Shear thickening in colloidal dispersions

Transcription

1 Seminar Shear thickening in colloidal dispersions Author: Andraž Krajnc Advisor: prof. dr. Rudolf Podgornik Ljubljana, February 2011 Abstract First of all, I will introduce the basics of Newtonian fluids, followed by differences in comparison to non-newtonian fluids. Following that, I will consider suspensions and focus on the shear thickening phenomena. Different theories behind that lead us to an order-disorder system and a hydro-clusters formulation. Finally, I will present its applications and prospects for the future.

2 Contents 1 Introduction Newtonian Fluids Non-Newtonian Fluids Rheology of Suspensions Zero-Shear Viscosity Viscosity under Shear Hydrodynamics Lubrication force Stokesian dynamics Applications Liquid Armor Protective Equipment Other Applications Conclusion Literature

3 1 Introduction Although shampoos, paints, cements and blood are known for a very long time, there has been only little interest for the exploration of their physics. The main liquid has always been water and its simple physics, which obey Newton s laws, satisfied many scientists in history until 20th century. After digging deeper and deeper many different phenomena were discovered. One of them, known as shear thickening, is the topic of this seminar. 2 Newtonian Fluids It would be useful to begin with the definition of a Newtonian fluid. Viscosity is the main attribute of all liquids. It is the measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress. The less viscous a liquid is, the greater its ease of movement. An ideal fluid has no resistance to shear stress and therefore its viscosity is equal to zero. Viscosity is influenced only by temperature and atmospheric pressure, so under stable conditions the stress versus strain rate curve is linear and passes through the origin [1]. We define Newtonian fluids with (1) where is the shear stress exerted by the fluid, the fluid viscosity and the velocity gradient perpendicular to the direction of the shear, or shorter, the strain rate. A schematic representation of a unidirectional shearing flow is presented in Figure 1a. In equation (1) the simplest case is presented; it only includes one non-zero component of velocity,. a) b) Figure 1 (a) A schematic representation of flow in shear. (b) Stress components in three dimensions [1]. In a three dimensional flow there are six shearing and three normal components of stress tensor, (Figure 1b). We usually split the total stress into where is an isotropic part (pressure) and the traceless deviatoric part of the shear stress. Because of the physical requirement of symmetry, there are only three independent shear components. For Newtonian fluids in Cartesian coordinates, these (2) 3

4 components are related linearly to the rate of deformation tensor components via scalar viscosity. For instance, let us observe components acting on the x-plane ( ) (3) ( ) (4) We can see that in a simple shear diagonal components of shear tensor are equal to zero, because velocity only varies in the y-direction. To satisfy Navier-Stokes equations we require the complete definition of Newtonian fluids rather than simply exhibiting a constant value of shear viscosity. 3 Non-Newtonian Fluids There exist also some fluids whose viscosity changes with the applied stress and they are called non-newtonian. In such cases, stress versus strain rate curve is not straight and/or does not pass through the origin (Figure 2). In other words, Navier-Stokes equations do not have solutions for such fluids. But viscosity is not only a function of flow conditions (geometry, rate of shear, etc.). It depends also on the kinematic history of the fluid. (5) Figure 2 Graph shows shear stress versus shear rate for different fluids. Curve for Newtonian fluid is straight and goes through the origin, whereas curve for non-newtonian fluids is not straight and/or does not pass through the origin [2]. According to the stress-rate curve, we differentiate different types of non-newtonian fluids (Table 1). One of them is the shear thickening fluid, which we will talk about in detail later. Shear thickening itself is a time independent process, in which viscosity increases with increased stress. Shear thinning usually happens in the same fluids but at lower shear stress. 4

