Time-temperature
superposition
Storage modulus
mapping
Contrast inversion inmaps of loss tangent

Application Note #154
Measuring Nanoscale Viscoelastic Properties
with AFM-Based Nanoscale DMA
The development of heterogeneous materials like polymer
composites, blends, and multilayers are of considerable
importance in the chemicals industry. Bulk viscoelastic
measurements are routine in establishing structure-property
relationship for these materials. However, materials R&D
often produces composites that contain nano-sized portions
that do not exist in the bulk or that have properties influenced
by the proximity of other components. A quantitative meansof measuring the viscoelastic properties of such materials
at the nanoscale has been a long-standing and elusive goal
for atomic force microscopy (AFM). While AFM has thesensitivity and resolution needed to do the measurement,
traditional AFM-based approaches are hampered by difficult
calibration, poorly defined measurement frequency, andinadequate modeling of the tip-sample interaction. Now,
with the development of the AFM-nDMA mode, thesepitfalls can be avoided and the frequency and temperaturedependence of viscoelastic properties in rheologically
relevant ranges can be directly measured with 10nm spatial
resolution. This application note explains the technology
behind this technique and details the ways it enables
investigation of previously inaccessible microscopic domains
in heterogeneous polymeric materials.

Viscoelastic Measurements
Nanomechanical measurements have become among
the most important measurements conducted with
AFM, aside from topography. The inherent mechanicalinteraction between probe and sample allows an atomic
force microscope to measure a sample’s various mechanicalproperties, while the size of the contact allows themeasurement to be localized at the nanometer scale.
The most popular nanomechanical applications include
measuring elastic modulus, adhesion, and friction.
Viscoelastic materials, such as polymers and biologicalmaterials, have elastic properties that exhibit timedependent behavior such as creep and stress relaxation.
The viscoelastic properties are storage modulus, loss
modulus, and loss tangent (tan δ). Nanoscale viscoelastic
measurements are of interest primarily due to theirinfluence on macroscopic material behavior and function.
An important segment of the chemicals industry is devoted
to developing heterogeneous materials, such as polymer
composites, blends, and multilayers. Bulk viscoelastic
measurements are routine in establishing structureproperty relationship for these materials, especially inthe development of a material’s mechanical properties.
However, material research and development often
produces composites or blends that contain nano-sizeddomains of components that do not exist in the bulk or
have properties that are influenced by the proximity of othercomponents (interphase). To fine-tune the characteristics ofsuch a heterogeneous material, the viscoelastic propertiesof the different components must be well-characterized.
Thus, a tool to measure such properties on the nanoscaleis essential for effective development of structure-property
relationships in next-generation materials development.
The stress-deformation relationship of materials can bemeasured with either shear or uniaxial loads, resulting inshear modulus (G) or tensile modulus (E), respectively. For
viscoelastic materials, the moduli are complex and haveboth real and imaginary parts. The real part (e.g., tensilestorage modulus, E’) is the elastic or in-phase component of
the response, and is given by (stress/strain)*cosδ where δ isthe phase shift between the stress and strain. The imaginary
part (e.g., tensile loss modulus, E”) represents the viscousor out-of-phase component of the response, and is givenby (stress/strain)*sinδ. The loss tangent (tan δ) is the ratio
between loss modulus and storage modulus.
Viscoelastic moduli often vary significantly with frequency
and/or temperature. For example, in rubber-like polymers,
the storage modulus typically has low values at low
frequencies. As the frequency increases, the storage
modulus increases dramatically, reaching a plateau at themodulus value of the glassy state at high frequencies.
For the same materials, the loss modulus is low (primarily
elastic) both at low and at high frequencies. The loss
modulus peaks in the frequency mid-range, reflecting thepresence of a glass transition between the rubbery andglassy states. Since the loss tangent is just the ratio of theloss modulus over the storage modulus (tan δ = E’’/E’), it
follows a similar trajectory to the loss modulus. The loss
tangent is a practical and useful viscoelastic parameter, as it
can pinpoint thermal and structural transitions in a material
when measured as a function of temperature or frequency.
There is an equivalence between time (frequency) andtemperature behavior in viscoelastic materials that arises
from the relaxation behavior of the molecules within thesample.1 This correspondence is formally expressed interms of the time-temperature superposition principle
or TTS. When modulus is measured over a range of
frequencies, similar curves are obtained when themeasurement is done at higher temperatures as are
obtained at lower temperatures and a lower range of
frequencies. Because these curves have the same shape,
they can be superimposed onto one another to generatea “master curve” over a wide range of frequencies.
Put another way, raising the temperature is equivalent
to shifting to a lower frequency, while lowering thetemperature is equivalent to shifting to a higher frequency.
Finally, the shift factors used to generate the master curve
can be analyzed via different models (WLF or Arrhenius)
to parameterize this time-temperature relationship. In
this way, data measured at one set of temperaturesand frequencies can be used to determine behavior at adifferent set of temperatures and frequencies, making TTS
and generation of the master curve very useful.
2

