NANOSCALE DYNAMIC MECHANICAL ANALYSIS VIA ATOMIC FORCE MICROSCOPY (AFM-nDMA)

Abstract

To allow for determining mechanical properties of a surface of a viscoelastic sample.SOLUTION: Disclosed are atomic-force-microscope-based apparatus and method, including hardware and software, configured to dynamically collect and analyze data representing mechanical properties of soft materials on a nanoscale, and to map viscoelastic properties of a soft-material sample. A use of the apparatus as an addition to the existing atomic-force microscope device is also disclosed.SELECTED DRAWING: Figure 1

Description

Technical Field

[0001] Cross - reference to Related Applications This application claims the benefit and priority of U.S. Provisional Patent Application No. 62 / 715,166, filed on Aug. 6, 2018, and U.S. Provisional Patent Application No. 62 / 769,905, filed on Nov. 20, 2018. The disclosures of each of the above - referenced applications are hereby incorporated by reference into this specification.

[0002] The present invention generally relates to a method for determining the dynamic mechanical properties of materials, and in particular, to the nanoscale rheology of materials performed using an atomic force microscope in a specific range of frequencies, namely, a low - frequency range substantially related to the rheology of soft materials.

Background Art

[0003] Dynamic mechanical analysis (DMA) is a measurement method designed to reveal the viscoelastic mechanical properties of various materials (such as metals, composites, polymers, elastomers, etc.).

[0004] Viscoelasticity is recognized as a property of materials that exhibits both viscous and elastic characteristics when deformed. A viscous material under stress usually resists shear flow and deforms linearly with time. An elastomer develops strain when stretched and returns to its original state immediately when the stress is removed. Considering viscoelasticity, the deformation (strain) exhibited by a solid material in response to a load (stress) generally has a time - dependence: such deformation (strain) depends not only on the magnitude of the load (stress) but also on the loading rate (~ loading speed) and relaxation time.

[0005] According to the characterization procedure of macroscopic (or bulk) DMA rheology, typically, periodic (harmonic) tensile stress, compressive stress, bending stress, or shear stress is applied to a material sample, causing excitation of the sample as a result of such loading. Subsequently, the mechanical response of the material (e.g., the amplitude and phase of such a response) is analyzed at the excitation frequency (excitation frequency). This analysis is usually performed using a lock-in amplifier. The DMA method is established to measure the storage modulus (E′) and the loss modulus (E″) of a material, and the ratio of these moduli, E″ / E′ (also referred to as the "tangent delta" (tan-delt) and also known as the "loss factor", "loss tangent", or "damping"), is typically expressed in MPa or GPa. These material properties are characterized as a function of frequency, temperature, time, stress or load, environmental conditions, or a combination of the above. (Alternative term - dynamic mechanical thermal analysis, DMTA, may be used to emphasize the temperature dimension or dependence of DMA measurement results).

[0006] Considering the mechanical properties of soft materials, the low-frequency mechanical properties (i.e., the mechanical properties at frequencies up to a few hundred Hz, e.g., up to 300 Hz) are recognized as being most relevant to the typical physiological motions of biological materials and cells. The ability to determine the low-frequency mechanical properties of biological materials and cells would significantly expand the current knowledge of soft materials. Also, a detailed understanding of the low-frequency performance of various other materials is desired. For example, it is well known that databases of the storage loss factor of polymers and rubbers currently used in industry are substantially lacking in micro- and nanoscale data.

[0007] However, existing DMA techniques (such as the use of nanoindentation systems for materials, which are almost universally used in related technologies) have been recognized to have limited spatial resolution when used with soft materials, which restricts or even prevents the use of such techniques to study the mechanics of soft materials at the length scales at which AFM-based devices operate. For example, some of the currently existing DMA techniques (which do not use equipment such as AFM by definition and use techniques such as nanoindentation systems that are recognized as such in related technologies (see, for example, Pharr, G.M., Oliver, W.C., & Brotzen, F.R., Journal of Materials Research 7, 613-617, 1992; S.A. Syed. Asif and J.P. P. Pethica, 505,103, 1997 Symposium "NN - Thin Film Stress and Mechanical Properties VII"; S.A. Syed Asif et al., Applied Physics Letters, 90, 3, 2001; Herbert, E.G. et al., Journal of Physics D: Applied Physics 41, 2008.)) are theoretically structured to enable the execution of such measurements. However, for example, it has been pointed out that existing DMA nanoindentation methods have several factors that substantially prevent actual measurements from being performed at the length scales at which AFM can perform measurements on soft materials such as biological materials and cells. Among these factors, to name just a few, there are non-linear elastic responses and very high adhesion.

[0008] Current AFM-based viscoelastic measurement techniques are fundamentally limited in that they cannot creep-relax materials using these techniques, which necessarily affects the quality and stability of the contact obtained between the tip of the probe and the sample during measurement, and as a result, adversely affects the accuracy of the measurement. Due to such limitations, for example, the system described in U.S. Pat. No. 9,417,170 is configured to explicitly avoid (stay away from, not permit) waiting for relaxation of the contact between the indenter probe and the surface of interest during measurement, whereby the described system and method are substantially non-practical in both quantitative measurement and mapping at (low-range) frequencies relevant to rheology.

[0009] Skilled artisans readily appreciate that there remains a strong need for AFM-based DMA techniques devised to measure the dynamic elastic modulus of nanoscale soft materials at low frequencies.

SUMMARY OF THE INVENTION

MEANS FOR SOLVING THE PROBLEM

[0010] Embodiments of the present invention are carefully configured to perform AFM-based nanoscale measurements (i.e., measurements on the nanometer geometric scale) of the mechanical response of soft materials at low frequencies (as defined herein) using a carefully configured atomic force microscopy method,

[0011] maintaining at least one of the average sample loading force and the average contact between the tip and the sample substantially constant. In one non-limiting embodiment, for example, the DC component of the sample loading force is maintained substantially constant, while the AC component of the sample loading force is preferably maintained variable,

[0012] performing dual-channel demodulation for up-to-date calibration of sample excitation,

[0013] In contrast to the related art, intentionally taking into account and compensating for sample drift / creep caused by preloading the sample, and / or achieving relaxation of the initial drift / creep of the material caused by preloading the sample,

[0014] correcting the contact radius (e.g., via the contact stiffness at a reference frequency).

[0015] Embodiments of the present invention provide an AFM-based system configured to determine the mechanical properties of the surface of a viscoelastic sample. Such a system includes a signal generator configured to generate a first vibration signal at at least one frequency, and a mechanical subsystem operably cooperating with the signal generator. Here, the mechanical system: i) relocates one of the sample and the AFM cantilever probe of the system relative to the other until the cantilever of the probe deflects by a predetermined amount from the nominal direction of the cantilever, ii) maintains the probe at a predetermined position relative to the surface of the sample, at which position at least one of 1) the average sample loading force generated by the probe and 2) the contact area between the tip of the probe and the surface is maintained substantially constant, and iii) is configured to generate mechanical vibrations of one of the sample and the probe relative to the other as a result of transfer of the first vibration signal at the signal frequency to the mechanical system. The system further includes a position detection system configured to detect the deflection of the cantilever as a function of at least one of the temporal and spatial factors characterizing the operation of the system.

[0016] The system further includes a programmable processor programmed to communicate electrically with the mechanical subsystem and transfer a first vibration signal from the signal generator to the mechanical subsystem, interrupting the operation of the mechanical subsystem for a time sufficient to relax creep of the surface (caused by repositioning one of the sample and the AFM cantilever probe relative to the other of the sample and the probe); and acquiring data from the position detection system to determine the viscoelastic parameters of the surface after the relaxation period has elapsed. Here, the relaxation period is a period sufficient to relax the creep (of the surface) caused by repositioning one of the sample and the AFM cantilever probe relative to the other of the sample and the cantilever probe. In certain embodiments, the system may further include an electronic circuit configured to measure the viscoelastic parameters of the surface at a predefined set of frequencies while compensating for creep of the surface, and / or a recording device operably communicating with the processor and configured to generate an output representing the viscoelastic parameters as a function of at least one variable condition of the viscoelastic parameter measurement process that is perceptible to the user. In any embodiment, the signal generator may be intentionally configured to generate the first vibration signal at only a single frequency.

[0017] Embodiments further provide a method for determining the mechanical properties of a soft viscoelastic sample using an atomic force microscope (AFM)-based system. The method includes: 1) repositioning the cantilever probe of the system towards the surface of the sample until the cantilever of the probe deflects by a predetermined amount from the nominal direction of the cantilever; and 2) modifying the repositioning process to maintain at least one of i) the average sample loading force generated by the probe and ii) the contact area between the tip of the probe and the surface, which is substantially constant. The method further includes measuring the viscoelastic parameters of the surface at a predefined set of frequencies while compensating or correcting for at least one of surface creep and system spatial drift, and generating an output representing the viscoelastic parameters as a function of at least one of the variable conditions of the measurement process, which is perceptible to the user. In one embodiment, the step of measuring the viscoelastic parameters may be performed simultaneously at a plurality of frequencies from a predefined set of frequencies. In any embodiment, the step of modifying the repositioning process may include modulating the sample loading force applied to the sample by the probe at a predetermined excitation frequency from a predefined set of frequencies. (In the latter specific case, the step of modulating the sample loading force is performed by adjusting the amplitude and phase of each oscillator component of the sample loading force to their respective target values from a predefined set of frequencies, and such adjustment is performed depending on the response of the material of the sample to which the modulated sample loading force is applied). In substantially any embodiment, the process of modifying the repositioning may include maintaining the average sample loading force substantially constant while the separation between the surface of the probe and the base is being modified. In substantially any embodiment, the step of measuring the viscoelastic parameters is performed by: (a) simultaneously measuring both the excitation force applied to the sample by the probe and the surface deformation caused by the excitation force; and (b) performing the dual-channel demodulation operation of the system to achieve at least one of avoiding / preventing the repeated calibration of the system.(In certain cases, the step of performing such dual-channel demodulation may include combining first data and second data received from a first sensor of the electronic circuitry of the system and a second sensor of the electronic circuitry of the system, respectively, during the step of measuring. Here, the first data represents the position of the probe relative to the surface, and the second data represents the degree of deflection of the probe's cantilever from its nominal direction). In any embodiment, the method may further include pausing or halting the operation of the system for a time sufficient to relax creep of the surface caused by the process of repositioning. In substantially any embodiment, the step of performing dual-channel demodulation may include introducing a correction for at least one of drift-induced changes and creep-induced changes in signal data received from at least one of the first channel and the second channel).

