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Elastic Modulus Measurement of Hydrogels

  • Donghee Lee
  • Haipeng Zhang
  • Sangjin RyuEmail author
Living reference work entry
Part of the Polymers and Polymeric Composites: A Reference Series book series (POPOC)

Abstract

Hydrogels have been employed for a wide variety of applications, and their mechanical properties need to be modulated based on the applications. In particular, the Young’s modulus, or elastic modulus, of hydrogels is a critical property for understanding their mechanical behaviors. In principle, the Young’s modulus of a hydrogel can be measured by finding a relationship between a force applied to the hydrogel and the resultant deformation of the hydrogel. On a macroscale, Young’s modulus is usually obtained by measuring the stress-strain curves of a hydrogel specimen through the compression method or the tensile method and then finding the slope of the curve. Also, the shear modulus of a hydrogel is measured using a rheometer with parallel plates and then converted into Young’s modulus considering Poisson’s ratio. On a mesoscale, the elastic modulus can be measured by the imaging-based indentation methods which measure the indentation depth of a hydrogel sample deformed by a static ball indenter on the gel. The measured indentation depth is converted to the Young’s modulus of the hydrogel via a contact mechanics model. The mesoscale indentation method and pipette aspiration method are also available. On a microscale, the elastic modulus is usually measured using the atomic force microscopy (AFM)-based indentation method. A hydrogel specimen is locally indented by a sharp or colloidal tip of an AFM probe, and the Young’s modulus of the hydrogel is obtained by fitting an appropriate indentation model against the recorded force-distance curves. An appropriate elastic modulus measurement method needs to be chosen depending on the application, length scale and expected elastic property of the hydrogels.

Keywords

Young’s modulus Shear modulus Atomic force microscopy Indentation Rheometer Compression test 

1 Introduction

Hydrogels consist of water-swollen, cross-linked hydrophilic polymers, and they are employed for a wide variety of applications in which their mechanical properties are modulated on purpose [1, 2, 3, 4]. In order to understand the basic mechanical or elastic properties related with hydrogels, let’s think about a gelatin dessert such as Jell-O. When we enjoy a piece of gelatin dessert and touch it gently with a spoon, the gel changes its shape or deforms. The spoon exerts a force to the gel, which results in the deformation of the gel. When the spoon is removed from the gel, the dessert restores to its initial shape because the gelatin dessert is resilient or elastic. This elastic property or elasticity of the gelatin gel is quantified by its Young’s modulus and shear modulus, and these moduli describe the relationships between the stress that the spoon applies to the gel and the strain that the gel shows through deformation.

Defining these mechanics concepts needs us to have some assumptions about our dessert. Our piece of dessert is shaped like a brick, and our spoon is flat and large enough to cover the entire top surface of the gel. Now let’s press the spoon against the gel (Fig. 1a). Then, normal stress (σ) is defined as the applied compressive force (Fc; unit: N [Newton]) divided with the gel’s top surface area where the force is applied (A; unit: m2), i.e., σ = Fc/A. The unit of stress is Pa (Pascal; 1 Pa = 1 N/m2). The definition of stress shows that stress increases either as the applied force increases or as the area decreases. With the applied compressive stress, the gel piece deforms from its original height (Ho) to the decreased height (H), and the resulting normal strain (ε) is defined as the height change (ΔH = HoH) per original height, i.e., ε = ΔH/Ho. Strain has no dimension because it is a ratio of two length dimensions. Since stress and strain do not contain information on the dimensions of our gelatin gel, they are effective ways to compare compressive force on and resultant deformation of gelatin gel pieces of different sizes.
Fig. 1

