阅读说明：本技术 经由原子力显微镜进行纳米级动态力学分析(afm-ndma) (Nanoscale dynamic mechanical analysis (AFM-NDMA) via atomic force microscope ) 是由 谢尔盖·奥斯钦斯基 安东尼奥斯·鲁伊特 B·皮腾杰 赛义德-阿西夫·赛义德-阿玛努拉 于 2019-08-02 设计创作，主要内容包括：一种基于原子力显微镜的设备和方法包括硬件和软件,其被配置为以动态方式收集并分析表示软材料在纳米级上的力学性质的数据,以绘制软材料样品的粘弹性性质。使用该设备作为现有原子力显微镜装置的补充。(An atomic force microscope-based apparatus and method includes hardware and software configured to dynamically collect and analyze data representing mechanical properties of a soft material on a nanometer scale to map viscoelastic properties of a soft material sample. The apparatus is used as a supplement to existing atomic force microscope devices.)

1. A method for determining mechanical properties of a soft viscoelastic sample using an Atomic Force Microscope (AFM) -based system, the method comprising:

repositioning a probe of the system toward a surface of the sample until a cantilever of the probe is deflected a predetermined amount relative to a nominal orientation of the cantilever;

modifying the repositioning to maintain at least one of the following substantially constant:

i) an average sample loading force generated by the probe, an

ii) a contact area between a tip of the probe and the surface;

measuring a viscoelastic parameter of the surface at a set of predefined frequencies while compensating for at least one of creep of the surface and spatial drift of the system; and

generating an output perceptible to a user and representative of the viscoelastic parameter based on at least one of the measured variable conditions.

2. The method of claim 1, wherein the measurements are made simultaneously at multiple frequencies in the set of predefined frequencies.

3. The method according to one of claims 1 and 2, wherein the modifying comprises modulating a sample loading force applied by the probe to the sample at a given excitation frequency of the set of predefined frequencies.

4. The method of one of claims 1 and 2, wherein the modifying comprises maintaining the average sample loading force substantially constant while modulating a separation between the surface and a base of the probe.

5. The method of one of claims 1, 2, 3, and 4, wherein measuring the viscoelastic parameter comprises performing a two-channel demodulation of operation of the system to perform at least one of:

-simultaneously measuring both an excitation force exerted by the probe on the sample and a deformation of the surface caused by the excitation force, and

-avoiding recalibration of the system.

6. The method of one of claims 1 to 5, further comprising suspending operation of the system for a period of time sufficient to cause creep relaxation of the surface caused by the repositioning.

7. The method of claim 5, wherein the performing dual channel demodulation comprises: combining first and second data received during the measurement from a first sensor of an electronic circuit of the system and a second sensor of the electronic circuit of the system, respectively,

wherein 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 relative to the nominal orientation.

8. The method of one of claims 5 and 7, wherein the performing dual channel demodulation comprises introducing correction for at least one of drift-induced variations and creep-induced variations in signal data that has been received from at least one of the two channels.

9. The method of one of claims 1 to 6, further comprising continuously monitoring operation of the system at a reference frequency with at least one of the first and second electronic circuits of the system to compensate/correct for changes in the contact area caused by creep of the surface.

10. The method of claim 9, wherein the continuously monitoring comprises continuously monitoring with only one of the first and second electronic circuits, and further comprising collecting calibration data representing a signal from the other of the first and second electronic circuits obtained from a hard calibration sample.

11. The method according to one of claims 9 and 10, further comprising compensating for changes in the contact area caused by creep of the surface, wherein the compensation comprises at least one of:

i) taking into account the change in the contact area when calculating the viscoelastic parameter with a programmable processor of the system, the programmable processor being operatively connected with the AFM; and

ii) repositioning the probe to compensate for the change.

12. The method of one of claims 9, 10 and 11, wherein the reference frequency is not included in the set of frequencies.

13. The method of one of claims 1 to 12, wherein the measuring comprises acquiring, during a first time period, a first set of electrical signals at frequencies of the set of frequencies from a sensor of an electronic circuit of the system to determine a degree of indentation of the surface with a tip of the probe, and

acquiring a second set of electrical signals at a reference frequency from a sensor of an electronic circuit of the system during a second time period to compensate for changes in the contact area caused by creep of the surface,

wherein the sensor comprises at least one of a deflection sensor (118) and a sensor configured to measure a position of the probe relative to the surface.

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

15. The method of one of claims 13 and 14, wherein said acquiring the first set of electrical signals and said acquiring the second set of electrical signals are interleaved with each other.

16. The method of one of claims 13, 14 and 15, further comprising compensating for the change in contact area based on determining a change in dynamic stiffness of contact between the probe and the sample.

17. The method of claim 3, wherein the modulating the sample loading force is performed by: adjusting the amplitude and phase of each oscillator component of the sample loading force to respectively corresponding target values at each given excitation frequency of the set of predefined frequencies, while the adjustment depends on the response of the material of the sample to which the modulated sample loading force is applied.

18. A system configured to determine a mechanical property of a surface of a viscoelastic sample using Atomic Force Microscope (AFM) hardware, the system comprising:

a signal generator configured to generate a first oscillating signal at least one frequency;

a mechanical subsystem operatively cooperating with the signal generator, the mechanical system configured to:

repositioning one of the sample and a cantilevered probe of the AFM relative to the other of the sample and the probe until a point wherein a cantilever of the probe is deflected a predetermined amount relative to a nominal orientation of the cantilever;

maintaining a probe in position relative to a surface of the sample, wherein at least one of the following remains substantially constant:

i) an average sample loading force generated by the probe, an

ii) a contact area between a tip of the probe and the surface;

to cause mechanical oscillation of one of the sample and a probe of the AFM relative to the other of the sample and the probe as a result of the first oscillating signal at the signal frequency being delivered to the mechanical system;

a position detection system configured to detect deflection of the cantilever according to at least one of a temporal factor and a spatial factor characterizing operation of the system

A programmable processor in electrical communication with the mechanical subsystem, the programmable processor programmed to:

conveying the first oscillating signal from the signal generator to the mechanical subsystem,

suspending operation of the mechanical subsystem for a period of time sufficient to cause creep relaxation of the surface caused by repositioning of one of the sample and a cantilevered probe of the AFM relative to the other of the sample and the probe; and

collecting data from the position detection system to determine a viscoelastic parameter of the surface after a relaxation time period has elapsed, wherein the relaxation time period is a time period sufficient to relax creep of the surface that has been caused by repositioning of one of the sample and a cantilevered probe of the AFM relative to the other of the sample and the probe.

19. The system of claim 18, further comprising

Electronic circuitry configured to measure the viscoelastic parameter of the surface at a set of predefined frequencies while compensating for creep of the surface; and

a recording device in operable communication with the processor and configured to generate an output perceptible to a user and representative of the viscoelastic parameter as a function of at least one of the measured variable conditions.

20. The system of one of claims 18 and 19, wherein the signal generator is configured to generate the first oscillating signal at a single frequency.

Technical Field

The present invention relates generally to methods of determining dynamic mechanical properties of materials, and more particularly to nano-scale rheology of materials performed using atomic force microscopy within a specific frequency range, actually within a low frequency range associated with the rheology of soft materials.

Background

Dynamic Mechanical Analysis (DMA) is a measurement method designed to characterize the viscoelastic mechanical properties of different materials, such as metals, composites, polymers, elastomers, etc.

Viscoelasticity is considered to be a property of those materials that exhibit both viscosity and elasticity when undergoing deformation. Viscous materials under stress generally resist shear flow and strain linearly over time. Elastic materials strain when stretched and, once the stress is removed, they quickly return to their original state. In view of viscoelasticity, the deformation (strain) exhibited by a solid material in response to a load force (stress) is generally time-dependent: this deformation (strain) depends not only on the magnitude of the load (stress), but also on the rate of loading (loading rate) and relaxation time.

According to the macroscopic (or global) DMA rheological characterization process, periodic (harmonic) tensile, compressive, bending or shear stresses are typically applied to a material sample, resulting in excitation of the sample due to the loading. The mechanical response of the material (e.g., the amplitude and phase of the response) is then analyzed at the frequency of the excitation (the excitation frequency). This analysis is typically performed using a lock-in amplifier. DMA methods have been established to measure the storage modulus (E ') and the material loss modulus (E ") of materials, usually expressed in MPa or GPa and the ratio of these moduli E"/E' (known as "angle tangent", also known as "loss factor", "loss tangent or" damping "). These material properties are characterized as a function of frequency, temperature, time, stress or loading, environmental conditions, or a combination of the above (an alternative term-dynamic mechanical thermal analysis, DMTA-sometimes used to emphasize the temperature aspect or correlation of DMA measurements).

When considering the mechanical properties of soft materials, low frequency mechanical properties (i.e. mechanical properties with frequencies up to several hundred Hz, e.g. up to 300Hz) are considered to be most relevant for the typical physiological movements of biological materials and cells. The ability to determine the low frequency mechanical properties of biological materials and cells would greatly expand the current understanding of soft materials. In addition, detailed knowledge of the low frequency performance of various other materials is also desired-for example, it is now well known that storage and loss modulus databases of polymers and rubbers used in the industry are substantially devoid of microscopic and nanoscale data.

However, existing DMA techniques (such as nanoindentation techniques of materials that are almost universally used in the related art) are considered to have limited spatial resolution when used on soft materials, which limits or even prevents the mechanics of soft materials that such techniques use to study the operation of AFM-based instruments on length scale. For example, while some of the currently existing DMA technologies, such as those that employ nano-indenter systems that do not employ any type of AFM instrument by definition and are considered as such in the related art (see, for example, Pharr, G.M., Oliver, W.C., and Brotzen, F.R., Journal of Materials Research 7, 613-617, 1992; S.A.Syed Assif. and JP Pethica, 505, 103, 1997, in Symphium NN-Thins-Streesses & Mechanical Properties VII; S.A.Syed Assif. et al, Journal of Applied Physics, 90, 3, Herbert, E.G. et al, Journal of Physics D-Applied Physics 41, 2008), are demonstrably configured to allow such measurements to be performed, several factors have been pointed out that virtually preclude existing DMA nanoimprint methods from actually measuring on the length scale where AFM can perform measurements on soft materials, such as biological materials or cells. Among these factors are the nonlinear elastic response and the generally significant adhesive forces, to name a few.