[0018] In substantially any embodiment, the method may further include continuously monitoring the operation of the system at a reference frequency to compensate / correct for changes in contact area due to creep of the surface, using at least one of a first and a second electronic circuit of the system. (In a latter specific embodiment, the continuously monitoring step may be performed by continuously monitoring using only one of the first and the second electronic circuit, and further includes obtaining calibration data representing a signal from the other of the first and second electronic circuits obtained from a sample of hard calibration.) Alternatively or additionally, the method may include a step of compensating for changes in contact area due to creep of the surface, and such a compensating step includes: i) calculating viscoelastic parameters using a programmable processor of the system while calculating the change in contact area, the calculating step in which the programmable processor is operably connected to the AFM, and ii) at least one of the steps of repositioning the probe to compensate for such a change. In any of the above cases, the selection of the reference frequency may include selecting a reference frequency that is not part of a predefined set of frequencies.

[0019] In the method of any embodiment, the measuring step may be configured to include: i) a process of obtaining, from a sensor of the electronic circuit of the system, a first set of electrical signals at a frequency from a set of frequencies (during a first period) to determine the degree of indentation of the surface having the tip of the probe; and ii) a process of obtaining, from a sensor of the electronic circuit of the system, a second set of electrical signals at a reference frequency (during a second period) to compensate for changes in the contact area due to creep of the surface. In such a case, the sensor includes at least one of a deflection sensor and a sensor configured to measure the position of the probe relative to the surface. (In a particular embodiment of the latter, the reference frequency may be selected to be not included in the set of frequencies.) The process of obtaining the first set of electrical signals and the process of obtaining the second set of electrical signals may be arranged such that such acquisitions are interleaved with each other. The method may further include a step of compensating for changes in the contact area based on determining changes in the dynamic stiffness of the contact between the probe and the sample.

[0020] The present invention will be more fully understood by reference to the following detailed description of specific embodiments in conjunction with the accompanying drawings, which are not to scale.

Brief Description of the Drawings

[0021]

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DETAILED DESCRIPTION OF THE INVENTION

[0022] Generally, the sizes and relative scales of elements in the drawings may be set differently from the actual objects in order to appropriately facilitate the simplicity, clarity, and understanding of the drawings. For the same reason, not all elements present in one drawing are necessarily shown in another drawing.

[0023] Embodiments of the system of the present invention (including, but not limited to, hardware, firmware, and software) are implemented based on a state-of-the-art atomic force microscope (AFM) apparatus with a digital controller and a programmable processor (computer system). Nano-scale dynamic mechanical analysis (NDMA) of a sample is performed using a cantilever probe that interacts with the sample surface with a controlled force and presses the sample, and the force is composed of both a quasi-static (DC) component and a dynamic (AC) vibration component.

[0024] The frequency (or frequencies) of the vibration component of the force is within a range that substantially matches the frequency range generally of interest in the bulk volume of soft materials and various polymers, i.e., within a range of several decades of sub-Hz and low-Hz frequencies (e.g., from 0.01 Hz to about 200 - 300 Hz as specified above). In the process of applying such a controlled force (including both the quasi-static interaction part and the vibration interaction part) between the cantilever probe and the sample under test (SUT), the movement of the surface (vertically, for brevity of notation) of the sample at the base of the cantilever probe relative to the normal of the sample is detected and measured in order to determine both the quasi-static force component (which results in at least loading and unloading deformations) and the dynamic vibration force component (which results in viscoelastic deformations under the vibrating load) and the mechanical response of the sample to each.

[0025] In the following disclosure, the term "soft material" refers to a material whose elastic modulus (Young's modulus) does not exceed 10 GPa. (In contrast, for example, a hard sample used for calibration purposes can be defined as having an elastic modulus in the range of 100 GPa. As approximate values of the elastic modulus of a plurality of hard materials, sapphire is about 350 GPa, silicon is about 50 GPa, mica is about 30 GPa or more, aluminum is about 70 GPa or more, and copper is 110 GPa or more.). Alternatively or additionally, whether a given material is soft may be defined in comparison to the material of the AFM probe used for measurement: Since AFM probes are often made of silicon or silicon nitride, a "soft material" having an elastic modulus of less than about 10 GPa will have an elastic modulus less than 10% of the elastic modulus of the tip material. In this case, the hard calibration sample may be defined as having an elastic modulus with a value substantially equal to the elastic modulus of the material of the tip of the AFM used during operation of the system, preferably a value of about 0% or more.

[0026] Embodiments of the system and method of the present invention (hereinafter generally referred to as "AFM-nDMA") utilize a cantilever AFM probe having a specifically defined particular geometric configuration of the tip that applies a dynamic vibration loading force to a material sample under test, referred to as the system under test (SUT), in a low frequency range that is practically important for the rheology of soft materials (in embodiments, defined as a sub-hertz frequency range, for example, a frequency range from several hertz up to 10 hertz or less; or in certain embodiments, defined as a frequency range of about 300 hertz or less, see also below), to enable measurement of the SUT localized on the nanoscale dynamic response. In other words, the use of embodiments of the present invention enables measurement of the nanoscale dynamic complex deformation of the SUT. For the purposes of this disclosure and the claims, the term "nanoscale" refers to the dimensions of the probe-sample contact that are sub-micron and is used to represent it.

[0027] In particular, in this embodiment, it is possible to characterize the nanoscale dynamic response of the SUT defined in terms of dimensions as a coating thin film or composite material. In contrast to related techniques, the measurement method is specially devised to take into account the relaxation of the creep of the material.

[0028] As a partial result of the measurement of the nanoscale dynamic response configured in this way, the viscoelastic storage and loss elastic moduli of the material SUT are determined in the low-frequency region particularly relevant to the rheological analysis of soft materials in a way that enables a direct comparison with the material properties measured using the conventional DMA method configured for the macroscopic (volume) analysis of the material properties. The presented method generally facilitates measurements at several decades of frequencies (sub-hertz, several hertz, dozens of hertz, about 100 hertz, and several hundreds of hertz such as up to 300 hertz) and is particularly useful. For example, according to an embodiment, an embodiment of the present invention provides operational advantages for measurements at frequencies in the range of 0.001 Hz to 1000 Hz, preferably in the range of 0.01 Hz to 300 Hz, more preferably in the range of 0.1 Hz to 150 Hz, and most preferably in the range of 0.1 Hz to 100 Hz.

[0029] One skilled in the art will readily recognize that the long-term stability of the system and probe-sample contact is important because the measurement time (required for measurements in the low sub-hertz frequency range) is relatively long compared to the measurement time performed at higher frequencies.

[0030] To meet the requirements for the long-term stability of the measurement system and to address the problems arising from the use of conventional embodiments of the DMA system for the measurement of the viscoelasticity of soft samples at low frequencies, embodiments of the present invention employ an atomic force microscope and related techniques (in contrast to systems such as different nanoindenters known in the art) and are carefully configured to specifically perform AFM-nDMA measurements over a long measurement time (seconds to minutes) at a predefined set of frequencies (the set is defined to include at least one, preferably a plurality of frequencies).

[0031] The use of the force setpoint modulation method is such that the AFM-nDMA system of the present invention maintains at least one of the probe-sample contacts having a substantially constant dimension (or an invariant dimension) during operation, regardless of (i) a predetermined level of preload gravity exerted on the SUT and (ii) the presence of thermal drift and / or creep, and also regardless of such drift and / or creep occurring during measurement.

[0032] In one embodiment, these operating characteristics are achieved by modulating the force applied to the sample by the probe (at a predetermined excitation frequency) while maintaining the vibration of the probe caused by the AC component of the force to enable dynamic measurement, and at the same time, maintaining the DC component of such force at a substantially constant level (through feedback by the electronic circuitry associated with the use of the probe). This is implemented in contrast to related AFM-based techniques that employ modulation of the displacement between the AFM probe and the sample under test.

[0033] Using an AFM-based dual-channel demodulation scheme configured as a measurement subsystem, data / information obtained from two measurement channels of data acquisition electronic circuitry (abbreviated as the Z-sensor and the deflection sensor) are combined to enable simultaneous and instantaneous measurement of both the excitation force of the SUT and the resulting deformation. This is done contrary to the established use of a single measurement channel in the art. (As will be readily appreciated by those skilled in the art, the single measurement channel of the prior art is configured such that continuous use of the system requires repeated calibration of the system).

[0034] The Z-channel of one embodiment of the present invention is configured to measure the separation between the base of the probe and the SUT in order to extract information regarding both the amplitude and phase of a signal representing the interaction between the tip and the SUT. (The base of the probe corresponds to the end of the probe on the side opposite the end that holds or includes the tip; it is typically the base of the probe that is fixed to the fixture of the AFM probe holder). Thus, the use of the dual-channel scheme makes it possible (and liberates during operation) in embodiments not to rely on keeping the calibration of the amplitude and phase of the excitation force up to date during potentially long measurement times.

[0035] A typical deflection sensor is realized using a laser light source configured to emit a light beam, which is focused on the upper surface of the probe lever and then reflected towards a four-quadrant photodetector. The resulting change in deflection of the probe cantilever is converted into a change in the angle of the reflected laser beam and the position of the reflected beam on the photodetector. As is known in the art, the differential electrical signal from a carefully defined four-quadrant photodetector circuit is amplified and functions as a signal representing the vertical deflection of the probe. With appropriate calibration, the deflection signal can be used to detect deflections on the nanometer scale or forces exerted by the probe on the nanonewton scale.

[0036] Thus, one embodiment of the present invention includes a method of performing dual-channel demodulation that combines first and second data received from a first sensor of an electronic circuit of a system and a second sensor of the electronic circuit of the system during measurement. Here, the first data represents the position of the probe relative to the surface, and the second data represents the degree of deflection of the probe's cantilever from its nominal direction (not deflected, not affected). (More generally, for example, if the Z-sensor channel is not used or unavailable, such as in the case of excitation of a sample actuator, one can compensate for the missing channel by using calibration on a calibration sample of a hard surface using the first channel. Such calibration can be provided by measuring the deflection of a sapphire sample when it is found that the Z-sensor is not available and then using the result as the Z-amplitude and phase of the target sample).