Measurement of the elastic moduli of a brick-shaped gelatin dessert gel piece of top area A. (a) Gel deformation under a compressive force (Fc): the height of the gel decreases from Ho to H while its width increases from Wo to W. The normal stress and strain of the gel are σ = Fc/A and ε = (HoH)/Ho, respectively. (b) Gel deformation under a tensile force (Ft): the height of the gel increases from Ho to H while its width decreases from Wo to W. The normal stress and strain of the gel are σ = Ft/A and ε = (HHo)/Ho, respectively. (c) Gel deformation under a shear force (Fs): the top surface of the gel is displaced by L while its bottom is fixed. The shear stress and strain of the gel are τ = Fs/A and γ = θ (= L/Ho if θ is small enough). (d) Stress-strain curve and elastic modulus. As applied stress increases, the gel deforms more with an increased strain. If the stress is removed, the gel restores to its initial shape due to its elasticity. If the gel is a linear elastic material, its strain-stress curve will be straight, and its elastic modulus will be determined from the slope of the curve

Now we are about to measure the Young’s modulus of our gelatin dessert. In this compression test, we increase the compressive force on the gel while measuring the resultant deformation of the gel. It is easily expected that the strain of our dessert gel will increase as we push it harder with the spoon. Once a curve showing this behavior is obtained, which is called a stress-strain curve, the Young’s modulus (E) of the gelatin gel is determined by the gradient of the curve (Fig. 1d). In the case that our dessert is a linear elastic material, its stress-strain curve is a straight line, and its Young’s modulus is measured from the slope of the stress-strain curve. If our gelatin gel is made to be soft, it deforms more easily (higher strain) with the same degree of stress, and thus its Young’s modulus is low. If the gel is stiffer, it resists deformation more (lower strain), and thus its Young’s modulus is high. Therefore, Young’s modulus is a measure of the elasticity or stiffness of our gelatin gel.

While being pressed down by the spoon, the gel also deforms in the horizontal direction (Fig. 1a). This lateral deformation is determined by the Poisson’s ratio of the gel. Poisson’s ratio (ν) is a ratio between the lateral strain and the normal strain of the gel, i.e., ν = (ΔW/Wo)/(ΔH/Ho). In other words, Poisson’s ratio illustrates a relationship between the gel deformation in the direction of the applied stress and the deformation in the direction perpendicular to the stress. If our gelatin dessert is incompressible, which means that its volume does not change with applied stress, its Poisson’s ratio is 0.5, i.e., ν = 0.5.

Similarly, Young’s modulus measurements can be done by stretching our dessert gel, i.e., applying a tensile stress, and Young’s modulus is determined in the same way (Fig. 1b). Let’s assume that we can attach the spoon to the top of the gel piece (possibly using surface tension or adhesion force) and that the bottom of the gel is fixed on the plate. As we lift the spoon applying a tensile force (Ft) to the gel, the gel is elongated in the direction of the force from Ho to H while its lateral dimension decreases from Wo to W according to its Poisson’s ratio. Then, the tensile stress on and resultant strain of the gel are σ = Ft/A and ε = (HHo)/Ho, respectively. It is expected that the gel will be stretched more as the tensile force increases. Therefore, we can measure a stress-strain curve of the gel by applying a tensile force and then measure the Young’s modulus of the gel in this tensile test. It needs to be noted that stress-strain curves from the compressive test and the tensile test can be different.

Now let’s change the direction of the stress: we gently place our spoon on the gel, not resulting in normal strain, and then move the spoon in the lateral direction (Fig. 1c). The spoon is assumed again to be attached to the gel. Then, the top surface of the gel moves in the direction of the applied shear force (Fs), while the bottom of the gel is fixed on the plate. Therefore, the gel shows shear deformation, which means that one part of the gel slides past the neighboring part. In this case, the applied stress is shear stress (τ = Fs/A; unit: Pa), and the resultant shear strain (γ) is quantified by the angle θ shown in Fig. 1c. If deformation is small enough, θ can be equated to a ratio of the lateral displacement of the gel’s top surface (L) to the height of the gel (Ho), i.e., γ = L/Ho. Similar to the compressive and tensile tests, we can obtain a curve of shear stress and shear strain, and the gradient is the shear modulus (G) (Fig. 1d). Again, in the case that the gel is linearly elastic, the curve is straight, and its slope is the shear modulus of the gel. If our gelatin gel is isotropic, which means that the gel’s properties do not depend on directions, the measured Young’s modulus and shear modulus are related via Poisson’s ratio: E = 2G(1 + ν).