In the bulk, viscoelastic properties are measured withdynamic mechanical analysis, DMA (also known as dynamic
mechanical spectroscopy, DMS). This measurement involves
application of an oscillating stress to a macroscopic sample,
with the response of the entire sample measured as afunction of oscillation frequency. Depending on the sample
properties and geometry, DMA can be performed in several
different sample-mounting configurations, including threepoint bending, shear, compression, and tension. Holding
the sample in tension for dynamic mechanical tensile
analysis (DMTA) offers the most direct comparison to the
motion of an AFM tip and sample.
Previous AFM measurements of viscoelastic properties
have suffered for several reasons. The first two challenges
involve frequency space. Rheologists working with
DMA typically work at frequencies of less than 200Hz,
while AFM imaging modes are typically at much higher
frequencies (kilohertz and higher) as they are optimized toprovide images quickly. Additionally, in resonance-based
techniques, the frequency of the AFM measurement is
at very high frequencies as dictated by the cantileverdimensions. These frequencies are discrete (not tunable),and widely spaced (e.g., the second free eigenmode of asimple beam is 6.3 times the first).2-4
Perhaps even more important than the frequency mismatch,the imaging speed requirements of AFM have driven
adoption of methods where the tip plunges into the sample
and rips away from it at high speed, spending the minimum
amount of time possible in contact. When the tip is making
and breaking contact with the sample every cycle, thetip-sample interaction is very non-linear and the frequencyof the measurement is not well defined but includes many
harmonics of the nominal operation frequency.5
The final challenge to design an effective technique tomeasure viscoelastic properties revolves around the models
used to extract the storage and loss modulus. The mostchallenging component of this involves measurementof and compensation for adhesion. In AFM of polymers,
the adhesion between tip and sample is relatively large
compared to the other tip-sample forces, and it often varies
significantly across heterogeneous polymer samples. The
adhesion is not measurable in many resonance-based AFM
modes, making consideration of this important parameter,
and ultimately its compensation, impossible. The primarycontact mechanics models available for modeling adhesive
tip-sample interactions for parabolic AFM tips are DerjaguinMuller-Toporov (DMT) and Johnson-Kendall-Roberts (JKR).
These models represent two limits in the range of possible
adhesive behaviors, with the DMT model accounting for
weak long-range adhesion (including outside the tip-sample
contact area) while the JKR model accounts for stronger
adhesion inside the tip-sample contact area.6 Unfortunately,
both models are purely elastic with no viscous component.
Models that can extract both storage and loss modulus
typically ignore adhesion and are therefore not appropriate

for application to AFM of polymers in air.7, 8 These challenges
make it clear that a different approach is required – one thatprovides well-defined rheological frequencies as a small
perturbation to a carefully controlled preload.

measurements (Figure 1c) were then performed with aramp scripting routine that used a series of segments tocontrol preload, relaxation, modulation, and calculation ofcontact radius.

A Different Approach for
Viscoelastic Measurements with AFM

The frequency range available with the AFM-nDMA spansmany orders of magnitude from 0.1Hz to 20,000Hz. The
lowest frequency measurements are recommended
only for those studies that really require properties to bemeasured in this frequency space, as these measurements
are relatively slow. Measurements above 300Hz require an
additional actuator (HFA) to expand the range of accessible
frequencies. The HFA provides Z modulation that is
independent of frequency and quite reproducible, enabling
the wide spectrum DMA measurements.