[0037] The use of the "software lock-in" method is for performing demodulation of signals received from a dual-channel measurement subsystem. Here, the signal trace (buffered or recorded for online or offline processing) is processed by an algorithm that combines steps of drift / creep correction and / or subtraction to compensate for changes in the signal data received from at least one of the channels, and "lock-in-like" demodulation of such signals, at a single or multiple frequencies in a lock-in-like manner.

[0038] Hardware lock-in is known to perform a simple process without flexibility. On the other hand, according to the idea of the present invention, a more sophisticated algorithm can be executed using digitized accumulated signals or buffered signals: in one example, a trend line of the estimated drift is subtracted from the accumulated signal or buffered signal using a moving average filter, leaving only the oscillatory component for the lock-in demodulation function. As a result, the error introduced into the lock-in amplitude (and / or especially the phase) by drift / creep is dramatically reduced.

[0039] Such demodulation is performed in contrast to the use of hardware lock-in and / or Fourier transform-based (FFT / DFT) spectral analysis in related art that does not allow correction of material drift or creep (thereby introducing inevitable errors in the determination of the amplitude and phase values of the excitation force, which are particularly prominent on a frequency scale substantially equal to the range of the material drift frequency). One embodiment of the present invention operates at a single excitation frequency (according to one embodiment of the force point setting modulation setup), but the proposed algorithm facilitates (enables, permits) the process of simultaneous demodulation of such signals at multiple excitation frequencies, as needed, thereby providing the advantages of parallel processing and shortening the overall measurement time. For example, as empirically verified, simultaneous demodulation at two frequencies (0.1 Hz and 0.18 Hz, 20 cycles) shortened the overall measurement time by about 36%, and simultaneous demodulation at five frequencies shortened it by about 51%.

[0040] Optionally, in one embodiment, an electronic circuit configured to continuously monitor the operation of the system at a reference frequency can be further used to correct for changes in the contact area between the tip of the probe and the sample due to material creep or adhesive creep. Here, to perform such monitoring, the excitation of the probe at a preselected reference frequency is continuously mixed with, interleaved with, or complemented by the excitation of the probe at other measurement (excitation) frequencies, so that the dynamic stiffness of the probe-sample contact at the reference frequency can be measured substantially in parallel with and concurrently with other measurements performed at such other excitation frequencies.

[0041] From the change in the dynamic stiffness (determined at the reference frequency), the relative change in the contact area / size is inferred, and the corresponding correction is applied to the AFM-nDMA measurements at other excitation frequencies (in practice, including measurements at low frequencies where material drift / creep may be more prominent).

[0042] As an alternative AFM-based embodiment, instead of continuous monitoring at a reference frequency, "interleaved" (time interval) reference frequency measurements can be used (i.e., the reference frequency segments can be interleaved among other frequency measurements). For the purposes of this disclosure and the appended claims, when a first step and a second step of measurement (e.g., steps A and B) are interleaved with each other, these two steps do not necessarily have to be executed regularly alternately to form a substantially arbitrary continuous sequence in which both A and B are present multiple times, such as ABABABAB AABABBA..., ABABABABABA..., etc., and are arranged to be generally executed.

[0043] It should be noted that the described reference frequency monitoring-based correction is typically not required (but is optionally possible) in connection with force setpoint modulation in embodiments where the contact area can be maintained substantially invariant (substantially constant) via force feedback. In some specific examples, for example, when creep / flow of the material is substantially significant (e.g., in the case of measuring a polymer lacking crosslinking chains), the proposed correction based on the reference frequency may be advantageously implemented in addition to force setpoint modulation.

[0044] Since the viscoelastic body exhibits relaxation in response to a step function load, it is recognized in the art that the initial creep of the material occurs immediately after the preloading of the sample using a probe, for example, at the initial load stage of the indentation cycle. Here, attention is required. Those skilled in the art are in a position to select a time scale where the drift is not operationally substantial, although they cannot choose to completely avoid the drift. (And some technical investigations related to AFM were, at that time, limited to measurements at higher frequencies with that technique. In other words, low-frequency drift was avoided by performing measurements at different high frequencies.) A skilled person in the art will easily understand that it is not possible to avoid the relaxation (creep) and drift of the material and that it must be addressed at a time scale corresponding to the frequency range where the embodiments of the inventive concept provide operational advantages. (Embodiments employing either the dual-channel demodulation approach and / or the reference frequency technique described later will successfully address the separation of creep from drift in the low-frequency range).

[0045] Accordingly, embodiments of the present invention address the problems caused by the presence of initial creep and, as an operation step of the AFM-nDMA system, explicitly include a waiting period or time segment (e.g., a duration of about 10 seconds, a duration of about 20 seconds, or a duration of about 30 seconds depending on a particular embodiment) prior to the step where the AFM-nDMA performs the operations of the excitation step and the measurement step, thereby improving the overall measurement accuracy of the viscoelastic properties of the material at low frequencies and enabling relaxation of the material under the applied preload step. In embodiments of the present invention, a flexible ramp script method is employed to seamlessly allocate such an initial relaxation "waiting segment", and in fact, the steps of nDMA measurement at only a single frequency or a plurality of mixed frequencies can be continued in parallel for this purpose.

[0046] The present invention provides an AFM-based nanoscale DMA (AFM-nDMA) method configured to extend a classical macroscopic volume DMA approach to a spatial scale of less than 1 micrometer. Note that nanoDMA (registered trademark) is a trademark for a viscoelastic property measurement technique on an instrumented nanoindenter (not AFM-based) platform (Bruker-Hysitron), not AFM-based. Therefore, to avoid confusion with the above trademarked "nanoDMA" technology name specific to techniques implemented using nanoindenter instruments, the present invention and the general scope of the present disclosure are further referred to as "AFM-nDMA", which is nano (scale) DMA specifically implemented on an AFM platform.

[0047] Examples of AFM-nDMA hardware and measurement methods Embodiments of the AFM-nDMA system

[0048] Generally, in nanoscale dynamic mechanical analysis, mechanical excitation of a "flat drive" without resonance is required, which can be difficult to achieve at the upper kHz end of the frequency range. Embodiments of the present invention address this issue by using a specially designed sample actuator having a high natural resonance frequency. Further, the present invention employs a special sample mounting scheme that does not significantly affect the resonance characteristics of the actuator and enables calibration of the amplitude and phase response of the actuator by in-situ measurement using an AFM system. In contrast to other designs using a probe holder actuator, this sample actuator does not excite or "backdrive" the resonance of the AFM scanner. In contrast to other designs based on electrical or magnetic forces or photothermal driving, the present invention can use a conventional AFM probe and does not require an AFM probe with a special (driving) lever.

[0049] In embodiments of the AFM-nDMA system, to achieve the above objectives, appropriately modified / deformed / extended AFM equipment is included.

[0050] Generally, an AFM-nDMA apparatus (and related methods for characterizing an SUT) is based on an AFM platform with a closed-loop scanner. The AFM scanner (alternatively referred to as a scanner head or AFM head) is a piezo-based actuator (having three orthogonal axes of motion: the x-axis, y-axis, and z-axis) appropriately programmed for positioning and scanning an AFM probe having a tip relative to a sample, and / or a sample scanner or actuator configured to position and scan a sample relative to the AFM probe. While the separation distance (along the z-axis) between the probe and the sample is maintained by a dedicated electronic circuit, the probe can be positioned or scanned in the sample plane (e.g., the xy plane). The scanner is configured to perform a push-in Z ramp (a “ramp & hold”) having a hold period (hold time) on or over the sample surface, where a predetermined level of loading force (alternatively referred to as a trigger force, a preload force, or a push-in force) is reached by the time the hold period is initiated. The interaction force between the tip and the sample is determined by the (vertical) deflection of the lever of the AFM probe, which is tracked by optical means and sensed by a quadrant photodetector. Optionally, a detector channel that records the lateral (e.g., horizontal) deflection of the probe lever during operation is configured to provide information regarding rolling or sliding of the tip of the probe. (Such a detector channel may be referred to as a “friction” channel).

[0051] Structurally, an AFM probe includes a flexible lever member (or simply lever) characterized by a spring constant or stiffness k (and measured, for example, in Newtons per meter, N / m) having a nanoscale tip (having a tip radius, a nominal dimension in [nm], and whose shape is typically approximated by a combination of a cone and a sphere). The lever member is attached to the "tip" (sized in millimeters) of a substrate that can be attached to various types of probe holders known in the art by spring clips or other means. The probe holder is then dimensioned to be attached to the XYZ scanner device of the AFM head (e.g., via an attachment member including a plurality of leaf spring sockets and metal pins).

[0052] In one embodiment, the AFM apparatus is preferably operably connected to a special (programmable) controller circuit including a digital signal processor (DSP) and a field programmable gate array (FPGA) configured to establish and maintain real-time control and digital feedback during operation of the apparatus, and the computer processor executes application code and communicates with the AFM controller circuit.

[0053] As schematically shown in FIG. 1, for example, embodiment 100 includes an atomic force microscope (AFM, shown in a simplified version as a combination of a probe 104 and a tip 104A disposed on the surface of a SUT 108 during operation). The instantaneous position of the flexible probe 104 and / or its deviation from a reference position (as a result of the interaction between the tip 104A and the SUT 108) is evaluated based on (a) the deviation in the reflection of a beam 110 from a laser light source 114 (usually configured to generate visible light) from the surface of the probe 104, and (b) recorded after such reflected beam 110 is received by a position-sensitive detector 118. (Other embodiments for determining the position of the probe can be implemented as known in the related art).