In summary, we can determine the Young’s modulus and shear modulus of our gelatin dessert by measuring its stress-strain curves and then interconvert its Young’s modulus and shear modulus using its Poisson’s ratio. Although very simple (thus not rigorous), the concept of the gel example is applied to the elastic modulus measurements of various types of hydrogel. In the following sections, we briefly review experimental methods to measure the elastic modulus of hydrogels depending on their measurement scale: macroscale (scale larger than ~1 mm), mesoscale (sub-mm scale), and microscale (scale smaller than ~100 μm). We also introduce our recent studies on elastic modulus measurement of alginate gels and polyacrylamide gels.

2 Macroscale Measurements

2.1 Compression Method

A traditionally well-established method for hydrogel elasticity measurement is the compression test [5, 6, 7]. In this method, hydrogel samples are usually prepared in a disc form, and they are compressed (either confined or unconfined) with a controlled force while their deformation is measured. Then the applied force and resultant gel deformation are converted to compressive stress and strain. Finally, the Young’s modulus of the gel specimen is determined from the slope of the obtained stress-strain curve.

One example introduced here is our recent measurement of the Young’s modulus of alginate hydrogels. Alginate gel is a biocompatible hydrogel extracted from brown seaweed [8], and its cross-linking is induced by calcium ions. Depending on the concentration and molecular weight of the alginate, cross-linked alginate gels have different elastic moduli. We tested alginate gels made of two different alginate products (Pronova UP MVG, NovaMatrix [molecular weight: >200 kDa] and Alginic acid sodium salt from brown algae, Sigma-Aldrich [molecular weight: 12–80 kDa]) at concentrations of 0.5%, 1.0%, and 1.5%. Alginate gel disc specimens of 9.7 mm in diameter and 6.2 mm in height were cast using agarose gel molds containing calcium ions, and they were immersed in a calcium solution for further cross-linking. Stress-strain curves of the prepared alginate gel specimens were obtained using Instron 5944 (Norwood, MA; loading rate = 1 mm/min) [9]. Their Young’s modulus values were obtained by fitting a line against a linear region of the data (Fig. 2a), and the Young’s modulus of alginate gels was found to increase with the molecular weight and concentration of alginate (Fig. 2b).
Fig. 2

Young’s modulus measurement of alginate hydrogels using the compression method. (a) Example of the measured stress-strain data (1.5% Pronova UP MVG alginate). Young’s modulus was determined to be E = 54.6 kPa from the linear region of the data (red line). (b) Measured Young’s modulus values of alginate gels increase with the molecular weight and concentration of the alginate. Tested alginate: Pronova UP MVG alginate (molecular weight: >200 kDa) and Sigma-Aldrich Alginic acid sodium salt from brown algae (molecular weight: 12–80 kDa). Error bar: standard deviation (the number of tested specimens per case: 5–6)

The compression method is also applicable to measuring the elastic modulus of hydrogel beads. Ouwerx et al. compressed alginate gel beads and measured their Young’s modulus using
$$ E=0.7956\frac{F_c}{\sqrt{h^3d}}, $$
(1)
where h and d are the deformation and diameter of the beads, respectively [10]. Later, Chan et al. employed a similar compressive approach to measure the elastic modulus of ~2-mm-diameter alginate gel beads [11]. They used the following Hertz theory, which is valid up to 10% strain for hydrogel beads and is similar to Eq. (1):
$$ E=\frac{3\left(1-{\nu}^2\right){F}_c}{\sqrt{h^3d}}. $$
(2)

The compressive method has an advantage in that the method has no limit on the shape of the hydrogel specimens [2, 4, 12] although cylindrical or disc-like specimens are usually used. In the compression method, preload can be applied to the specimen before stress-strain curve measurements because direct contact between the loading platens and the specimen is important. Furthermore, lubrication can be necessary between the specimen and the platens because their full contact may cause friction which can result in bulging or shear deformation of the specimen [12].

