} to the Einstein result. For the typical case of *η*_{d} ≅ 1, we have *η*_{r} = 1 + 7*φ*/4. These results are in rather good agreement with experiment but only in the absence of adsorbed species.

With a larger volume fraction of particles, the encounters between the individual particles cause additional energy dissipation. Because the number of encounters is proportional to the concentrations of both particles involved in such an encounter, this extra dissipation must be second order in *φ*. Moreover, in shear flow pairs of particles may exhibit trajectories that are closed orbits. Batchelor and Green [12] derived for dispersions of hard monodisperse spheres that

10.7

but where the value of *K*_{2} depends on the conditions. For shear flow the value of *K*_{2} depends on the Péclet number *Pe* measuring the ratio of the Brownian motion time scale and the viscous transport time scale. The former is characterized by the diffusion coefficient *D* for a particle of radius *a* in a liquid with viscosity *η* given by *D* = *kT*/6π*ηa*, thus leading to a characteristic diffusion time *t*_{dif} ≅ *a*^{2}/*D* for a diffusion distance equal to *a*. The latter is characterized by a characteristic shear time , so that (note that in the literature *Pe* is also defined including the factor 6π). For *Pe* ≪ 1 (low relative shear rate), *K*_{2} = 6.2, while for *Pe* ≫ 1 (high relative shear rate), *K*_{2} = 5.2. For elongation flow there are no closed orbits, and the effect of Brownian motion is unimportant (essentially *Pe* ≫ 1 always), leading to *K*_{2} = 7.6 (Figure 10.4a). The second‐order expression with *K*_{2} = 6.2 has been verified experimentally for *φ* ≤ 0.10.

Electrical double layers affect these results. The *primary electroviscous effect* is due to the electrical potential that arises upon transport of charged particles. This potential causes a backflow of electrolyte, deforming the electrical double layer and leading to an effective increase in *η*. The intrinsic viscosity [*η*] is given, for *ξ* ≤ 25 mV and both *Pe* ≪ 1 and *Pe* ≫ 1 [13], by

10.8

Here *ε* is the permittivity, *ξ* is the zeta potential, and *σ* is the specific conductivity of the fluid. Although it has been stated that the effect is often small and difficult to distinguish from variations in *K*_{1} (which for monodisperse particle should be 5/2), values up to [*η*] = 20 can result under certain conditions [14]. The *secondary electroviscous effect* refers to the change in [*η*] due to the interaction of the electrical double layers. It is expedient to define a parameter *α* ≡ (4π*εξ*^{2}*a*^{2}*κ*)exp(2*κa*) representing the ratio of electrostatic to thermal energy [13]. Analysis for *α* ≫ 1 (electrostatic energy dominant) and *Pe* ≪ 1 leads to the effective diameter given by *d*_{eff} ≅ *κ*^{−1}{ln *α*/ln[*α*/ln(*α*/…)]}, where in practice the concatenation of logarithms can be truncated after the third. The effective volume thus becomes *φ*_{eff} = *φ*(*d*_{eff}/2*a*)^{3}, while for the coefficient *K*_{2} the result is [15]

Here [*η*] should include the primary effect. It follows that for *ξ* → 0, *K*_{2} → 5/2. However, we have noticed already that for hard spheres (*Pe* ≪ 1), *K*_{2} = 6.2, and this difference is due to using an approximate hydrodynamic force field taking into account the flow around a single particle only. Hence, it seems better to use the factor 6.2 instead of 5/2 in the expression for *K*_{2}. The *tertiary electroviscous effect* refers to intramolecular electrical double layer effects for particles stabilized by polyelectrolytes, either grafted or adsorbed. Assuming a size *δ* for the polyelectrolyte layer thickness (or any other stabilizing polymer for that matter), the volume fraction for particles with radius *a* effectively increases with a factor (1 + *δ*/*a*)^{3}, in reasonable agreement with experiment [16].

At still larger volume fraction, the analysis of Batchelor and Green fails because of the occurrence of many‐particle encounters. A very simple, yet fairly accurate theory by Frankel and Acrivos [17] leads for monodisperse spheres with radius *a* at high volume fraction to

10.10

The parameter *h* is the average distance between particle surfaces, while the ratio *a*/*h* is calculated assuming a cubical arrangement of particles with *φ*_{m} = π/6 ≅ 0.52. One might expect that *φ*_{m} ≅ 0.64, corresponding to random close packing, should be used. While the authors claimed good agreement with their experimental data, others have used *φ*_{m} = 0.64 indeed. The expression has the correct asymptotic behavior for *φ* → *φ*_{m}, but for *φ* → 0 does not agree with Einstein's result. This theory is based on the assumption that the viscosity of concentrated suspensions is controlled by energy dissipation in the narrow gaps between the particles.

