Rheology. Definition of viscosity. Non-newtonian behaviour.

Rheology. Definition of viscosity. Non-newtonian behaviour.

Rheology Rheology is the science of the flow and deformation of matter (liquid or soft solid) under the effect of an applied force Deformation change of the shape and the size of a body due to applied forces (external forces and internal forces) Flow irreversible deformation (matter is not reverted to the original state when the force is removed) Elasticity reversible deformation (matter is reverted to the original form after stress is removed)

Applications of rheology Understanding the fundamental nature of a system (basic science) Quality control (raw materials and products, processes) Study of the effect of different parameters on the quality of a product Tuning rheological properties of a system has many applications in every day's life Pharmaceutics Cosmetics Chemical industry Oil-drilling etc

Deformation Solids or liquids in rest keep their shape (=form) unchanged When forces act on these bodies, deformation can occur if the force exerted is larger than the internal forces holding the body in its original form Deformation is the transient or permanent shape change of a given body transient or reversible deformation (elasticity): when the force acting upon the body ends, the shape reverts to its original state and the deformation work (=energy) is recovered permanent or irreversible deformation (flow): shape does not revert to its original state, the deformation energy can not be recovered

Deformation forces The deformation forces (also often called loading) which act on a solid body or a liquid can be Static: the force is acting constantly and its direction and magnitude are constant (constant loading) Dynamic: the magnitude and/or direction of the force(s) are variable as a function of time (variable loading) cyclic acyclic

Deformation forces

Definitions Strain: deformation in term of relative displacement of the particles composing the body Stress: measure of internal forces acting within a (deformable) body Shear: deformation of a body in one direction only (resulting from the action of a force per unit area τ=shear stress) and having a given perpendicular gradient (γ=shear strain)

Ideal and real bodies Ideal bodies 1. Ideally elastic: Hookean body (only reversible deformation, linear relation between stress and strain) 2. Ideally viscous: Newtonian fluids (continuous irreversible deformation, flow) 3. Ideally plastic: (no permanent deformation below the yield stress, and continuous shear rate at and above the yield stress.) Real bodies (combination of the properties above) 1+2: viscoelastic materials 2+3: plastic materials

Elastic deformation, ideally elastic bodies For ideally elastic bodies, there is a linear relationship between the relative deformation and the applied force (observation of R. Hooke on springs) h = h 0 Relative deformation (=strain): ε= Δ l Hooke's law: τ=εe l 0 Shear stress: F τ = (in N/m 2 = Pa) A yz (without unit) E is Young's modulus (in Pa), the measure of the stiffness of an isotropic elastic material. For e.g. rubber: E = 0.01 GPa = 1 10 4 Pa steel: E = 200 GPa = 2 10 8 Pa

Shearing deformation of solids If a tangential force is acting on the upper plane of a body fixed at its base a shearing deformation will result h < h 0 Shear stress: F τ = (in N/m 2 = Pa) A xz The deformation will vary perpendicularly with the distance from the base to the maximal shear plane: dx = f(y) and dx max = f(h) The gradient of the shear in this perpendicular direction is called shear strain: γ = dx dy = dx max (without unit) h

Shearing deformation of liquids In liquids, a constant shear will cause the liquid to flow (viscous deformation). If the flow is laminar (there are no turbulences) the liquid flows as layers parallel to the wall of the vessel. The velocity of these layers is decreasing from a maximal value to zero in the direction perpendicular to the wall (the layer adsorbed at the wall does not move). The gradient of the shear in this perpendicular direction is also called shear strain: γ = dx (without unit) dy But as the layers of liquid are constantly moving (dx is not constant) we can define a velocity gradient from the bulk to the wall called shear rate: D = dx /dt dy = dv x dy (unit: 1 s = s 1 )

Newtonian liquids In Newtonian liquids shear rate (D) is linearly proportional to shear stress (τ): τ = ηd The proportionality coefficient η (called viscosity) is constant in the case of Newtonian liquids: η = const. Viscosity is the measure of resistance against flow. η (Pa s) τ (Pa) η = tg α = τ/d α τ (Pa) Viscosity curve Flow curve D (s -1 )

Ideally plastic bodies Ideally plastic bodies would behave as rigid bodies until a yield value of shear and flow as Newtonian liquids above the yield value: τ = τ 0 +ηd These bodies are termed ideal Bingham bodies. They are practically non-existent. τ (Pa) A mechanical analogue to plastic deformation is the frictional resistance to sliding of a block on a plane. No displacement occurs until the applied stress reaches the frictional resistance. α τ 0 No flow until the yield stress Viscosity curve D (s -1 )

