Viscosity and molecular properties

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On a liquid, a tensile stress decreases pressure; a shear stress causes flow: Def Shear-rate.png
Recall that shear is defined as as the response to a shearing stress (the bold arrow in the figure): Shear.PNG
A linear relation between shear stress and shear rate is called Newtonian flow: Newtonian liquid.png
However, non-linear behavior is common and interesting: Non-Newtonian Flow.png
  • Relationship between velocity gradient and strain rate: Viscosity can be understood from the following thought experiment. Say that a liquid is sandwiched between a static plane (z=0) and a plane (z=L) moving at a speed v. In a steady state, the liquid immediately near the moving plane will be moving at v, while the liquid near the static plane will not be moving. Assuming that the fluid is well-behaved (i.e. "linear"), the velocity will be a linear function of z: v(z) = \dot{\gamma} z, where \dot{\gamma} is the shear rate (in this case, v/L). Thus we can see that the shear rate, in addition to being the time-derivative of the shear, is dv/dz. To see the connection between dv/dz and d\gamma/dt, consider looking at a cube of liquid in the flow. Because the top of the cube is flowing faster than the bottom, the cube will be stretched into a parallelpiped. This stretching will increase with time; the top will be moving at speed h*(dv/dz) with respect to the bottom, thus after time t the stretching is t*h*(dv/dz). The strain is then t*(dv/dz), and thus the strain rate is dv/dz as claimed.

Viscosity and energy

For Newtonian liquids, the viscosity is just an empirical constant: Newtonian liquid.png
But a calculation of work done per time on a Newtonian liquid is illuminating. The viscosity is related to molecular processes (Witten, p. 28): WorkPerTime NewtonianLiquids.png

Viscosity and molecular properties

Think of flow as repeated steps, stretch then relax: AffineTransition.png
The stress per step: Linear modulus Eqn.png
The elastic energy stored per unit volume is: ElasticEnergyPerVolume Eqn.png
The work rate is the work divided by a relaxation time: ElasticWorkRate Eqn.png

We have calculated the work rate per unit volume from the viscosity and from the elasticity, including a characteristic time. Comparing them gives:

Work rate from the viscosity: Work rate from the elascity: Combining gives:
ElasticWorkRateFromViscosity Eqn.png ElasticWorkRate Eqn.png ViscosityFromElasticityRelaxation Eqn.png

The viscosity is related to the energy per unit volume and a relaxation time.

Glass Transition Temperature: "...the viscosity grows too large to be measurable as the liquid cools towards a characteristic temperature Tg...A liquid near its glass transition cannot equilibriate from an imposed distortion in a few atomic collision times." Witten p.31

"In the technical sense, glass is an inorganic product of fusion which has been cooled to a rigid condition without crystallizing"

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