In designing a technique for the nanoscale measurementof viscoelastic properties (AFM-nDMA), there are somekey considerations that need to be incorporated. First,
the AFM tip-sample interaction must be operated at low,
well-controlled loads. The modulation needs to occur inthe linear tip-sample interaction regime, requiring evensmaller perturbations in force exerted on the sample by
the tip. Finally, the sample needs to be deformed at welldefined rheologically relevant frequencies. This refers to
the low frequency regime of 0.1Hz to 100Hz, far from thetraditional kilohertz and megahertz regime where atomic
force microscopes usually operate.
Due to the low frequency of the measurements,
AFM-nDMA measurements take time. It is therefore
preferred to locate the region of interest with a fastermethod of imaging that also provides mechanical property
contrast. PeakForce QNM® is particularly well suited to thisfunction because it provides maps of modulus, dissipation,
deformation, and adhesion relatively quickly and with highresolution.9 Probes that are good for AFM-nDMA also
typically work well with PeakForce QNM on the samematerials, so there is no need to switch probes between
measurements. Once the region or feature of interest
has been identified (Figure 1a), single frequency imagesof viscoelastic moduli images can be collected throughFASTForce Volume™ (FFV), as shown in Figure 1b. In
this case, the storage modulus was mapped at 100Hz(frequencies of 50Hz or greater are recommended
for reasonable acquisition times). Frequency sweepab

The first step in an AFM-nDMA spectroscopy measurement
is to bring the tip into contact with the sample and apply
a preload of a known force set by the user (see A portionof Figure 2). Once in contact, modulation occurs at a low
amplitude and well-defined frequencies (see C and E)
to maintain a linear tip-sample interaction regime. Bothamplitudes and frequencies are set by the user. Finally, thetip is pulled away from the sample and the adhesion and full
retract curve are measured (see D).
On viscoelastic samples, the sample deformation
will continue to increase for the full duration of themeasurement, causing the contact area to slowly increase.
Learning lessons from the dynamic nanoindentation
community, this is addressed in three ways. First, after
applying the preload a segment is included to allow for
sample relaxation (see B portion of Figure 2). Second,reference segments at a given frequency are collectedperiodically throughout the measurement as well as justbefore the tip is pulled away from the sample, to allow
for later estimation of the contact radius throughout
the measurement (see two E segments). Finally, thecFigure 1. Comparison of mechanical property maps of a four-component polymer blend composed of COC (red), PP (blue), LLDPE (green), andelastomer (yellow) generated with PeakForce QNM (a) and AFM-nDMA (b) at 100Hz, and spectra at selected points on the sample (c). The
spectra show that COC grows in stiffness most rapidly with frequency, followed by PP, LLDPE, and elastomer. This PeakForce QNM 512x512
image (2µm scale bar) was collected in about 10 minutes, while the 100Hz AFM-nDMA 70x70 map took 30 minutes. Each spectrum took about
10 minutes to collect.
3

Figure 2. Typical AFM-nDMA spectroscopy force (orange) and Z position (blue) data. The tip is brought into contact and a preload is applied(marked A); the system waits for creep relaxation (B); low-amplitude modulation occurs at various frequencies, including reference frequencies(C and E); and the tip is retracted from the sample (D). Inset shows the key parameters to be extracted from the modulation segments: thedeflection amplitude, D1, the Z amplitude, Z1, and the phase shift between Z and deflection (ϕ - ψ ).

frequency order of the measurements is shuffled to avoid
misinterpretation of property variation with contact time
as a frequency dependence. Spectroscopy measurements
can require several minutes to acquire a single point
measurement, depending on the frequencies involved.
For single-frequency mapping measurements, themeasurement at each point is kept as simple as possible toreduce the total time of acquisition, and includes only threesegments: approach, modulate, and retract.