Semiempirical expressions are widely used for the description of the viscosity over the whole range of accessible volume fractions. A popular approach is to consider the fluid as an effective medium and to assume that an increase in particle volume fraction d*φ* leads to an increase in *η*(*φ*), described by d*η* = [*η*]*η*(*φ*)d*φ*/(1 − *φ*/*φ*_{m}). The factor (1 − *φ*/*φ*_{m}) is added to account for the maximum volume fraction *φ*_{m}. Integration leads to the *Krieger–Dougherty relation* [18] reading

10.11

with as fit parameters *φ*_{m} and [*η*]. In this description the value for *η*_{r} increases rapidly with increasing value of *φ* and diverges at *φ* = *φ*_{m} (Figure 10.4a). For *φ*_{m} one expects a maximum value of 0.74 for close packing, but with increasing particle size dispersity, the *φ*_{m}‐value rises, up to 0.8 or even 0.9. For agglomerated particles the factor *φ*/*φ*_{m} can be approximated by *φ*/*φ*_{m}*φ*_{a}, where *φ*_{a} is the packing fraction of the particles within the agglomerates, since the agglomerates move as a whole. For multimodal size distributions, much higher volume fractions of solid material can be reached than for a unimodal size distribution at the same viscosity [19]. Expanding the Krieger–Dougherty expression leads to *η*_{r} = 1 + [*η*]*φ* + ½[*η*]([*η*] + *φ*_{m}^{−1})*φ*^{2} + …. To first order the Einstein expression is recovered, but in practice [*η*] has a slightly higher value, *K*_{1} ≅ 2.7, dependent on the size distribution of the particles [20]. The second‐order term using [*η*] = 5/2 leads to 4.8 and 5.2 for *φ*_{m} = 0.60 and *φ*_{m} = 0.74, respectively, somewhat smaller than the Batchelor values. By assuming that *K*_{2} = ½[*η*]([*η*] + *φ*_{m}^{−1}) and that either *φ*_{m} or 1/*φ*_{m} increases linearly with *φ*, van de Ven [21] showed that a correct expansion to second order is obtained without introducing another parameter. Moreover, using his expressions, *φ*_{m} ≅ 0.74 is obtained for both *η*_{0} and *η*_{∞} (see below). The structural implication of his assumption is that in principle all many‐particle effects are incorporated. Maron and Pierce [22] derived an equation similar to Eq. 10.9 on the basis of Eyring's activated complex theory applied to flow, but in which the exponent [*η*]*φ*_{m} is replaced by 2. In fact, Metzner [23] recommended to use *η*_{r} = (1 − *φ*/*A*)^{−2} with *A* an empirical parameter having for smooth spherical particles the value *A* ≅ 0.68 and for rough, more or less isometric, particles *A* ≅ 0.44. The Krieger–Dougherty expression can also be used for fluids containing charged spheres but with *φ* replaced by *φ*_{eff} and *φ*_{m} typically having value of about 0.15. Above that value these materials behave as viscoelastic solids.

It appears that both *η*_{0} and *η*_{∞} can be described by the Krieger–Dougherty relation but with different values for *φ*_{m} and [*η*]. From data of Krieger [24] and their own, de Kruif et al. [25] obtained *φ*_{m} = 0.63 and [*η*] = 3.1 for *η*_{0} and *φ*_{m} = 0.71 and [*η*] = 2.7 for *η*_{∞} [5, 26]. These *φ*_{m} values suggest that random close packing with *φ*_{m} ≅ 0.64 is approached at low shear rate, while at high shear rate the fluids order approaching the value *φ*_{m} ≅ 0.74 for dense close packing. However, these values are sensitive to the deformability of the stabilization layers, either adsorbed or grafted on the particles [23]. For values of *η* between *η*_{0} and *η*_{∞}, the suspension shows an apparent yield strength *τ*_{Y} for which a Casson equation can be used. The values for *τ*_{Y} so obtained often differ from the true yield strength. Krieger [24] has suggested that scaling of is best done by instead of *η*_{s}, and since represents the shear stress *τ*, one can define a (nondimensional) characteristic shear stress *τ*_{char} = *τa*^{3}/*kT*. This implies that *η*_{r} is a universal function of *φ* and *τ*_{char}, and a simple but effective semiempirical relation describing *η* at constant *φ* but variable *τ* experimentally very well is