Current AFM-based viscoelastic measurement techniques are fundamentally limited because the use of these techniques does not allow for material creep relaxation which inevitably affects the quality and/or stability of the contact achieved between the probe tip and the sample during measurement and thus adversely affects the accuracy of the measurement. Because of this type(s) of limitation, the system described in US 9,417,170, for example, is configured to specifically avoid (avoid and disallow) waiting for the indenter probe to loose contact with the surface of interest during measurement, thereby making the described system and method impractical for quantitative measurements and mapping at rheologically-relevant (low-range) frequencies.

The skilled person readily realises that there is still a strong need for AFM-based DMA techniques designed to measure the dynamic modulus of soft materials at low frequencies in the nanometer range.

SUMMARY

Embodiments of the present invention are suitably configured to perform AFM-based nanoscale measurements (i.e., measurements on the geometric scale of nanometers) of the mechanical response of soft materials at low frequencies (as defined herein) using an atomic microscope modality suitably configured to

-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;

-performing a two-channel demodulation for a latest calibration of the excitation of the sample;

in clear contrast to the related art, the drift/creep of the sample caused by the sample preloading is purposefully taken into account and compensated, and/or the relaxation of the initial drift/creep of the material caused by the sample preloading is achieved; and

correction of contact radius (e.g., via contact stiffness at a reference frequency).

Embodiments of the present invention provide an AFM-based system configured to determine mechanical properties of a viscoelastic sample surface. The system includes a signal generator configured to generate a first oscillating signal at least one frequency and a mechanical subsystem operatively cooperating with the signal generator. Here, the mechanical system is configured to: i) repositioning one of the sample and a cantilevered probe of an AFM of the system relative to the other until a cantilever of the probe is deflected by a predetermined amount of point relative to a nominal orientation of the cantilever; ii) maintaining the probe in a position relative to the sample surface in which at least one of the following remains substantially constant: 1) an average sample loading force generated by the probe, and 2) a contact area between the tip and the surface of the probe; iii) causing mechanical oscillation of one of the sample and probe relative to the other as a result of delivering a first oscillating signal at the signal frequency to the mechanical system. The system also includes a position detection system configured to detect deflection of the cantilever based on at least one of a temporal factor and a spatial factor characterizing operation of the system.

The system additionally includes a programmable processor in electrical communication with the mechanical subsystem, and programmed to deliver a first oscillating signal from the signal generator to the mechanical subsystem to halt operation of the mechanical subsystem for a period of time sufficient to cause creep relaxation of the surface (caused by repositioning one of the sample and the cantilevered probe of the AFM relative to the other of the sample and the probe); and collecting data from the position detection system to determine a viscoelastic parameter of the surface after a relaxation time period has elapsed. Here, the relaxation time period is a time period sufficient to relax the (superficial) creep that has been caused by repositioning one of the sample and the cantilever probe of the AFM relative to the other of the sample and the cantilever probe. In one particular implementation, the system may additionally include: an electronic circuit configured to measure a viscoelastic parameter of the surface at a set of predefined frequencies while compensating for creep of the surface; and/or a recording device in operative communication with the processor and configured to generate a user perceptible output and the output is indicative of the viscoelastic parameter in accordance with at least one of the variable conditions of the measurement process of the viscoelastic parameter. In any implementation, the signal generator may be intentionally configured to generate the first oscillating signal at a unique single frequency.

Embodiments additionally provide a method for determining mechanical properties of a soft viscoelastic sample with an Atomic Force Microscope (AFM) based system. The method comprises the following steps: 1) repositioning the cantilevered probe of the system toward the sample surface until the cantilever of the probe is deflected a predetermined amount relative to the nominal orientation of the cantilever; and 2) modifying the process of relocation to maintain at least one of the following substantially constant: i) an average sample loading force generated by the probe, and ii) a contact area between the tip of the probe and the surface. The method further comprises the following steps: measuring a viscoelastic parameter of the surface at a set of predefined frequencies while compensating or correcting for at least one of creep of the surface and spatial drift of the system; and generating an output perceptible to a user and representative of the viscoelastic parameter according to at least one of the variable conditions of the measurement process. In one implementation, the process of measuring the viscoelastic parameter may be performed simultaneously at a plurality of frequencies in the set of predefined frequencies. In any implementation, the step of modifying the relocation process may include: modulating a sample loading force applied by the probe to the sample at a given excitation frequency in the set of predefined frequencies. (in the latter particular case, modulating the sample loading force may be performed by adjusting the amplitude and phase of each oscillator component of the sample loading force to a respectively corresponding target value at each given excitation frequency in the set of predefined frequencies, while the adjustment is made as a function of the response of the sample material to which the modulated sample loading force is applied.) in substantially any implementation, the process of modifying the repositioning may include maintaining the average sample loading force substantially constant while modulating the separation between the surface and the base of the probe. In substantially any embodiment, measuring the viscoelastic parameter may be performed by performing a two-channel demodulation of the system operation to achieve at least one of: (a) simultaneously measuring both the excitation force exerted by the probe on the sample and the surface deformation caused by the excitation force, and (b) avoiding/preventing repeated calibration of the system. (in certain instances, such performing dual channel demodulation may include combining first and second data received from first and second sensors of electronic circuitry of the system, respectively, during the measuring step. here, 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 relative to a nominal orientation.) in any embodiment, the method may additionally include interrupting or suspending operation of the system for a period of time sufficient to relax creep of the surface caused by the repositioning process. In substantially any embodiment, the step of performing two-channel demodulation can include correcting at least one of drift-induced variations and creep-induced variations in signal data that has been received from at least one of the first channel and the second channel.

In substantially any embodiment, the method may further comprise the steps of: the operation of the system at the reference frequency is continuously monitored with at least one of the first and second electronic circuits of the system to compensate/correct for changes in contact area caused by creep of the surface. (in certain implementations of the latter, the continuous monitoring may be performed by continuous monitoring using only one of the first and second electronic circuits, and further comprising the step of acquiring calibration data representing a signal obtained from a hard calibration sample from the other of the first and second electronic circuits.) alternatively or additionally, the method may comprise the steps of: compensating for changes in contact area caused by creep of the surface, wherein the compensation comprises at least one of: i) taking into account the change in contact area when calculating the viscoelastic parameter with 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. In any of the above cases, the selection of the reference frequency may include selecting a reference frequency that is not part of the set of predefined frequencies.

In any embodiment of the method, the measuring step may be configured to include: i) a process of acquiring (during a first time period) a first set of electrical signals at a frequency of the set of frequencies from the sensors of the electronic circuit of the system to determine the degree of indentation of the surface with the tip of the probe, and ii) a second set of electrical signals at a reference frequency from the sensors of the electronic circuit of the system (during a second time period) to compensate for the change in contact area caused by the creep of the surface. In this case, the sensor includes at least one of: a deflection sensor and a sensor configured to measure a position of the probe relative to the surface. (in a particular implementation of the latter, a reference frequency may be selected that is not included in the set of frequencies). The acquisition of the first set of electrical signals and the acquisition of the second set of electrical signals may be organized such that the acquisitions are interleaved with each other. A method may additionally include the steps of: compensating for the change in contact area based on determining a change in dynamic stiffness of the contact between the probe and the sample.

Brief Description of Drawings

The 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, and in which:

FIG. 1 provides a schematic diagram of an embodiment of an AFM-nDMA system of the present invention, the operation of which includes the cooperation of the process of a force set point modulation modality and the operation of Z-scanner modulation electronics;

FIG. 2 is a schematic diagram of a particular version of the embodiment of FIG. 1 configured to implement a force set point modulation modality;

3A, 3B, 3C, 3D, 3E, and 3F provide examples of signal traces, vertical deflection, and Z-sensors;

fig. 4A and 4B show the dependence of the storage modulus and loss modulus at low frequency (at a fixed temperature) of Polydimethylsiloxane (PDMS). A comparison is provided between measurements performed with the AFM-nDMA based embodiment of the present invention and those performed with the conventional global DMA method;

FIGS. 5A, 5B, 5C, 5D, 5E, and 5F show the results of measurements of storage modulus and loss modulus of Fluorinated Ethylene Propylene (FEP) material as a function of temperature (at three different fixed low frequencies: 0.1Hz, 1.0Hz, and 5.6 Hz). A comparison is provided between the results of the measurements of viscoelastic properties performed with the AFM-nDMA-based embodiments of the present invention (on geometric scales of 1 micron and below 1 micron) and those performed with the traditional (used by the related art) global DMA method (on scales of about 1mm and larger);

fig. 6A shows the experimentally defined dependence of the ratio of storage modulus to loss modulus of FEP with respect to frequency. Presented via time-temperature superposition (TTS). A comparison is provided between measurements performed with the AFM-nDMA based embodiment of the present invention and those performed with the conventional global DMA method;

fig. 6B, 6C present the correlation of storage modulus and loss modulus with frequency, respectively (corresponding to the graph of fig. 6A). Presented via time-temperature superposition (TTS). A comparison is provided between measurements performed with the AFM-nDMA based embodiment of the present invention and those performed with the conventional global DMA method;

fig. 6D shows an example of time-temperature superposition (TTS) of shift factors, which includes a comparison between measurements performed using an embodiment of the present invention and those performed with a conventional global DMA method.

