Research Article | Open Access
Nanoscale Deformation and Nanomechanical Properties of Soft Matter Study Cases: Polydimethylsiloxane, Cells and Tissues
Nanoindentation technique was used to investigate the nanomechanical behaviour of different soft materials. Polydimethylsiloxane (PDMS), cells and tissues were examined. The nanomechanical properties (with loading rate and creep study), namely, the hardness () and the elastic modulus () of PDMS, were determined. A classical Hertzian contact analysis was also performed in order to obtain values of . Moreover, the plastic deformation where no load had yet been applied to PDMS was investigated (zero load plastic deformation). Finally, the difficulties of measuring the nanomechanical properties (&) of cells and tissues were evaluated, showing the need for a modification of the current experimental protocols for preparing and mechanically testing in a mode that maintains their structure and their biological functioning in order to make indentation results more reproducible. Additionally, finite element method is used in order to simulate the nanoindentation of PDMS in correlation with experimental data.
Soft matter (or soft condensed matter) refers to a group of systems that includes polymers, colloids, amphiphiles, membranes, micelles, emulsions, dendrimers, liquid crystals, polyelectrolytes, and their mixtures. Soft matter structures usually have structural scales in the region from a nanometer to several hundred nanometers and thus fall within the domain of “nanotechnology,” The soft matter length scales are often characterized by interactions that are of the order of thermal energies so that relatively small perturbations can result in dramatic stfructural changes. From an industrial point of view, soft matter is important since it exhibits promising industrial growth and helps in understanding the intricacies of the life processes (paints, surfactants, porous media, plastics, pharmaceuticals, ceramic precursors, textiles, proteins, polysaccharides, blood, etc.). Additionally, soft materials have a wide range of applications in bioengineering, electronics, and medicine .
Determination of the mechanical properties of soft materials is a complicated process due to their heterogeneous nature. Therefore, standard characterization methods, such as compression and tensile tests, may not be applicable due to inadequate sample dimensions. In this regard, techniques that allow the examination of the mechanical properties on a much reduced scale are needed. Nanoindentation is a technique which has been widely used to characterize the mechanical properties of materials at surface or subsurface. More specifically, through the nanoindentation experiment, the hardness (), the elastic modulus (), and other mechanical properties can be determined from very small volumes of materials. The high spatial resolution of nanoindentation allows local testing of mechanical properties of soft matter that is not possible using macroscale techniques. For this reason, the application of nanoindentation to soft materials has increased over recent years .
Polymers, a major category of soft materials, are typically much more compliant compared to hard materials, with ranging from a few GPa for common glassy polymers to a few MPa or lower for rubbery polymers . Considerable uncertainties arise when applying the nanoindentation technique to polymers with below 5 MPa or adhesive polymers (e.g., Polydimethylsiloxane (PDMS)) . Polydimethylsiloxane, a clear elastomer, is a common material used in many applications in bioengineering, electronics, and microelectromechanical systems because it is biologically inert, gas permeable, insulator, and good for rapid prototyping of devices . The advantages of PDMS also include its good electrical properties, low surface energy, isotropic and homogeneous properties. Polydimethylsiloxane is also thermally stable and performs well over a wide range of temperatures . Its nanomechanical properties are of great interest, for example in applications involving pressure support or fluid manipulation, and have to be carefully validated, because exposure to chemicals and temperature changes may cause their variation.
Biological materials, such as cells and tissues, can be classifiable as soft materials. The mechanical characterization of cells and tissues is of utmost importance in clinical medicine and the field of biomaterials. Considerable research relating to the links between the biomechanical properties of cells and human diseases has been carried out [7–11]. A change in the mechanical properties of cells has recently emerged as an indication of disease, although malignant and benign cells have similar morphological features. Thus, mechanical elucidation is crucial for realizing advances in diagnostics and therapeutics .