Separating the real and imaginary parts of equation 3 resultsin the storage stiffness (real) and loss stiffness (imaginary)components. The loss tangent (tan δ) is simply the ratio of
the two:S' =KcD1cos(ϕ - ψ)-D1/ Z1Z1 (cos(ϕ - ψ)-D1/ Z1)2+(sin(ϕ - ψ))2

(4)

S" =

KcD1
sin(ϕ - ψ)Z1 (cos(ϕ - ψ)-D1/ Z1)2+(sin(ϕ - ψ))2(5)

sin(ϕ - ψ)cos(ϕ - ψ)-(D1/ Z1)(6)

AFM-nDMA Data Analysis
The amplitude and phase of the cantilever deflection andZ-piezo motion collected during modulation segments likethose shown at C in Figure 2 can be analyzed to obtainthe sample viscoelastic properties at the modulation
frequency.10, 11 The Z-piezo motion is described byz(t) = Z1 sin(ωt+ψ)(1)

while the cantilever deflection can be described byd(t) = D1 sin(ωt+φ)(2)

If the amplitude ratio (D1 /Z1 ) and phase shift (ϕ-ψ) are
extracted (Figure 2, inset), one can calculate the complex“dynamic stiffness” or force divided by deformation:
S* = S'+iS" =

4

force
KcD1eiφ=deformation(Z1eiψ-D1eiφ)(3)tan δ = S"/S' =The relationship between dynamic stiffness and the storageand loss modulus are simply:
E'=

S'
S"
; and E" =
2ac
2ac

(7)

where the contact radius (ac ) is determined from the final
step in either the FFV ramp or the ramp script where thetip is pulled away from the sample (see D in Figure 2). This
segment is fit with an adhesive contact mechanics modelsuch as DMT or JKR to obtain the contact radius at the
beginning of the retract.

Calibration and Probes
Quantitative nanomechanical measurements with AFM
have long struggled with the need for accurate calibrations
involving the probe and system, specifically the tip shape,
cantilever spring constant, and detector sensitivity.
Bruker now offers pre-calibrated probes with rounded,well-defined tips, providing confidence in their shape anddimensions.12 These tips have spherical apices with radiiof either 33nm or 125nm (individually measured via SEM),
providing a controlled contact area for indentation depths up
to 100nm or 500nm, respectively. Tip shape is especially
important for measurements on heterogeneous samples
where a constant load will result in different indentations
on the different components based on the component’s
material properties. With these probes, it is possible tocalculate the contact area for a wide range of indentation
depths. The rounded geometry of the probe shape further
promotes linear deformation mechanisms, which are theonly ones accounted for in elastic models.13

Finally, there are a few calibration parameters that cannot
be pre-calibrated because they are system dependent. The
photodiode deflection sensitivity can be calibrated either
through a thermal tune, or by conducting a force curve on a
stiff sample, such as sapphire. The amplitude and phase of
the Z position also need to be calibrated, where the phasecan depend on the probe and probe (or sample) mounting.
These parameters are calibrated on sapphire by running theidentical AFM-nDMA or force volume script with the sameprobe as is used to measure the sample.

The spring constant of each probe is also pre-calibrated
with a laser Doppler vibrometer (LDV), providing an
accurate measurement of this important cantileverparameter and obviating the user from this step. These
probes are offered in a range of spring constants from
0.25N/m to 200N/m, appropriate for a wide range of sample
stiffness. Table 1 lists the available pre-calibrated probe
types along with nominal spring constant, tip radius, andrecommended range of storage modulus to be measured.
Note, however, that AFM-nDMA is not limited to only theprobe choices listed in this table. The 125nm tips provide
improved signal-to-noise level over the 33nm tips, reducing
the chance of plastic deformation, while the sharper tips
provide better resolution for samples with small features.
The pre-calibrated parameters are easily read into thesystem by scanning the QR code on each cantilever box,
which contains the spring constant, resonance frequency,
quality factor, tip radius, and tip half-angle for eachindividual cantilever. Once the QR code is scanned, there isno need for adjustment or “fudging” of the parameters, andno sample of known stiffness is required.

