The interfacial stress rheometer (ISR), uses the oscillations of a magnetic needle suspended on an interface to characterize the dynamic moduli of thin films. Mathematical theories to interpret the device have developed slowly because of the strong coupling between the stresses in the surface and the bulk subphase. In this work, we simplify the equations of motion by introducing new length scales and reinterpreting the dimensionless numbers. Several Green's functions are developed for typical ISR geometries, leading to a set of boundary element methods for the full numerical solution of the equations of motion. Using Taylor series, a multipole expansion is extracted from the boundary integral equations, and we show that both numerical methods converge in under five elements. Analytical theories are developed for the cases of small and large interfacial stress, proving that the finite size of the needle has an O(1) effect and reinforcing the physics behind the length scales and dimensionless groupings. We directly compare our numerical and analytical solutions to published interfacial velocity data, showing good agreement, and discuss the implications of our results.

1.
Abramowitz
,
M.
, and
I.
Stegun
,
Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables
, 9th ed. (
Dover Publications, Inc.
,
New York
,
1965
).
2.
Anseth
,
J.
,
A.
Goffin
,
G.
Fuller
,
A.
Ghio
,
P.
Kao
, and
D.
Upadhyay
, “
Lung surfactant gelation induced by epithelial cells exposed to air pollution or oxidative stress
,”
Am. J. Respir. Cell Mol. Biol.
33
,
161
168
(
2005
).
3.
Brooks
,
C.
,
G.
Fuller
,
C.
Frank
, and
C.
Robertson
, “
An interfacial stress rheometer to study rheological transitions in monolayers at the air-water interface
,”
Langmuir
15
,
2450
2459
(
1999
).
4.
Choi
,
S.
,
S.
Steltenkamp
,
J.
Sasadzinski
, and
T.
Squires
, “
Active microrheology and simultaneous visualization of sheared phospholipid monolayers
,”
Nat. Commun.
2
,
1
6
(
2011
).
5.
Ding
,
J.
,
H.
Warriner
, and
A.
Zasadzinski
, “
Magnetic needle viscometer for Langmuir monolayers
,”
Langmuir
18
,
2800
2806
(
2002
).
6.
Erni
,
P.
,
E.
Windhab
,
R.
Gunde
,
M.
Graber
,
B.
Pfister
,
A.
Parker
, and
P.
Fischer
, “
Interfacial rheology of surface-active biopolymers: Acacia senegal gum versus hydrophobically modified starch
,”
Biomacromolecules
8
,
3458
3466
(
2007
).
7.
Fischer
,
T.
, “
The drag on needles moving in a Langmuir monolayer
,”
JFM
498
,
123
137
(
2004
).
8.
Fischer
,
T.
,
P.
Dhar
, and
P.
Heinig
, “
The viscous drag of spheres and filaments moving in membranes or monolayers
,”
JFM
558
,
451
475
(
2006
).
9.
Freer
,
E.
,
T.
Svitova
, and
C.
Radke
, “
The role of interfacial rheology in reservoir mixed wettability
,”
Engineering
39
,
137
158
(
2003
).
10.
Gradshteyn
,
I.
, and
I.
Ryzhik
,
Table of Integrals Series and Products
, 4th ed. (
Academic Press Inc.
,
New York
,
1965
).
12.
Kim
,
Y.
,
J.
Pyun
,
J.
Frechet
,
C.
Hawker
, and
C.
Frank
, “
The dramatic effect of architecture on self-assembly of block copolymers at interfaces
,”
Langmuir
21
,
10444
10458
(
2005
).
13.
Leal
,
G.
,
Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes
(
Cambridge University Press
,
New York
,
2007
).
14.
Levine
,
A.
,
T.
Liverpool
, and
F.
MacKintosh
, “
Dynamics of rigid and flexible extended bodies in viscous films and membranes
,”
Phys. Rev. Lett.
93
,
038102
(
2004
).
15.
Ooura
,
T.
, “
A double exponential formula for the Fourier transforms
,”
Publ. RIMS
41
,
971
977
(
2005
).
16.
Reynaert
,
S.
,
C.
Brooks
,
P.
Moldenaers
,
J.
Vermant
, and
G.
Fuller
, “
Analysis of the magnetic rod interfacial stress rheometer
,”
J. Rheol.
52
,
261
285
(
2008
).
17.
Saffman
,
P.
, “
Brownian motion in thin sheets of viscous fluid
,”
JFM
73
,
593
602
(
1976
).
18.
Scriven
,
L. E.
, “
Dynamics of a fluid interface: Equation of motion for Newtonian surface fluid
,”
Chem. Eng. Sci.
12
,
98
108
(
1960
).
19.
Shlomovitz
,
R.
,
A.
Evans
,
T.
Boatwright
,
M.
Dennin
, and
A.
Levine
, “
Measurement of monolayer viscosity using noncontact microrheology
,”
Phys. Rev. Lett.
110
,
137802
(
2013
).
20.
Stone
,
H.
, and
A.
Ajdari
, “
Hydrodynamics of particles embedded in a flat surfactant layer overlying a subphase of finite depth
,”
JFM
369
,
151
173
(
1998
).
21.
Vandebril
,
S.
,
A.
Franck
,
G.
Fuller
,
P.
Moldenaers
, and
J.
Vermant
, “
A double wall-ring geometry for interfacial shear rheometry
,”
Rheol. Acta
49
,
131
144
(
2010
).
22.
Verwijlen
,
T.
,
P.
Moldenaers
,
H.
Stone
, and
J.
Vermant
, “
Study of the flow field in the magnetic rod interfacial stress rheometer
,”
Langmuir
27
,
9345
9358
(
2011
).
23.
Wilks
,
G.
, “
The flow around a semi-infinite oscillating plate and the skin friction on arbitrarily cross-sectioned infinite cylinders oscillating parallel to their length
,”
Proc. Cambridge Philos. Soc.
66
,
163
(
1969
).
24.
Yim
,
K.
,
B.
Rahaii
, and
G.
Fuller
, “
Surface rheological transitions in Langmuir monolayers of bi-competitive fatty acids
,”
Langmuir
18
,
6597
6601
(
2002
).
You do not currently have access to this content.