There are critical issues associated with applying nanoindentation to soft materials. Soft materials exhibit viscoelastic behavior, and their mechanical response is strongly dependent on loading rate and time . The effect of viscoelasticity, which is most observed on nanoindentation, is creep, that is, sinking of the tip into the sample under a constant load . Adhesion between the tip of the nanoindenter and the sample’s surface has a big influence on soft materials and can add up to a significant percentage of the overall maximum load . Major sources of difficulty in applying nanoindentation to soft materials also include evaluating the appropriateness of mechanical models  and defining proper calibration materials . As the majority of biological materials have water contents up to 80%, it is of paramount importance that nanoindentation tests are carried out in the appropriate medium. It is desirable to perform tests in fluid environment rather than to attempt to infer from measurements on dry (or 50% relative humidity, RH) samples in order to understand their properties and behaviour. To meet this need, the testing capability of the nanomechanical instrument is extended by the development of a fluid cell allowing nanoindentation, nanoscratch, and nanowear testing of samples fully immersed in fluids. This development allows testing under 25–90% RH. The fluid cell works with the existing pendulum design, and the horizontal loading has several key advantages for testing in fluid . In general, for quantitatively accurate and reproducible results, nanoindentation of soft materials requires further validation and suitable modification. It is necessary to modify the standard testing protocols for stiff materials in order to take into account their viscoelasticity, their roughness and accurately calculate the surface position .
In the present study, nanoindentation is used to investigate the nanomechanical behaviour of different soft materials. PDMS, cells and tissues are examined. The nanomechanical properties ( and ) of PDMS are estimated by the Oliver and Pharr (O&P) model performed on the unloading data of the load-displacement curves. The influence of polymer’s viscosity and the effect of loading rate and creep on and values of PDMS are also discussed. A classical Hertzian contact analysis is performed in order to obtain values of . Moreover, the nanoscale deformation of PDMS is investigated, and the so-called phenomenon referred in literature as zero load plastic deformation  (i.e., the phenomenon during which when the indenter tip is pulled away from the sample, an adhesive attraction deforms the soft polymer along the direction of the tip motion and causes a negative indentation) is evaluated. Adhesion energy at the tip-PDMS surface sample interface is a significantly important parameter and needs to be taken into account for consistent determination of soft materials by nanoindentation. Finally, the difficulties of the measurement of the nanomechanical properties ( and ) of cells and tissues are analyzed showing the need for a modification of the current experimental protocol. Finally, analysis based on the finite element method (FEM) is used to simulate the nanoindentation of PDMS in correlation with experimental data.
2.1. PDMS Sample Preparation
PDMS elastomer sample has been prepared by mixing 30 g siloxane base with 4.5 g crosslinking agent. An increase in viscosity is observed during the preparation of the sample. The addition of the crosslinking agent is followed by the addition of catalyst. Its final shape (3 mm thickness with smooth surface at top and rough surface at bottom) is due to the casting process in aluminum pot/pipkin, where the smooth surface is exposed to air and the rough one is in contact with the bottom of the pot (vulcanization time about 24 h). The surface of the sample has been cleaned with water prior to testing to avoid contamination .
The nanoindentation analysis in this work has been performed using a nanomechanical test instrument which allows the application of loads from 1 to 10000 μN and the recording of penetration depths as a function of applied loads with a high load resolution (1 nN) and a high displacement resolution. The nanomechanical test instrument employed in this study is equipped with a Scanning Probe Microscope (SPM), in which the sharp probe tip moves in a raster scan pattern across a sample surface using a three-axis piezo positioner. In all depth-sensing tests a total of 10 indents are averaged to determine the mean values for statistical purposes, with a spacing of 50 μm (~45% RH, 23°C). After the indenter tip makes contact with the specimen surface, the indentation applied load and displacement are recorded simultaneously. Feedback control is performed for its ability to operate under closed loop load or displacement control (this capability provides the means of performing traditional tests utilized for viscoelastic property measurements of biomaterials and polymers, such as creep and stress relaxation tests) . All nanoindentation measurements have been performed with the standard three-sided pyramidal Berkovich probe, with an average radius of curvature of about 100 nm.