10.12

with *b* ≈ 2–3 a parameter and *τ*_{cri} = *kT*/*a*^{2}*b*. The critical stress *τ*_{cri} increases approximately linearly from 0 to about 3 Pa at *φ* ≅ 0.5 and thereafter decreases again to 0 at *φ*_{m} [25]. This maximum suggests the disorder–order transition for hard sphere fluids. Some further details can be found in the monographs [7, 21, 27], while experimental data have been reviewed by Mellema [28]. Nanofluid viscosity is reviewed in [29].

So far, the particles were considered to be spherical. When a dilute suspension of nonspherical particles is subjected to linear shear flow, each particle describes a complex pattern of rotations induced by the shear field, as analyzed by Jeffery [30]. For a spheroid with particle aspect ratio *w* = *a*/*b*, where *a* and *b* are its major and minor semiaxes, in a flow field given by , the axis of revolution of the particle rotates with angle *φ* around the *x*‐axis according to tan *φ* = *w* tan(2π*t*/*T*) with the period of rotation . For a cylindrical rod with aspect ratio *l*/*d*, *w* ≅ 0.71(*l*/*d*).These predictions have been confirmed by experiment. For these spheroids the flow resistance is larger than for spheres, and the shear viscosity *η* decreases with increasing shear rate . Analytical theories are rather complex, but the limiting viscosities at low and high Péclet numbers for spheroids with arbitrary aspect ratio *w* = *a*/*b* have been calculated (see Figure 10.5 [31]).

Brownian motion affects the rotary motion of particles with different shapes differently. While for spheres with radius *a* the rotary Brownian diffusion coefficient is given by *D*_{rot} = *kT*/8π*ηa*^{3}, for disks *D*_{rot} = 3*kT*/32*ηb*^{3} where *b* is their radius. Brenner [32] provides data for many shapes. He also describes the effect of *D*_{rot} on [*η*] (see Figure 10.6).

A practically accessible route is to employ one of the available semiempirical equations such as the Krieger–Dougherty relation and to replace the parameters [*η*] and *φ*_{m} by optimized, larger values. For fibers having a length *l* and diameter *d* for an aspect ratio *w* = *l*/*d* between 5 and 30, *η*_{r} = (1 − *φ*/*A*)^{−2} can be used with *A* approximately given by *A* = 0.55 − 0.013*w*. For spheres (*w* = 1), this yields *A* = 0.54, a value between that of a smooth sphere with *A* = 0.68 and a rough particle with *A* = 0.44, indicating that this result should be used with caution. For dilute suspensions fairly detailed models are available. Here we just indicate somewhat more precisely what *dilute* implies, namely, that the particles can rotate freely without hindrance from neighbors. Hence, for a particle the distance between particles Δ should be Δ > *l*, so that a volume of *l*^{3} contains only one particle. For *φ* therefore we have *φ* ≅ *d*^{2}*l*/Δ^{3} ≅ *d*^{2}*l*/*l*^{3} or *φ*(*l*/*d*)^{2} < 1. The regime between 1 < *φ*(*l*/*d*)^{2} < *l*/*d* is called *semiconcentrated*, while suspensions with *φ*(*l*/*d*)^{2} > 1 are called *concentrated*. In the latter regime particle–particle interactions are essential, leading to rather involved models. Some further details are given in [21], while a more detailed overview on colloidal stabilization can be found in [33].

A useful discussion of shape anisotropy effects is given by Mueller et al. [34] although the size of the particles used, about 100 µm, is quite somewhat larger than practically used in coatings. These authors considered the viscosity of various materials in silicone oil to prevent electroviscous effects and to minimize particle settling. For spherical particles they assumed Bingham behavior. Their data for spheres (measured for a wide range of volume fractions and shear rates) correspond well with earlier data up to *φ* ≅ 0.35 with *τ*_{0} ≅ 0 and *n* ≅ 1 and at higher volume fractions position themselves nicely in middle of other available data. The Krieger–Dougherty relation fits the data well with [*η*] = 3.27 and *φ*_{m} = 0.641 (*R*^{2} = 0.998), leading to [*η*]*φ*_{m} = 2.10, close to the Maron–Pierce value of 2, so that also the Maron–Pierce equation actually fits the data well (*R*^{2} = 0.997). Shear thinning becomes important for *φ* ≥ 0.25, and for *φ* ≥ 0.5 the suspensions exhibit a yield strength, which is well described by a modified Maron–Pierce relationship reading