5 Kelvin material Viscoelastic Anelastic Rheopectic Time-dependent viscosity Thixotropic Shear thickening (dilatant) Time-independent viscosity Shear thinning (pseudoplastic) Table 1 Presentation of different types of non-newtonian fluids. [2]. "Parallel" linear combination of elastic and viscous effects Material returns to a welldefined "rest shape" Apparent viscosity increases with duration of stress Apparent viscosity decreases with duration of stress Apparent viscosity increases with increased stress Apparent viscosity decreases with increased stress 3.1 Rheology of Suspensions The addition of dispersed solid phase to Newtonian fluid forms suspension, which can lead to the introduction of all kinds of non-newtonian behavior. If the solid phase is made of small particles, such a suspension is often called colloidal suspension. The paramount concern in any multiphase liquid is stability. Thermodynamics drives clumping of dispersed particles, and this is sometimes enhanced by flow. First of all, particles must be small enough to prevent settling under gravity, unless their density matches that of the suspending medium, or the suspending medium is very viscous. Secondly, we need to prevent particles to clump and therefore need to increase the repulsive interactions between them. Brownian motion of small particle movement promotes particle collisions, which could lead to aggregation and consequently to gravitational settling [3]. Electrostatic and steric stabilization are the most common. There is a possibility for another problem if the particles are big enough inhomogeneities due to its migration with a flow. Particle size should be kept below about 1 µm Zero-Shear Viscosity In 1906, Einstein in his PhD thesis came up with a simple formula for dilute suspensions at very low volume fractions ( 0.03). It was the first analytical solution for hydrodynamics around an isolated sphere. Relative viscosity is expressed as where is the Einstein coefficient, and takes the value. This coefficient is still not incontrovertibly validated, and varies between 1.5 and 5 in different researches [4]. Equation (6) is only valid for suspensions dilute enough for the flow field around one sphere not to be influenced by the presence of neighboring spheres. If two spheres are close enough, they experience hydrodynamic interactions, which leads to a contribution to viscosity that is proportional to. When there are more neighboring inter-particle interactions, higher orders of contribute to viscosity. Such a polynomial dependency can suitably describe semi-dilute suspensions ( 0.25) with the equation (6) (7) 5

6 where we find values of varying from 7.35 to 14.1 derived from the consideration of particle-particle interactions and the value can be even lower when Brownian motion and inertia have a big enough influence. For concentrated suspension ( > 0.25), relationship (7) typically describes viscosity behavior poorly. At the densest possible packing for monodisperse spherical particles, that is, we expect viscosity to be infinite. Therefore, a much more appropriate choice is the Krieger & Dougherty equation ( ) (8) where B is the Einstein coefficient and the maximum packing volume fraction. Binomial expansion of equation (8) also recovers the polynomial given in equation (7). The product usually remains in the range [5]. As the particle aspect ratio increases, increases and decreases. Viscosity is very sensitive to volume fraction at large, thus small errors of could lead to large errors in the value of. When we obtain a steric stabilization of particles, the hard-sphere radius and the volume fraction must be adjusted to avoid these errors (Figure 3a). If the volume fraction is not too close to the maximum packing fraction, the grafted layer can be considered as hard coating and can be corrected with the formula ( ) (9) where is the volume fraction of uncoated particles, the thickness of coating and the particle radius. At high values, the particle-size distribution has a strong effect on viscosity. Figure 3b shows how relative viscosity changes with the fraction of large particles in bidisperse suspension with a 5:1 ratio of diameters. For a total volume fraction greater than 0.60, the viscosity drops by more than a decade as the fraction of large particles increases from 0 to 0.6. This is a consequence of the packing of smaller particles into the interstices between the large ones. a) b) Figure 3 (a) Hard sphere particle grafted with a layer of polymer can be approximated by a hard sphere with bigger radius [6]. (b) Relative viscosity versus fraction of large particles in a bimodal distribution of particle sizes with a 5:1 ratio of diameters is shown at different total volume particle fractions [5]. 6

7 3.1.2 Viscosity under Shear For colloidal dispersions with a volume fraction > 0.3, viscosity becomes sensitive to shear rate (or equivalently shear stress). This occurs when the shear rate is high enough to disturb the distribution of interparticle spacings from equilibrium [7]. At high particle concentrations, suspension has a much greater value of viscosity and it noticeably flows under yield stress. The region where viscosity drops with shear rate is called shear thinning. Hard sphere colloids are similar to atoms, unless they are much bigger and thus well suited for optical microscopy and scattering experiments. That makes the dispersions ideal models for exploring equilibrium and near-equilibrium phenomena of interest in atomic and molecular physics. However, its relevance to atomic scale breaks down for highly non-equilibrium regime. In suspensions, at high shear rates, we witness the growth of viscosity, which is known as shear thickening. Figure 4 illustrates both effects. Figure 4 Viscosity versus shear stress for colloidal latex dispersions at various volume fractions. At high critical yield stress must be applied to flow and at higher stress viscosity increases (shear thickening) [7]. 4 Hydrodynamics In zero-shear-rate suspension there are particles that move with Brownian motion. Diffusivity in dilute solution is given by where k is the Boltzmann constant, T the temperature, the solvent viscosity and r the radius of the particle. It takes seconds for the particle to diffuse the radius distance. Instead of the shear rate, it is more appropriate to define the dimensionless Pécklet number (10) (11) 7

