Optimizing the Accuracy of Viscoelastic Characterization with AFM Force -Distance Experiments in the Time and Frequency Domains Marshall R. McCraw Berkin Uluutku Halen D. Solomon Megan S. Anderson Kausik Sarkar and Santiago

2025-04-29 0 0 2.54MB 51 页 10玖币
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Optimizing the Accuracy of Viscoelastic Characterization with AFM Force-Distance Experiments in the
Time and Frequency Domains
Marshall R. McCraw, Berkin Uluutku, Halen D. Solomon, Megan S. Anderson, Kausik Sarkar, and Santiago
D. Solares*
Department of Mechanical and Aerospace Engineering, The George Washington University School of
Engineering and Applied Science, Washington, District of Columbia, USA
*Corresponding Author: Santiago D. Solares – ssolares@gwu.edu
KEYWORDS: Atomic force microscopy; viscoelasticity; material properties; force spectroscopy.
Abstract
We demonstrate that the method of characterizing viscoelastic materials with Atomic
Force Microscopy (AFM) by fitting analytical models to force-distance (FD) curves often yields
conflicting and physically unrealistic results. Because this method involves specifying a
constitutive time-dependent viscoelastic model and then fitting said model to the experimental
data, we show that the inconsistencies in this method are due to a lack of sensitivity of the model
with respect to its parameters. Using approaches from information theory, this lack of sensitivity
can be interpreted as a narrowed distribution of information which is obtained from the
experiment. Furthermore, the equivalent representation of the problem in the frequency domain,
achieved via modified Fourier transformation, offers an enhanced sensitivity through a widening
of the information distribution. Using these distributions, we then define restrictions for the
timescales which can be reliably accessed in both the time and frequency domains, which leads
to the conclusion that the analysis of experiments in the time domain can frequently lead to
inaccuracies. Finally, we provide an example where we use these restrictions as a guide to
optimally design an experiment to characterize a polydimethylsiloxane (PDMS) polymer sample.
1. Introduction
The field of mechanobiology has provided a number of powerful insights into the rich
behaviour of soft, biological materials
1–6
. Within this field, colloid based rheology measurements
have allowed researchers to probe both the passive and active time-dependent responses of
biological materials across a large range of timescales, making it possible to directly measure the
contribution of individual bio-polymers to a cell’s global behaviour
7–15
. Although these micro-
rheological techniques are among the most prevalent in the mechanobiology community, Atomic
2
Force Microscopy (AFM) and, more generally, Scanning Probe Microscopy (SPM), have become
popular alternatives due to their uniquely versatile range of in vitro measurements and non-
destructive nature
2,3,16–21
. Although these methods are typically incapable of achieving a similarly
broad characterization of the viscoelastic timescales of a material, attempts have been made to
close this gap
22–24
. Still, the most common viscoelastic characterization technique among AFM
practitioners is the force-indentation (or force-distance, FD) experiment. Due to its simplicity and
short duration, FD experiments are an attractive option for those seeking high throughput data
acquisition to probe the heterogeneities inherent in biological materials
19,25
. Despite previous
attempts to leverage the wide-band nature of thermal excitations in AFM based measurements,
the traditional method used to characterize the viscoelastic properties of a material is still the
force-indentation (or force-distance FD) experiment, due to its simplicity and short duration –
allowing high throughput data acquisition.
Within AFM, other methods such as band excitation, stress relaxation (creep testing), and
dynamic mechanical analysis can also be performed; however, such experiments typically take
longer, thus limiting the amount of data that can be obtained from an experiment when measuring
sensitive, active materials like cells, for example
19,26,27
.
In force-indentation experiments, the surface of the material is indented with the AFM
probe while the indentation depth
and the resulting interaction force
are recorded in a way
that is analogous to macro-scale tensile and compression test experiments. Using contact
mechanics, one can relate the force and indentation in terms of a constitutive stress-strain
equation which, for the case of a linear viscoelastic material, is generally a convolution integral as
shown in Eqn. 1 and 1a
28–35
. Here, the terms
and
depend on the geometry of the AFM probe
and can be found in further sources
19,28,30–32
.

(1)

(1a)

(2)
In the state-of-the-art, the viscoelastic modulus
can then be obtained by first assuming
a functional form for
and then fitting the specific constitutive equation to the obtained force-
indentation or stress-strain data in what we will refer to as the time domain approach
18,19,36–40
.
Alternatively, one can use certain integral transforms, for which the convolution theorem is valid,
to ‘undo’ the convolution and directly obtain the modulus from the transformed force-indentation
data without prescribing a functional form to
as seen in Eqn. 2 (here, we will use the modified
Fourier transform with the notation
denoting the transform of
)
41,42
. While this ‘frequency
domain approach’ allows an independence from model prescription, obtaining a model-based
parameterization is often necessary for communication and comparison purposes. Thus, just as
is done in the time domain approach, models for
can be chosen and then fit to the
transformed data.
While these methods offer straightforward solutions for obtaining parameterized
descriptions of viscoelastic materials, their accuracy and reliability has recently been
questioned
40,43
. Specifically, the time domain approach often yields unphysical or even conflicting
3
values of the model parameters which depend on the initialization of the fitting algorithm
44
.
Furthermore, a recent work by Vemaganti et al demonstrates that relaxation times can only be
reliably obtained from stress relaxation experiments if they are less than

of the total
experiment length
45
. Despite these potentially major inconsistencies, the results obtained from
the time domain approach often are in close agreement with the data and require a close
inspection to validate their physical sensibility. However, such an approach would be completely
impractical in high throughput applications which often involve