AFM-nDMA is designed to measure localized viscoelastic
properties as a function of frequency or temperature on
heterogeneous samples. The first step in the measurement
is to image the sample with a conventional AFM imaging
method, such as PeakForce QNM. Figure 1a shows a
PeakForce QNM image of a four-component polymer
blend where all components are easily differentiated. This
512x512 image was acquired in ten minutes and provides
a good overview to allow selection of areas of interest for
further investigation. A FASTForce volume AFM-nDMA
map was then collected at 100Hz (the resulting storage
modulus map is shown in Figure 1b). Although thesemaps can be collected at the same frequencies as thoseavailable in ramp scripting measurements, frequencies lessthan about 50Hz are less practical for image acquisition
due to the amount of time needed, and the risk of drift
during the measurement. This 100Hz 70x70 map wascollected in about 30 minutes. Finally, single points wereselected for individual ramp-scripting measurements overa wide range of frequencies, as shown in Figure 1c forthe storage modulus plotted as a function of frequency
in the range of 0.1 to 200Hz. The blend was composedof cyclic olefin copolymer (indicated with a red circle),
polypropylene (blue), linear low-density polyethylene (green),
and elastomer (yellow). Comparison of the spectra showsthat COC has the largest storage modulus followed by PP,
LLDPE, and elastomer. The storage modulus of the COC andLLDPE nearly doubles over the frequency range measured,
while the other components vary less dramatically.

Spring Const.
(N/m)

Tip Radius(nm)

min E'(MPa)

max E'(MPa)

SAA-HPI-DC-125

0.25

125

0.1

10

RTESPA150-125

5

125

2

200

RTESPA300-125

40

125

20

2,000

RFESPA-40-30

0.9

33

0.8

80

RTESPA150-30

5

33

4

400

RTESPA300-30

40

33

40

4,000

RTESPA525-30

200

33

200

20,000Probe Type

Table 1. AFM-nDMA optimized probes and the corresponding
storage modulus ranges for which these probes are best suited.
These probes are supplied with pre-calibrated spring constants(via LDV) and tip radii (via SEM). Note that AFM-nDMA is not limited
to the probe choices listed here.

5

Results

To check for consistency between methods, viscoelastic
measurements by AFM-nDMA were compared withbulk DMA, the industry standard for such measurements.
Figure 3 compares the storage modulus (E’) and loss
modulus (E”) of polydimethylsiloxane (PDMS) as measured
by three methods: 1) bulk tensile strain modulation
DMA (green lines); 2) an instrumented nanoindenter(light blue ‘x’); and 3) AFM-nDMA in a Santa Barbaralab (dark blue ‘○’) and 4) AFM-nDMA in a collaborator’slab (orange ‘∆’) (performed on a different AFM systemthan the one at the Santa Barbara lab, with a different
probe and different AFM operator). There is excellent
agreement among all these methods for both E’ and E”:
about 5% RMS deviation between bulk and AFM-nDMA
for E’, and about 25% for E’’. It is important to note thedifference in frequency ranges accessible by the twomethods. While DMA and indenter methods can typically
only measure moduli at frequencies up to around 200Hz,
the AFM-nDMA method enables measurements at up to
20,000Hz with the addition of an external actuator.

Figure 4. AFM-nDMA storage modulus (left axis) and loss tangent(right axis) of fluorinated ethylene propylene. The blue squares arestorage modulus versus temperature plots at 0.1, 1, 10, and 100Hz(darkest to lightest blue respectively), while the brown circles are theloss tangent plots at 0.1, 1, 10, and 100Hz (darkest to lightest brown.
Figure 3. Storage modulus (E’) and loss modulus (E’’) spectra ofpolydimethylsiloxane at 25°C. The green dashed and solid lines arestorage and loss modulus as measured with bulk DMA. The light
blue ‘x’ were from an instrumented nanoindenter, while the dark
blue ‘○’ and orange ‘∆’ used AFM-nDMA in either the Santa Barbara
lab or in a collaborator’s lab.