3. Results and Discussion
3.1. Polydimethylsiloxane (PDMS)
3.1.1. Hardness and Elastic Modulus of PDMS
A typical nanoindentation test provides load-displacement data, which are the deformation response of a material. Figure 1 presents typical load-displacement curves, where the applied load is plotted in accordance to the displacement of the indenter. It is obvious that PDMS has a small amount of plastic deformation. The adhesion effect during nanoindentation experiments of soft polymers leads to overestimation of the modulus [2, 23–25]. One effective method to ensure that the zero displacement position is accurately determined is the correct determination of the contact area between the indenter and the sample being tested. One method to define the surface position of soft materials correctly is to perform displacement-controlled experiments as described in Cao et al. . In the present work, the area for the nanoindentation testing was carefully chosen using the SPM, in order to obtain low values of the surface roughness of the sample, that is, 3-4 nm. Thus, it was ensured that no negative load values would appear in the load-displacement curve. The and of the PDMS in accordance to the displacement of the tip are presented elsewhere , showing the decrease of and as the displacement of the tip increased, until they reached a constant value for bulk PDMS. O&P model  was used to estimate the and values using the unloading data of the load-displacement curves. The obtained results revealed the heterogeneity of PDMS, that is, the existence of two regions: (1) surface/near surface ~250 nm thick ( MPa & MPa) and (2) bulk ( MPa & MPa) region. Higher and values in the surface area are caused probably by higher crosslink density in the low displacement range (0–300 nm) and the exposure of the PDMS samples in atmospheric air. Several studies on PDMS samples via nanoindentation have revealed various nanomechanical properties [29–34]. Hou et al.  investigated PDMS samples (surface MPa, bulk region MPa) with average surface roughness ~15 nm. Greenwood et al.  confirmed the increase in of PDMS with higher crosslinking agent concentrations due to polymerization degree decrease ( MPa).
3.1.2. Time-Dependent Behaviour of PDMS
The mechanical behaviour of polymers is usually described by viscoelastic and/or viscoplastic characteristics, depending on time (or frequency) as well as temperature, which makes the mechanical response of the polymer systems complicated. An analysis of the extension load-displacement curves reveals the viscoelastic relaxation processes in the bulk polymer that causes the monotonic rate dependence of the adhesion-induced indentation. The viscoelastic relaxation processes in the bulk polymer have different components based on the molecular weight between crosslinks. An analysis of the retraction load-displacement curves shows that besides the viscoelastic processes in the bulk, relaxation processes also occur at the interface of the tip-polymer surface. Consideration of the adhesion energy at the interface of the tip-PDMS surface is a significantly important parameter and needs to be taken into account for consistent determination of soft materials by nanoindentation.
When materials are subjected to a constant applied load for a period of time (hold time), creep (a time-dependent deformation) is observed. Creep is an important phenomenon which describes viscoelasticity of polymers. A better understanding of creep would clearly improve the ability to enhance the service performance of polymers. In the present work, the effect of loading rate and creep on and is studied.
(1) Loading Rate
Regarding the influence of polymer’s viscosity [35, 36], the effect of loading rate on and of PDMS is investigated. The effect of loading rate on and values is studied for loading rates 0.1 μN/s, 0.8 μN/s, 1.2 μN/s, 1.5 μN/s, and 8 μN/s in PDMS with a maximum load of 500 μN through a three-segment process (hold time 3 s between load and unload segments). and values are relatively high (3.3 and 9.8 MPa, resp.) for a loading rate of 0.1 μN/s, then decreasing with the increase of loading rate and reaching constant values (~2 and 5.6 MPa, resp.). The unloading rate is the same with that of loading in each case. The and values have been estimated by O&P model  and are presented in Table 1. and values versus loading-unloading rate are presented in Figure 2.
Since the same force profile is applied during a longer time interval at low loading-unloading rates than that of high loading-unloading rates, a larger amount of kinetic energy is applied to the PDMS at low loading rate than that at high loading rate, resulting in differences in nanomechanical properties because of the time-dependent viscoelastic behaviour of PDMS. This is because in case of PDMS, which is a viscoelastic material whose glass transition temperature is far below room temperature, when the push-in rate is low, the PDMS molecules are able to move (deformation and subsequent recovery to their equilibrium conformation) in response to the applied deformation. The mechanical response of elastomers under applied loading is one of the major issues in many applications such as tunable microdoublet lenses  and dielectric elastomer actuators .
One of the commonly used techniques taking into account the viscoelastic properties of polymeric materials during nanoindentation is to hold the indenter at the maximum load, for a period of time . This, however, leads to creep effects during indentation, in which the measured and are dependent to the hold period, maximum load and loading/unloading rates . The term creep is often used to describe a delayed response to an applied stress or strain that may be a result of viscoelastic or viscoplastic deformation . Nevertheless, creep influences the maximum displacement and the unloading curve in a way that nonnegligible errors of and calculations may occur . In the present study, the nanoindentation creep behaviour of PDMS is investigated. Different hold rates are selected, and the nanomechanical properties ( and ) are calculated. In Figure 3, and values as a function of hold period are presented.