Real materials In practice only a few materials have an ideal flow behavior Usually rheological properties are a combination of viscous, elastic, and plastic properties Moreover these properties change most often non-linearly Sometimes the sample is subject to breakdown if sheared, in this case small dynamic strain or stress is applied during rheological measurements Oscillation: small oscillating τ is applied Creep: small constant τ is applied and watch strain increase Relaxation: small strain is applied and watch the decay of τ

Non-newtonian viscosity If the relation between shear stress and shear rate is not linear: non-newtonian viscosity Viscosity varies with the shear: η = f(τ) or η = f(d) Most viscous materials are non-newtonian Non-newtonian behavior depends on the micro- or nanostructure of the material (breakdown, arrangement, or entanglement) SHEAR-THINNING SHEAR-THICKENING τ (Pa) η (Pa s) τ (Pa) η (Pa s) D (s -1 ) D (s -1 ) D (s -1 ) D (s -1 )

The Weissenberg effect A spinning rod is placed in a polymer solution composed of long chains Polymer chains are drawn towards the rod Long polymers get wrapped around the rod Entanglement of the polymer chains make the wrapped chains to stretch The stretched chains pull the free polymers towards the rod Low viscosity High viscosity Newtonian liquid Viscoelastic liquid

Influences on the viscosity η (c,t, p,t ) = τ D Viscosity depends on: concentration (c) temperature (T) pressure (p) time (t) shear rate (D) If the shear rate changes during an application, the internal structure of the sample will change and the change in stress or viscosity can then be seen.

Apparent viscosity The ratio of stress to rate of strain, calculated from measurements of forces and velocities as though the liquid were Newtonian. IUPAC definition η = (τ τ 0) n D Nonlinearity factor This is a general equation valid also for systems having a yield stress value (τ 0 ).

Shear-thinning behavior Structural changes due to the forces changes in viscosity: ordering of molecules or particles ( τ ) n η = n<1 D

Shear-thickening behavior Structural changes due to the forces changes in viscosity, disordering of the particles or molecules E.g. wet sand or mixture of water and cornstarch ( τ ) n η = n>1 D http://video.google.com/videoplay?docid=-4684348427588167444&ei=4jfvstqgi86z-abyhtgrcg&hl=hu#

Example of shear-thickening system Hydrogel: 5% PVA + 5% sodium borate Force~0 : viscous fluid weak force : plastic medium force, : elastic Very strong force, rigid solid http://www.youtube.com/watch?v=f2xq97xhjvw&feature=related

Yield stress τ (Pa) η (Pa s) η = ( τ τ ) n 0 τ D 0 Viscosity curve D (s -1 ) τ 0 Flow curve τ (Pa) Everyday's example: a cardhouse Below the yield value the sample keeps its shape and behaves as a solid body. Above the yield value the structure breaks down and sample start to flow. The yield value shows how strong the structure is.

Explanation of yield value gel The height of the energy barrier indicates how stable the system is. Vmax>>kT kinetically stable sol ~ yield value In a secondary minimum a much weaker and potentially reversible adhesion between particles exists in a gel structure. These weak flocs are sufficiently stable not to be broken up by Brownian motion, but may dissociate under an externally applied force such as vigorous agitation

Time-dependent effects When viscosity at a given shear depends on time, the system is Thixotropic: constant shear causes a decrease in viscosity very common property (e.g. ketchup, yoghurt, paints, etc.) Rheopectic: constant shear causes an increase in viscosity very few materials are rheopectic (gypsum paste, printer ink) If time-dependent effects are significant, flow and viscosity curves present a hysteresis loop (curves measured by increasing shear do not coincide with curves measured by decreasing shear) These effects are caused by the breakdown or buildup of ordered structures within the flowing matter

Hysteresis loop Flow curve of thixotropic systems with and without yield stress τ (Pa) Viscoplastic Hysteresis loops τ 0 Red: with increasing shear rate, system is breaking down Viscous Blue: with decreasing shear rate, system is building up D (s -1 )

Flow curves

Polymer solutions Dilute polymer solutions have generally shearthinning properties Viscosity of these solutions increases with increasing molar weight hydrodynamic radius of the polymer coil increases with molar weight larger radius means more pronounced interaction with solvent molecules (= friction ) increase in viscosity Empirical relation between (intrinsic) viscosity and molecular weight: the Mark-Houwink equation

Molar weight determination by viscosity Relative viscosity η r = η solution η solvent Specific viscosity η sp = η r 1 = η solution η solvent 1 250 Graphical determination of [η] 200 150 100 50 0 η spec /c ln η rel /c 0 0.02 0.04 0.06 c, g/ml Mark-Houwink equation [η] = K M a [η]: intrinsic viscosity K: empirical constant M: molar mass a: solvent-polymer interaction parameter

Dynamic measurements Stress relaxation (recoil, loosen up, be tired out) Small oscillation stress and strain shift D Elastic term in phase (δ=0), viscous term out of phase (δ=90 ), viscoelastic (δ~45 )