Abstract
A rheological model has been derived for the linear-viscoelastic behaviour of a dispersion of transversely rigid spherical capsules. The model incorporates finite thickness of the elastic shell of the capsules, anisotropy of the mechanical properties of the interface and finite volume fraction. The dynamic viscosity of the dispersion is calculated. The influence of the microstructural parameters is considered and the results are compared with those of other models. The model shows that finite thickness of the shell can strongly influence the relaxation times.
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Abbreviations
- a :
-
particle radius
- A :
-
6 × 6 matrix
- b :
-
radius of the cell
- B :
-
6 × 6 matrix
- C :
-
6 × 1 matrix
- D :
-
rate of strain tensor
- e :
-
unit vector
- E :
-
Young's modulus
- f :
-
correction factor
- F λ :
-
force vector
- g λ :
-
scalar quantity defined in eq. (20)
- G :
-
constant proportional to the applied rate of strain
- h :
-
shell thickness
- k :
-
particle index
- L :
-
(=h/2a m ) relative shell thickness
- m λ :
-
resultant surface moment per unit surface
- M λ :
-
resultant bending moment per unit length
- M λµ :
-
resultant twisting moment per unit length
- n :
-
normal vector
- N λ :
-
resultant normal force per unit length
- N λµ :
-
resultant shear force per unit length
- p :
-
pressure
- p 0 :
-
equilibrium pressure
- q λ :
-
resultant loading force per unit surface
- Q λ :
-
resultant shear force per unit length
- r :
-
spherical coordinate
- r :
-
position vector
- R :
-
(=b/a m ) relative cell radius
- R λ :
-
radius of curvature
- s :
-
displacement vector
- s λ :
-
component ofs
- S :
-
area
- t :
-
time
- T :
-
stress tensor in the fluid
- T rr ,T ør :
-
components ofT
- u :
-
velocity vector
- u r ,u ø :
-
spherical components ofu
- u 0 :
-
velocity vector at the sample boundary
- V :
-
(= η(i)/η(e)) viscosity ratio
- V c :
-
cell volume
- ∂V c :
-
cell surface
- V p :
-
particle volume
- ∂V p :
-
particle surface
- V s :
-
sample volume
- ∂V s :
-
sample surface
- X 1, ⋯X 6 :
-
functions ofR, L, V andv
- Y :
-
6 × 1 matrix
- Y 1, ⋯Y 6 :
-
components ofY
- Z :
-
(= 2ω η (e) a m /(Eh))
- α, β :
-
curvilinear surface coordinates
- y λµ :
-
strain component
- ε λ :
-
strain component
- ζ :
-
distance along the normal to the middle surface
- η :
-
viscosity
- η * :
-
(=η′ − i η″) complex viscosity
- η *spec :
-
(= (η * − η(e))/η(e) = η ′spec −i η ″spec ) specific complex viscosity
- θ :
-
spherical coordinate
- θ λ :
-
strain component
- ϰ:
-
surface dilatational modulus
- K λ :
-
strain component
- µ :
-
surface shear modulus
- v :
-
Poisson ratio
- σ λ :
-
normal force per unit surface
- τ λµ :
-
shear force per unit surface
- φ :
-
spherical coordinate
- Φ :
-
volume concentration
- ω :
-
angular frequency
- ext:
-
including shell volume
- int:
-
excluding shell volume
- l :
-
longest
- m :
-
middle surface
- s :
-
shortest
- α, β, ζ :
-
component inα, β, ζ direction
- e :
-
external fluid
- i :
-
internal fluid
- ζ :
-
quantity at a distanceζ from the middle surface
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de Bruijn, R.A., Mellema, J. The linear-viscoelastic behaviour of a dispersion of transversely rigid spherical capsules. Rheol Acta 24, 159–174 (1985). https://doi.org/10.1007/BF01333244
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DOI: https://doi.org/10.1007/BF01333244