’s of separate force curves
within a single measurement.
2. Theoretical Background
Although not an exhaustive list, researchers tend to choose from Maxwell, Kelvin-Voigt,
power-law, and fractional calculus-based models for the parametrization of the viscoelastic
modulus
18,19,36–38,46
. Here, we will focus specifically on the generalized Maxwell model

seen
in Fig. 1b and defined by Eqn. 3 and 3a for the time and frequency domains, respectively (note
that the modified Fourier domain representation is used in Eqn. 3a; further details can be found
in the appendix). As seen in Fig. 1b, the generalized Maxwell model is comprised of a series of
‘Maxwell arms’ in parallel with a single elastic element
. Each arm has a spring element
contributing an elasticity
and a dashpot (damper) element contributing a viscosity
. The
ratio of these arm components gives a characteristic relaxation time

which governs
the rate at which the stress in the

arm relaxes, with larger values corresponding to slower
stress relaxation and vice-versa. By increasing the number of Maxwell arms
, one can describe
a material with an intricate relaxation process. For example, a two-armed generalized Maxwell
model was used to describe the ‘fast’ and ‘slow’ cytoskeletal rearrangement of cancer cells from
stress-relaxation experiments
47
.

(3)



(3a)
4
Figure 1:
a) illustrations of a Standard Linear Solid (SLS) model and b) a generalized Maxwell (GM)
model comprised of an arbitrary number of Maxwell arms, seen in dashed red.
As previously mentioned, we have found that the behavior of the generalized Maxwell
model is relatively insensitive to changes in its parameter set
(i.e.,
). In the time
domain, for instance, while a model fitted to a force-indentation curve may agree with the true
data obtained from an experiment, the fitted values of
may vary significantly compared to the
true values for the material. Such discrepancies are demonstrated in Fig. 2a where 100
generalized Maxwell models were fit to a simulated force-indentation curve (seen in blue).
Although there are a few fits (seen in grey) which visually disagree with the true force-indentation
data, the average (seen in red) follows the data quite well. If this were representative of a typical
characterization experiment of the simulated material, the averaged data would be considered to
have successfully described the material due to its agreement with the data. However, the
discrepancies in the resulting fitted
’s become apparent when plotting the storage and loss
moduli (real and imaginary parts of

) for the fits, their average, and the data as seen in Fig.
2b and c, respectively. Here, we see that values of the relaxation times (inverse of the frequency
index of the peak in the loss modulus) disagree by 1-2 orders of magnitude between the fits and
the data. Furthermore, as models with
between 1 and 4 were fit to the data, models with
different numbers of relaxation times seem to equally describe the same force-indentation
behavior in the time domain, despite the qualitative differences in the physics. As a result of this
insensitivity to the number of relaxation times, the common practice for materials to be fitted
with an arbitrary
, raises questions regarding overfitting the data and regarding fitting the data
to incorrect physical behaviors. Although the comparison of the storage and loss moduli of the
fits and the material seems to allow a more accurate assessment of the fit accuracy, such data is
not directly obtained from the time domain fitting approach, thus making this assessment
5
impossible if one fits in the time domain. However, the results suggest that the frequency domain
approach should offer an advantage in more accurately parameterizing materials from force-
indentation data.
Figure 2
: a) 100 randomly initialized (N between 1 and 4) fit attempts (grey) to a simulated force-
indentation curve (N=1) (blue) with the average behavior of the fit attempts shown in dashed red.
b) Storage moduli for each parameter set obtained from fitting and averaging. c) Loss moduli for
each parameter set obtained from fitting and averaging.
3.
Analysis
a.
Insensitivity of the
Norm
To compare the capabilities of these two approaches in determining an accurate
for a
material, we will first use simplified, theoretical cases, treating force-indentation curves as linear
strain inputs

. While the indentation in AFM experiments is not typically
linear, the ‘strain’ factor
in the contact mechanics models often closely follows a line as seen
in the technical appendix. With this simplification, the stress-strain behavior for the linearized
force-indentation model can be obtained by solving the Volterra integral in Eqn. 1 for the given
strain

and modulus


. We further express both the time
and the
relaxation times
as fractions of the full experiment length
as well as normalize each arm
elasticity
by the glassy (or instantaneous) modulus
to remove the dependence
on
(the equilibrium modulus) and thus, reduce the dimensionality of
. A similar dimensional
analysis was performed for the modified Fourier domain representation of the modulus, yielding
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OptimizingtheAccuracyofViscoelasticCharacterizationwithAFMForce-DistanceExperimentsintheTimeandFrequencyDomainsMarshallR.McCraw,BerkinUluutku,HalenD.Solomon,MeganS.Anderson,KausikSarkar,andSantiagoD.Solares*DepartmentofMechanicalandAerospaceEngineering,TheGeorgeWashingtonUniversitySchoolofEngineerin...

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