[0054] The AFM controller electronic circuit system 122 includes a special control module that enables delivery of a specific type of excitation signal to the AFM feedback electronic circuit 130 (configured to control the operation of the system 100 in the force setpoint modulation regime) and / or to the Z scanner programmable electronic circuit module 134 (configured to change and / or modulate the position of the probe 104 by use of the Z repositioner 140 and / or the position of the sample 108 by use of the Z repositioner 144 along the normal direction to the surface of the sample 108 during operation of the system 100). Among the specific excitation signals are at least low-frequency excitation signals (respectively generated by the electronic circuits of the electronic blocks 126A, 126B, and 126C), multi-frequency (in one embodiment, dual) excitation signals, and mixed-frequency sine-wave excitation signals. DDS: Direct Digital Synthesizer (a specific form of a digitally implemented waveform generator as known in the relevant art). The process of changing and / or modulating the position of either or both of the sample 108 and the probe 104 along the normal direction to the surface of the sample 108 (as indicated by the Z-axis) is generally referred to herein as "Z modulation". Examples of repositioners include electronically controlled micro and sub-micro stepping positioning devices known in the relevant art.

[0055] Here, it is as shown below.

[0056] In the case of low-frequency excitation provided by the Z scanner module 134, the frequency of excitation of the movement of the probe and / or the sample during operation of the system 100 is from sub-Hz frequencies up to several hundred Hz (specifically, from 0.001 Hz to 1,000 Hz, preferably from 0.01 Hz to 300 Hz, more preferably from 0.1 Hz to 150 Hz).

[0057] In the case of dual - frequency excitation, the signal provided to at least one of the repositioners 140, 144 by the Z - scanner module 134 includes a mixed - wave signal that combines a low - frequency signal and a signal of a predetermined reference frequency (which may be a frequency higher than the low - frequency).

[0058] In the case of mixed - frequency excitation, the module 134 is configured to generate a signal that combines several (preferably 10 or more) sine waves with different frequencies, amplitudes, and (optionally) phase relationships corresponding to each for driving. The selected plurality of frequencies of the individual sine - wave components in the mixed - frequency excitation can cover one or several decades in the frequency range. The information summarized in Tables 1, 2 and FIGS. 7A, 7B provides examples of such mixed - frequency excitation of the probe of embodiments of the present invention.

[0059] [Table 1]

[0060] [Table 2]

[0061] In one embodiment (see Table 1 and FIG. 7A), the probe is driven by an electrical signal that combines nine frequency components, where component #1 is considered the fundamental wave, and the frequencies of the remaining components are harmonics of the frequency of component #1. The amplitude of each of the frequency components is selected to vary from - 1 to + 1. Further, phase shifts (as specified) are introduced between and among the itemized components of the signal that drives the probe. FIG. 7A shows, in dashed lines, nine plots 704 representing each of the drive sub - signals at these frequency components, and in a solid line (710), the resultant collective excitation force (or, similarly, the resultant collective displacement signal delivered to the probe by the overall system electronics) applied to the probe to displace the probe.

[0062] For comparison, Table 2 and FIG. 7B illustrate the situation where the probe is driven by a mixture of nine signals representing the harmonics of a selected fundamental frequency (of component #1 in Table 2), and these harmonic signals are applied to the probe simultaneously without a predetermined phase shift, i.e., actually in the same phase. FIG. 7B shows nine plots 714 (shown as dashed lines) representing the harmonic components, and the curve 720 shows the resulting excitation force applied to the tip of the probe.

[0063] In one embodiment, the special electronic circuit control module 122 can be implemented in firmware, generally using, for example, a field programmable gate array (FPGA) and a digital signal processor (DSP), based on existing flexible AFM control. Further, the excitation signal that is provided to at least one of the probe's Z repositioner 140 and the sample's Z repositioner 144 (as indicated by lines 140A, 144A) and that governs its operation (preferably at a frequency in the range of about 100 Hz to 100 kHz) can also be routed by the AFM-nDMA control module 122 to a special "high-frequency" sample actuator / sample heater 148, as indicated by line 148A.

[0064] The AFM digital feedback electronic module 122 (implemented by a DSP, FPGA, or a combination thereof in one embodiment) may be configured using a PID (Proportional Integral Derivative) or PI (Proportional Integral) electronic circuit controller, which, during operation, receives as input a deflection signal (shown as 150) or a signal from a Z sensor (shown as 140 or 144), and generates a control output for positioning the Z scanner aimed at minimizing the difference (error signal) between the input and the setpoint. (The setpoint is understood as the desired value of the signal controlled in the feedback loop). For example, when a deflection signal is input, the AFM holds and maintains the load at a selected level. When an AC signal is mixed with the setpoint (a situation called setpoint modulation), the AFM digital feedback follows a signal representing both the DC and AC parts of the setpoint, for example, in a force setpoint modulation regime.

[0065] The signal routing control electronic circuit 160 includes a digital control multiplexer configured to control outputs and signal inputs, which is intended to implement various AFM control schemes and / or operating modes such as, for example, force setpoint modulation, z setpoint modulation, and z modulation. In the operation of the system, this module 160 connects input signals, setpoint signals, setpoint modulation signals to the AFM digital feedback module 130, and routes waveforms from appropriate DDSs (Direct Digital Synthesizers, vibration waveform generators) 126A, 126B, and / or 126C, and input signals (150, a deflection signal not shown in FIG. 1, and signals from Z sensors or height sensors) for acquisition and lock-in processing.

[0066] FIG. 2 is a simplified schematic diagram 100' of a specific version of the embodiment 100 of FIG. 1 configured to implement the force setpoint modulation mode of operation of the system. Here, the deflection setpoint represents the desired value, or target value, of the deflection signal controlled by the AFM feedback loop. The deflection setpoint including the modulation component (setpoint modulation) is shown as 152.

[0067] Mounting of the Sample

[0068] The sample (SUT under test shown as 108 in FIG. 1) measured in the AFMnDMA embodiment of the present invention can be sized as a thin (a few microns thick) cross-section or slice of the material, or alternatively, as a bulk piece having a cryo-sectioned surface (substantially flat block surface) (e.g., up to 3 mm thick). For the purpose of implementing the idea of the present invention, the term "substantially flat" identifies a surface characterized in that the average unevenness difference of its spatial profile is 20 nm or less, more preferably 10 nm or less.

[0069] Such a substantially flat or substantially planar surface can be prepared by cryomicrotome sectioning, by casting a thermosetting polymer onto a mica surface, or by spin-casting a dissolved polymer onto the surface of the sample. The prepared sample section is attached onto a selected substrate (e.g., in one embodiment, a sapphire or stainless steel disk having a diameter of about 10 mm to 12 mm and a thickness of less than 1 mm) to form a sample substrate assembly. Thereafter, the sample substrate assembly can be fixed to the heating and cooling device using magnetic attachment or thermal compound grease.

[0070] Example of System Configuration for Temperature-Dependent Measurement

[0071] It is recognized that AFM-nDMA measurement of the sample at variable temperature may generally require a heating and cooling device for the sample (shown as 148; referred to synonymously with the heating and cooling device for simplicity). It is desirable to utilize a sample heater specially designed for highly spatially localized / focused confinement of the thermal gradient only in the region of the sample, in which case only the sample is heated, as opposed to heating the mechanical structure of the entire sample and the AFM stage to minimize overall thermal drift.

[0072] According to the idea of the present invention, such a carefully designed sample heating and cooling circuit is configured to achieve a low level of thermal drift rate in the lateral and vertical spatial directions (x-direction, y-direction, and z-direction with reference to the local coordinate system in FIG. 1) in the equilibrium state of the thermal gradient. (The term "low drift rate" is defined and referred to as such a drift observed during the measurement time when its spatial value is small compared to the measured dimensions of the nanoscale features. To determine the value of the lateral drift, it can be compared with the contact radius / contact size, and the value of the vertical drift can be compared with the penetration depth / deformation depth of the sample.) Next, such practical results advantageously enable measuring the properties of the sample at a specific target position on or in the sample, thus affecting the quality and / or accuracy of the spatially resolved AFM-nDMA measurement.

[0073] (Those skilled in the art will easily understand that the thermal drift of the cooler / heater represents the lateral (XY) or vertical (Z) drift of the relative position of the probe and the sample. The drift rate is measured as the change in position per unit time. The lateral drift represents the speed at which the XY position of the probe relative to the sample changes. The vertical drift or simply the Z drift will indicate how fast the vertical position of the probe and the sample is changing.)

[0074] In addition to the sample heater device, a dedicated heater for the probe is used to facilitate the localization of the thermal gradient and prevent the probe lever 104 from accumulating condensation deposits. Thus, in one embodiment, the probe heater device 154 may include first and second heater plates arranged in cooperation with the upper and lower surfaces of the sample.

[0075] Alternatively or additionally, calibration of the surface temperature of the sample (the temperature measured by the set point of the heater 148 and a dedicated sensor built into the heating element of the heater 148) can be actually achieved using a small thermocouple attached to the surface of a sample carrier (not shown in FIG. 1 and configured as, for example, a sapphire disk or a steel pack with a diameter of 10 mm in one example) that mechanically supports and transports the sample in proximity to the position of the sample 108 on the sample carrier.

[0076] As a non-limiting example

[0077] To measure the viscoelastic properties of a material as a function of temperature (e.g., in a range from room temperature RT to the upper limit of the range, e.g., 250 °C), an embodiment 100 of the AFM-nDMA system can optionally include a sample heater holder / sample actuator 148 that includes an electronic circuit designed to ensure low thermal drift (on the order of 2 nm / min or less) of the reference surface of the holder 148 in the X, Y, and Z directions. When using such a sample heater holder, the sample 108 cooperates with the reference surface, and the temperature of the heater of the holder 148 is controlled by a thermal controller (not shown for simplicity of illustration) that establishes a programmable temperature setpoint and feedback (e.g., PID feedback, or proportional-integral-derivative controller / feedback). The use of such a sample holder with a carefully designed electronic heating circuit facilitates the measurement of the viscoelastic properties of the sample 108 at a substantially predetermined temperature within the temperature range spanning through the glass transition of a specific polymer material (e.g., at the glass transition temperature Tg decreased from RT to 250 °C in the temperature space of the heater). For example, in the case of a polymethyl methacrylate (PMMA) material with a glass transition temperature of about 105 °C, the preferred predetermined temperature is within the range from room temperature (about 25 °C) to the upper limit of about 140 °C to 150 °C.

[0078] Preferably, the system 100 may also include a top plate heater or a probe heater 154 configured to maintain a low thermal gradient within the probe-sample space.