8 This number is a direct indicator of the interparticle hydrodynamic forces in comparison to the Brownian force, which tries to establish equilibrium. For low, Brownian motion dominates, but with large numbers deformation occurs so fast that Brownian motion cannot restore it [7]. Shear thinning is evident for Pécklet numbers around 1, but on the other hand, a much higher triggers the onset of shear thickening. 4.1 Lubrication force The key mechanism that overcomes Brownian motion is lubrication hydrodynamics. Every particle motion must displace an incompressible fluid, which results in a longrange transmission of forces via suspending medium. All particles collectively contribute to local flow field disturbance through hydrodynamic interactions [8]. This is not present in atomic fluids, where the intervening medium is a vacuum and from that point of view it is understandable that we should not observe shear thickening at atomic scale. Let us, for example, take a look at hard-sphere colloids approaching each other in a suspended fluid. If the gap between particle surfaces is much smaller than the particle diameter, lubrication force comes into effect [9]. The average distance between neighboring particles in concentrated suspension is given as follows ( ) (12) where h is the shortest distance between two particles, the radii of the particle and the particle loading. For the average distance between two particles is times the radius. For small gaps ( ) we can apply the lubrication approximation to the Stokes equations. We take into account the leading singular terms from the expansion in the narrow gap, which can be considered as different modes [10]. In Figure 5a, there are two modes with the greatest impact on lubrication force. Squeeze mode can be determined by the leading order of the force on particle as ( ) (13) where we sum over nearest neighbors, is the surface separation between particles, and are the particle velocities, and is the unit vector along the line of the centers between two particles. Equation (13) can also be derived from the Stefan force, which is the interaction between two surfaces in a viscous fluid at small separation [11]. There are also other lubrication modes (shear, pump, and twist). Shear mode has the second greatest impact on lubrication force. It diverges logarithmically with the interparticle gap as we see in ( ) ( ) ( ) (14) 8

9 Here is the unit matrix. The lubrication force increases inversely with the distance between the surfaces of the particles and diverges to a singularity as we see in Figure 5b. The Navier-Stokes equations are time reversible; consequentially the lubrication hydrodynamics have the same effect on particles separation. a) b) Figure 5 (a) In squeeze mode particles move along the line of the particle centers, rotation is not included. Squeeze force acts so as to oppose the relative motion [10]. (b) Lubrication force increases inversely with the distance between particle surfaces and diverge on contact [7]. Very common is the polymer stabilization of the particles, which changes lubrication force and also affects shear thickening characteristics. A polymer coated particle feels an enhanced dissipative force due to the flow of the solvent within the coat. By solving the Brinkman equation (Stokes eqn. extension for porous medium) within the coat (15) where is the flow penetration depth into the brush, we get (see [12] for details) (16) where U is the relative velocity of the particles; the function of penetration depth, length of polymer coat and the distance between hard spheres [12]. At high shear rates, particles are driven into close proximity and remain strongly correlated. The flow-induced density fluctuations are known as hydroclusters. Clustering leads to an increase in energy dissipation and consequently to a higher viscosity [7]. Stokesian dynamics gives the correct weight to the forces in a colloidal motion mechanism. 4.2 Stokesian dynamics The simulation of a colloidal system presents a significant challenge. For a single sphere it can be handled analytically, for two spheres semianalytically and for more particles it requires numerical solution of the Navier-Stokes equation without inertia [7]. There are (17) 9