10.13

where *τ*^{*} is a parameter describing the (apparent) yield strength at *φ* = *φ*_{m}(1 − ½√2). A reasonable fit (*R*^{2} = 0.977) was obtained with *φ*_{m} = 0.633, resulting in *τ*^{*} = 0.153 Pa, but with *φ*_{m} left free, an excellent fit (*R*^{2} = 0.999) was obtained with *φ*_{m} = 0.611 and *τ*^{*} = 0.0483 Pa. Due to the higher number of particle–particle contacts per unit volume, the *τ*^{*} values for suspensions with smaller particles are generally larger [35] (1.5 µm: *τ*^{*} = 3.1 Pa; 2.5 µm; *τ*^{*} = 0.20 Pa; 50 µm: *τ*^{*} = 0.048 Pa).

For anisometric particles, Mueller et al. [34] considered particles as (prolate or oblate) ellipsoids and expressed their results in the ratio *w* = *a*/*b*. Experimental data for a wide range of particle aspect ratios (0.13 < *w* < 9.2) were used up to a maximum of *φ*/*φ*_{m} ≅ 0.8. It appeared that the Maron–Pierce exponent relation [*η*]*φ*_{m} = 2 was well obeyed also for ellipsoids. This led to

10.14

the latter expression indicating that, as compared with spherical particles, shear thinning is more important. This might have been expected as rodlike particles tend to assume parallel orientations in a shear field. The fact that scaling of both [*η*] and *n* with *φ*_{m} leads to a master curve indicates that the ratio *φ*/*φ*_{m} represents a measure of the typical minimum separation dominating hydrodynamics that is independent of the particle aspect ratio.

When dissolving polymer molecules in water or an organic solvent, the polymer molecules usually unfold and take a fluctuating coil‐like conformation (see Chapter 2). This results for an individual polymer molecule in a chain segment density (concentration) having on average a maximum at its geometric center, which decays with increasing distance from that center. These polymer chains can be considered as being swollen coils with *radius of gyration**R*_{g} = *αr*/6^{1/2} = *αC*^{1/2}*n*^{1/2}*l*/6^{1/2}. Here *r* = *C*^{1/2}*n*^{1/2}*l* is the root mean square end‐to‐end distance under theta conditions, *α* is the solvent expansion factor, *C* is the characteristic ratio, and *n* is the number of backbone bonds with (average) length *l* (see Chapter 2). It may be useful to recall that at theta conditions *α* = 1, but that for good solvents with *α* ≫ 1, from *α*^{5} − *α*^{3} = *A*(1 − *θ*/*T*)*M*^{1/2}, we obtain the following scaling laws: *α* ∼ *M*^{1/10} and *αr* ∼ *M*^{1/10}*M*^{1/2} ∼ *M*^{3/5}. The number of coils per unit volume is defined by *n*_{coil} = *cN*_{A}/*M*, where *c* is the concentration of polymer with molecular weight *M* and, as usual, *N*_{A} represents Avogadro's constant. These polymer coils are subject to Brownian motion, thus continuously readjusting their conformations.

Distinguishing three regimes, namely, the *dilute regime* with *c* < *c**, the *semidilute regime* with *c** < *c* < *c*^{‡} (this is actually the regime where Flory–Huggins theory applies), and the *concentrated regime* with *c* > *c*^{‡}, let us describe what happens with increasing the concentration of the polymer. Describing the viscosity of a polymeric solution similar as for spherical particles, we have *η*_{r} = 1 + [*η*]*c* + *K*_{H}[*η*]^{2}*c*^{2} + ⋯. The *intrinsic viscosity* [*η*] for a polymer solution is defined by [*η*] ≡ (*η*_{r} − 1)/*c* for *c* → 0 and is experimentally described by the (Staudinger)–*Mark–Houwink*–(Sakurada) *equation*, where the exponent *ν* ranges from 0.5 to 0.8. The parameter *M*_{v} is the *viscosity average molar weight**M*_{v} = (Σ_{j}*n*_{j}*M*_{j}^{ν + 1}/Σ_{j}*n*_{j}*M*_{j})^{1/ν} with *n*_{j} the number of molecules with molar weight *M*_{j}. The parameter *K*_{2}, for polymers usually denoted as the *Huggins constant*, appears to be approximately constant. For good solvents, *K*_{H} ≅ 0.3–0.4, while for theta solvents *K*_{H} ≅ 0.5–0.8. In the coil overlap region 1 < [*η*] < 10, the viscosity is reasonable well described by the *Martin equation* (*η*_{r} − 1)/*c* = [*η*]exp(*kc*[*η*]), with the constant *k* often close to *K*_{H}.