Fig. 7A and 7B provide an illustration of the mode of operation of the system in which the probe is driven with a signal whose frequency spectrum combines several selected frequencies.

In general, the sizes and relative dimensions of elements in the drawings may be set differently than actual sizes and relative dimensions to facilitate ease, clarity and understanding of the drawings as appropriate. For the same reason, all elements present in one drawing are not necessarily shown in another drawing.

Detailed Description

Embodiments of the system of the present invention, including but not limited to hardware, firmware, and software, are implemented based on the most advanced Atomic Force Microscope (AFM) instruments equipped with a digital controller and a programmable processor (computer system). Nanoscale Dynamic Mechanical Analysis (NDMA) of a sample is performed with the aid of a cantilevered probe that interacts with and indents a sample surface with a controlled force that includes both a quasi-static (DC) component and a dynamic (AC) oscillatory component.

The frequency (or frequencies) of the oscillating component of the force may be in a range that substantially matches the frequency range typically of interest in bulk, macroscopic DMA studies of soft materials and various polymers, i.e., within the limits of the tens of sub-Hz and low Hz frequency ranges (e.g., from 0.01Hz to about 200-300Hz, as described above). During the application of this controlled force between the cantilevered probe and the Sample Under Test (SUT), which includes both quasi-static and oscillatory interaction portions, the normal (referred to simply as vertical) motion of the sample surface of the base of the cantilevered probe is detected and measured in order to determine the mechanical response of the sample to each and both of the static force component (causing at least loading and unloading deformation) and the dynamic oscillatory force component (causing viscoelastic deformation under oscillatory loading).

In the following disclosure, the term "soft material" refers to a material whose elastic modulus (young's modulus) does not exceed 10 GPa. (conversely, hard samples-e.g., for calibration purposes-may be defined as having an elastic modulus in the range of 100 GPa. approximate modulus values for some of the hard materials include about 350GPa for sapphire, about 150GPa for silicon, greater than or equal to about 30GPa for mica, greater than or equal to about 70GPa for aluminum, greater than or equal to 110GPa for copper), alternatively or additionally, whether a given material is soft may be defined as compared to the material of the AFM probe used in the measurement: AFM probes are typically made of silicon or silicon nitride, so a "soft material" with an elastic modulus of less than about 10GPa will be < 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 or preferably 50% or more greater than the elastic modulus of the material of the AFM tip used during system operation.

Embodiments of the system and method of the present invention, hereinafter generally referred to as "AFM-nDMA", utilize a cantilevered AFM probe with a well-defined specific geometric configuration of the tip that applies a dynamic oscillatory loading force to a measured material sample, referred to as SUT (i.e., exposing such material sample to dynamic stress) within a low frequency range (defined, according to implementation, as a sub-hertz frequency range, or, for example, as a frequency range up to several hertz below 10Hz, or, in a specific implementation, as a frequency range below 300 Hz; see also below) of practical significance to the rheology of soft materials to localize the measurement of SUT at the nanoscale dynamic response. In other words, using embodiments of the present invention enables the measurement of nanoscale dynamic complex-valued deformation of the SUT. For the purposes of the present disclosure and claims, the term "nanoscale" refers to and is used to refer to the dimension of sub-micron probe(s) -sample contact.

Embodiments particularly allow for characterization of nanoscale dynamic responses of SUTs sized as coatings or thin films of composite materials. In contrast to the related art, this measurement method is specifically designed to take into account relaxation of material creep.

Part of the results of the nanoscale dynamic response measurements so configured is that the viscoelastic storage and loss moduli of the material SUT are determined in the low frequency range that is particularly relevant to rheological analysis of soft materials in the following manner: allowing direct comparison with material properties measured using conventional DMA methods configured for macroscopic (global) analysis of material properties. The proposed method generally facilitates measurements at several frequencies decimal (at sub-hertz, several hertz, tens of hertz, about 100 hertz, and hundreds of hertz, such as up to 300 hertz and is particularly useful for it, for example, depending on the implementation, embodiments of the invention provide operational advantages for measurements at frequencies in the range of from 0.001Hz to 1000Hz, preferably in the range of 0.01Hz to 300Hz, more preferably in the range of 0.1Hz to 150Hz, and most preferably in the range of 0.1Hz to 100 Hz.

Those skilled in the art will readily recognize that long term stability of the system and probe sample contact remain important because the measurement time (the time required to perform the operations associated with low sub-hertz frequency range determination) is relatively long compared to the time to perform measurements at higher frequencies.

In order to meet the long-term stability requirements of the measurement system and to solve the problems arising from the use of conventional embodiments of DMA systems for measuring the viscoelasticity of soft samples at low frequencies, embodiments of the present invention are suitably configured to employ atomic force microscopy and related techniques (e.g. unlike nanoindentation-like systems known in the art) and to perform AFM-nDMA measurements over long measurement times (from seconds to several minutes) exclusively at a set of predefined frequencies (the set being defined as comprising at least one frequency and preferably a plurality of frequencies) due to a combination of the following technical features:

method of using force set point modulationAccording to the method, the AFM-nDMA system of the present invention is configured to maintain, in operation, at least one of: (i) a specified level of preload force applied to the SUT, and (ii) probe-sample contact, whether or not and despite thermal drift and material creep-when such drift and/or creep occurs during a measurement, the probe-sample contact remains with substantially constant dimension(s) (in one case, dimensions do not change).

In one implementation, these operational characteristics are achieved by modulating the force applied to the sample by the probe (at a given excitation frequency), while maintaining the oscillation of the probe caused by the AC component of the force to achieve dynamic measurements, and while maintaining the DC component of such force at a substantially constant level (via feedback using the electronic circuitry of the probe). This is in contrast to AFM-based techniques that employ correlation of displacement modulation between the probe of the AFM and the sample being measured.

Using AFM-based dual channel demodulation scheme，Dual-channel demodulation scheme based on AFMConfigured as a measurement subsystem to combine data/information acquired from two measurement channels (Z-sensor and deflection sensor for short) of the data acquisition electronics to allow simultaneous and instantaneous measurement of both the excitation force and the resulting SUT deformation. This is in contrast to the accepted utilization of a single measurement channel that is unique in the art. (as the skilled person will readily appreciate, the single unique measurement channel of the prior art is configured such that continued use of the system requires iterative calibration of the system).

The Z channel of embodiments of the present invention is configured to measure the separation between the base of the probe and the SUT to extract information about both the amplitude and phase of the signal representing the interaction between the tip and the SUT. (the base of the probe corresponds to the end of the probe opposite the end with or containing the tip; it is the base of the probe that is typically attached in an AFM probe holder device.) thus, the use of a dual channel scheme allows embodiments to keep up-to-date the calibration of the amplitude and phase of the excitation force independent of (and free from in operation) during potentially long measurement times.

A typical deflection sensor is implemented by using a laser source configured to deliver and focus a light beam on the upper surface of a probe stem, and then reflect the light beam toward a four-quadrant photodetector. The resulting change in deflection of the probe cantilever translates into a change in the angle of the reflected laser beam and a change in the position of the reflected beam on the photodetector. As is known in the art, it is well defined that the differential electronic signals from the four quadrant photodetector circuits are amplified and used as signals representative of the vertical deflection of the probe. With proper calibration, the deflection signal can be used to detect nano-scale deflection or nano-newton forces applied by the probe.

Accordingly, an embodiment of the present invention encompasses a method wherein performing dual channel demodulation comprises: during the measurement, first data and second data received from a first sensor of the electronic circuit of the system and a second sensor of the electronic circuit of the system, respectively, are combined. 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 cantilever of the probe relative to a nominal (undeflected, unaffected) orientation. (it will be appreciated that more generally, such calibration is provided by measuring the deflection on the sapphire sample and later using the result as the Z amplitude and phase on the target sample (if the Z sensor proves to be unavailable), for example, if the Z sensor channel is not used or unavailable, such as in the case of sample actuator excitation, the missing channel may be compensated for by using the calibration on the hard surface calibration sample with the first channel.

Using a "software locking" methodTo effect demodulation of the signal received from the dual channel measurement subsystem. Here, the signal traces (buffered or recorded for on-line or off-line processing) are processed by an algorithm that combines the steps of drift/creep correction and/or subtraction of "lock-like" demodulation of the signal at a single or multiple frequencies in a lock-like manner to compensate for variations in the signal data received from at least one of the channels.

Hardware locks are known to be used to perform direct processing, which is inflexible. On the other hand, by using digitized stored or buffered signals, more complex algorithms can be run according to the inventive idea: in one example, the drift trend line estimated with a moving average filter is subtracted from the stored or buffered signal, leaving only the oscillation component for locking the demodulation function. As a result, errors introduced into the lock amplitude (and/or in particular the phase) by drift/creep are greatly reduced.

This demodulation is achieved contrary to the use in the related art of hardware locking and/or spectral analysis based on fourier transform (FFT/DFT) which do not allow to correct the drift or creep of the material (thereby causing an error inevitable in determining each of the amplitude and phase values of the excitation force, which error is particularly noticeable at a frequency scale substantially equal to the range of material drift frequencies). Although one embodiment of the present invention operates at a single excitation frequency (one implementation set according to force set point modulation), the proposed algorithm facilitates (enables, allows) if desired the simultaneous demodulation process of such signals at multiple excitation frequencies, thereby providing parallel processing advantages and reducing overall measurement time. For example, according to empirical verification, the total measurement time is reduced by about 36% by means of simultaneous demodulation at two frequencies (0.1Hz and 0.18Hz, 20 cycles); by means of simultaneous demodulation at five frequencies-the total measurement time is reduced by about 51%.

Alternatively, the embodiments may additionally employAn electronic circuit configured to be continuous at a reference frequency Monitoring operation of the system to correct for interface between the tip of the probe and the sample due to material or adhesive creep Variation of contact area. Here, to achieve such monitoring, probe excitation at a preselected reference frequency is continuously mixed or interleaved or complementary with probe excitation at other measurement (excitation) frequencies, so that the dynamic stiffness of the probe sample contact at the reference frequency can be measured in parallel and substantially continuously during other measurements performed at such other excitation frequencies.