10 many different methods for describing fluids, most of them start with the Langevin equation for N-particle dynamics, where is 6N 6N mass and moment-of-inertia matrix (generalized mass), is 6Ndimensional particle velocity vector, represents the interparticle and external forces, hydrodynamic forces and Brownian forces. This equation must be solved at each time step to determine the particle velocities. In Stokes flow the hydrodynamic forces are linearly related to the particle velocities as where is the hydrodynamic resistance matrix, the elements of which contain details about the hydrodynamic interactions between particles. In a simulation we need inversion of the resistance matrix which is operation [10]. To overcome this, one can approximately describe the evolution of the system by just including two body terms with divergent lubrication interactions between close particle surfaces, within the elements of the matrix,. Now inversion of the resistance matrix becomes an process and the results of simulation reveal the colloidal microstructure associated with a particular shear viscosity (Figure 6). (18) (19) a) b) Figure 6 (a) Microstructure in hard-sphere colloidal suspensions and relationship between viscosity and shear stress is shown. In equilibrium, random collision prevents flow, increasing shear rate forces particles to organize into layers and viscosity lowers. At yet a higher shear rate interparticle interactions dominate over stochastic ones and there occurs the formulation of hydroclusters, which increases viscosity [7]. (b) The contribution of Brownian and hydrodynamic particle movement to total viscosity shows that at low Pecklet number Brownian force is dominated and at high Pe the lubrication forces [5]. At equilibrium Brownian force prevails, colloids are not correlated. The resistance to flow is naturally high, because of shearing, the random distribution of particles causes them to frequently collide [8]. At low shear stress Brownian contribution disappears, leaving only a hydrodynamic contribution (see Figure 6b). Simulations show the formation of layers of particles parallel to the flow direction [13], but they are not necessarily rigorously ordered, and layer thickness can range from one to multiple particle diameters. The flow becomes streamlined and the ease of movement of 10

11 colloidal particles reduces the viscosity of the system. At the onset of shear thickening, a hydrodynamic instability causes particles to be driven out of these layers. Particles interact through clustering with lubrication and frictional forces involved. The difficulty of particles flowing around each other results in a higher rate of energy dissipation and increase in viscosity. Rheo-optical measurements also confirmed the predictions of the simulation [14; 15]. 5 Applications 5.1 Liquid Armor During the last 10 years the US Army has shown great interest in shear thickening fluids (STF). They came to the conclusion that impregnating STF into Kevlar fabrics improved its performance. It seems that the most suitable STF for such application is the suspension of silica particles in ethylene glycol. The average particle diameter in research letters was measured by dynamic light scattering and determined to be 446 nm [16]. Ethylene glycol is an organic compound widely used as automotive antifreeze. It was chosen as a solvent due to its volatility and thermal stability. The silica particles were, for better dispersibility, predispersed in methanol and then blended with ethylene glycol. Afterwads the dispersion was treated by homogenization for 1h and sonication for 10h to further improve the dispersibility [17]. At the end density of the silica particles in solution was estimated to be 1.78 kg/l ( ). Experimental research suggests that the best way to include STF into body armor vest is to impregnate it into Kevlar fabrics. First of all, the fabrics are immersed in the prepared STF, and are then, when wet, squeezed by a 2-roll mangle to set the specific wet pick up and to improve the STF infiltration. At the end, the fabrics are dried in a vacuum oven at 65 C for 20 min. Figure 7 indicates the comparison of six different ways of STF inclusion tested. Figure 7 Six different setups of Kevlar and STF under study (upper-left). Penetration depth of the projectile in the clay witness (bottom-left) prefers Kevlar impregnated with STF to other configurations. In the right graph the energy dissipation percentage is calculated for each of the setups [16]. 11

12 Even though the dissipation energy does not vary much, there is another important factor for protection, that is penetration depth. Body armor standards require that penetration depth into a clay witness should not exceed 4.3 cm [16]. To satisfy the conditions, approximately layers of neat Kevlar fabric are required. Such a vest is heavy and not very flexible. Impregnation of STF into Kevlar is therefore the best option, because it not only absorbs more energy, but also spreads its impact across a wider area, resulting in a lower value of penetration depth. For comparison see Table 2. Table 2 Flexibility and thickness for pure Kevlar and STF impregnated Kevlar [16]. The other great advantage of STF impregnated over neat Kevlar is the protection against stabbing [17]. It is presumed that the STF, which fills the interstices between Kevlar filaments, keeps the arrangement of the fibers in the yarn, which induces an increase in the endurance of the yarn under pulling stress (Figure 8). a) b) Figure 8 (a) Photos of front and rear sides of neat Kevlar and STF-Kevlar after ice pick drop test [18]. (b) Experimental results of spike test with different loads confirm the improvement of armor with STF impregnated [17]. 5.2 Protective Equipment British company d3o Lab developed shear thickening material for use in protection purposes [19]. The structure of this material is unknown to the public. They are currently trying to negotiate a contract with the Ministry of Defense. The material is called d3o and is widely spread in the sports equipment industry. Ski helmets, footwear, gloves, jackets etc. are just a few products that contain d3o as a shock absorber. The most advertised feature is flexibility in contrast to hardness, for when an accident occurs. 12