[0079] It should be noted that in order to perform AFM-nDMA measurements on materials having a Tg below room temperature, it may be desirable to lower the temperature of the sample by cooling the sample. (Such examples include polypropylene having a glass transition temperature in the range of -20°C to -5°C, polyvinyl ethylene having a glass transition temperature near -20°C, or polymethylpolysiloxane having a glass transition temperature near -12°C). The option of heating and cooling hardware (details not shown) corresponds to a temperature space below RT, for example, a temperature space from RT to -35°C.

[0080] For operation by the above-described heater or heating and cooling hardware, the AFM scanning mechanism (depending on the specific embodiment, whether it is the Z scanner 144 of the sample and / or the Z scanner 140 of the AFM probe 104) needs to be sufficiently thermally insulated from the heating / cooling source. Otherwise, the performance of the scanner (drift, calibration, dynamics, etc.) may be variously affected throughout the temperature space. In some cases, the desired insulation can be achieved by using a special probe holder made of a material having a low thermal conductivity (for example, MACOR, a machinable ceramic material). On the other hand, at the same time, the tip 104A of the AFM is preferably maintained at a temperature close to the temperature at which the sample 108 is held simultaneously from the formation of condensation on the lever surface, the thermal bending of the lever, the local cooling of the sample, and the thermal gradient. (It should be noted that the systems of the related art are not known to balance these two characteristic requirements and conditions). Keeping the tip 104A at a temperature substantially equal to the temperature of the sample 108 can be achieved by using heater hardware at the tip within the probe holder that constitutes the heater element under a so-called probe nest (i.e., under the portion of the probe where the tip of the probe is attached to the probe holder by a spring clip or other means).

[0081] The use of the heating and cooling option in an embodiment of the AFM-nDMA system preferably further requires environmental control (humidity control - RH, inert atmosphere, e.g., dry nitrogen purge) to prevent oxidation and degradation of the sample surface due to moisture absorption. In the simplest case, such environmental control can be achieved by using a flexible seal sleeve attached to the probe holder, which forms an insulated local environment that can be purged with a low flow of dry nitrogen gas. Alternatively, a special sealed local environment cell (LEC, incorporating heating and cooling) can be used.

[0082] A sample holder actuator configured to measure only at room temperature

[0083] When the above “heating and cooling” hardware option is implemented and used, the system is configured to hold the sample fixed and stationary in space while mechanical excitation due to AFM-nDMA is delivered via the spatial actuation of the tip 104. (This is achieved by the use of either the AFM's Z scanner 140 or the piezo actuator of an additional probe holder). However, it is recognized that if the measurement is intended to be made only at room temperature, the mechanical excitation or mechanical actuation of the mutual orientation between the sample 108 and the tip 104 can alternatively be done via the spatial actuation of the sample holder (harmonic, small amplitude). Thus, the combination of the sample holder actuator 148 and the sample's Z scanner device 144 is carefully designed to operate by causing mechanical movement of the sample 108 at at least one frequency within a wide frequency range (e.g., from about 100 Hz to about 100 kHz). Unlike the AFM's Z scanner 140, the sample holder actuator 148 and / or the sample's Z scanner device 144 typically do not have an associated Z sensor configured to detect and provide a readout of the amplitude and phase of the mechanical vibrations of the mechanical movement. Instead, the amplitude and phase of the mechanical movement provided by devices 144 and / or l48 as a function of frequency can be calibrated in a separate additional reference “calibration” measurement by contacting the AFM probe with a hard reference sample and measuring the deflection of the probe (i.e., the amplitude and / or phase of such deflection).

[0084] AFM-nDMA Method: Operational Characteristics

[0085] One embodiment of the AFM-nDMA system of the present invention is configured to measure viscoelastic properties at a (user-selected) point location on the surface of a sample. (It should be noted that, unlike and in contrast to most conventional AFM modalities, embodiments of the AFM-nDMA method of the present invention enable an imaging mode with "mapping" of viscoelastic properties in a limited frequency range, but are generally not surface imaging techniques. In fact, the main target of the proposed method is point measurement / point spectroscopy (multi-frequency) applications, but can also be used for surface imaging / mapping at a single selected frequency or a limited number of selected frequencies).

[0086] Operating mode of the "ramp" (or forward ramp) At each point location on the surface of the sample under test, AFM nanoindentation measurements are performed. Here, the Z scanner 140 is spatially extended along the z-axis to draw the tip 104A of the probe 104 towards the surface of the sample 108 (ramp motion) until a specified and / or pre-set threshold in the deflection of the probe cantilever is detected using the PSD 118. The pre-setting of the cantilever deflection corresponds to a specific pre-load (usually) force (referred to as the trigger force) exerted on the sample 108 by the tip 104A of the probe, thereby enabling the system 100 to determine the deformation of the sample required under the corresponding load.

[0087] Operating mode of the "hold" After the pre-load force reaches the threshold, the operation of the Z ramp is aborted / stopped and the probe 104 is maintained (held) on the "hold" for a specified duration. (When AFM-nDMA excitation is present, such a duration is specified according to the number of cycles required at the frequency selected for the measurement. As a non-limiting example, the probe is held on the "hold" for 20 cycles at 0.1 Hz, or for 200 seconds). This is the operating segment while the modulation / excitation of AFM-nDMA is on. Some variations of the "hold" modality are within the scope of the present invention.

[0088] Operation mode of "holding force" Here, the AFM feedback electronic circuit keeps the deflection of the cantilever (the force applied to the cantilever) constant at a predetermined target value (typically, the value of the force before loading). On the other hand, the creep of the Z piezoelectric element, thermal drift, and material creep under load are compensated for as a result by adjusting the Z position of the probe 104 with the AFM feedback circuit.

[0089] Operation mode of "holding Z sensor" In this mode, the AFM feedback circuit uses the Z sensor associated with the Z scanner 140 to keep the mechanical extension of the Z scanner 140 (along the Z axis) substantially constant. The creep of the Z piezo is corrected dynamically, but the drift of the Z sensor, thermal drift, and material creep are not corrected. In this operation mode, the force applied to the sample is not necessarily kept constant. As a result, it is preferable not to use the "holding Z sensor" operation mode for a long time. (It should be noted that this operation mode may be useful in the case of adhesive creep or adhesion force creep. In such cases, keeping the position constant may function better than keeping the deflection / force constant. In the latter case, the adhesion force creep can "suck" the probe into the surface and cause deep indentation holes.)

[0090] Operation mode of "holding Z drive" Here, while the signal of the AFM feedback circuit is off, the Z piezo high voltage is kept constant. As a result, no compensation is provided. Therefore, this operation mode is preferably used for a short time to avoid piezo creep. This operation mode is aimed at high-speed measurements at relatively high frequencies (for example, in a region where other feedback-based operation modes may not be able to maintain / track the performance at the modulation frequency when the frequency exceeds 100 Hz).

[0091] Operation mode of "retract" (or reverse ramp) At the end of the "hold" segment of the operation, the probe 104 is retracted from the surface of the sample 108. The shrinkage curve is recorded using a programmable processor that operably cooperates with the PSD 118. For viscoelastic materials, the shrinkage rate is an important parameter that can affect the accuracy of the JKR model analysis (described below).

[0092] One of ordinary skill in the art will readily understand that the proposed AFM-nDMA method should not be confused with, and should not be, the "tapping" mode of operation of a conventional AFM system: the tapping mode is a different intermittently contacting AFM technique. In the AFM-nDMA modality of the present invention, the probe is brought close to and fully contacted with the surface of the sample, and after actually deforming / denting the surface, modulation, which is the oscillatory component of the force or Z displacement, is turned on. At the end of the hold operation period, the probe is retracted, and then the probe is laterally moved / transited to another point on the surface and used to perform another point measurement at another location on the surface.

[0093] "Force-Distance Curve" ("FDC") The AFM force-distance curve (also known as deflection vs. Z scanner extension) is recorded between a forward ramp and a reverse ramp (retraction of the probe). As will be readily recognized by those skilled in the art, the force-distance curve is a plot or trace of the deflection / force signal vs. the Z-separation signal obtained as the Z scanner 140 moves the tip 104A towards the sample surface (~approach curve), or away from a pre-contact with the sample surface (~retraction curve). The FDC is further important for calculating the elastic properties of the sample (such as reduced elastic modulus and Young's modulus), and, more importantly for embodiments of the AFM-nDMA of the present invention, for estimating the size of the contact area or "contact radius" of the tip with the sample, and can be analyzed using a contact mechanics model (such as those represented by any of the Hertzian, Johnson-Kendal-Roberts (JKR), Derjaguin-Muller-Toporove (DMT) models). (As references, for example, K. L. Johnson, K. Kendall, and A. D. Roberts, Surface energy and the contact of elastic solids, Proc. R. Soc. Lond. A 324 (1971) 301-313; or Deqaguin, B. V., Muller, V. M., and Toporov, Y. P., 1975, Effect of contact deformation on adhesion of particles, Journal of Colloid and Interface Science, 53(2), pp. 314-326)

[0094] When the sample is represented by a polymer material with strong adhesion, the JKR model provides the best results in fitting the experimental data. (In a preferred situation, the force-distance curve of a viscoelastic material should be analyzed using a contact mechanics model of the viscoelastic adhesive surface.) The contact radius calculated from the retraction curve is only applicable to the conditions at the end of the hold period and does not actually provide information regarding the contact radius at all instants during or within the hold period required to obtain accurate quantitative AFMnDMA results.

[0095] Due to creep of the sample, the contact area between the tip of the probe and the sample may change during the hold / measurement time. The dynamic stiffness (of the contact between the probe and the sample measured at a preselected "reference" frequency) is proportional to the contact radius. If the system is configured to monitor this dynamic stiffness throughout the hold / measurement time (in "interleave", which is a continuous or primary measurement mode), a determination of the relative change in the contact radius during the measurement can be performed. The contact radius is determined from the force-distance indentation curve (ramp) after or before the hold period, and this correction of the contact radius is applied at each specific instant of the hold period.