The relative change in contact area/size is then derived from the change in dynamic stiffness (determined at the reference frequency) and corresponding corrections are then applied to AFM-nDMA measurements at other excitation frequencies (including measurements at lower frequencies where drift/creep of the material may actually be more pronounced).

As an alternative AFM-based embodiment, "staggered" (time spaced) reference frequency measurements may be used rather than continuously monitoring at the reference frequency (i.e., the reference frequency bins may be staggered with other frequency measurements). As defined for purposes of this disclosure and the appended claims, when the first and second processes of measurement (e.g., processes a and B) are interleaved with each other, the two processes are arranged to be performed in a generally, but not necessarily, regular alternating manner to form substantially any continuous sequence in which both a and B occur multiple times, such as ababab.

Notably, the described reference frequency monitoring based correction may not generally be needed (but may alternatively be performed) in connection with implementing force set point modulation, wherein the contact area may be maintained virtually constant (substantially constant) via force feedback. In some specific cases, for example when the material creep/flow of the material is actually significant (e.g. in the case of measuring polymers lacking cross-linked chains), the proposed reference frequency based correction can advantageously be achieved in addition to the force set point modulation.

It is recognized in the art that initial creep of a material occurs immediately after preloading a sample with a probe-e.g., during the initial loading phase of an indentation cycle-because viscoelastic materials exhibit relaxation in response to step-function-like loading. Here, attention should be paid. The person skilled in the art cannot choose to avoid drift altogether, but to be in a position where a timescale on which drift is operationally insignificant can be chosen. (and some of the technical research related to AFM in so doing limit their technology to measurements at higher frequencies. in other words, avoid low frequency drift by performing measurements at different-higher-frequencies.) the skilled person will readily recognize that material relaxation (creep) and drift cannot be avoided and must be accounted for on a time scale corresponding to the frequency range in which implementations of the inventive concept provide operational advantages. (embodiments employing the dual channel demodulation method and/or the reference frequency technique discussed below successfully address the creep to drift separation in the low frequency range).

Thus, embodiments of the present invention address the problems caused by the presence of initial creep and improve the accuracy of the overall measurement of the viscoelastic properties of a material at low frequencies by explicitly including a wait period or time period (e.g., a duration of about 10 seconds, or a duration of about 20 seconds, or a duration of about 30 seconds, depending on the particular implementation) as an operational step in an AFM-nDMA system, prior to the step of the AFM-nDMA performing the actions of the excitation and measurement steps, to allow the material to relax under the applied pre-loading step. Embodiments of the present invention employ a flexible ramp-foot approach to seamlessly allocate this initial slack "waiting segment" which, in practice, may be followed by the step of nDMA measurements at only a single frequency or in parallel at multiple mixed frequencies.

The present invention provides an AFM-based nanoscale DMA (AFM-nDMA) method configured to extend classical macroscopic bulk DMA methods to spatial scales well below 1 micron. Note that NanoDMA^TMIs a trademark of the viscoelastic property measurement technique on an instrumented nanoindenter (not based on AFM) platform (Bruker-Hysitron). Thus, the present invention and the general scope of the present disclosure are nano (scale) DMAs implemented specifically on AFM platforms, further referred to as "AFM-nDMA" to avoid confusion with the above trademark "nanoDMA" technology name, the "nanoDMA" technology being specific to technologies used with and implemented by means of the use of nanoindenter instrumentation.

AFM-nDMA hardware and measurement method.

Embodiment of AFM-nDMA System。

In general, nanoscale dynamic mechanical analysis requires a "plane-driven" mechanical excitation without resonance, which can be challenging to achieve in the higher kHz end of the frequency range. Embodiments of the present invention address this challenge by using a specially designed sample actuator with a high natural resonant frequency. In addition, the present invention employs a particular sample mounting scheme that does not significantly affect the resonant properties of the actuator, and also allows the amplitude and phase response of the actuator to be calibrated by in situ measurements with the AFM system. In contrast to other designs that utilize a probe carriage actuator, the sample actuator does not excite or "back drive" the resonance of the AFM scanner. In contrast to other designs based on electrical or magnetic or photothermal actuation, the present invention can use conventional AFM probes and does not require AFM probes with specialized (actuating) rods.

Implementations of AFM-nDMA systems include AFM instruments that are appropriately modified/adapted/augmented to achieve the above-described goals.

Typically, AFM-nDMA devices (and associated SUT characterization methods) are based on AFM platforms with closed-loop scanners. An AFM scanner (interchangeably referred to as a scanner head or AFM head) employs a piezoelectric-based actuator (having three orthogonal axes of operation, an x-axis, a y-axis, and a z-axis) suitably programmed for positioning and scanning an AFM probe having a probe tip relative to a sample and/or a sample scanner or actuator configured to position and scan a sample relative to an AFM probe. The probe can be positioned or scanned in the sample plane (e.g., xy plane) while maintaining a separation distance (along the z axis) between the probe and the sample with dedicated electronic circuitry. The scanner is configured to perform an indentation Z-ramp with a holding time period (holding time) on or at the sample surface ("ramp and hold"), wherein a predetermined level of loading force (interchangeably referred to as trigger force, preload force, or indentation force) is reached at the beginning of the holding time period. The tip sample interaction force is determined by the (vertical) deflection of the AFM probe rod, which is tracked by optical means and sensed on a four quadrant photodetector. Optionally, detector channels that in operation record lateral (e.g., horizontal) deflection of the probe shaft are configured to provide information about roll or slide of the probe tip. (such a detector channel may be referred to as a "friction" channel).

Structurally, an AFM probe includes a flexible rod member (or simply rod) characterized by a spring constant or stiffness k (and measured, for example, in newtons per meter, N/m), having a tip (with a tip radius, nominally expressed in nm, dimensions, the shape of which is generally approximated by a cone and sphere combination) that is nanometer-sized. The rods are attached to a substrate "chip" (several millimeters in size), which may be spring-clamped or otherwise attached to various types of probe holders known in the art. The probe mount is then sized to attach to an XYZ scanner device of the AFM head (e.g., via an attachment member comprising several leaf spring receptacles and metal pins).

In one implementation, the AFM instrument is operably connected to a dedicated (programmable) controller circuit, which preferably contains 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 instrument; the computer processor runs application code and communicates with the AFM controller circuitry.

As schematically shown in fig. 1, for example, the embodiment 100 includes an atomic force microscope (AFM, shown in simplified version, as a combination of a probe 104 and a tip 104A, the tip 104A being disposed above a surface of the SUT 108 in operation). The instantaneous position of the flexible probe 104 and/or its deviation from a reference position (due to the interaction between the tip 104A and the SUT 108): (a) based on the deviation of the beam 110 relative to the laser source 114 (typically configured to generate visible light) when reflecting the beam from the surface of the probe head 104, and (b) recorded after the beam 110 so reflected has been received by the position sensitive detector 118. (other implementations of determination of probe position may be implemented as known in the relevant art).

The AFM controller electronics 122 is equipped with specialized control modules that allow the particular type(s) of excitation signal(s) to be delivered to the AFM feedback electronics 130 (which are configured to govern operation of the system 100 in a force set point modulation regime) and/or to the Z-scanner modulation programmable electronics module 134 (which is configured to change and/or modulate the position of the probe 104 by means of the Z-repositioner 140 and/or change and/or modulate the position of the sample 108 by means of the Z-repositioner 144 in a direction orthogonal to the surface of the sample 108 during operation of the system 100). Among the particular excitation signal(s) are at least a low frequency excitation signal, a multiple (in one embodiment-dual) frequency excitation signal, and a mixed frequency sinusoidal excitation signal (which are generated by the electronic circuitry of the electronic blocks 126A, 126B, and 126C, respectively). DDS: a direct digital synthesizer (e.g., a particular form of digitally implemented waveform generator known in the relevant art). The process of changing the position of the sample 108 or the probe 104 or both in a direction normal to the surface of the sample 108 (as shown-the Z-axis) and/or modulating the position of the sample 108 or the probe 104 or both is generally referred to herein as "Z-modulation". Examples of relocators include electronically controlled micro-step and sub-micro-step positioning equipment known in the relevant art.

In this case, the amount of the solvent to be used,

in the case of low frequency excitation provided by the Z-scanner module 134, the excitation frequency of the movement of the probe and/or sample during operation of the system 100 is from sub-Hz frequency to several hundred Hz (in particular, from 0.001Hz to 1,000 Hz; preferably from 0.01Hz to 300Hz, and even more preferably from 0.1Hz to 150 Hz).

In the case of dual frequency excitation, the signal provided by the Z-scanner module 134 to at least one of the relocators 140, 144 comprises a mixed wave signal combining a low frequency signal with a signal at a predetermined reference frequency (which may be higher than the low frequency).

In the case of mixed frequency excitation, the module 134 is configured to generate a drive signal that combines several (preferably ten or more) sine waves with correspondingly different frequencies, amplitudes and (optionally) phase relationships with respect to each other. The selected multiple frequencies of the individual sinusoidal components of the mixed frequency excitation may, in aggregate, cover one frequency or even tens of frequencies in the frequency range. The information summarized in table 1, table 2, and fig. 7A, 7B provide an illustration of such mixed frequency excitation for a probe of an embodiment of the present invention.

Table 1:

table 2:

in one example (see table 1 and fig. 7A), the probe is driven with an electrical signal that combines 9 frequency components, where component #1 is considered the fundamental harmonic 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. In addition, a (as specified) phase shift is introduced between and among the term-by-term components of the signal driving the probe. Figure 7A-in dashed lines-shows nine curves 704 representing each of the drive sub-signals at these frequency components, and in solid lines (710) -shows the resulting aggregate excitation force applied to the probe to displace it (or similarly, the resulting aggregate displacement signal is delivered to the probe by the electronic circuitry of the overall system).