As described above, master curves are important tools forpolymer scientists to explore viscoelastic properties overa wide range of frequencies and temperatures. The firststep in generating a master curve is to collect a series ofviscoelastic spectra at different temperatures. Figure 4
shows the storage modulus (blue) and loss tangent or
tan δ (orange) acquired via AFM-nDMA over a temperature
range from 25°C to 130°C and a frequency range of
0.1 to 100Hz on fluorinated ethylene propylene (FEP). As
the temperature increases, the material becomes softer as
demonstrated by a lowering of the storage modulus. The
90°C peak in the loss tangent is due to the material's glasstransition temperature (Tg). As expected, the softening andpeak in loss tangent occur at higher temperatures whenthe measurement is conducted at higher frequencies.

By shifting the series of curves from Figure 4, a mastercurve for FEP loss tangent is then created. Figure 5ashows that the loss tangent master curve from the
AFM-nDMA data (blue ‘○’) compares favorably withbulk DMA (green ‘∆’). These master curves span a very
wide range of frequencies of about 24 decades. Finally,
the activation energy can be calculated from the shift
factors used to generate the master curves (Figure 5b).
The activation energy from the AFM-nDMA master curve(490kJ/mol) matches very well that from the bulk DMA
master curve (489kJ/mol).

ab

Figure 5. Time-temperature superposition analysis of FEP: (a) losstangent master curves from AFM-nDMA (blue ‘○’) compared to thatfrom bulk DMA (green ‘∆’); (b) TTS shift factor versus 1/T plot for
Arrhenius activation energy analysis. Linear fits give activation energies
from bulk DMA and AFM-nDMA of 489 and 490kJ/mol, respectively.

6

a

b

Figure 6. Storage modulus (top row) and loss tangent (bottom row) of a PP-COC blend: (a) 100Hz AFM-nDMA maps with increasing
temperature (500nm scale bar); (b) plots of storage modulus and loss tangent at 10Hz versus temperature. Blue circles were measured within
the COC domain and red triangles were from the PP matrix.

One of the most exciting applications of AFM-nDMA isto probe viscoelastic properties of highly localized spatial
domains, a capability that is critical to many kinds ofmaterials, including blends and nanocomposites. A blendof a polypropylene (PP) matrix with a 1µm diameter domainof cyclic olefin copolymer (COC) is shown in Figure 6. The
progression of the viscoelastic properties of these twomaterials as a function of temperatures from 25°C-175°C
is shown in the series of 100Hz AFM-nDMA storage
modulus and loss tangent plots. The two materials start out
at equivalent E’ values at ambient conditions, but quickly
diverge as the temperature increases and two materials
approach their respective thermal transition points. The
loss tangent images reveal an inversion of relative loss
tangents between the two materials at 175°C. E’ and tan δ
values were calculated from their respective domains in
the image and plotted as a function of temperature in thecorresponding graphs adjacent to the maps.

7

Conclusions
Quantitative viscoelastic property measurements ofnanoscale domains in heterogeneous polymeric materials
are now possible. AFM-nDMA directly addresses thefrequency and temperature dependence of viscoelastic
properties in the rheologically relevant range, avoiding
the pitfalls of traditional AFM imaging modes. As seen in
the PDMS test case, E’ and E’’ values compare well withbulk DMA measurements from 0.1 to 200Hz (and canbe extended to 20,000Hz using external actuator). A full
TTS analysis of FEP results over a range of temperaturesand frequencies also agrees with bulk DMA, and yields
the expected Arrhenius activation energy. Additional
measurements on a blend of PP and COC demonstrateviscoelastic property mapping and the importance ofconsideration of the temperature and frequency tounderstand even qualitative material behavior. AFM-nDMA
adds a powerful, quantitative capability for measurementof viscoelastic properties, which compares well withestablished techniques while enabling investigationsof the microstructure as well as bulk properties ofheterogeneous polymers.

1.

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Authors
Bede Pittenger (Bruker Nano Surfaces), Sergey Osechinskiy (Bruker
Nano Surfaces), and Dalia Yablon (SurfaceChar, LLC)

Bruker Nano Surfaces Division
Santa Barbara, CA · USA
phone +1.805.767.1400
productinfo@bruker.com

www.bruker.com/nano8© 2019 Bruker Corporation. All rights reserved. are trademarks of Bruker Corporation. All other trademarks are the property of theirrespective companies. AN154, Rev. A0.

References