Extensional viscosity

The elongational viscosity ( English elongational viscosity ) with the unit Pascal second (Pa.s) is a measure of the resistance of a substance to the flow in a Dehn flow . ${\displaystyle \eta _{\mathrm {e} }}$

For fluids with Newtonian flow behavior , the extensional viscosity can be calculated from its constant ratio to the classically determined shear viscosity ( Trouton ratio). For non-Newtonian fluids, this ratio is not constant, and the extensional viscosity must be determined experimentally as an independent parameter with the aid of an extensional rheometer .

In practically all technical processes, expansion flows occur in addition to the shear flows. They are particularly useful in many manufacturing processes in the plastics industry , e.g. B. in fiber spinning , film blowing , deep drawing , foaming or hollow body blowing , the dominant flow form. Furthermore, there are expansion flows in the reduction or enlargement of cross-sections, also known as capillary inlet and outlet flows (corresponds to the Venturi effect ).

general definition

From a kinematic point of view, two extensional viscosities can be defined, which are generally independent of one another:

${\displaystyle \eta _{\mathrm {e} ,1}={\frac {\sigma _{11}-\sigma _{33}}{\dot {\varepsilon }}}}$is called planar and describes the pressure required to stretch the fluid in 1-direction

and

${\displaystyle \eta _{\mathrm {e} ,2}={\frac {\sigma _{22}-\sigma _{33}}{\dot {\varepsilon }}}}$describes the pressure required to prevent deformation perpendicular to the 1-direction.

With

• ${\displaystyle \sigma _{ii}}$the normal stress in the coordinate direction ${\displaystyle i}$
• ${\displaystyle {\dot {\varepsilon }}}$the Hencky expansion speed , i.e. the time derivative of the Henky expansion .

The coordinates are chosen so that the greatest stretch speed is. ${\displaystyle {\dot {\varepsilon }}=D_{11}\geq D_{22}\geq D_{33}}$

In general, the extensional viscosity depends on the type of elongation. For incompressible fluids , the conditions of Hauptdehngeschwindigkeiten by a parameter can correlating ${\displaystyle -0{,}5\leq m\leq 1}$

${\displaystyle D={\dot {\varepsilon }}\;\left({\begin{matrix}1&0&0\\0&m&0\\0&0&-(1+m)\end{matrix}}\right)}$, d. H. ${\displaystyle D_{11}={\dot {\varepsilon }}\qquad D_{22}=m\cdot {\dot {\varepsilon }}\qquad D_{33}=-(1+m)\cdot {\dot {\varepsilon }}=-(D_{11}+D_{22})}$

and characterize the elongation by the value of the parameter. Special forms of expansion are called

• uniaxial ${\displaystyle \quad (m=-0{,}5)}$
• equibiaxial ${\displaystyle \quad (m=1)}$
• planar ${\displaystyle \quad (m=0)}$

Uniaxial extensional viscosity

The uniaxial ( uniaxial ) expansion is the most important form of expansion in rheology . Here, two normal stresses are identical, so that only one of the two extensional viscosities has a value other than zero:

${\displaystyle \sigma _{22}=\sigma _{33}\Rightarrow \eta _{\mathrm {e,2} }=0}$

The uniaxial extensional viscosity is expediently often only abbreviated as :, but the Greek letter is also used as a formula symbol. ${\displaystyle \eta _{\mathrm {e} }}$ ${\displaystyle \eta _{\mathrm {e,1} }=\eta _{\mathrm {e} }}$ ${\displaystyle \mu }$

For Newtonian fluids, the uniaxial extensional viscosity is three times the shear viscosity , resulting in a Trouton ratio of: ${\displaystyle \eta _{\mathrm {s} }}$

${\displaystyle {\mathit {Tr}}={\frac {\eta _{\mathrm {e} }}{\eta _{\mathrm {s} }}}=3}$

This is partly the relationship between shear and tensile modulus for a Poisson's ratio , d. H. an incompressible fluid. ${\displaystyle \nu =-{\frac {D_{22}}{D_{11}}}=-{\frac {m\cdot {\dot {\varepsilon }}}{\dot {\varepsilon }}}=-m=0{,}5}$

In the case of non-Newtonian fluids, deviations from : ${\displaystyle {\mathit {Tr}}=3}$

• for materials with fillers one is often less Trouton ratio found what has been called Dehnentfestigung (in English strain softening designated):

${\displaystyle {\mathit {Tr}}<3}$

This effect is technically undesirable because it negatively affects the processing behavior of plastics.

• Materials with certain molecular properties, v. a. with long chain branches , however, have a higher extensional viscosity than would be expected from the Trouton ratio. This is known as strain hardening (Engl. Strain hardening ) shall apply:

${\displaystyle {\mathit {Tr}}>3}$

The strain hardening has a positive effect on plastics processing because it can lead to self-healing of inhomogeneities and thus to improved process stability.

Equibiaxial extensional viscosity

With equibiaxial expansion ( evenly in two axes ), both expansion viscosities and thus both tensions are identical:

${\displaystyle \eta _{\mathrm {e} ,1}=\eta _{\mathrm {e} ,2}=\eta _{\mathrm {e} }\Rightarrow \sigma _{11}=\sigma _{22}}$

The equibiaxial extensional viscosity of Newtian fluids is six times the shear viscosity and thus is the Trouton ratio

${\displaystyle {\mathit {Tr}}={\frac {\eta _{\mathrm {e} }}{\eta _{\mathrm {s} }}}=6}$.

Planar extensional viscosity

Only in the case of a planar elongational flow ( precisely , there ), two different values ​​different from zero result for the two elongational viscosities. For a Newtonian liquid: ${\displaystyle D_{22}=0}$

${\displaystyle \eta _{\mathrm {e} ,1}=2\cdot \eta _{\mathrm {e} ,2}}$

${\displaystyle {\mathit {Tr_{1}}}={\frac {\eta _{\mathrm {e} ,1}}{\eta _{\mathrm {s} }}}=4}$

${\displaystyle {\mathit {Tr_{2}}}={\frac {\eta _{\mathrm {e} ,2}}{\eta _{\mathrm {s} }}}=2}$.

literature

• Introduction of Henckydehnung: Hencky H (1928) About the form of the law of elasticity in ideally elastic materials. Journal for Technical Physics 9 215–220.
• First descriptions of today's common forms of uniaxial extensional rheometers :
  • Constant sample length: Meissner J (1969) Rheometer for the study of mechanical properties of deformation of plastic melts under definite tensile stress. Rheologica Acta 8 (1): 78-88.
  • Constant sample volume: Münstedt H (1979) New Universal Extensional Rheometer for Polymer Melts. Measurements on a Polystyrene Sample. Journal of Rheology 23 (4): 421-436.
• Introduction of the Trouton ratio: Trouton FT (1906) On the viscous traction and its relation to that of viscosity. Proc Roy Soc 77 426

Individual evidence

1. ↑ ^{a b} Christopher JS Petrie: Extensional viscosity: A critical discussion. In: Journal of Non-Newtonian Fluid Mechanics. 137, 2006, pp. 15-23, doi : 10.1016 / j.jnnfm.2006.01.011 .