For comparison, table 2 and fig. 7B show the situation when the probe is driven by a mix of 9 signals representing harmonics of the selected fundamental frequency (of component #1 of table 2), however these harmonic signals are applied to the probe simultaneously without any predetermined phase shift — in fact with the same phase. Fig. 7B shows nine curves 714 (dashed lines) representing harmonic components, while curve 720 shows the resulting excitation force applied to the tip of the probe.

In one embodiment, the specialized electronic circuit control module 122 may be typically implemented in firmware based on existing flexible AFM control, for example, by using Field Programmable Gate Arrays (FPGAs) and Digital Signal Processors (DSPs). In addition, the excitation signals provided to (as indicated by lines 140A, 144A) and governing the operation of at least one of the probe Z relocator 140 and the sample Z relocator 144 (preferably at a frequency in the range of about 100Hz to 100kHz) may also be routed by the AFM-nDMA control module 122 to a dedicated "high frequency" sample actuator/heater 148, as indicated by line 148A.

The AFM digital feedback electronics module 122 (implemented in one implementation, with a DSP, FPGA, or a combination of the two) may be configured by using a PID (proportional integral derivative) or PI (proportional integral) electronic circuit controller that, in 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 to the positioning Z scanner with the purpose of minimizing the difference (error signal) between the input and the setpoint (the setpoint being understood to be the desired value of the signal controlled in the feedback loop.) in the case of a deflection signal being input, for example, the AFM maintains the loading force at a selected level; in the case of a signal being input, the AFM maintains/maintains the Z position; when an AC signal is blended to the setpoint (this condition is referred to as setpoint modulation), the AFM digital feedback will follow signals representing both the DC and AC portions of the setpoint-e.g., under a force setpoint modulation scheme.

The signal routing control electronics 160 includes digitally controlled multiplexers configured to handle outputs and signal inputs, which are intended to implement various AFM control schemes and/or modes of operation: such as force set point modulation, z set point modulation, and z modulation. In operation of the system, this module 160 connects the input signal, the setpoint modulation signal to the AFM digital feedback module 130, and also routes waveforms from the appropriate DDS (direct digital synthesizer — generator of oscillating waveforms) 126A, 126B, and/or 126C and the input signal (150, deflection signal, and signals from Z or height sensors, not shown in fig. 1) for acquisition and lock-in processing.

Fig. 2 is a simplified schematic diagram 100' of a particular version of the embodiment 100 of fig. 1 configured to implement a force set point modulation mode of system operation. Here, the deflection setpoint represents a desired or target value of the deflection signal controlled by the AFM feedback loop. The deflection set point includes a modulation component (set point modulation), indicated at 152.

Sample mounting。

The sample (sample under test, SUT, shown as 108 in fig. 1) to be measured using the AFM-nDMA embodiment of the present invention may be dimensioned as a thin (on the order of a few microns in thickness) segment or sheet of material, or alternatively, may be dimensioned as a block (e.g., up to 3mm in thickness) having a cryo-sliced surface (a substantially flat block-face surface). For the purpose of implementing the inventive idea, the term "substantially flat" means that its surface spatial profile is characterized by an average peak-to-valley difference of not more than 20nm, more preferably not more than 10 nm.

For example, such substantially flat or substantially planar surfaces can be prepared by microtome slicing, or by casting a thermosetting polymer onto the mica surface, or by spin casting a dissolved polymer onto the sample surface. The prepared sample segments are mounted on a selected substrate (such as, for example, a sapphire or stainless steel disk in one instance, about 10mm to 12mm in diameter and less than 1mm in thickness) to form a sample-substrate assembly. The sample-substrate assembly can then be secured in the heater-cooler device by use of a magnetic attachment or a thermally conductive paste.

Examples of System configurations for temperature-dependent measurements。

It should be understood that AFM-nDMA measurements of a sample at variable temperatures may generally require a sample heater/cooler arrangement (shown as 148; interchangeably referred to as heater or heater arrangement for simplicity). It may be desirable to utilize a sample heater specifically designed to suppress thermal gradients only highly spatially localized/concentrated in the sample area, in which case only the sample is heated, rather than the entire sample and AFM stage mechanical structure, to minimize overall thermal drift).

In accordance with the inventive concept, such a properly designed sample heater/cooler circuit is configured to achieve a low level of thermal drift rate in the equilibrium state of the thermal gradients in the lateral and vertical spatial directions (x, y and z directions, with reference to the local coordinate system of fig. 1). (the term "low drift rate" is defined and refers to the fact that this drift, observed during the measurement time, is small in spatial value compared to the size of the measured nanoscale features. to determine the value of lateral drift, it can be compared to the contact radius/contact size, and the value of vertical drift can be compared to the indentation depth/sample deformation depth.) this actual result, in turn, favorably impacts the quality and/or accuracy of the spatially resolved AFM-nDMA measurement, as it allows the measurement of sample properties at a specific target location on or at the sample.

(those skilled in the art will readily recognize that thermal drift in the cooler/heater is manifested as lateral (XY) or vertical (Z) drift in the relative position of the probe with respect to the sample

In addition to the sample heater arrangement, a dedicated heater for the probe (probe heater, shown as 154) is also utilized to facilitate localization of the thermal gradient (upper and lower heating plates, one below and one above the sample, starting from both sides of the sample) and to prevent the probe stem 104 from accumulating condensation deposits. Thus, in one implementation, the probe heater apparatus 154 may include first and second heater plates disposed in cooperation with top and bottom surfaces of the sample.

Alternatively or additionally, calibration of the sample surface temperature (relative to the heater 148 set point and the temperature measured by a dedicated sensor inside the heating element of the heater 148) may be achieved in practice by means of a small thermocouple attached to the surface of a sample carrier (not shown in fig. 1; configured in one example as a sapphire or steel disc of 10mm diameter) that mechanically supports and carries the sample, close to the location of the sample 108 on the sample carrier.

As non-limiting examples：

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 end of the range, e.g., 250 degrees celsius), an embodiment 100 of the AFM-nDMA system may optionally be equipped with a sample heater holder/sample actuator 148 that includes a reference surface designed to ensure low thermal drift (about 2 nm/minute or less) of the holder 148 in the X, Y and Z directions. When such a sample heater fixture is used, the sample 108 cooperates with a reference surface and the temperature of the heater of the fixture 148 is controlled with a thermal controller (not shown for simplicity of illustration) that establishes programmable temperature set points and feedback (e.g., PID feedback or proportional-integral-derivative controller/feedback). The use of such a sample holder equipped with a properly designed electronic heating circuit facilitates measurement of viscoelastic properties of sample 108 at substantially any predetermined temperature within the temperature range across the glass transition of a particular polymeric material (where the glass transition temperature Tg falls within the temperature space of the heater, e.g., RT to 250 degrees celsius). For example, for a Polymethylmethacrylate (PMMA) material having a glass transition temperature of about 105℃, the preferred predetermined temperature will be in the range of room temperature (about 25C) to an upper limit of about 140C to 150C.

Preferably, the system 100 may also be equipped with a ceiling heater or probe heater 154 configured to maintain a low thermal gradient in the space of the probe sample.

Notably, in order to achieve AFM-nDMA measurements on materials with Tg below room temperature, it may be desirable to reduce the temperature of the sample by cooling. (e.g., polypropylene with a glass transition temperature in the range of-20C to-5C, polybutadiene with a glass transition temperature of about-20C, or polymethyltolylsiloxane with a glass transition temperature of about-12℃) heater-cooler hardware options (not shown in detail) address temperature spaces below RT, e.g., RT to-35 degrees Celsius.

To operate with the heater or heater-cooler hardware described above, the AFM scanning mechanism (which, depending on the particular implementation, is the Z scanner 144 of the sample and/or the Z scanner 140 of the AFM probe 104) must be thermally isolated from the heating/cooling source — otherwise, the scanner performance (drift, calibration, dynamics, etc.) may be adversely affected to varying degrees throughout the temperature space. In one case, the desired thermal insulation effect can be achieved by means of a special probe holder made of a material with low thermal conductivity (e.g. MACOR, machinable ceramic material). On the other hand, the AFM tip 104A should preferably be maintained at a temperature close to the temperature at which the sample 108 is held, at the same time, in order to prevent condensation from forming on the rod surface(s), thermal bending of the rod, caused by local cooling of the sample and thermal gradients. (notably, the related art systems fail to balance these two different requirements and conditions.) maintaining the tip 104A at a temperature substantially equal to the temperature of the sample 108 can be accomplished (for the heater) by using tip heater hardware in the probe holder that constitutes a heater element (and optionally a thermocouple or other temperature sensor) under the so-called probe cover (i.e., under the probe portion where the probe chip is spring-clamped or otherwise attached to the probe holder).

In embodiments of the AFM-nDMA system, the use of the heater-cooler option preferably additionally requires environmental control (humidity control-RH and inert atmosphere, such as a dry nitrogen purge) to prevent oxidation and deterioration of the sample surface due to moisture absorption. In the simplest case, such environmental control can be achieved by using a flexible sealing sleeve attached to the probe carrier that creates an insulated local environment that can be purged with a low flow rate of dry nitrogen. Alternatively, a special sealed local environment unit (LEC, combined heater-cooler) may be used.