2.2 Tensile Method

The tensile method is also well-established and widely used for measuring the Young’s modulus of hydrogels [6, 13, 14, 15, 16, 17, 18]. For tensile testing, hydrogel specimens are prepared in a dumbbell or dog-bone shape, and the ASTM D638 standard is often employed [12, 16, 19]. Because hydrogel specimens are hydrated and usually soft, it is not easy to grip them properly for tensile testing. For this purpose, handles are added to the ends of the specimen using cardboard tabs, double-sided tape and glue, which can facilitate easier clamping and prevent handling damage [3, 12, 14]. Once a clamped hydrogel specimen is stretched at a uniform deformation rate up to a certain level of strain, its Young’s modulus is determined by the slope of a certain linear region of the obtained stress-strain curve. Although widely applied, the tensile method has limitations in that the method requires specific geometries of hydrogel specimens, and specimen misalignment can cause measurement errors [4].

2.3 Rheometric Method

As aforementioned, the Young’s modulus and shear modulus of a hydrogel can be interconverted with its Poisson’s ratio if the gel is isotropic: E = 2G(1 + ν). Based on this principle, it is possible to measure the elastic modulus of hydrogels by measuring their shear modulus using a parallel-plate-type rheometer [20, 21, 22, 23]. A hydrogel specimen is cast between the top and base plates of the rheometer, and the top plate is oscillated at a desired frequency and shear strain (Fig. 3a). As the specimen is twisted and undergoes shear deformation, it exerts resistant shear force on the oscillating plate; and the rheometer measures the shear modulus of the gel based on the relationship between the shear stress applied to the gel and the resultant shear strain of the gel. Since the rheometric method involves shearing a hydrogel specimen, it is important to prevent slip between the hydrogel specimen and the plates.
Fig. 3

Rheometric measurement of polyacrylamide (PAAM) hydrogel elasticity. (a) Schematic of the rheometric measurement using the parallel-plate-type rheometer. A hydrogel specimen is cast between the parallel plates and twisted by the oscillating top plate. The shear modulus of the gel sample is measured from a relationship between the shear stress applied to the gel and the resistance shear strain of the gel. (b) Example shear modulus data of PAAM gels of two compositions. Measured shear moduli were converted to Young’s moduli: E = 18.9 kPa for PAAM gel of 10% acrylamide and 0.1% bis-acrylamide, and E = 32.0 kPa for PAAM gel of 10% acrylamide and 0.3% bis-acrylamide. (c) Young’s moduli of PAAM gels measured by the rheometric method [24, 25]. The elasticity of PAAM gels depends on their compositions

An exemplary case is our measurement of the Young’s modulus of polyacrylamide (PAAM; Bio-Rad, Hercules, CA) gels using a rheometer (AR 1500ex; TA Instruments, New Castle, DE) [24, 25]. The elasticity of the PAAM gel can be modulated by controlling the ratio between its monomer (acrylamide) and cross-linker (bis-acrylamide) [26, 27]. A PAAM gel specimen was cast between the rheometer base plate and a 25-mm-diameter stainless steel plate (Fig. 3a). After 1.5-h-long polymerization at room temperature, the shear modulus of the gel was measured using strain amplitude sweep (1 rad/s, 0.1–10% strain) while evaporation of water was reduced with a humidity chamber. The measured shear modulus was found to be almost constant over the applied strain range (Fig. 3b). Finally, the Young’s modulus of PAAM gels was calculated from the measured shear modulus values assuming that the PAAM gels were incompressible (ν = 0.5). As shown in Fig. 3c, PAAM gels showed different Young’s moduli depending on their ratios between acrylamide and bis-acrylamide.