From Figure 3, it is obvious that and values decrease as the hold period increases, until they reach almost constant values (~0.5 MPa and ~1.9 MPa, resp.).
3.1.3. Contact Analysis and Zero Load Plastic Deformation
A classical Hertzian contact analysis is performed in order to calculate the . value is ~70 MPa, which is lower than that obtained from the O&P method (130 MPa). This difference is due to the assumption given in the Hertz formulation stating that the contact depth () is related to the total penetration depth (), . Only for ideal elastic contact (), both analysis will compute the same moduli. The classical Hertzian model is a hard contact model that does not take adhesive interactions into account. Moreover, the plastic deformation when no load is applied to PDMS is examined in the present work. The dynamic load signal (dynamic contact stiffness) is used to provide much greater sensitivity to surface contact and surface stiffness in comparison with a change in quasistatic force or stiffness measurement. The dynamic load signal as the tip approaches the surface, snaps into contact because of surface forces, and measures the increase in the surface stiffness with displacement . Figure 4 presents the applied load versus displacement into PDMS surface . Although no load had been applied to the sample, there is an interaction between the tip and the sample. This phenomenon is frequently referred in the literature as zero load plastic deformation (ZLPD) . The point where the interaction becomes attractive is the point where the tip contacts the surface. From Figure 4, one can see that when the tip is pulled away from the sample, the adhesive attraction deforms the soft polymer along the direction of the tip motion and causes a negative indentation (i.e., polymer extension in a typical tensile stress test). At point “0,” where the tip is attracted to the sample surface because of the adhesive interaction, the stored elastic energy and the surface energy are balanced. Indentation between the point where the tip starts to contact the sample surface and point “0” is defined as the adhesion-induced indentation, while point “1” corresponds to the adherence force.
In many dynamic adhesion processes, it is found that more energy is required to separate two surfaces in adhesive contact in relation to the energy that is released when they come into contact. This is usually manifested as a hysteresis between the loading and unloading force versus separation force curves in a force measurement. The adhesion hysteresis is widely believed to be due to energy dissipation in the bulk material and at the crack tip. A number of mechanisms have been proposed to explain the adhesion hysteresis. For instance, adhesion hysteresis is influenced by viscoelastic or plastic bulk deformation, the formation of weak or strong dipolar attractions at the interface, and crosslinking or physical entangling of tethered chains across the interface; these mechanisms are usually coupled in a practical adhesion system. Considerable work has been performed in order to investigate the different mechanisms for the adhesion hysteresis qualitatively or quantitatively. Silberzan et al. , Choi et al. [45–47], Kim et al. , Mason et al. , and Perutz et al. [50, 51] studied the adhesion hysteresis of PDMS-PDMS self-adhesion systems and PDMS self-assembled monolayer systems using the macroscopic Johnson-Kendall-Roberts (JKR) method. In these experiments, the polymer crosslink density, sol fraction, surface functionality, loading and unloading rates, and dwell time of contact were varied. The contributions to the adhesion hysteresis from the viscoelasticity of the bulk material were ignored, because the loading and unloading processes were controlled to follow a quasiequilibrium process. Instead, adhesion hysteresis was found to be dominated by hydrogen bonds, crosslinking, and physical chain entanglements. Pickering and Vancso , Vakarelski et al. , Noël et al. , and Gillies et al.  used AFM to study the adhesion hysteresis of polymer systems and found that the viscoelasticity was the dominant factor.
The energy which has to be overcome is the energy required to bend the cantilever until it reaches the value of adhesion force (). The Derjaguin-Muller-Toporov (DMT)  and the JKR  theories establish a relationship between the adhesion force (), the Berkovich-PDMS interaction constant (), and the thermodynamic work of adhesion (), given by
The dependence of on the is presented in Figure 5. This dependence of on the is in good agreement with findings of other researchers [1, 6, 58, 59]. Song et al.  performed indentations to untreated and UV-treated PDMS and compared the obtained values (43.1 and 1.7 nN, resp.). Similar studies revealed dependence on the hydrophobicity of the PDMS surface ( nN [6, 59]) and the crosslinking density ( from 93.1 to 43.6 μN with decreasing crosslinker concentration ).