The behavior indicated above applies as long as the molecules are too small to make entanglements, that is, below a *critical entanglement molar mass M*_{cri}. The value of *M*_{cri} ranges from 2 × 10^{3} for polyethylene (PE) to 3 × 10^{4} for more complex polymers. Beyond *M*_{cri} the viscosity becomes very sensitive to the molar mass *M*, and *η* typically scales as *M*^{ν} with *ν* ≅ 3.4 (see Figure 10.4b). Section 9.4.2 provides a qualitative explanation for the entangled region.

If we consider a polymer coil as a sphere with radius *R*_{g}, the volume of the coil is *V*_{coil} = 4π*R*_{g}^{3}/3. Comparison of *η*_{r} − 1 ≅ [*η*]*c* for a polymer solution with *η*_{r} − 1 ≅ 2.5*φ* for particles suggests that

10.15

leading for the dilute regime to a slightly larger exponent as observed experimentally. Since *c* ∼ [*η*]^{−1}, the *overlap concentration c** ∼ *M*^{−4/5} in good solvents. For example, for polystyrene (PS) with *M* = 10^{6} in a theta solvent (*α* = 1), *c** = 0.02 g cm^{−3}.

The effect of polymer coils in the dilute regime can be theoretically described in more detail by the Kirkwood–Riseman theory [36], which predicts that [*η*] ∼ *n*^{−1}*R*_{g}^{3}*f*(*x*) with *x* = *nξ*/*η*_{s}*R*_{g} and *ξ* is the friction coefficient (of a segment). For the *free draining limit*, that is, the flow of the liquid is not interrupted by the coils, this theory results in *f*(*x*) → *x*. Hence,

10.16

For the *nondraining limit*, that is, the flow of the liquid leaves the interior segments of the coil unaffected, *f*(*x*) → constant, and thus

10.17

Evidently, the exponent *ν* is limited to 0.5 < *ν* < 1.0, and its precise value depends on the polymer–solvent combination.

With increasing *c* we enter the semidilute regime. Now we define the *correlation length ξ* as the mean distance between neighboring polymer segments. We might expect that for *c* > *c**, the molecular structure does not dependent on *M* as we are considering a small section of the polymer. Further, we expect that at *c* ≅ *c**, *ξ* ≅ *αr* and that for *c* > *c** the correlation length scales as *ξ* ≅ *αr*(*c**/*c*)^{m}. This leads to *ξ* ∼ *M*^{3/5}(*M*^{−4/5}/*M*^{0})^{m} and, as *ξ* should scale as *ξ* ∼ *M*^{0} according to our expectation, to *m* = ¾. Hence, *ξ* ∼ *c*^{−3/4}.

We can also estimate how the radius of the coil changes with *c*. To that purpose we consider a chain as a sequence of *n*_{b}*blobs*, each of molar mass *M*_{b} = *M*/*n*_{b}. Since the coils are swollen, the contour length of the coil within each blob relates to the size of the blob *ξ* ∼ *αn*_{b}^{1/2} ∼ *n*_{b}^{1/10}*n*_{b}^{1/2} ∼ *n*_{b}^{3/5}. Therefore, using *ξ* ∼ *c*^{−3/4}, we find *n*_{b} ∼ *c*^{−5/4}. The mean square end‐to‐end distance of the complete coil *α*^{2}*r*^{2} = [*n*/*n*_{b}(*c*)]*ξ*(*c*)^{2} thus becomes *α*^{2}*r*^{2} ∼ (*c*^{0}/*c*^{−5/4})(*c*^{−3/4})^{2} ∼ *c*^{−1/4}. The coil therefore shrinks in the semidilute regime, in essence due to the increasing segment–segment interaction.

