Rheology. Definition of viscosity. Non-newtonian behaviour.

Rheology

● Rheology is the science of the flow and deforma-tion of matter (liquid or “soft” solid) under the ef-fect 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 re-vert to its original state, the deformation energy can not be re-covered

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 displace-ment 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 deforma-tion, linear relation between stress and strain)

2. Ideally viscous: Newtonian fluids (continuous irre-versible 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)

Relative deformation (=strain): ε=Δ ll 0

(without unit)

Hooke's law:τ=εE

Shear stress:

τ = F

Ayz

(in N/m2 = Pa)

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 = 1·104 Pa steel: E = 200 GPa = 2·108 Pa

h = h0

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

The deformation will vary perpendicularly with the distance from the base to the maximal shear plane: dx = f(y) and dxmax= f(h)

The gradient of the shear in this perpendicular direction is called shear strain:

γ = dxdy

= dx max

h (without unit)

Shear stress:

τ = F

Axz

(in N/m2 = Pa)h < h

0

Shearing deformation of liquids

● In liquids, a constant shear will cause the liquid to flow (viscous de-formation).

● 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:

● 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:

γ = dxdy

(without unit)

D = dx /dt

dy =

dv x

dy (unit:

1s

= s−1 )

Newtonian liquids

● In Newtonian liquids shear rate (D) is linearly proportional to shear stress (τ):

● The proportionality coefficient η (called viscosity) is constant in the case of Newtonian liquids: η = const.

● Viscosity is the measure of resistance against flow.

τ = ηD

τ(Pa)

D (s-1)

α

η(Pa·s)

τ (Pa)

Viscosity curve Flow curve

η = tg α = τ/D

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:

● These bodies are termed ideal Bingham bodies. They are practically non-existent.

τ = τ0+ηD

τ(Pa)

D (s-1)

α

τ0

No flow untilthe yield stress

A mechanical analogue to plastic deforma-tion is the frictional resistance to sliding of a block on a plane. No displacement oc-curs until the applied stress reaches the frictional resistance.

Viscosity curve

Real materials

● In practice only a few materials have an ideal flow behavior● Usually rheological properties are a combination of viscous, elas-

tic, 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)

τ(Pa)

D (s-1)

η(Pa·s)

D (s-1)

τ(Pa)

D (s-1)

η(Pa·s)

D (s-1)

SHEAR-THINNING SHEAR-THICKENING

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

Newtonian liquid Viscoelastic liquid

Low viscosity High viscosity

Influences on the viscosity

η (c ,T , p , t ) = τD

Viscosity depends on:● concentration (c)● temperature (T)● pressure (p)● time (t)● shear rate (D)

If the shear rate changes during an ap-plication, the internal structure of the sample will change and the change in stress or vis-cosity can then be seen.

Apparent viscosity

η = (τ−τ0)

n

D

The ratio of stress to rate of strain, calculated from measure-ments of forces and velocities as though the liquid were Newto-nian. IUPAC definition

This is a general equation valid also for systems having a yield stress value (τ0).

Nonlinearity factor

Shear-thinning behavior

Structural changes due to the forces – changes in viscosity: ordering of molecules or particles

( ) nD

τη = n<1

Shear-thickening behavior

Structural changes due to the forces – changes in viscosity, disordering of the particles or molecules

( ) nD

τη = n>1

http://video.google.com/videoplay?docid=-4684348427588167444&ei=4JfVStqgI86z-AbYhtGrCg&hl=hu#

E.g. wet sand or mixture of water and cornstarch

Example of shear-thickening system

Very strong force, rigid solid

http://www.youtube.com/watch?v=f2XQ97XHjVw&feature=related

Hydrogel: 5% PVA + 5% sodium borate

Force~0 : viscous fluid

weak force : plastic

medium force, : elastic

Yield stress

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.

τ(Pa)

D (s-1)

τ0

η = ( τ−τ0)

n

D

η(Pa·s)

τ (Pa)τ0

Viscosity curve Flow curve

Explanation of yield value

~ yield value

The height of the energy bar-rier indicates how stable the system is.

Vmax>>kT kinetically sta-ble sol

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

gel

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

Hysteresis loops

Viscoplastic

Viscous

τ(Pa)

D (s-1)

τ0

Red: with increasing shear rate, sys-tem is breaking down

Blue: with decreasing shear rate, system is building up

Flow curves

Polymer solutions

● Dilute polymer solutions have generally shear-thinning properties

● Viscosity of these solutions increases with increas-ing 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

0

50

100

150

200

250

0 0.02 0.04 0.06c, g/mL

ηspec/c

ln ηrel/c

[η]: intrinsic viscosity

K: empirical constant

M: molar massa: solvent-polymer interaction parameter

Mark-Houwink equation

[η] = K Ma

ηsp = ηr−1 = ηsolutionηsolvent

−1

Specific viscosity

ηr = ηsolutionηsolvent

Relative viscosity

Graphical determination of [η]

Stress relaxation (recoil, loosen up, be tired out)

D

Small oscillation stress and strain

Elastic term in phase (δ=0),viscous term out of phase (δ=90°),viscoelastic (δ~45°)

shift

Dynamic measurements