13 5.3 Other Applications In the oil industry, many costly and dangerous problems can be encountered when drilling, such as gas influx, lost circulation or underground blowouts [20]. They often fail to seal the loss zone with lost circulation materials or cements. A blowout usually occurs when a drill reaches a gas pocket. When drilling, the shear thickening suspension is still liquid and can be pumped to the wellbore. STF reacts as an immediate patch due to the stress caused by a sudden blowout and it solidifies. Another wide-spread use of STF is traction control, which comes into play in the automotive industry [21]. To provide the power transfer between the front and the rear wheels, some all wheel drive systems use a viscous coupling unit. Imagine two coaxial cylinders rotating in the same directions, one at the speed of the front wheels and the other at the speed of the rear wheels. The space between them is full of STF. When the grip of all wheels is good, the relative motion is zero, thus low or no shear is presented. At the moment of the front wheel slip, STF is affected by a high shear stress, which leads to shear thickening. The power of the front wheels is transferred to the rear wheels which hopefully push the car out of the slippery zone. Since they used STF with very limited maximum viscosity, which limits the amount of torque that can be passed across the coupling, this system is more appropriate for on-road vehicles. 6 Conclusion There are many good things about shear thickening and there are also industrial problems especially for material treatment at high frequencies. In both cases it is very welcome to know the origin and background of this phenomenon. Even if it is not completely surveyed the science surrounding shear thickening in colloidal dispersions, it is a good start for further investigations and applicative usage in everyday life. 13

14 Literature [1] A. P. Deshpande, J. M. Krishnan and P. B. S. Kumar, Rheology of Complex Fluids (Springer, New York, 2010). [2] ( ). [3] C. Tropea, A. L. Yarin and J. F. Foss, Springer Handbook of Experimental Fluid Mechanics (Springer, New York, 2007). [4] S. Mueller, E. W. Llewellin and H. M. Mader, Phil. Trans. R. Soc. A 466, 1201 (2010). [5] R. G. Larson, The Structure and Rheology of Complex Fluids (Oxford, New York, 1999). [6] J. M. Brader et al., J. Phys.: Condens. Matter 22, (2010). [7] N. J. Wagner and J. F. Brady, Physics Today 62, 27 (2009). [8] R. L. Hoffman et al., J.Rheol. 42, 111 (1998). [9] S. W. Lee, S. H. Ryu and C. Y. Kim, J. Non-Newtonian Fluid Mech. 110, 1 (2003). [10] A. A. Catherall and J. R. Melrose, J. Rheol. 44, 1 (2000). [11] A. Borštnik, R. Podgornik and M. Vencelj, Rešene naloge iz mehanike kontinuov (DMFA, Ljubljana, 2001). [12] A. A. Potanin and W. B. Russel, Physical Review E 52, 730 (1995). [13] P. J. Mitchell and D. M. Heyes, Molecular Simulation 15, 361 (1995). [14] S. J. Lee and N. J. Wagner, Ind. Eng. Chem. Res. 45, 7015 (2006). [15] F. Ianni, Complex behavior of colloidal suspensions under shear: dynamic investigation through light scattering techniques (doctoral dissertation, 2006). [16] Y. S. Lee, E. D. Wetzel and N. J. Wagner, J. Mat. Sci. 38, 2825 (2003). [17] T. J. Kang, K. H. Hong and M. R. Yoo, Fibers and Polymers 11, 719 (2010). [18] ( ). [19] ( ). [20] C. Hamburger et al., J. Petrol. Tech. 37, 499 (1985). [21] ( ). 14