Approaching and retracting parts of force curves are often not identical due to various reasons (in liquids hydrodynamic drag on the cantilever is one of the reasons) . It causes a difference in the zero force line. In order to prevent a jump-in, contact the use of stiffer cantilevers is proposed (however sensitivity is lost). Due to this fact, alternative techniques have been performed to access the unstable regime such as the application of a force feedback (electric or magnetic) to balance the surface force [61–67], kinetic force experiments [68, 69], and using a microsphere rather than a microfabricated tip (the hydrodynamic drag can be used to compensate for the attraction) . Another instability occurs when tip or sample surface is not perfectly rigid but is deformed by the tip. For that case of elastic deformation, Pethica and Sutton showed that at sufficiently small separations, typically 1-2 Å, the tip and the sample will jump together, irrespective of apparatus construction . A similar jump occurs for nonelastic deformations. In this case, however, the jump depends on the rate of the plastic deformation . When retracting the tip adhesion often keeps it in contact with the surface until the cantilever force overcomes . This even happens in purely elastic, nondissipative situations. Another issue is the influence of interfacial adhesive forces on the indentation contact area determination. The limitations in using nanoindentation for quantitatively determining the PDMS properties are uncertainties associated with the initial tip-sample contact point and tip shape at large depths. Gupta et al.  investigated the effects of adhesion on the for soft PDMS samples, including the adhesive forces to the analysis of the nanoindentation data in order to elucidate the apparent linearity of the load-displacement curves with the differences in values obtained from Hertzian model observed at the different applied loads (applied loads from 5 to 20 μN, values from ~1.8 to 1.4 MPa, resp.). Carrillo et al.  performed adhesion experiments in aqueous solution in order to study the influence of experimental environment on the adhesion work. Their results indicated a significant decrease in the adhesive force when testing was performed in distilled water. Since the contribution of electrostatic attraction to the total adhesive force is relatively large for PDMS (in addition to the dispersive interactions due to the nonpolar behavior of PDMS), the decrease found in the measured is probably the result of the weakening of the electrostatic charge under wet conditions. Moreover, the values do not seem to be converging to zero (repellant interaction of PDMS (an inherent hydrophobic material mainly due to the presence of –CH3 groups) with water, and constant presence of the dispersive van der Waals forces). Thus, the current calculation methods, though seemingly adequate for adhesive polymeric materials, need proper modification of the current experimental protocol in order to capture the dynamic response of materials for which viscous or rubbery flow conditions exist (i.e., where energy loss is a more significant factor) and the adhesion tip-sample interactions .
3.2. Cells and Tissues
The mechanical characterization of tissues and other biological materials is of outmost importance in clinical medicine and the field of biomaterials. Soft hydrated materials, such as vascular tissues and other biomaterials, are a challenging class of materials as they are composed of hierarchical structures with important features down to the nanometer or micrometer scale. Because of its small probe size, nanoindentation can be used to measure local material properties and deformation in small, thin, and heterogeneous samples. This allows testing of tissue specimens that are unsuitable for traditional mechanical testing, including biopsy specimens. Nanoindentation is also useful for measuring mechanical properties or microstructural features within bulk samples, characterizing the properties of individual constituents within composite or heterogeneous samples, or mapping mechanical properties across a sample surface. As a result, nanoindentation is emerging as a valuable mechanical testing technique for cells and tissues .
Because of the considerable differences in the mechanical responses when compared to hard solids, different testing and analysis protocols must be performed for the extraction of meaningful indentation data from soft biological specimens. The goal to make indentation testing of soft biological tissues more reproducible and also capable of evaluating the effects of changes in fine structure on tissues stiffness—both mean values and site-to-site variability—will result in a new protocol for preparing and mechanically testing tissues and cells in a mode that maintains their bona fide structure and their biological functioning. Some commonly used indentation testing protocols include the use of preloads before loading and/or long hold periods before unloading to reduce time-dependent effects on the resulting data. However, preloading of soft biological tissues would inevitably move the water within the tissue. Inherent in the analyses of tissues and cells, is the need to precisely locate the initial point of contact between the tip and the specimen surface . For soft biological tissues, the accurate determination was described by A-Hassan et al.  to be “one of the most vexing problems.”