2.4 Macroscopic Indentation Method

The macroscopic indentation method measures the Young’s modulus of a hydrogel specimen by indenting its surface using an indenter that is smaller than the specimen. This method depends on the tip geometry of the indenter used and the relative dimension of the indenter to the hydrogel specimen [7]. Hemispherical indenters are known to minimize the stress concentration and sample damage. In contrast, flat-ended cylindrical indenters are known to simplify theoretical analysis of the gel-indenter interaction because the contact area between the indenter and the gel specimen can be assumed to be constant. However, using flat-ended indenters requires indenting the specimen perpendicularly to its surface [18]. For instance, Ross and Scanlon used flat-ended cylindrical indenters with 0.5–4 mm diameters and measured the Young’s modulus of agar gels using
$$ E=\frac{\left(1-{\nu}^2\right){F}_c}{2\delta r}, $$
(3)
where δ is the indentation depth of the hydrogel and r is the indenter radius [28]. They found that the elastic modulus value from the indentation method was about 50% higher than that from the compression test.

2.5 Bending Method

It is also possible to measure the elastic modulus of hydrogels based on the gravity-driven bending of cylindrical hydrogel specimens. Peng et al. prepared cylindrical specimens of PAAM gel by pushing the gel out of a syringe pinhead; then they imaged the bent hydrogel specimens [29]. When a hydrogel cylinder was held horizontally, it was deflected downward because of its own weight. Then, the Young’s modulus of the hydrogel was measured by fitting a mechanics model against the curvature of the bent cylinder.

3 Mesoscale Measurements

3.1 Microscopy Indentation Method

Conventional microscopy indentation methods for hydrogel elastic modulus measurement consist of the following steps (Fig. 4) [18, 19, 30, 31, 32, 33, 34, 35, 36, 37]. First, micron-sized beads are embedded in a hydrogel sample to visualize the gel’s top surface, and a sphere or ball indenter of 0.5–1 mm in diameter is placed on the hydrogel in a liquid. The indentation force (F) from the indenter, which is the weight of the indenter in the liquid, deforms the gel surface locally. Then, the beads at the bottom of the indented surface are identified with an optical microscope. After the indenter is removed and the hydrogel has restored elastically, the beads identified in the previous step are focused again. Finally, the indentation depth (δ) of the gel is measured from the displacement of the focal plane of the microscope. The indentation depth can also be measured by measuring the distance between the equator of the ball indenter and the undeformed top surface of the hydrogel and then subtracting this distance from the radius of the indenter [36, 37].
Fig. 4

Schematic of the microscopy indentation method. A hydrogel specimen is prepared with micron-sized beads embedded, and a ball indenter is placed on the specimen. A microbead near the center of the indented gel surface is focused using a microscope, and its vertical location is traced upon removal of the indenter to measure the indentation depth (δ) of the hydrogel. Then, Young’s modulus is determined based on the measured indentation depth, the indenter’s size (2r) and indentation force, and an appropriate contact mechanics model

Then, the Young’s modulus of the hydrogel is calculated from the indentation force and radius (r) of the indenter, and the indentation depth of the gel, using a proper indentation model, such as the following Hertz model for spherical indenters:
$$ E=\frac{3\left(1-{\nu}^2\right)F}{4\sqrt{r{\delta}^3}}. $$
(4)

It is known that Eq. (4) is valid for the parameter regime of 0 < δ < min(0.3r, 0.2h) [34, 38] and a/r < 0.1 [38, 39, 40, 41]. Here, h is the gel thickness, and a is the contact radius between the indenter and the gel. Refer to the work of Long et al. [35] for a detailed discussion on Hertz theory and the effect of hydrogel specimen thickness on the microscopy indentation method.

The microscopy indentation method is cost-effective because it does not require special instruments and it can be easily adopted in research settings with an optical microscope. Also, the method has been validated with tensile measurement [18, 19]. However, the current microscopy indentation method has a limitation that it cannot visualize the deformed surface of hydrogels. Such information is important for understanding gel-indenter interactions and selecting an appropriate contact mechanics model for Young’s modulus determination.