Shrinking stops when, upon further increasing *c*, *α* decreases and theta conditions are reached at the *entanglement concentration c*^{‡} where *α* = 1. At *c* = *c**, we have *α*^{2}*r*^{2} ∼ (*c**)^{−1/4}, while at *c* = *c*^{‡} we have *r*^{2} ∼ (*c*^{‡})^{−1/4}. Thus, *α*^{2} = (*c*^{‡}/*c**)^{1/4} or *c*^{‡} = *c***α*^{8}. This leads to *c*^{‡} ∼ *M*^{−4/5}(*M*^{1/10})^{8} ∼ *M*^{0}, and *c*^{‡} is independent of *M*. Its precise value can be calculated from *c*^{‡} = *c***α*^{8}, *c** = *M*/*α*^{3}*r*^{3}, and *α*^{5} − *α*^{3} = *A*(1 − *θ*/*T*)*M*^{1/2}. The overall viscosity behavior is shown in Figure 10.7a, while Figure 10.7b provides a schematic of the various regimes.

Let us now discuss the behavior of a polymer solution in shear flow in qualitative terms. Suppose a polymer solution is being sheared between two parallel plates and, for convenience, that the polymer coils are organized in layers. When the upper plate is displaced with respect to the lower, these layers will slide over each other.

First, consider the dilute regime. In that case, the resistance to flow for the solutions is slightly larger than that of the pure liquid, like for solid spheres. In fact, *η* can be interpreted as following approximately Einstein's law, provided the interpretation of *φ* is taken as the volume fraction of swollen coils. If we stick to the original interpretation, that is, that *φ* represents the volume occupied by the *dry* polymer, the factor [*η*] in *η*_{r} = 1 + [*η*]*φ* is actually a measure of the degree of swelling.

Above the *overlap concentration c**, the behavior depends on the ability of coils to interpenetrate and create physical entanglements. With small‐sized coils, that is, with low molar mass polymers, the behavior to some extent resembles the Krieger–Dougherty behavior, be it that *φ*_{max} is not a fixed value due to compression of the coils at increasingly higher concentration. With large molar masses, the random‐coil nature allows the coils to interpenetrate. This implies that upon shearing these entanglements have to be disentangled, involving a force that usually is orders of magnitude larger than that for dilute solutions and is the reason why nondilute polymer solutions (say, >1%) and polymer melts can have an abnormal high viscosity. This behavior is described by the reptation theory as described by Doi and Edwards [37] (see also [38]).

When shearing a solution of interpenetrating polymer coils, determines the time scale for the polymer chains to adjust to a deformation. For a small value of , the disentanglement of coils requires a large amount of energy. When one reference coil A passes over another coil B at low speed, the Brownian diffusion of coil A allows it to penetrate B and become fully entangled. On the verge of leaving coil B and arriving at a third coil C, again a large amount of work has to be done to disentangle. The overall result is that the value of *η* is constantly large.

For a large value of , the reference coil needs for leaving the first coil a large amount of energy, as indicated before. However, because of the high shear rate, the residence time in the vicinity of the second coil is insufficient to effectively interact with it, with a consequence that for the reference coil to leave the second coil, in order to arrive at the third coil, much less energy is needed. Thus, although the onset viscosity in both cases may be similar, the final (*steady‐state*) viscosities differ considerably, being much lower at high shear rates. Polymer melts and nondilute polymer solutions are typically shear thinning or even perfectly plastic in exceptional cases. As the entanglements referred to before also induce elasticity to the liquid system, the liquid becomes viscoelastic.

A great deal of research has been devoted to quantifying and rationalizing the effect of molar mass *M* on the viscosity*η* for polymer solutions (see, e.g. [39]) as well as for pure polymer systems (see, e.g. [40]).

As discussed before, the rheology of dispersed systems can be controlled via thickening by adding particles or dissolvable polymers. The question is: in which cases there is need for such adjustment with paints? We consider just two cases: powder coatings and alkyd solution paints.

With powder coatings the rheology is important in view of the melting of the individual particles that should lead to a coherent, flat film. Most powder coating materials are almost perfectly Newtonian, as the molecular mass of the molecules is still fairly low, say, between 10^{3} and 10^{4} Da. This simplifies their theoretical treatment considerably. The viscosity *η* for a typical polyester composition as a function of angular frequency for various temperatures is shown in Figure 10.8a [41]. The viscosity remains fairly constant with shear rate near *η*_{0}, but deviations occur, approaching the melting point. With decreasing temperature the curve shows an increasing decrease in *η* at high rate. Figure 10.8a also indicates that the viscosity is sensitive to temperature changes. In this particular case a change of 70 °C leads to a viscosity change by almost three decades.