There is currently no well-accepted technique for viscoelastic or poroelastic analysis of indentation data in soft hydrated materials, a deficiency that makes the quantitative values measured through this technique unreliable . By using the combination of hydration system, proper tip geometry, and control material, this limitation can be addressed, broadening the applicability of nanoindentation in the study of diverse biomaterials. Tip selection is also a critical issue in the indentation of soft materials. For commonly used indentation of hard materials, the Berkovich tip and the cube-corner tip, three-sided pyramidal tips that come to a sharp point, are commonly used. This tip geometry has also been used in the mechanical testing of mineralized tissues and many glassy polymers [76–78]. However, spherical tips are commonly used when indenting soft polymers [23, 79–81]. This geometry leads to a larger contact area and hence a larger contact stiffness at the surface. Additionally, the lack of sharp edges minimizes plastic deformation and stress concentrations in the indented material [81–85]. A third option on occasion utilized in the study of viscoelastic materials is the flat punch [79, 86, 87]. The flat tip has the characteristic of having a constant, known contact area as a function of depth. However, for this contact area to be valid full contact is required with the surface, and thus the flat punch is optimal only for very flat surfaces or deep depths, while spherical tips will contact the surface with an orb shape even if there is a slight tilt to the sample or small amounts of surface roughness. In Figure 6 , a comparison of contact sizes of different types of tips during indentation is made.
The full assessment of mechanical properties of cells and tissues is limited by the complex nature of their structure and physiological environment. Hydration is a major issue for biological materials, since most of them are 80% water based. Attempts of measuring the nanomechanical properties of an endometrial tissue have been made, but there were no successful results since the sample was destroyed before the end of the measurement due to lack of hydration.
A liquid tip has been recently introduced in order to perform tests at in vitro conditions. The liquid tip allows a liquid film of up to 3 mm thick on the sample surface (Figure 7), [88, 89]. The longer shaft of the liquid tip allows machine calibrations in the testing media and changes in damping behaviour. Testing in liquids also causes meniscus forces on the indenter. Therefore it is important for the indenter to be fully submerged in the liquid prior to testing. A minimization of the deformation caused by the surface approach and detection is achieved by using a two-step approach. In the first step a high setpoint is used to penetrate the liquid. The deformation before the actual indent is reduced by using a lower contact force in the second step .
Akhtar et al.  used the nanoindentation technique to determine trabecular stiffness of bovine bone, both dehydrated and rehydrated. Testing was conducted using a customized liquid cell. Values of were obtained for dehydrated and rehydrated trabeculae (18.3 GPa and 14.3 GPa, resp.). Hence, dehydration led to an increase of approximately 22% in .
Rico et al.  developed a contact model of a blunted pyramidal tip and assessed the suitability of pyramidal tips for probing mechanical properties of soft gels and living cells (blunted pyramidal tip indenting an elastic half-space). The measurement of and complex shear modulus of agarose gels and A549 alveolar epithelial cells with pyramidal tips and the comparison with those obtained with spherical tips led to the conclusion that the gels exhibited an elastic behaviour with almost coincident loading and unloading force curves and negligible values of . fell sharply with indentation up to ~300 nm, showing a linear regime for deeper indentations. A similar indentation dependence of with twofold lower values at the linear regime was obtained with the spherical tip fitted (Hertz’s model fit). The dependence of on indentation in cells was parallel to that found in gels. Cells exhibited viscoelastic behaviour with ratio ~1/4. In Figure 8, the scanning electron micrographs of pyramidal and spherical tips used in agarose gels are presented.
Daphalapurkar et al.  used nanoindentation techniques to measure the viscoelastic functions of four quadrants of human tympanic membrane (TM). TM (or eardrum) is composed of three membrane layers with collagen fibers oriented in the radial and circumferential directions and exhibits viscoelastic behaviour with membrane (or in-plane) properties different from through-thickness (or out-of-plane) properties. Due to the interaction of bundled fibers and ground substance, which is inhomogeneous, these properties could vary with locations. For measurement of in-plane Young’s relaxation modulus, a sectioned quadrant of the TM on a circular hole was fixed, and a spherical nanoindenter tip was used to apply load at the center of the suspended circular portion of the specimen. An inverse problem-solving methodology was employed using finite element analysis to determine the average in-plane Young’s relaxation modulus of the TM quadrant. Results indicated that the in-plane steady-state Young’s relaxation modulus for four quadrants of the TM did not change significantly. However, a variation of the modulus from 25.7 MPa to 37.8 MPa was observed with measurements from different individuals. For measurement of Young’s relaxation modulus in the through-thickness direction a spherical indenter tip was used to indent into different locations on the surface of the TM specimen supported by a substrate. Viscoelastic contact mechanics analysis of the load-displacement data, representative primarily of the through-thickness stiffness of the TM, was conducted to extract the Young’s relaxation modulus in the out-of-plane direction. Results indicate a wide range of steady-state Young’s relaxation modulus values, from 2 MPa to 15 MPa, in the through-thickness direction over the TM.
