REVIEW OF SCIENTIFIC INSTRUMENTS 83, 043908 (2012)
Rheology of ﬂuids measured by correlation force spectroscopy
Christopher D. F. Honig,
John Y. Walz,
Mark R. Paul,
and William A. Ducker
Department of Chemical Engineering, Virginia Tech, Blacksburg, Virginia 24060, USA
Department of Mechanical Engineering, Virginia Tech, Blacksburg, Virginia 24060, USA
(Received 5 January 2012; accepted 1 April 2012; published online 23 April 2012)
We describe a method, correlation force spectrometry (CFS), which characterizes ﬂuids through
measurement of the correlations between the thermally stimulated vibrations of two closely spaced
micrometer-scale cantilevers in ﬂuid. We discuss a major application: measurement of the rheological
properties of ﬂuids at high frequency and high spatial resolution. Use of CFS as a rheometer is vali-
dated by comparison between experimental data and ﬁnite element modeling of the deterministic ring-
down of cantilevers using the known viscosity of ﬂuids. The data can also be accurately ﬁtted using
a harmonic oscillator model, which can be used for rapid rheometric measurements after calibration.
The method is non-invasive, uses a very small amount of ﬂuid, and has no actively moving parts. It
can also be used to analyze the rheology of complex ﬂuids. We use CFS to show that (non-Newtonian)
aqueous polyethylene oxide solution can be modeled approximately by incorporating an elastic spring
between the cantilevers. © 2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.4704085]
The time response of individual and collective motions
of molecules and particles can vary over many orders of mag-
nitude, thereby introducing considerable complexity into the
study of polymeric solutions and particulate suspensions.
These complex ﬂuids are found in thin ﬁlms as lubricants, in
separation processes such as chromatography and ﬁltration,
and in many personal products such as shaving cream and
toothpaste. Complex ﬂuids exhibit a non-trivial response to
an applied strain that is neither Newtonian nor purely elas-
tic. One of the most common techniques for measuring the
rheological properties of materials is the traditional stress-
controlled or strain-controlled mechanical shear-rheometer.
Although the technique is extremely versatile, it suffers from
a number of limitations. For example, the instruments typ-
ically require several milliliters of sample, and the spacing
between the opposing plates is of order 1 cm. As a result, the
measured response obtained is an average of the bulk response
that does not provide information about the local dynamics of
Moreover, conventional rheometers
have a relatively limited range of oscillation frequencies that
can be accessed. Speciﬁcally, the ﬁnite mass of the device
(i.e., parallel plate and cone-and-plate) sets an upper limit of
frequencies to about 100 kHz.
Recently, microrheology techniques were developed
that overcome many of the limitations of traditional
The most common of these are the one-
point (or one-particle)
and two-point (or two-particle)
Although very successful both one-point
and two-point microrheologies have limits of applicability
that suggest the need for complementary techniques. The
video tracking technique used to measure particle displace-
ments is limited to frequencies less than 30 Hz (half the
Obtaining higher frequency response
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information has only been achieved using diffusing wave
spectroscopy (DWS), which requires a concentrated particle
suspension, or laser trapping of the two probe particles in
conjunction with fast photodiodes.
4, 12, 13
This can be a severe
disadvantage for materials that have a broad range of charac-
teristic time scales. The video tracking techniques typically
can only resolve displacements of ∼20 nm.
particle separation distance in two-point microrheology must
typically be of order 1 μm to allow accurate resolution. The
video techniques also require that particles are large enough
to be visualized (>0.5 μm). The technique is most effective
for “soft” systems with elastic moduli < 100 Pa.
There are parallel efforts in microrheology measurements
based on atomic force microscopy (AFM) cantilevers,
following on recent success in understanding the dynam-
ics of cantilevers in ﬂuid.
In the current work we make
the extension from one cantilever rheology to two can-
tilever rheology, which we describe as correlation force spec-
troscopy (CFS). We are motivated to use two cantilevers
rather than one for the same reasons that others used two-
particle rheology rather than one-particle microrheology.
First, the correlation in ﬂuctuations of two particles depends
on the ﬂuid structure between the particles,
microrheology has been used to examine different length
scales of structure in the immersion ﬂuid, such as chemi-
cally or physically cross-linked polymers, micellar solutions,
and colloidal suspensions.
By varying the separation be-
tween two cantilevers, we expect to be able to examine
different length scales in solution. A second advantage of
using two cantilevers instead of one cantilever is for single-
molecule force spectroscopy. The thermal noise, which lim-
its resolution in force microscopy, is expected to be much
smaller in the energy spectrum of the cross-correlation than
in the auto-correlation.
As a precursor to such single-
molecule studies, it is necessary to understand the correlations
of the two cantilevers in ﬂuid without the connecting single
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