Viscosity describes the ability of a liquid to resist strain. For a liquid a constant applied stress will result in a time-dependent strain.
A liquid is imagined to be placed between parallel plates of area A separated by a distance y.
The shear stress σ is
σ = F/A
and the shear strain e is
e = Δx/y.
The shear modulus is
G = σ /e.
If the plates are moved with a relative velocity v, the force F resisting the
relative motion of the plates is given by
F = A η v/y,
where η is the viscosity of the fluid.
The shear stress is in terms of the viscosty:
σ = η de/dt.
Hooke law σ(t) = G γ(t), where G is the shear modulus
Newtonian behavior σ(t) = η dγ(t)/dt, where η
is the shear viscosity
M. Kiss, N. Hagemeister, A. Levasseur, J. Fernandes, B. Lussier, Y. Petit. A low-cost thermoelectrically cooled tissue clamp for in vitro cyclic loading and load-to-failure testing of muscles and tendons. Medical Engineering & Physics, Volume 31, Issue 9, Pages 1182-1186 http://linkinghub.elsevier.com/retrieve/pii/S1350453309001271
Perfectly elastic solid is sometimes called a Hookean solid. It has the following properties:
Newtonian fluid
Properties of perfectly viscous liquid include:
Non-Newtonian fluid is a fluid in which the viscosity changes with the applied shear force. As a result, non-Newtonian fluids may not have a well-defined viscosity.
http://en.wikipedia.org/wiki/Non-Newtonian_fluid
Example.
J. Park and JE Butler. Analysis of the Migration of Rigid Polymers and Nanorods in a Rotating Viscometric Flow. Macromolecules, Article ASAP. http://pubs.acs.org/doi/abs/10.1021/ma901369a
D. W. Mead and R. G. Larson.A Molecular Theory for Fast Flows of Entangled Polymers. Macromolecules, 1998, 31 (22), pp 7895–7914 http://pubs.acs.org/doi/abs/10.1021/ma980127x
Viscoelastic material shows both viscous and elastic behavior. The response of a viscoelastic material depends on time.
Examples of viscoelastic materials: polymeric liquids
Properties of viscoelastic materials:
The time dependent, or viscoelastic, behavior of soft tissue can be attributed to a variety of mechanisms:
The time T is called the relaxation time.
Polymeric liquids are known as viscoelastic materials.
There is dissipation of energy – and irreversible shape changes – associated
with the flow.
In contrast most solids exhibit pure elasticity.
Energy is stored as elastic energy.
Material returns to original shape once stress removed.
For a viscous liquid with the viscosity η the equation relating stress σ to strain ε is
σ = η dε/dt.
The shape of the object can change irreversibly.
Here a particle in a viscous liquid is considered. Radius of the particle is denoted a. There is external force f on a particle or atom.
Velocity may be written as
v = μ f
where μ is called the mobility. The Einstein relation connects mobility and diffusion coefficient D as
μ = D/kT.
Stokes law says that f = 6πη a v and the viscosity is
η = kT/(6πaD).
As D increases the viscosity decreases.
Effective viscosity may depend on the shear rate. Then in the formula of the shear stress,
σ = η de/dt,
the viscosity η is no more
constant.
Shear thinning fluid becomes easier to make flow as the shear rate gets larger.
A shear thickening fluid flows easily when a low shear rate is applied, but becomes more resistant to flow when sheared at high rate.
Many biological materials - including blood vessels, lung parenchyma, cornea, and blood clots - stiffen as they are strained, preventing large deformations that could threaten tissue integrity.
Storm et al. Nonlinear elasticity in biological gels. Nature vol 435, 12 May
2005, 191-194.
Experiment: Material is subjected to a constant deformation i.e. small strain
γ0 is imposed
in the material. This strain is kept constant and stress decay is monitored as a function of
time.
Shear stress relaxation modulus is defined as
G(t) = σ(t) / γ0
If there is viscous flow the stress can drop to zero.
J(t) = γ(t)/
σ.
Boltzmann superposition principle: For small loading steps one assumes that each loading step makes an independent contribution to the total loading history. The final deformation is the sum of deformations at each step.
Creep is a function of the whole sample loading history.
Each loading step makes independent contribution to total loading history.
Total final deformation is the sum of each contribution.