Zysset et al.  performed nanoindentation experiments on wet bone tissue. A total of 39 indentations failed probably due to the invasion of water on the surface of the tissue and were attributed to zero values for both and . From the remaining indentations, the ranged from GPa in trabecular tissue from the femoral neck of a 74-yr-old female up to GPa in interstitial tissue from the diaphyseal cortex of a 69-yr-old female. followed a similar distribution as , but with lower statistical contrast.
In Figure 9, typical load-displacement curves of cortical and trabecular bone are presented indicating a different response of cortical and trabecular tissue.
Nanoindentation has the ability to detect changes in mechanical properties associated with small defects of the tissue. Considerable research relating to the links between the biomechanical properties of cells and human diseases has been carried out [7–11]. Nanoindentation can be used to compare the bone tissue properties in healthy bones with those of diseased and genetically modified ones. Both studies aid in understanding the role of local tissue properties and hierarchical structure in macroscale fracture and deformation mechanisms of bone . Figure 10 presents the mapping of the through a cartilage defect in a rabbit knee joint using nanoindentation. It is obvious that values of healthy and diseased regions of the cartilage are different. Mechanical parameters in specific regions of rabbit finger joint cartilage can be associated with structural parameters such as cell density and thickness of the superficial zone .
In conclusion, nanoindentation technique has recently emerged as a powerful tool for measuring nanomechanical properties of cells and tissues. It can play a valuable role in the assessment of material properties of healthy, diseased, and tissue-engineered soft tissues. Because of the differences in the mechanical responses compared to hard solids, different analysis protocols should be used to extract meaningful indentation data from biological specimens and make indentation testing more reproducible. Recently, heating/cooling stages have been introduced. Heating/cooling stages allow the whole range of soft matter phase behaviours to be explored with SPM. Several approaches have been taken [93–96], either using peltier devices (which are easy to implement) or resistive heaters (which provide greater temperature range), and heater/coolers are now available for most commercial instruments .
3.3. Finite Element Modeling of Nanoindentation of Soft Matter
Sharp indentation of soft matter is widely investigated numerically using the finite element method. Large deformation theory is relied upon for accuracy. In recent studies, cone and Vickers indentation is considered, and in particular the study focuses on second-order effects on relevant indentation quantities in the microindentation regime. The second-order effects include indenter tip radius, friction, in general, effects due to different values on the included angle of the cone indenter, showing that frictional effects as well as effects due to indenter tip roundness are small at cone indentation but friction can substantially influence the results in case of Vickers indentation (most likely due to frictional effects at the edges of the indenter); such behaviour is not accurately described using conventional theory of friction, so in a recent work a numerical approach for this purpose is discussed . Lim and Chaudhri [99, 100] performed careful and comprehensive experimental investigations; five highly elastic polymers and rubber materials were indented by cone and Vickers (sharp pyramid indenter with a quadratic base) indenters, and the main conclusion was that deviations between theory and experiments existed in case of Vickers indentation. This is certainly an alarming result taking into account that existing equations form the basis for many standard formulae designed for material characterization using indentation. There are many possible explanations for this discrepancy such as friction, remote boundary effects, contained plastic deformation, and tip radius effects. It is suggested that frictional effects at the edges of the Vickers indenter might contribute, but it could be argued that at elastic indentation, for example, tip roundness effects are of importance. It is the purpose of the present paper to investigate this important issue further.