[0096] "Temperature step" Lamp and hold-based measurements are performed using an embodiment of the present invention's AFM-nDMA at a substantially constant temperature after reaching thermal equilibrium in the system of the instrument sample heater. On the other hand, AFM-nDMA measurements as a function of temperature are performed by sequentially considering a list of temperature setpoints / steps (according to a specific temperature program), waiting to reach thermal equilibrium at each temperature setpoint, and then performing AFM-nDMA lamp and hold point measurements at that temperature point. The degree of thermal equilibrium reached before starting the lamp and hold measurements can be evaluated, for example, by measuring the thermal drift rate in the Z direction until a numerical value of the desired (low) drift rate is reached while waiting for and performing a "zero-size" scan on the surface in the tapping AFM mode of the peak force (optionally, performing scans other than zero-size to evaluate XY drift by tracking topographical and boundary features of material property maps such as DMT modulus, adhesiveness, deformation, etc.).

[0097] While waiting for thermal equilibrium and remaining within the AFM feedback loop on the surface, the Z scanner 140 may reach the limits of the Z piezo elongation or contraction due to thermal drift and thermal expansion / contraction / flow of the material. Therefore, it is preferable to continuously recenter the Z scanner by stepping the Z engage motor up and down so that the Z position of the scanner is maintained at the center of the dynamic range of the piezo.

[0098] Figures 3A, 3B, and 3C show examples of signal traces for force setpoint modulation. Here, in the operating mode of force setpoint modulation, the AFM feedback electronics module tracks both the static ("DC") component and the dynamic, oscillatory ("AC") component of the setpoint. The error signal trace shown in Figure 3A (provided by the AFM feedback control electronics circuit 130 of the system 100 of Figure 1) has high-frequency noise and a small residual vibration (AC) error (these data examples were acquired at a modulation frequency of 5.6 Hz), and the residual has a high level of noise. These results indicate that the AFM feedback circuit 130 is tracking the oscillatory setpoint component and providing modulation of the load force exerted on the sample 108 by the probe 104 (constituting the "force setpoint modulation" mode of operation). The actual vertical deflection trace of Figure 3B shows the oscillatory (AC) component, i.e., the modulation of the force (here, the normal force exerted by the probe on the sample is equal to the vertical deflection of the lever multiplied by the spring constant of the lever). The Z sensor or "height" signal trace is shown in Figure 3C. Here, the overall downward slope of the trace line may be due to thermal drift of the system and / or viscoelastic creep of the sample material. (Drift correction techniques described in the software lock-in processing method are useful for mitigating the detrimental effects of such gradients / tendencies on the accuracy of signal amplitude and phase measurements).

[0099] The data shown in FIGS. 3D, 3E, and 3F are similar to the data in FIGS. 3A, 3B, and 3C, but show examples corresponding to measurements at a different, lower modulation frequency of -0.32 Hz (compared to 5.6 Hz in FIGS. 3A, 3B, and 3C). Notably, the residual vibration (AC) component in the error signal trace of FIG. 3D is substantially indistinguishable in the noise, in contrast to that of FIG. 3A. This fact is due to the AFM feedback control frequency response and the effectively superior feedback tracking at low frequencies compared to high frequencies. The “height” signal trace of the Z sensor in FIG. 3F compared to FIG. 3C shows a more pronounced downward drift / creep trend because the measurement time at the low frequency of 0.32 Hz (about 60 seconds) is much longer compared to the short measurement time (about 4.5 seconds) at 5.6 Hz in FIG. 3C. This indicates that the drift correction process described by the software lock-in method is particularly important at low frequencies that require a significant measurement time.

[0100] FIGS. 4A and 4B show the results of experimental AFM-nDMA measurements performed in an embodiment of the system of the present invention. For a sample of polydimethylsiloxane (PDMS) material, FIG. 4A shows the relationship of the storage (E’) modulus data versus the measurement frequency (at fixed, room temperature), and FIG. 4B shows the relationship of the loss (E”) modulus data versus the measurement frequency (at fixed, room temperature). The comparison of the AFM-nDMA results (red, crosshairs, 410, and 420) and the bulk DMA measurements (green, dashed lines, 415, and 425) for samples from the same material shows a substantial agreement between the storage and loss moduli measured at the nanoscale (AFM-nDMA) and the moduli measured using the bulk macroscopic method (DMA) (the latter is the underlying truth and is used in related art as a reference to verify other results).

[0101] Figures 5A, 5B, 5C, 5D, 5E, and 5F show the results of experimental AFM-nDMA measurements of the storage modulus and loss modulus of fluorinated ethylene propylene (FEP) material as a function of temperature (at three different fixed low frequencies: 0.1 Hz, 1.0 Hz, and 5.6 Hz). A comparison is provided between the measurement results obtained using the AFM-nDMA-based embodiments of the present invention and those obtained using conventional bulk DMA methods. Figures 5A, 5B, and 5C show the dependence of the storage modulus on temperature, and Figures 5D, 5E, and 5F show the dependence of the loss modulus on temperature of FEP material samples measured at three different frequencies (in the range from 0.1 Hz to 10 Hz) using the nanoscale AFM-nDMA method and the bulk macroscopic DMA technique according to the present invention. The comparison of the AFM-nDMA data (red, cross, 530, 540, 550, 560, 570, 580) and the bulk DMA data (green, dashed line, 535, 545, 555, 565, 575, 585) shows that both methods detect a significant decrease in the storage modulus value with increasing temperature, and as the measurement frequency increases in accordance with the expected rheological behavior of the FEP material, the peak of the loss modulus is detected to shift towards higher temperatures. Thus, the glass transition of FEP can be detected by both the conventional bulk DMA method and the proposed AFM-nDMA method.

[0102] Figure 6A shows the experimentally defined dependence of the loss tangent (the ratio of the loss modulus to the storage modulus) of FEP on frequency, presented via time-temperature superposition (TTS). A comparison is provided between the measurement results obtained with the AFMnDMA-based embodiments (610) of the present invention and the measurement results obtained with conventional bulk DMA methods (615). Figures 6B and 6C show the dependence of the storage modulus and the loss modulus, respectively, on frequency (corresponding to the graph of Figure 6A), presented via time-temperature superposition (TTS). Here, a comparison is provided between the results of measurements performed using the AFM-nDMA-based embodiments (620, 630) of the present invention and the results of measurements performed using conventional bulk DMA methods (625, 635). Figure 6D shows an example of time-temperature superposition (TTS) of the shift factor (well known to those skilled in the art) by comparison between the results of measurements performed using the use of embodiments of the present invention (open circles; 640) and the results of measurements performed with conventional bulk DMA methods (solid-line circles; 645).

[0103] Referring to FIGS. 6A, 6B, 6C, and 6D, the data were measured using both the nanoscale AFM-nDMA method configured in accordance with the concepts of the present invention and the conventional macroscopic volume DMA method (for the same fluorinated ethylene propylene, FEP, material as in the examples of FIGS. 5A-5F described above) over a frequency range from 0.1 Hz to beyond 100 Hz and a temperature range from room temperature to beyond 120°C. The data for all these temperatures and frequencies were superimposed via time-temperature superposition (TTS), a rheological data analysis technique commonly used for macroscopic measurements, and plotted on a scale of "TTS-shifted" frequencies. FIG. 6A shows a TTS-plot of loss tangent ("tangent delta") versus shifted frequency, and FIGS. 6A and 6B show TTS-plots of storage modulus and loss modulus, respectively. FIG. 6D shows the "shift factor" of TTS applied to the frequency during TTS processing. As is apparent from FIGS. 6A-6D, the disclosed AFM-nDMA method provides novel and unique performance (with respect to nanoscale measurements at low frequencies) that enables direct comparison with results obtained via conventional volume macroscopic rheology techniques and methods such as DMA and TTS. To the best of the inventors' knowledge, these AFM-nDMA results are the first example of time-temperature superposition data at the nanoscale via AFM.

[0104] Addendum: AFM-nDMA modulation / excitation. Frequency space addressing.

[0105] Operation in frequency range: 0.1 Hz to 10 Hz (force point setting modulation) Measurements at low frequencies are associated with long measurement times (covering 10 or more cycles of the excitation frequency). Therefore, a hold force mode that can keep the load state constant even in the presence of drift or creep is preferred. On the other hand, active AFM feedback will cancel out the modulation introduced into the Z channel, effectively disabling the mechanical excitation and rendering the measurement useless. Instead of Z modulation, force setpoint modulation should be used at low frequencies (within the range of the active AFM feedback bandwidth). In force setpoint modulation, the AFM feedback tracks both the DC preload force setpoint and the AC periodic modulation component, providing the necessary mechanical excitation. The force setpoint modulation scheme can be implemented in the FPGA firmware by adding a low-frequency direct digital synthesis (DDS) component. The amplitudes and phases of the force and displacement (deformation) can be measured by capturing / recording and demodulating the signal trace using a hardware lock-in in the FPGA or a "software lock-in" method (the "software lock-in") that corrects for drift and creep implemented in software via two channels of a flexure sensor and a Z sensor.

[0106] Operation in the frequency range of approximately 10 Hz to 100 Hz (modulation of the force setpoint and / or the Z scanner) The hold period in this frequency range can be made relatively short, and a hold Z drive mode (or hold Z sensor with AFM feedback that tracks both the DC position and the AC modulation of the Z sensor channel) can be tolerated for low or moderate drift and creep rates. Therefore, Z scanner modulation (with a hold Z sensor or hold Z drive) can be used. Alternatively, force setpoint modulation can be used since the AFM feedback can have sufficient bandwidth to track the AC setpoint in this frequency range.

[0107] Operation in the frequency range of approximately 100 Hz to approximately 1000 Hz ("Z modulation") AFM feedback has problems tracking AC setpoints at frequencies above 100 Hz and can cause inefficient excitation (where a significant portion of the modulation amplitude ends up as a residual error signal). Z modulation is preferred.

[0108] (Reference frequency techniques for correcting (or in some cases subtracting) creep in the contact area and / or reference frequency techniques for tracking the contact area during the associated creep correction process) In the case of AFM-nDMA, the elastic modulus of the material at a particular frequency is measured (monitored) over the entire duration of the hold segment in parallel with AFM-nDMA measurements at other frequencies, and then the contact radius calculated from the JKR fit of the shrinkage curve at the end of the hold can be corrected. This requires exciting at least two frequencies simultaneously. Alternatively, the measurements at the reference frequency can be interleaved with the measurements at other frequencies.