Strain may be written as
ε(t) = ∫-∞ t J(t - τ) dσ(t)
Stress is
σ(t) = ∫-∞ t G(t - τ) dε(t)/dτ dτ
Zero shear viscosity is the integral
η0 = ∫0∞ G(t – τ) dτ
Hooke
law:
σ1 = E ε1
Newtonian behavior: σ2 = η dε2/dt
Serial σ1 = σ2 = σ and ε = ε1 + ε2
dε/dt = dε1/dt + dε2/dt = 1/E dσ1/dt + σ2/η
dε/dt = 1/E dσ /dt + σ/η
Assume ε(t) = ε0 = constant, thus
dε/dt = 0 and
0 = 1/E dσ /dt + σ/η
Integrating ∫σ0σ dσ/σ = -E/η ∫0t dt
one obtains
σ(t) = σ0 exp(-E/η t).
Maxwell model describes nicely relaxation of stress.
Other models exist, e.g. Voight-Kelvin ”parallel” σ1 + σ2 = σ and ε = ε1 = ε2.
This model describes nicely creep.
Particles vibrate. In deformation, particles move in the liquid to a more high energy positions. In a liquid, motions of individual particles may reduce the stress.
Define instantaneous modulus G0 at t < relaxation time τ, for the viscosity holds approximately
γ ≈ G0 τ.
Frequence of the particle vibration is denoted by v. Time needed for a particle to escape from its high energy position with a barrier height u may be estimated as
1/τ = v exp(-u/kT).
Arrhenius behaviour (high temperatures):
η = G0/v exp(u/kT)
where u energy barrier height, T temperature, and G0 instantaneous shear modulus.
Relaxation time it the time that atom needs to change its position to a lower
energy configuration
Empirical Vogel-Fulcher law for viscosity η:
η = η(0) exp(B/(T-T(0)))
where B is constant and T is temperature.
Oscillatory shear: liquid is strained sinusoidally at some frequency ω in the
linear region with small enough strain amplitude γ0.
γ(t) = γ0 sin ωt input
σ(t) /γ0 = G’(ω) sin ωt + G’’(ω) cos ωt output
Dynamic storage modulus G’(ω)
Dynamic loss modulus G’’(ω)
For
Hookean solid G’(ω) = G and G’’(ω) = 0.
For Newtonian liquid G’(ω) = 0 and G’’(ω) = ηω.
G’(ω) = ω ∫G(ω) sinωt dt and
G’’(ω) = ω ∫G(ω) cosωt dt
Analogy to LCR circuit where voltage V and current I are out of phase for
oscillating signals.
Viscoleastic material: stress and strain are out of phase.
Complex modulus is defined as G = G’ + G’’,
where the real part is the storage modulus G’ and imaginary part the loss modulus G’’.
Let the phase angle be δ and the frequency ω.
Stress σ = σ0 exp(iωt) and strain ε = ε0 exp(iωt)
The modulus can also be given in exponential form
G = σ/ε = σ0/ε0 exp(iωt) .
Energy loss in the cycle ΔE = Re ∫σ dε
The modulus may vary greatly with frequency or time scale of the experiment.
When a usual material is subjected to stress, it resists deformation. Any stable material resists compression—even liquids. Solids resist simple shear deformations that conserve volume. Under shear most materials show positive normal stress: a tendency to expand in the direction perpendicular to the applied shear stress.
Networks of semiflexible biopolymers exhibit the opposite tendency: When sheared between two plates, they tend to pull the plates together. These negative normal stresses can be as large as the shear stress. This property is directly related to the nonlinear strain-stiffening behaviour of biopolymer gels.Gels formed by semiflexible polymers that become stiffer the more they are deformed, generate normal forces that are both larger in magnitude and of the opposite sign.
PA Janmey et al. Negative normal stress in semiflexible biopolymer gels. Nature Materials 6, 48 - 51 (2007) http://www.nature.com/nmat/journal/v6/n1/full/nmat1810.htmlRheology means studies of the flow of matter. The materials may be liquids or soft solids. Solids under conditions in which they flow rather than deform elastically may also be studied.
Examples. Substances which have a complex structure, including polymers, suspensions, many foods, and many biological materials.
Instruments used to characterize the rheological properties of materials are called rheometers. These instruments impose a specific stress field or deformation to the fluid, and monitor the deformation or stress. Instruments can be run in steady flow or oscillatory flow, in both shear and extension.
Liquid is forced through a tube of constant cross-section under conditions of laminar flow. Either the flow-rate or the pressure drop are fixed and the other measured. Knowing the dimensions of the tube, the flow-rate can be converted into a value for the shear rate and the pressure drop into a value for the shear stress.
Rheometer at synchrotron or neutron beam line
Examples of rheology studies
DMA http://en.wikipedia.org/wiki/Dynamic_mechanical_analysis