Sample holder actuator configured for measurements only at room temperature

When the "heater-cooler" hardware option discussed above is implemented and used, the system is configured to hold the sample stationary, fixed in space, while delivering mechanical excitation caused by AFM-nDMA via spatial actuation of the tip 104. (this is achieved by means of using the AFM Z scanner 140 or with another probe-holder piezoelectric actuator.) however, it has been realized that the mechanical excitation or actuation of the mutual orientation between the sample 108 and the tip 104 can be performed alternately via (harmonic, small-amplitude) spatial actuation of the sample holder when it is intended to measure only at room temperature. Thus, the combination of sample holder actuator 148 and sample Z scanner device 144 is suitably designed to operate by: the mechanical movement of the sample 108 is caused at least at one frequency within a wide frequency range (e.g., about 100Hz to about 100 kHz). Unlike the AFM Z scanner 140, (the sample holder actuator 148 and/or the sample Z scanner device 144 typically do not have an associated Z sensor configured to detect and provide readings of mechanical vibration amplitude and mechanical movement phase.) instead, the amplitude and phase of the mechanical movement provided by the device 144 and/or 148 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 the deflection).

AFM-nDMA method: operating characteristics

Embodiments of the AFM-nDMA system of the present invention are configured to measure viscoelastic properties at (user) selected point locations on the sample surface. (notably, unlike and contrary to most conventional AFM modalities, embodiments of the AFM-nDMA method of the present invention are not surface imaging techniques in general, although imaging modes with viscoelastic property "mapping" over a limited frequency range are possible

A "ramp" (or forward ramp) mode of operation. At each point location on the surface of the sample being measured, an AFM nanoindentation measurement is performed. Here, the Z scanner 140 extends spatially along the Z axis to direct the tip 104A of the probe 104 towards the surface of the sample 108 (tilt motion) until a specified and/or predetermined threshold in the cantilever deflection of the probe is reached, as detected by using the PSD 118. The presetting of cantilever deflection corresponds to a particular pre-load (normal) force (referred to as a trigger force) applied by the probe tip 104A on the sample 108, which in turn allows the system 100 to determine the desired sample deformation under the corresponding loading.

- "hold" mode of operation. After the preload force threshold is reached, the Z ramp activity ends/stops, and the probe 104 remains (stays) "held" for a specified duration. (in the presence of AFM-nDMA excitation, the duration is specified in terms of the number of cycles required at the frequency selected for measurement. As a non-limiting example, the probe may "hold" 20 cycles, or 200 seconds, at a frequency of 0.1 Hz.) this is the operating segment to turn on the AFM-nDMA modulation/excitation. Several variations of the "hold" mode are within the scope of the invention:

- "holding force" mode of operation. Here, the AFM feedback electronics maintains the cantilever deflection (the force applied to the cantilever) constant at a specified target value (typically at a pre-load force value), while the creep of the Z piezo element, thermal drift, material creep under load are compensated for by and by adjusting the Z position of the probe 104 with the AFM feedback circuitry.

- "hold Z sensor" operating mode. In this mode, the AFM feedback circuit maintains the mechanical extension (along the Z-axis) of the Z-scanner 140 substantially constant through the use of a Z-sensor associated with the Z-scanner 140. The Z piezoelectric creep is dynamically compensated, while the Z sensor drift, thermal drift, material creep are not compensated. In this mode of operation, the force applied to the sample does not necessarily remain constant, and therefore, in no longer periods of time, it is preferable to use a "hold Z sensor" mode of operation. (notably, this mode of operation may be used in the case of bond creep or bond creep in which it may be better to maintain position constancy than to maintain deflection/force constancy, where bond creep may "suck" the probe into the surface, resulting in deep indentation holes.)

- "hold Z drive" mode of operation. Here, the Z piezo high voltage remains constant while the AFM feedback circuit signal is off; as a result, no compensation is provided and therefore it is preferred to use this mode of operation during short periods of time to avoid piezoelectric creep. This mode of operation is intended for fast measurements at relatively high frequencies (e.g., at frequencies in excess of 100Hz, that is, other feedback-based modes of operation may not be able to maintain/track performance at the modulation frequency under this mechanism).

A "retract" (or "reverse ramp") mode of operation. At the end of the "hold" segment of operation, the probe 104 is retracted from the surface of the sample 108. The retraction profile is recorded by use of a programmable processor in operative cooperation with the PSD 118. For viscoelastic materials, the rate of retraction is an important parameter that may affect the accuracy of the JKR model analysis (as described below).

The skilled person will also readily recognize that the proposed AFM-nDMA method should not and cannot be confused with the "tapping" mode of operation of conventional AFM systems: tapping mode is a different intermittent contact AFM technique. In the AFM-nDMA mode of the present invention, the probe is brought close to the sample surface to fully contact the surface and actually deform/indent the surface, and then modulated — the oscillatory component of force or Z displacement. At the end of the holding period of operation, the probe is retracted; the probe can then be moved/translated laterally to another point on the surface and used to perform another point measurement at another location on the surface.

"moment curve" ("FDC"). AFM force moment curves (also known as deflection versus Z scanner extension) were recorded during the forward and reverse ramps (probe retraction). As the skilled person will immediately recognize, the force-moment curve is a curve or trajectory of deflection/force signals versus Z-interval signals acquired when the Z-scanner 140 moves the tip 104A toward the sample surface (-extension curve) or moves the tip 104A away from previous contact with the surface (-retraction curve). The FDC can be analyzed with contact mechanics models (such as the model represented by any of the Hertzian, Johnson-Kendall-roberts (jkr), Derjaguin-Muller-toporov (dmt) models) to calculate the elastic properties of the sample (such as the reduced modulus and young's modulus), and more importantly, for the AFM-nDMA embodiments of the present invention, the size of the tip sample contact area or "contact radius" is estimated. (for reference, the reader is referred to, for example, KL Johnson and K.Kendall and AD Roberts, Surface energy and the contact of electronic sources, Proc.R.Soc.Lond.A 324(1971)301-

In the case where the samples are represented by polymeric materials with strong adhesion, the JKR model generally provides the best results in fitting the experimental data. (in the preferred case, the force-moment curve on the 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 really provide information about the contact radius during the entire hold period or at each moment in the hold period, which may be necessary for accurate quantification of AFM-nDMA results.

During the holding/measuring time, the contact area (between the tip and the sample) may change due to sample creep. 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 (continuously or "staggered" from the primary measurement modality) throughout the hold/measurement time, then a determination of the relative change in contact radius can be made during the measurement. The contact radius is determined from the moment indentation curve (ramp) after or before the holding period; this correction of the contact radius is then applied to each particular instant of the holding period.

"temperature step". After thermal equilibrium in the instrument-sample-heater system is reached, ramp and hold based measurements are taken at a substantially constant temperature by using the AFM-nDMA embodiment of the present invention. On the other hand, AFM-nDMA measurements as a function of temperature were performed by the following method: sequentially tabulated by temperature set point/step (according to a specific temperature program) and waiting to reach thermal equilibrium at the temperature set point before performing AFM-nDMA ramp and hold point measurements at each temperature point. The degree of thermal equilibrium that has been reached before the ramp and hold measurements can be started can be evaluated by measuring the thermal drift rate in the Z direction, e.g., while waiting and performing a "zero size" scan on the surface in peak force tapping AFM mode (and optionally evaluating XY drift by performing a non-zero size scan and tracking topographical or boundary features such as DMT modulus or adhesion or deformation in the material property map) until the desired (low) drift rate figure is reached.

While waiting for the heat to equilibrate and settle in the AFM feedback loop on the surface, the Z scanner 140 may reach the limits of extension or retraction of the Z piezo due to thermal drift and material thermal expansion/contraction/flow. Therefore, it is preferable to have the Z scanner continuously re-centered by moving the Z engagement motor up and down in steps so that the Z position of the scanner is maintained at the center of the piezoelectric dynamic range.

Fig. 3A, 3B, and 3C provide examples of signal traces for force set point modulation. Here, in the force setpoint modulation mode of operation, the AFM feedback electronics module tracks both static ("DC") and dynamic, oscillating ("AC") components of the setpoint. An error signal trace (provided by the AFM feedback control electronics 130 of the system 100 of fig. 1) and shown in fig. 3A, with high frequency noise and very small residual oscillation (AC) errors (these data examples were acquired at a modulation frequency of 5.6Hz), and a high level of noise remaining. These results demonstrate that the AFM feedback circuit 130 is tracking the oscillation setpoint component and providing modulation of the loading force applied by the probe 104 to the sample 108 (which constitutes a "force setpoint modulation" mode of operation). The actual vertical deflection curve in fig. 3B shows the oscillation (AC) component, i.e., the force modulation (here, the normal force exerted by the probe on the sample is equal to the vertical deflection of the rod times the spring constant of the rod). The Z-sensor or "height" signal trace is shown in fig. 3C. Here, the overall downward slope in the trace may be attributable to thermal drift of the system and/or viscoelastic creep in the sample material (drift correction techniques described in software lock-in processing methods may be used to mitigate the deleterious effects of such slope/trend on the accuracy of the amplitude and phase measurements of the signal).

The data presented in fig. 3D, 3E, and 3F are similar to those of fig. 3A, 3B, and 3C, but represent examples corresponding to measurements at a different, lower modulation frequency of 0.32Hz (relative to 5.6Hz of fig. 3A, 3B, 3C). Notably, the residual oscillation (AC) component in the error signal trace in fig. 3D is practically indistinguishable in noise as compared to fig. 3A. This fact is due to the AFM feedback control frequency response (in terms of PID feedback gain) and better feedback tracking that is effective at lower frequencies than higher frequencies. The Z-sensor "height" signal trace in FIG. 3F shows a more pronounced tendency to drift/creep downward as compared to FIG. 3C, due to the much longer measurement time (about 60 seconds) at the lower frequency of 0.32Hz, compared to the shorter measurement time (about 4.5 seconds) at 5.6Hz in FIG. 3C. This shows that the drift correction process described in the software locking method is particularly important at low frequencies, where a large amount of measurement time is required.