[0109] Multi-frequency excitation Multi-frequency excitation can reduce the measurement time of AFM-nDMA. When the principle of superposition is applied, the results of multi-frequency excitation should be equivalent to sequential measurements (where the provided drift and creep are appropriately taken into account). In the case of non-linearity (essentially present at the tip-sample contact), "cross-talk" between frequencies can occur during multi-frequency excitation.

[0110] AFM-nDMA model equation ((Dynamic stiffness in harmonic excitation))

[0111] As described above, AFM-nDMA is a nano-scale dynamic mechanical analysis of a sample, performed by pushing the probe of a cantilever into the surface of the sample with a controlled force that includes both a quasi-static (DC) component and a dynamic (AC) component, a vibrational component. The frequency (or frequencies) of the vibrational component of the force applied to the sample is carefully selected to match the low-frequency range - from sub-hertz to several hundred hertz - typically associated with bulk macroscopic DMA of soft materials and various polymers.

[0112] Embodiments of the AFM-nDMA system of the present invention are configured to operate in a plurality of different regimes.

[0113] 1. Force setpoint modulation regime or mode: Here, when the AFM feedback is turned on, the deflection of the probe cantilever is monitored to maintain both the DC force (“preload force”) and the AC vibration modulation component. This regime is suitable for low-frequency AFM-nDMA experiments (sub-hertz to several hundred hertz) because it can maintain a stable state despite drift and creep of the sample. The vibration displacement is measured by the AFM height sensor. Note that for a fixed predetermined amplitude of force modulation, the amplitude of the Z scanner displacement (height sensor) generally depends on the viscoelastic properties of the sample.

[0114] 2. Displacement mode (or Z modulation) regime: The push-in ramp is stopped at a predetermined trigger force (preload force), but the AFM feedback is not activated. The Z displacement of the scanner is modulated at a fixed predetermined amplitude; therefore, the amplitude of the force AC component (deflection) depends on the viscoelastic properties of the sample material. This regime is most suitable for high-speed force volume AFM-nDMA experiments where the contact time duration is relatively short, and the measurement frequency can range from about one hundred to several hundred hertz.

[0115] 3. External actuator mode regime: Here, the sample is attached to a high-frequency actuator (HFA) stage. The AFM probe is tilted towards the sample surface and held at a position where a predetermined preload force exists (within the closed-loop feedback of the height sensor signal). The actuator is excited to provide modulation of the vertical (Z) displacement of the sample surface relative to the AFM probe base held at the stationary position via the height sensor feedback loop. The AFM vertical deflection signal is recorded and provides information about the oscillatory part of the force at the contact between the probe and the sample in response to the modulation of the separation distance (between the probe base and the sample surface). Calibration must be used because there is no sensor to measure the amplitude and phase of the actuator vibration.

[0116] For all these system configurations, the operation of the embodiments of the present invention results in an evaluation of the viscoelastic material properties of the sample material using a common theoretical framework (referred to as "dynamic stiffness in harmonic excitation") based on the equations of the inverse and forward problems of nanoscale dynamic mechanical analysis. The equation of the inverse problem makes it possible to calculate the dynamic stiffness of the contact from material properties such as the storage modulus, loss modulus, and tangent difference (loss tangent, attenuation coefficient), and from the results of AFMn-DMA measurements providing the amplitude and phase of the acquired signal. The equation of the forward problem can be used for the optimization of experiments by calculating the desired excitation amplitude and preload force.

[0117] For simplicity, a common set of notations (for the equations describing the theory and calculations of AFM-nDMA) is used to describe all three regimes introduced above using Z modulation via either the Z scanner or the sample actuator. Assume that the probe-based Z displacement is described by a harmonic signal (in complex number form):

[0118] Z(t)=Z 1 e i(ωt+Ψ) +Z 0 (Equation 1)

[0119] where Z 1 , Ψ are the amplitude and phase of the displacement vibration component at frequency ω = 2πf. The probe is calibrated and known, and is considered to have a spring constant K c . Assuming that the system is linear (in the sense of producing a harmonic response to harmonic excitation), the deflection of the AFM probe can be described by a harmonic signal (which is the measured value, the vertical deflection signal):

[0120] d(t)=D 1 e i(ωt+φ) +D 0 (Equation 2)

[0121] where D 1 , φ are the amplitude and phase of the deflection vibration component at frequency ω = 2πf, respectively.

[0122] The general formula for AFM-nDMA calculations can be derived using the definition of dynamic stiffness (the contact between the probe and the sample), which is simply an extension of the definition of stiffness to the case of harmonics and complex numbers: the dynamic stiffness S * (in units of Newtons / meter) is defined as the ratio of the complex force to the complex deformation caused by this force:

[0123] S * =F * / L * (Equation 3)

[0124] And the oscillatory deformation can be determined as the difference between the displacement and the deflection of the probe, or in complex form:

[0125] L * =Z 1 e i(ωt+Ψ) -D 1 e i(ωt+φ) (Equation 4)

[0126] When the spring constant (K c ) of the cantilever is known, the oscillatory force is determined from the deflection:

[0127] F * =K c D 1 e i(ωt+φ) (Equation 5)

[0128] Therefore, Equation 3 for dynamic stiffness can be rewritten as follows.

[0129] S * =S’+iS’’=K c D 1 e i(ωt+φ) / [Z 1 e i(ωt+Ψ) -D 1 e i(ωt+φ) (Equation 6)

[0130] Applying algebraic operations to the complex-valued equation and separating the real and imaginary parts, Equation 6 becomes as follows.

[0131] [Number] (Equation 7.1, Storage Rigidity)

[0132] [Number] (Equation 7.2, Loss Rigidity)

[0133] [Number] (Equation 7.3, Loss Tangent, Tangent Difference)

[0134] The numerical determination of viscoelastic material properties from dynamic rigidity experimental data requires knowledge of the contact size.

[0135] [Number] (Equation 8.1, Storage Modulus)

[0136] [Number] (Equation 8.2, Loss Modulus)

[0137] [Number] (Equation 8.3, Loss Tangent, Tangent Difference)

[0138] Here, a c is the contact radius between the tip of the probe and the sample.

[0139] In practice, the size of the nanoscale contact cannot be easily visualized or directly measured. However, it can be determined from the analysis of the indentation force-distance curve, for example, by applying the Johnson-Kendall-Roberts (JKR) contact mechanics model to the retract part of the force-distance curve.

[0140] Therefore, embodiments of the AFM-nDMA method of the present invention utilize the Johnson-Kendall-Roberts (JKR) contact mechanics model, which is well-known for calculating contact size, in conjunction with the use of a probe having a well-characterized spherical tip shape. (It should be noted that the JKR model is widely used and well-accepted for analyzing the mechanical properties of soft materials such as polymers, but it is also well-recognized that the JKR model was formulated for linearly elastic materials and is not strictly suitable for accurately describing viscoelastic materials. KJ Wahl et al., Journal of Colloid and Interface Science, 296 (1), 178-188, 2006 extended this analysis to viscoelastic materials for oscillatory adhesive contacts). To reduce the suspicion of discrepancies in the results of the JKR model for viscoelastic materials (e.g., in experiments using long hold periods for AFM-nDMA measurements at low frequencies), embodiments of the present invention utilize a special "punch probe" AFM tip with a known contact area size and additionally provide a verification method that does not depend on a specific contact mechanics model for calculating material properties from dynamic stiffness. Such features are not known or used in AFM-related technologies. Notably, referring to equations 8.1, 8.2, and 8.3, the determination of the storage modulus and loss modulus requires the inference of the contact radius from the mechanical analysis of indentation contact. In contrast, the determination of the loss tangent (tangent delta) does not require knowledge of the contact radius and is calculated directly from the ratio of the amplitude of deflection to displacement and the phase difference between deflection and displacement.

[0141] Reference Frequency Technique - Correction of Creep of Contact Radius in AFM-Based Measurements

[0142] As described above, the contact radius required for the calculation of the viscoelastic storage amount and loss elastic modulus is determined from the retract portion of the force-distance curve after all hold segments in the AFM-nDMA measurement at all predetermined frequencies are completed. This contact radius value estimated at the end of the indentation hold needs to be applied to the calculation across all frequency segments, which may precede the retract event by a longer time (possibly several minutes at sub-hertz frequencies). Before the frequency measurement segment, when the creep relaxation of the sample (under preload) is achieved (in other words, when the first relaxation waiting segment is long enough), and the DC force (such as in the force setpoint modulation regime) is maintained sufficiently constant during the hold and there is substantially no creep of the adhesion force, the contact radius can continue to be substantially constant over the entire duration of the AFM-nDMA measurement hold.

[0143] However, in most other cases, there is some creep in the contact radius during the hold, and a simple adaptation of the contact radius value of the retract curve may introduce systematic errors in the AFMnDMA calculation. Related techniques do not address these problems.

[0144] To compensate for the uncertainty due to the possible creep of the contact area during the hold, embodiments of the present invention utilize a special reference frequency approach described below. Rearranging Equation 8.1 gives the following:

[0145] S’ = 2a c E’ (Equation 9)

[0146] Those skilled in the art will readily understand that it is substantially and operationally reasonable to assume experimental conditions (such as temperature, humidity, etc.) that maintain the substantial stability, continuity, immutability of the material properties and the storage modulus E’ at 3 / 4 at a specific frequency. And the storage stiffness S’ 0 measured at a predetermined frequency f c is proportional to the contact radius a i . The storage stiffness S’(t 1 ), during the indentation hold {t 2, t 3 ...} at different instants at a predetermined frequency f 0 measured, and further measured as S’(t r ) at the end of the hold immediately before the probe retract time tr, the contact radius value a c (t i ) can be reconstructed (or, further, interpolated) during the entire time of the AFM-nDMA measurement.

[0147]

Number

[0148] Here, the value of the contact radius a c (t r ) is determined from the retract curve.

[0149] In particular, the loss stiffness S” (Equation 7.2) can in principle be used in a similar way for creep compensation of the contact radius, but considering the signal-to-noise ratio, it is more practical to use the storage stiffness S’.