The nanoindentation of soft matter (PDMS in this work) is simulated with FEM analysis  using ANSYS , and the load-displacement curve through this analysis is obtained and compared with the analytical solution provided by Sneddon’ s relationship
The three curves are presented in Figure 11 (experimental load-displacement curve for 50 μN of applied load and 2500 nm of displacement). It is obvious that the FEM analysis curve and the curve based on Sneddon’s analysis are in good agreement with the experimental curve (Figures 11 and 12). Sneddon’s solution assumes that the indenter is perfectly sharp and rigid. Frictional effects are not included in these analyses. For displacements lower than ~1250 nm, the three curves are almost identical. Small deviations are observed due to the fact that the indenter tip is never perfectly sharp. For displacements higher than ~1250 nm, the simulated curves deviate from the experimental one. FEM analysis results are higher due to large deformation effects. This phenomenon has also been reported by Kamran and Larsson  and Xu and Li  in similar experiments.
In further investigation, a quantitative analysis of the problem will be conducted, and then nonlocal frictional models have to be implemented into the numerical scheme. In commercial finite element packages, this can be achieved by using a so-called “elastic slip” description, compare or confer for example ABAQUS . In the “elastic slip” description the Coulomb condition is not applied pointwise but weighted over an area region with a so-called mollifying function, Zhong . Specifically, the characteristic length of the area region then constitutes the length scale in the indentation problem. It should be mentioned that the present results, although first and foremost intended for an investigation of sharp elastic indentation problems, are relevant also for other types of contact problems. Typically, such problems would include for example nanoindentation methods for determination of the elastic stiffness of elastoplastic materials [28, 98–100] and scratching of polymers [106–109].
In the present study, nanoscale deformation and nanomechanical properties of polydimethylsiloxane, cells and tissues are presented. More specifically, (i)the nanomechanical properties ( and ) of PDMS were estimated with the Oliver and Pharr (O&P) model and determined the heterogeneity of PDMS, namely: a surface/near surface region, ~250 nm thick ( MPa & MPa), and a bulk region ( MPa & MPa). The surface/near surface region with enhanced nanomechanical properties was attributed to exposure of the samples to atmospheric air and higher concentration of the crosslinking agent in the surface;(ii)a number of problems arise with the application of nanoindentation technique to polymeric materials. In general, tends to increase with decreasing penetration depth, often referred to as an indentation size effect. Comparisons of modulus values are most appropriate for polymer samples with identical chemistry, molecular weight, and processing history, because quoted values of for many polymer systems can cover a large range due to potential variations in microstructure, semicrystalline morphology, anisotropy, molecular weight, crosslink density, and so forth; (iii)the time-dependent behaviour of PDMS was examined. The influence of polymer’s viscosity and the effect of loading rate and creep on and values of PDMS were studied and should be taken into account. From the creep measurements on PDMS, and were found to decrease as the hold time increased, until they reached a constant value for bulk PDMS. The analysis of ZLPD of PDMS revealed that after the indenter tip was pulled away from the sample, the adhesive attraction deformed the soft polymer along the direction of the tip motion and caused a negative indentation (for applied load ~4.5 μN ). Thus, consideration of the adhesion energy at the tip-sample interface is requisite for determining accurate of PDMS samples and other soft, elastomeric materials from nanoindentation experiments, since soft material samples are expected to have significant adhesive forces;(iv)the nanomechanical properties ( and ) of cells and tissues were reviewed revealing the need for a modification of the current experimental protocol. As soft tissues have water contents up to 80%, it is important that tests are carried out in the appropriate medium. For this reason, a liquid tip can be used, which allows a liquid film of up to 3 mm on the sample surface. Considering the hydration system, proper tip geometry, testing protocols and improved analysis methods, and calibration materials (with similar properties to the probed materials) could broaden the applicability of nanoindentation in the quantitative assessment of mechanical behaviour of materials that were previously unattainable, such as cells and tissues; (v)load-displacement curve from FEM analysis is compared with the experimental data; the one based on Sneddon’s relationship showing that the results are in good correlation for lower than ~1250 nm displacements. For higher displacements, the FEM analysis results are higher due to large deformation effects of PDMS.
While new techniques for conventional nanoindentation continue to evolve, scientists are investigating the system’s ability to measure force in the nanonewton range in order to quantify molecular binding interactions. Binding forces can also be quantified when a small number of molecular interactions occur, providing piconewton binding measurements for individual molecular pairs. There are many exciting possibilities for scanning probe microscopy and nanoindentation providing convenience and flexibility at the nanoscale.
The author wishes to acknowledge Ms. V. P. Tsikourkitoudi, Mr. E. P. Koumoulos, and Mr. D. A. Dragatogiannis for their comments that helped to improve the paper.
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