After spraying the powder on a substrate, the coated substrate is placed in an oven at temperature *T*_{2}, and the coating viscosity will change according to curve a in Figure 10.8b if the powder does not contain, in the usual jargon, a *dryer* (a catalyst for crosslinking of the binder). If a dryer is added, the viscosity profile changes to curve b, where we see that at high temperature crosslinking leads to an ever‐increasing viscosity, eventually leading to a solid network. If the viscosity of the molten powder is too low, this may lead to sagging and a higher level of catalyst can be employed, leading to curve c. If the crosslinking process is happening too fast, the melting together of the individual particles cannot be completed, and in such cases the amount of catalyst used can be reduced, giving profile d. One could try to bake at a higher temperature in order to shorten the process time. The control of the process will now become more difficult. Curve f shows this process with a somewhat lower amount of catalyst, leading to a too low viscosity. Increasing the catalyst level as a remedy may lead to very quick crosslinking, *freezing* the material already before the sample arrives at the set temperature (curve g). As not all parts of the object to be coated heat up equally quickly, different parts of the coating will not have the same appearance.

In the case described, the temperature and catalyst level provide sufficient possibilities to manipulate the process for obtaining a good coating. Additionally, the molecular mass of the resin can be changed for influencing the viscosity and therefore the viscosity profile during baking. It will be clear that adding thickeners in the form of high molecular mass molecules is not done in the case of powder coatings.

Powder coatings usually contain pigments. These pigments are normally not colloidally stabilized. They are just added as agglomerated particles to the resin and intensely mixed by extruding the mixture (including other additives, like catalysts). It will be clear that this mixing action will also start the crosslinking. The limited time available is sufficient for breaking up the agglomerates in a laboratory‐scale extruder, but complete dispersing usually does not occur with full‐scale extruders because of a lack of colloidal stability. In practice, however, they will stay apart in the curing step largely because the high viscosity requires long reagglomeration times, comparable with or longer than the time for the polymer in the molten state.

One of the problems with waterborne paints is their pseudoplastic behavior. Such paints are based on a dispersion in water, with a viscosity not much larger than that of water (*η* ≅ 1 mPa s). In contrast, a traditional alkyd paint typically has a viscosity of 1 Pa s. In order to bridge this gap, one uses so‐called thickeners, polymers dissolved in the water phase. Examples are polyethylene glycol (PEG), polyethylene oxide (PEO), and cellulose thickeners, as shown in Figure 10.9. These *conventional thickener**s* (CTs) thicken just the water phase, even at low concentration considerably, as shown in Figure 10.10a (solid and open triangles). However, at useful concentrations such thickeners also induce considerable shear thinning at sufficiently high shear rate to the paint, as shown schematically in Figure 10.10b.

Alternatively, one uses *associative thickener**s* (ATs) [42]. Such thickeners contain hydrophobic groups that are chemically attached to the hydrophilic backbone. A specific example of a *hydrophobically end‐capped urethane* (HEUR) is shown in Figure 10.11, where also it is indicated how these molecules can associate in aqueous solution toward a very weak network. Usually the hydrophobes are alkyl chains (C_{n}H_{2n+1}), normally with a molar mass smaller than for their conventional counterparts. Consequently, their thickening effect in a pure polymeric solution is less at the same concentration (compare open and closed triangles in Figure 10.10).

When added to latex, ATs show an exceptional behavior [43]. While a CT just thickens the water phase between the latex particles (the solid circles and squares in Figure 10.10a), the ATs give rise to a large increase in the viscosity (open circles and squares in Figure 10.10a). This is due to their interaction with the latex particles. They adsorb on the latex surface, thereby increasing the effective volume fraction of the dispersed phase, up to the maximum packing value or beyond. This gives rise to a tremendous increase in viscosity. Any shear thinning occurs at higher shear rates. This thinning is due to the fact that at high shear stresses the adsorbed layer gives way, thereby reducing the viscosity.

In conclusion, ATs can be applied in lower concentrations, making the dried films less vulnerable to water. Additionally, shear thinning occurs only at higher shear rates; the rheology at low rates is more Newtonian, which is advantageous for good film formation (leveling).

When a paint has been applied, for example, by a brush, the solvent will evaporate and the binder will be left, together with pigment and other minor additives. A major drawback with waterborne paints is the low open time. The *open time* is the time between application and the moment that no readjustments can be made without permanent damage. This is due to the fact that the volume fraction of the relatively quickly evaporating water has only to reduce from 60% to 40%. In contrast, with a solvent‐based alkyd paint, the film keeps some mobility even when the slower evaporating solvent has arrived at a volume fraction of 30% or less.