[0150] To effectively perform the operations of the embodiments of the present invention, careful use of a processor controlled by specific instructions of an application stored in a tangible memory element may be required. Those skilled in the art should readily understand that the required algorithmic functions, operations, and decisions may be implemented as computer program instructions, software, hardware, firmware, or a combination thereof. Those skilled in the art should readily understand that the instructions or programs defining the functions and elements of the present invention may be distributed to the processor in many forms, including but not limited to information permanently stored in a non-writable storage medium (such as a read-only memory device in a computer like ROM, or a device readable by a computer I / O attachment like a CDROM or DVD disk), information changeably stored in a writable storage medium (such as a floppy disk, removable flash memory, and hard drive), or information transmitted to the computer via a communication medium including a wired or wireless computer network. Further, the present invention may be embodied in software, but the functions necessary to implement the present invention may be partially or wholly optionally or alternatively embodied using firmware and / or hardware components such as combinatorial logic, application specific integrated circuits (ASICs), field programmable gate arrays (FPGAs), other hardware, or some combination of hardware, software, and / or firmware components.

[0151] Throughout this specification, references to "one embodiment", "an embodiment", "a related embodiment", or similar language mean that a particular feature, structure, or characteristic described in connection with that "embodiment" is included in at least one embodiment of the present invention. Thus, appearances of these phrases and terms are likely to refer to the same embodiment, but not necessarily. It should be understood that no part of the disclosure taken alone and / or with reference to the drawings is intended to provide a complete description of all features of the present invention.

[0152] Also, it should be understood that no single drawing is intended to support a complete description of all features of the present invention. In other words, a given drawing generally describes only some, but not all, of the features of the present invention. The related portions of the disclosure, including a given drawing and the description referring to such drawing, generally do not include all elements of a particular view or all features that may be presented, for the purpose of simplifying the given drawing and directing discussion to the specific elements characterized in this drawing. One of ordinary skill in the art will understand that the present invention may be practiced without one or more of the particular features, elements, components, structures, details, or features, or using other methods, components, materials, etc. Thus, specific details of a particular embodiment of the present invention may not necessarily be shown in each drawing that describes such an embodiment, but the presence of this detail in the drawings may be implied, unless the context of the specification requires otherwise. In other instances, well-known structures, details, materials, or operations may not be shown in a given drawing or described in detail to avoid obscuring aspects of the embodiments of the present invention being discussed.

[0153] The invention described in the claims appended to this disclosure is intended to be evaluated in light of the disclosure as a whole and includes the features described in the claims and disclosed in the prior art referred to.

[0154] For the purposes of this disclosure and the appended claims, the use of the terms "substantially," "approximately," "about," and similar terms in relation to a nearby value, element, characteristic, or descriptor of a characteristic is intended to emphasize that the value, element, characteristic, or descriptor being referred to is not necessarily precisely described, but nevertheless is considered by those skilled in the art to be so for practical purposes. These terms, when applied to a particular characteristic or quality descriptor, reasonably represent approximate language as would be understood by a normal person skilled in the art, and also mean "mostly," "mainly," "considerably," "by and large," "essentially," "to a great or significant extent," "largely but not necessarily wholly the same" for the purpose of describing a particular characteristic or descriptor. In a specific case, the terms "approximately," "substantially," and "about," when used in relation to a numerical value, represent a range of plus or minus 20% with respect to the specified value, more preferably plus or minus 10%, even more preferably plus or minus 5%, and most preferably plus or minus 2%. By way of non-limiting example, for two values to be "substantially equal" to each other means that the difference between the two values may be within the range of ±20% of the value itself, preferably within the range of ±10% of the value itself, more preferably within the range of ±5% of the value itself, and even more preferably within the range of ±2% or less of the value itself.

[0155] The use of these terms in describing a selected feature or concept does not imply or provide a basis for grounds of indefiniteness or for imposing numerical limitations on a specified property or descriptor. As will be understood by those skilled in the art, the exact value or the actual deviation of such a value, element, or property departs from that described and may vary within a numerical range defined by the typical experimental measurement errors when using measurement methods recognized in the art for such purposes.

[0156] Changes to and variations of the illustrated embodiments may be made without departing from the concepts of the invention disclosed herein. Further, the disclosed aspects or portions of these aspects may be combined in ways not described above. Accordingly, the invention should not be regarded as limited to the disclosed embodiments as described herein. Further, the terms used herein are for the purpose of describing particular embodiments only and are not intended to limit the scope of the invention.

Claims

1. 1. A method for determining mechanical properties of a soft viscoelastic sample using an atomic force microscope (AFM) based system, comprising: repositioning a probe of the system toward the surface of the sample until a cantilever of the probe is deflected a predetermined amount from a nominal orientation of the cantilever; Modifying the rearranging step, comprising: i) the average sample loading force generated by the probe; and ii) the contact area between the tip of the probe and the surface; maintaining at least one of the following substantially constant; measuring viscoelastic parameters of the surface at a set of predefined frequencies while correcting for at least one of creep of the surface and spatial drift of the system; generating an output that is perceptible to a user and represents the viscoelastic parameter as a function of at least one variable of the measuring step; The method includes:

2. the measuring step is performed simultaneously at multiple frequencies from the set of predefined frequencies. The method of claim 1.

3. the modifying step includes modulating a sample loading force applied to the sample by the probe with a predetermined excitation frequency from the set of predefined frequencies. The method according to claim 1 or 2.

4. the modifying step includes maintaining the average sample loading force substantially constant while a separation between a surface and a base of the probe is modulated. The method according to claim 1 or 2.

5. The step of measuring the viscoelastic parameter comprises: simultaneously measuring both an excitation force applied to the sample by the probe and a deformation of the surface caused by the excitation force; avoiding repeated calibration of the system; performing a dual channel demodulation operation of the system to perform at least one of 5. The method according to any one of claims 1 to 4.

6. and maintaining operation of the system for a time sufficient to mitigate any creep of the surface resulting from the repositioning step.

6. The method according to any one of claims 1 to 5.

7. the step of performing dual channel demodulation includes combining first and second data received from a first sensor of the system electronics and a second sensor of the system electronics, respectively, during the measuring step; the first data represents a position of the probe relative to the surface, and the second data represents a degree of deflection of a cantilever of the probe from a nominal orientation; The method according to claim 5.

8. and performing the dual channel demodulation step includes introducing a correction for at least one of drift induced changes and creep induced changes in signal data received from at least one of the two channels. The method according to claim 5 or 7.

9. and continuously monitoring operation of the system at a reference frequency using at least one of a first electronic circuit and a second electronic circuit of the system to compensate / correct for changes in the contact area due to creep of the surface.

7. The method according to any one of claims 1 to 6.

10. the step of continuously monitoring includes continuously monitoring only one of the first electronic circuit and the second electronic circuit, and further comprising the step of obtaining calibration data representative of signals from the other of the first electronic circuit and the second electronic circuit obtained from a sample of the hard calibration.

10. The method of claim 9.

11. further comprising the step of compensating for changes in the contact area due to creep of the surface; The compensating step includes: i) calculating the change in contact area while calculating the viscoelastic parameters using a programmable processor of the system, the programmable processor being operatively connected to the AFM; and ii) repositioning the probe to compensate for the change; 11. The method according to claim 9 or 10.

12. the reference frequency is not included in the set of frequencies; 12. The method according to any one of claims 9 to 11.

13. the measuring step includes obtaining, during a first time period, from a sensor in the system's electronic circuitry, a first set of electrical signals at frequencies from the set of frequencies to determine an extent of indentation of the surface with the tip of the probe; acquiring, during a second period of time, a second set of electrical signals at a reference frequency to compensate for changes in the contact area due to creep of the surface; the sensors include at least one of a deflection sensor (118) and a sensor configured to measure a position of the probe relative to the surface; 13. The method according to any one of claims 1 to 12.

14. the reference frequency is not included in the set of frequencies; The method of claim 13.

15. the steps of acquiring the first set of electrical signals and acquiring the second set of electrical signals are interleaved with each other.

15. The method according to claim 13 or 14.

16. and compensating for the change in contact area based on determining a change in dynamic stiffness of contact between the probe and the sample.

16. The method according to any one of claims 13 to 15.

17. modulating the sample loading force is performed by adjusting the amplitude and phase of each oscillator component of the sample loading force from the set of predefined frequencies to a corresponding target value, the adjustment being dependent on a response of the material of the sample to which the modulated sample loading force is applied. The method according to claim 3.

18. 1. A system configured to determine surface mechanical properties of a viscoelastic sample using atomic force microscope (AFM) hardware, comprising: a signal generator configured to generate a first vibration signal at at least one frequency; a mechanical subsystem operable in cooperation with the signal generator, repositioning one of the sample and the AFM cantilever probe relative to the other until the cantilever of the probe is deflected a predetermined amount from the nominal orientation of the cantilever; maintaining the probe at a predetermined position relative to the surface of the sample, where at least one of: 1) an average sample loading force generated by the probe; and 2) a contact area between the tip of the probe and the surface is maintained substantially constant; a mechanical subsystem configured to generate mechanical vibration of one of the sample and the probe relative to the other of the sample and the probe as a result of transfer of the first vibration signal at the signal frequency to the mechanical subsystem; a position detection system configured to detect deflection of the cantilever as a function of at least one of temporal and spatial factors that characterize the operation of the system; a programmable processor in electrical communication with the mechanical subsystem, transferring the first vibration signal from the signal generator to the mechanical subsystem; caused by repositioning one of the sample and the AFM cantilever probe relative to the other of the sample and the probe by ceasing operation of the mechanical subsystem for a period of time sufficient to relax creep of the surface; a programmable processor programmed to acquire data from the position detection system to determine viscoelastic parameters of the surface after a relaxation period sufficient for relaxation of creep of the surface caused by repositioning one of the sample and the AFM cantilever probe relative to the other of the sample and the cantilever probe; A system comprising:

19. an electronic circuit configured to measure the viscoelastic parameters of the surface at the set of predefined frequencies while compensating for creep of the surface; a recorder in operative communication with the processor, the recorder configured to generate an output perceptible to a user and representative of the viscoelastic parameter as a function of at least one variable of the measuring step.

20. The system of claim 18.

20. the signal generator is configured to generate the first vibration signal at a single frequency.

20. A system according to claim 18 or 19.

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