Fig. 4A, 4B show results of experimental AFM-nDMA measurements performed with an embodiment of the system of the present invention. Fig. 4A shows the stored (E') modulus data versus the measured frequency (at a fixed room temperature) for a sample of Polydimethylsiloxane (PDMS) material, and fig. 4B shows the loss (E ") modulus data versus the measured frequency. Comparison of the AFM-nDMA results (red, crosses, 410 and 420) with the global DMA measurements (green, dashed lines, 415, 425) on samples from the same material shows that the storage and loss moduli measured at the nanoscale (AFM-nDMA) and using the global macroscopic method (DMA) are essentially identical (the latter is used as a ground truth in the related art, being a reference to validate other results).

FIGS. 5A, 5B, 5C, 5D, 5E, 5F show experimental AFM-nDMA measurements of storage and loss moduli of Fluorinated Ethylene Propylene (FEP) material as a function of temperature (at three different fixed low frequencies: 0.1Hz, 1.0Hz, and 5.6 Hz). A comparison is provided between measurements performed with the AFM-nDMA based embodiment of the present invention and measurements performed with the conventional global DMA method. Fig. 5A, 5B, 5C show the storage modulus dependence on temperature, and fig. 5D, 5E and 5F show the loss modulus dependence on temperature for samples of FEP material, measured at three different frequencies (in the range of 0.1Hz to 10 Hz) by using the nanoscale AFM-nDMA method and the bulk macroscopic DMA technique according to the present invention. Comparison of the AFM-nDMA (red, cross, 530, 540, 550, 560, 570, 580) and bulk DMA (green, dashed, 535, 545, 555, 565, 575, 585) data shows that both techniques detect a significant drop in the storage modulus values with increasing temperature and also detect a shift in the loss modulus peak towards higher temperatures with increasing frequency of measurement-consistent with the expected rheological behavior of the FEP material. Thus, the glass transition of FEP can and is detected with both the traditional integral DMA technique and the proposed AFM-nDMA technique.

Fig. 6A shows an experimentally defined correlation of loss tangent (ratio of loss modulus to storage modulus) of FEP with frequency as presented via time-temperature superposition (TTS). A comparison between measurements (610) performed with the AFM-nDMA based embodiment of the present invention and those (615) performed with the conventional global DMA method is provided. Fig. 6B, 6C show the dependence of the storage modulus and the loss modulus, respectively, on the frequency (corresponding to the graph of fig. 6A), which is represented via a time-temperature superposition (TTS). Here, a comparison is provided between the measurements (620, 630) performed with the AFM-nDMA based embodiment of the present invention and those (625, 635) performed with the conventional global DMA method. Fig. 6D shows an example of time-temperature superposition (TTS) of the shift factors (well known to those skilled in the art) and a comparison between measurements performed with embodiments of the present invention (open circles; 640) and those performed with conventional global DMA methods (filled circles; 645).

Referring to fig. 6A, 6B, 6C and 6D, a nanoscale AFM-nDMA method configured according to the concepts of the present invention and a conventional macroscopic bulk DMA method (the same as in the examples of fig. 5A to 5F described above for the same fluorinated ethylene propylene FEP material) are used in a frequency range from 0.1Hz to over 100Hz, and in a temperature range from room temperature to a temperature range exceeding 120 degrees. Those data for all temperatures and frequencies are superimposed via time-temperature superposition (TTS), which is a rheological data analysis technique commonly used for macroscopic measurements, and plotted on the scale of the "TTS shift" frequency. FIG. 6A shows a TTS plot of loss tangent ("tangent delta") versus shift frequency; fig. 6A and 6B show TTS plots of storage and loss modulus, respectively. FIG. 6D shows a TTS "shift factor" applied to a frequency during TTS processing. As shown in fig. 6A-6D, the disclosed AFM-nDMA method provides novel and unique capabilities (with respect to nanoscale measurements at low frequencies) that allow for direct comparison with results obtained via conventional bulk macrorheological techniques and methods, such as DMA and TTS. To the inventors' knowledge, these AFM-nDMA results are via AFM) the first example of time-temperature overlay data at the nanometer scale.

Additional description: AFM-nDMA modulation/excitation. The frequency space is addressed.

Frequency range: operation within 0.1Hz-10Hz (force set point modulation). The low frequency measurement is associated with a long measurement time (covering at least ten excitation frequency cycles). Therefore, the holding force mode is preferred, which maintains the loading force condition constant despite drift and creep. On the other hand, effective AFM feedback will cancel out any modulation introduced in the Z channel, effectively eliminating mechanical excitation and rendering the measurement useless. Instead of Z modulation, force set point modulation should be used at low frequenciesSystem (falling within the range of effective AFM feedback bandwidth). In force set point modulation, AFM feedback tracks both the DC preload force set point and the AC periodic modulation component, which provides the necessary mechanical excitation. The force set point modulation scheme can be implemented in FPGA firmware by adding a low frequency Direct Digital Synthesis (DDS) component. The magnitude and phase of the force and displacement (deformation) can be measured via two channels (deflection and Z-sensors), via hardware lock in FPGAs, or demodulated by capturing/recording signal traces and using a "software lock" method that is drift and creep corrected, which is implemented in software ("software lock").

At about 10Hz-100 Operating in the frequency range of Hz (force set point and/or Z-scanner modulation). The hold period in this frequency range may be relatively short, and holding the Z drive mode (or holding the Z sensor with AFM feedback that tracks both DC position and AC modulation in the Z sensor channel) may be acceptable for low or moderate drift and creep rates. Therefore, Z-scanner modulation (with hold Z-sensor or hold Z-drive) can be used. Alternatively, force set point modulation may be used, as the AFM feedback may have sufficient bandwidth to track the AC set point within this frequency range.

Operating in a frequency range of about 100Hz to about 1000Hz (Z modulation). AFM feedback may not be able to track the AC set point at frequencies above 100Hz, resulting in excitation inefficiencies (a significant portion of the modulation amplitude ultimately appears in the residual error signal). Z modulation is preferred.

Reference frequency techniqueFor correcting (in one case, subtracting its value) creep in the contact area and/or tracking the contact area during the process of correcting the associated creep, in the case of AFM-nDMA, the modulus of the material at one particular frequency is measured (monitored) over the duration of the hold period in parallel with the AFM-nDMA measurements at the other frequencies, and then the contact radius calculated from the JKR fit of the pull-back curve at the end of the hold can be corrected (creep over the contact size during hold can be taken into account). This requires excitation at least two frequencies simultaneously.Optionally, measurements at the reference frequency may be interleaved with measurements at other frequencies.

Excitation in multiple frequencies.The multi-frequency excitation can shorten the AFM-nDMA measurement time. When applying the superposition principle, the result of the multi-frequency excitation should be equivalent to a sequential measurement (in the latter the provided drift and creep are properly taken into account). In the case of non-linearity, which is inherent in tip sample contact, "cross talk" between frequencies may occur during multi-frequency excitation.

AFM-nDMA model equation (dynamic stiffness in harmonic excitation')

As already mentioned above, AFM-nDMA is a nanoscale dynamic mechanical analysis of a sample performed by means of a cantilever probe with indentation of the sample surface with a controlled force comprising both a quasi-static (DC) component and a dynamic (AC) oscillation component. One or more frequencies of the oscillating component of the force applied to the sample are properly selected to match the low frequency range, which is typically of interest for bulk macroscopic DMA of soft materials and various polymers-from sub-hertz to hundreds of hertz.

Embodiments of the AFM-nDMA system of the present invention are configured to operate in several different mechanisms:

1. mechanism or mode of force set point modulation: here, AFM feedback is turned on to monitor the deflection of the probe cantilever and maintain both DC force ("preload force") and AC oscillation modulation components. This mechanism is ideal for low frequency AFM-nDMA experiments (sub-hertz to hundreds of hertz) because it allows the user to maintain stable conditions of tip sample contact even in the presence of drift and sample creep. The oscillatory displacement is measured by an AFM height sensor. It should be noted that for a fixed predetermined amplitude of force modulation, the amplitude of the Z-scanner displacement (height sensor) is typically dependent on the viscoelastic properties of the sample.

2. Mechanism of displacement pattern (or "Z" modulation): the indentation ramp is stopped at a predetermined trigger force (preload), but the AFM feedback is not activated. The scanner Z displacement is modulated at a fixed predetermined amplitude; thus, the magnitude of the force AC component (deflection) depends on the viscoelastic properties of the sample material. This mechanism is best suited for rapid force magnitude AFM-nDMA experiments, where the measurement frequency can be in the range of about one hundred to several hundred hertz and the contact duration is relatively short.

3. Mechanism of external actuator mode: here, the sample is attached to a High Frequency Actuator (HFA) stage. The AFM probe is tilted towards the sample surface and held in a position (in closed loop feedback on the height sensor signal) in which a predetermined preload force is present. The actuator is energized and provides modulation of vertical (Z) displacement of the sample surface relative to the AFM probe base, which is held in a static position via a feedback loop through the height sensor. The AFM vertical deflection signal is recorded and provides information about the oscillating portion of the force in the probe-sample contact in response to modulation of the separation distance (separation between the probe base and the sample surface). No sensors measure the amplitude and phase of the actuator vibrations and therefore calibration must be used.

For all of these system configurations, the operation of embodiments of the present invention leads to equations based on both the positive and negative problems of nanoscale dynamic mechanical analysis, using a common theoretical framework (referred to as "dynamic stiffness in harmonic excitation") to evaluate viscoelastic material properties of sample materials. The inverse problem equation allows the dynamic stiffness of the contact to be calculated and calculated from material properties such as storage and loss modulus and tangent delta (loss tangent, damping coefficient) derived from AFM-nDMA measurements that provide the amplitude and phase of the acquired signal. By calculating the desired excitation amplitude and preload force, a positive problem equation can be used to optimize the experiment.