Formulations of solventborne or solvent‐free paints usually contain oligomers rather than high molar mass, pure polymers, typically with a molecular mass of a few thousand Da. Paints usually start as a liquid, possibly pass through the rubbery phase, and eventually arrive in the glassy state. The key concept here is the free volume as the thermal mobility at high temperature *T* creates an appreciable free volume *v*_{f}, in excess to the minimum, *own* volume of the polymer (see Section 9.3.5). This gives mobility to segments of a chain, leading to a relatively small elastic modulus for the rubbery state. On reducing *T*, at some moment *v*_{f} becomes so small that segment mobility is lost, leading to a dramatic rise in the shear modulus *G*. This is the *glass transition temperature**T*_{g}. While for polymers with high molar mass *M* the glass transition temperature *T*_{g} hardly depends on *M*, for oligomers this dependency for oligomers is appreciable. The exact position of *T*_{g} is important because:

- It can greatly influence the viscosity and therefore the application and leveling properties of a paint.
- It influences the degree of crosslinking and possibly inclusion of solvent because once a system vitrifies and molecular segments become immobile, crosslinking is almost stopped and not yet evaporated solvent can remain entrapped.

Thus, a prediction of the viscosity *η* near *T*_{g} and the value of *T*_{g} itself is useful. The *T*_{g}(*M*) can be described by an expression akin to the Fox equation (see Section 2.1.4):

10.18

where the exponent *v* varies between 1 and 0.5 and *T*_{g}^{∞} represents *T*_{g} for *M* = ∞. In analogy with high *M* polymers, the WLF equation for *η*(*T*, *T*_{g}) can be used in an approximate way. As an example, consider the design of a crosslinking paint. During crosslinking the molar mass rises and therefore also *T*_{g} rises. Suppose that the paint becomes tack‐free once the viscosity rises beyond 10^{3} Pa s at 25 °C. Even without further information, using the WLF equation with the *universal* constants, one can deduce that this requires a degree of crosslinking corresponding to *T*_{g} = −29 °C. In a similar way the degree of crosslinking can be assessed that is required for a paint to become nonsticky on loading the paint surface, for example, when a series of panels is painted and one wants to stack them at 25 °C as soon as possible. For a required stress of 1.4 × 10^{5} Pa, experience indicates that this requires a viscosity as high as 10^{7} Pa s. On the basis of the WLF equation, the *T*_{g} should have risen to 4 °C.

The situation becomes more complicated if one also considers the effect of solvents on *T*_{g}. For mixtures of a polymer and a solvent, *T*_{g} can be described by

10.19

where *C*_{polymer} and *C*_{solvent} are polymer‐ and solvent‐specific constants and *W*_{solvent} the mass fraction of solvent. Figure 10.12a shows how sensitive the viscosity of an oligomer solution is for the solvent level and the temperature. This again indicates the complexity of modeling the drying process of paints, especially because temperature and thus the evaporation rate, may be so variable.

Finally, we focus somewhat closer on the way how early vitrification may cause insufficient crosslinking, as shown in the *temperature–time–transformation* (TTT) diagram in Figure 10.12b. In such a diagram the reaction path can be indicated as a function of temperature and time.

If the entire curing process takes place below the glass transition temperature *T*_{g0} of the mixture of reactive components, the system continuously is in the vitrified state, slowing down crosslinking to a large extent.

If curing takes place between *T*_{g}^{0} and *T*_{g}^{gel}, the molar mass of the oligomers rises to some extent. However, before the reaction has proceeded sufficiently to allow network formation, the system vitrifies and the reaction stops. In this case the end product contains high(er) molar mass polymers, but no network will be formed.

When processing between *T*_{g}^{gel} and *T*_{g}^{∞}, by the higher temperature the vitrification is delayed sufficiently to let the polymerization proceed so far that network formation occurs. Again, on vitrification the reaction virtually stops. Now the final coating is crosslinked. Note that in this case *later on* devitrification occurs. This means, as noted before, that a *T*_{g} value is not unique, but depends to some extent on the time scale of the process, in such a way that with larger time scales, *T*_{g} decreases. Thus, once having passed *T*_{g} for short time scales, long time scale processes may still be able to proceed slowly, and some reaction is still possible on this larger time scale.

When curing at or beyond *T*_{g}^{∞}, the *T*_{g} of a fully crosslinked network, crosslinking is not frozen by vitrification. However, this usually needs a high temperature with risk of char formation.

In summary, the best is to let the process occur just beyond *T*_{g}^{∞}.

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