For simplicity, one common set of symbols (equations used to describe AFM-nDMA theory and calculations) is used to describe all three mechanisms introduced above, where Z modulation is performed via a Z scanner or sample actuator. Suppose that the Z-displacement of the probe base (relative to the sample surface; sometimes also referred to as the "standoff distance") is described by a harmonic signal (in complex valued form):

z(t)＝Z₁ e^i(ωt+ψ)+Z₀[ equation 1]

Wherein Z₁And ψ is the amplitude and phase of the displacement oscillation component at the frequency ω ═ 2 π f. The probeIs considered to be calibrated and known and has the spring constant K of the probe_c. Further assuming that the system is linear (because it produces a harmonic response to harmonic excitation), the AFM probe deflection can also be described by a harmonic signal (this is a measurement; vertical deflection signal):

here, D₁、The amplitude and phase of the deflection oscillation component are deflected at a frequency ω ═ 2 π f, respectively.

The general equation for AFM-nDMA calculation can be derived using the definition of dynamic stiffness (of the contact between the probe and the sample), which simply extends the definition of stiffness to the harmonic, complex-valued case: dynamic stiffness S (in newtons/meter) is defined as the ratio of a complex-valued force to the complex-valued deformation caused by that force:

s ═ F ═ L [ eq3 ]

The oscillatory deformation can then be determined as the difference between displacement and probe deflection, or as a complex value:

as cantilever spring constant (K)_c) When known, the oscillating force is determined from the deflection:

thus, equation 3 for dynamic stiffness can be rewritten as:

applying some algebraic operations to the complex-valued expressions and separating the real and imaginary parts, equation 6 yields:

experimental data for determining viscoelastic material properties (storage and loss modulus) from dynamic stiffness values (probe-sample contact) requires knowledge of the magnitude of the contact: if contact (contact area, contact radius) is characterized, these parameters are determined from the storage and loss stiffnesses as follows:

wherein a is_cIs the radius of contact between the probe tip and the sample.

In practice, the size of the nanoscale contact cannot be easily shown or measured directly, but can be determined from analysis of the indentation moment curve, for example, by applying a Johnson-Kendall-Roberts (JKR) contact mechanics model to the retracted portion of the moment curve.

Thus, embodiments of the AFM-nDMA method of the present invention utilize the well-known Johnson-Kendall-Roberts (JKR) contact mechanics model for calculating contact size in conjunction with the use of a probe with well-characterized spherical tip geometry. (notably, while the JKR model is widely used for mechanical property analysis of soft materials such as polymers and is well-accepted, it is also well known that the JKR model is formulated for linear elastic materials and, strictly speaking, is not suitable for accurate description of viscoelastic materials. KJ Wahl et al, Journal of colloidal and interface science, 296(1), pp. 178 and 188, 2006, which extends this analysis to viscoelastic materials for oscillatory adhesive contact.) embodiments of the present invention additionally provide a validation method for mitigating suspected differences in viscoelastic material in JKR model results (e.g., in experiments with longer AFM-nDMA measurement hold periods at low frequencies), the validation method utilizes a specialized "punch probe" AFM tip with a contact area of known size and does not rely on any particular contact mechanics model to calculate material properties from dynamic stiffness. Such features are not known or used in AFM related art. ) Notably-and with reference to equations 8.1, 8.2 and 8.3, determining storage and loss moduli requires inferring the contact radius from indentation contact mechanical analysis. 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 deflection and displacement amplitudes and the phase difference between the deflection and displacement.

Reference frequency technique-compensation for creep of contact radius in AFM-based measurements.

As described above, the contact radius required for the calculation of the viscoelastic storage and loss modulus values after all holding segments with AFM-nDMA measurements at all predetermined frequencies have been completed-determined from the retracted portion of the force-moment curve. This contact radius value at the last estimate of indentation hold will need to be applied in the calculation across all frequency bands that may exist for a long time (perhaps several minutes for sub-hertz frequencies) before the retraction event. If the sample creep relaxation (under the preload force) is reached before the frequency measurement segment (in other words if the first relaxation waiting segment is long enough), and if the DC force remains sufficiently constant during the hold (as in the force set point modulation scheme), and if there is essentially no creep of the adhesion force, the contact radius can remain nearly constant for the entire duration of the AFM-nDMA measurement hold.

However, in most other cases, there is some creep in the contact radius during hold, and direct adaptation of the retract curve contact radius value may result in systematic errors in the AFM-nDMA calculation. The related art does not solve these problems.

To compensate for uncertainty due to possible creep of the contact area during hold, embodiments of the present invention utilize a special reference frequency method discussed below. Rewriting equation 8.1 again yields:

S'＝2a_ce' [ Eq.9]

One skilled in the art will readily recognize that under maintained experimental conditions (such as temperature, humidity, etc.), at a particular frequency f₀The underlying assumptions of basic stability, continuity, material property invariance, storage modulus E' are reasonable in practice and operation. Then, at a predetermined frequency f₀Lower measured stored stiffness S' and contact radius a_cAnd (4) in proportion. If during indentation hold { t }₁,t₂,t₃... } at different times t_iAt a predetermined frequency f₀(referred to herein as the "reference frequency") measuring the stored stiffness S' (t)_i) And at the end of the hold, immediately following the probe retraction time t_rPreviously, it was also measured as S' (t)_r) The contact radius value a can then be reconstructed (or further interpolated) during the entire time of the AFM-nDMA measurement_c(t_i)

S'(ti)/S'(tr)＝a_c(ti)/a_c(tr) [ equation 10]

Wherein the contact radius a_c(t_r) The values are determined from the retraction curve.

It is noted that in principle the loss stiffness S "(equation 7.2) can also be used in a similar way for contact radius creep compensation; however, the use of the stored stiffness S' is more practical in view of signal to noise ratio.

To achieve operation of an embodiment of the present invention, it may be desirable to properly use a processor controlled by dedicated instructions stored in a tangible memory element. Those skilled in the art will readily appreciate that the required algorithmic functions, operations, and decisions may be implemented as computer program instructions, software, hardware, firmware, or combinations thereof. Those skilled in the art will also readily appreciate that the instructions or programs defining the functions and elements of the present invention can be delivered to a processor in a variety of forms, including, but not limited to, information permanently stored on non-writable storage media (e.g., read-only memory devices within a computer such as ROM or devices readable by a computer I/O accessory such as CD-ROM or DVD disks), information alterably stored on writable storage media (e.g., floppy disks, removable flash memory and hard disk drives), or information conveyed to a computer by a communication medium including a wired or wireless computer network. Additionally, although the present invention may be embodied in software, the functionality necessary to implement the invention may alternatively or in part be embodied in firmware and/or hardware components such as combinational logic, an Application Specific Integrated Circuit (ASIC), a Field Programmable Gate Array (FPGA) or other hardware or some combination of hardware, software, and/or firmware components.

Reference throughout this specification to "one embodiment," "an embodiment," "a related embodiment," or similar language means that a particular feature, structure, or characteristic described in connection with the "embodiment" is included in at least one embodiment of the present invention. Thus, appearances of such phrases and terms may, but do not necessarily, all refer to the same implementation. It should be understood that any portion of this disclosure, by itself and possibly in combination with the accompanying drawings, is not intended to provide a complete description of all of the features of the invention.

It should also be understood that no single drawing is intended to support a complete description of all the features of the invention. In other words, a given figure generally describes only some, and often not all, features of the invention. For purposes of simplifying a given figure and discussion, and in order to direct the discussion to specific elements illustrated in the figure, a given figure and an associated portion of the disclosure that contains the description with reference to that figure typically do not contain all elements of a particular view or all features that such view may present. The skilled artisan will recognize that the invention may be practiced without one or more of the specific features, elements, components, structures, details, or characteristics, or with the use of other methods, components, materials, etc. Thus, although specific details of an embodiment of the invention may not necessarily be shown in each figure describing such embodiment, unless the context of the specification otherwise requires, the presence of such details in the figures may be implied. In other instances, well-known structures, details, materials, or operations may not be shown in a given figure or described in detail to avoid obscuring aspects of the embodiments of the invention being discussed.

The invention as described in the claims appended to this disclosure is intended to be assessed as a whole by reference to the present disclosure, including the features as recited in the claims and as disclosed in the prior art.

For the purposes of this disclosure and the appended claims, the use of the terms "substantially," "about," and similar terms with reference to a descriptor of a value, element, property, or characteristic at hand is intended to emphasize that the value, element, property, or characteristic, although not necessarily exactly as stated above, will be considered as being for practical purposes by those skilled in the art. These terms, as applied to a given characteristic or quality descriptor, mean "substantially," "predominantly," "comparable to," "substantially," "to a large or significant degree," and "largely but not necessarily all identical" to, such as reasonably indicate approximate language and to describe the characteristic or descriptor specified, so that those skilled in the art will understand its scope. In a particular instance, the terms "about", "substantially" and "about", when used in reference to a numerical value, mean a range of plus or minus 20%, more preferably plus or minus 10%, even more preferably plus or minus 5%, and most preferably plus or minus 2% relative to the specified value. By way of non-limiting example, two values being "substantially equal" to each other means that the difference between the two values may be within +/-20% of the value itself, preferably within +/-10% of the value itself, more preferably within +/-5% of the value itself, and even more preferably within +/-2% or less of the value itself.

The use of these terms in describing selected characteristics or concepts is not meant to be an uncertainty and implies or provides any basis for adding numerical limitations to the specified characteristics or descriptors. As will be understood by those of skill, the actual deviation of an exact value or characteristic of such value, element or property from that stated, is within and can vary from the numerical range defined by the experimental measurement errors which are typical of using accepted measurement methods in the art for such purposes.

Modifications and variations may be made to the illustrated embodiments without departing from the inventive concepts disclosed herein. Further, the disclosed aspects or portions of these aspects may be combined in ways not listed above. Accordingly, the invention should not be construed as being limited to the disclosed embodiment(s). In addition, the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the scope of the present invention.