3.2 Confocal Microscopy Indentation Method

We recently proposed an improved microscopy indentation method that uses confocal laser fluorescence microscopy and automated image processing (Fig. 5a) [24]. Briefly, allylamine particles (Alfa Aesar, Ward Hill, MA) were included in a PAAM gel specimen during gel preparation, and the gel was fluorescently labeled using a fluorescent dye (Alexa Fluor 488; Life Technologies, Carlsbad, CA). Then, a submillimeter-sized ball indenter was placed on the hydrogel specimen immersed in a liquid buffer, and a three-dimensional (3D) image of the indented gel was obtained using a confocal fluorescence microscope (LSM 510; Carl Zeiss, Jena, Germany). The indentation depth was measured by automated image processing of the 3D image using MATLAB (MathWorks, MathWorks, Natick, MA). Finally, Young’s modulus was calculated using the contact mechanics model proposed by Dimitriadis et al. [42] because the size of the ball indenters was comparable to the gel thickness.
Fig. 5

Confocal microscopy indentation method (reproduced with permission from [24]). (a) Schematic of the method. Top: Reconstructed 3D image of a fluorescently labeled PAAM gel specimen indented by a spherical ball indenter. Bottom: Cross-sectional view of the 3D gel image showing the indented surface of the gel. Indentation depth (δ) was determined from this cross section. (b) The Young’s modulus values of tested PAAM gels compared with those measured with the rheometric method. A good agreement between the two methods validated the confocal microscopy indentation method. The color of the dots indicates different diameters (400, 670, 794, and 1000 μm) and densities (3.84, 7.67, and 14.95 g/cm3) of five ball indenters. Regardless of different levels of the applied indentation force, the confocal microscopy indentation method resulted in similar Young’s moduli for a certain PAAM composition

The confocal microscopy indentation method was tested using four different PAAM gel compositions and five ball indenters of different sizes (400, 670, 794, and 1000 μm) and densities (3.84, 7.67, and 14.95 g/cm3). All the gel specimens showed an approximate twofold change in the indentation depth responding to increased indentation forces from heavier indenters, but the measured Young’s modulus values showed negligible differences (Fig. 5b). Then, the method was validated with the rheometric method, and the measured E values showed good agreement between the two methods (Fig. 5b).

The confocal microscopy indentation method has the following advantages. First, the method does not require indenter removal for indentation depth measurement. Second, the method enables user-independent measurement of the indentation depth based on its automated image processing technique. Third, the method can measure the contact radius between the indenter and the gel from the visualized indented gel profile, which enables selecting an appropriate contact mechanics model. Last, the method can be applied to any kind of hydrogel that can be fluorescently stained for confocal imaging.

3.3 Optical Coherence Tomography-Based Method

Similar to the confocal microscopy indentation method, optical coherence tomography (OCT) was employed for the indentation test in order to image the cross section of indented hydrogel specimens [2, 43, 44]. OCT is an interferometric imaging method that enables 3D and noninvasive imaging of soft materials at a micrometer-level resolution. Therefore, the OCT-based indentation method can measure the indentation depth of a ball indenter into a hydrogel specimen, as well as the thickness and geometry of the specimen. Then, the elastic modulus of the hydrogel sample can be determined based on the obtained dimensions.

3.4 Mesoscale Indentation Method

Because the indentation methods introduced in Sects. 3.1, 3.2, and 3.3 rely on the gravitational force on the ball indenters, they are limited in modulating indentation force and thus local deformation of hydrogels. Instead, a calibrated glass cantilever with a spherical tip of 100–200 μm in diameter was used to indent hydrogels with a controllable indentation force [34, 45, 46, 47], which is similar to the atomic force microscopy (AFM)-based indentation method introduced in Sect. 4.1. Especially, Jacot et al. validated the method based on comparisons with the tensile method using PAAM gels [45]. They found that the elastic modulus measured by the mesoscale indentation method was ~20% lower than that by the tensile test. Similar to the macroscale indentation method, mesoscale indenters with a flat-ended tip were also used [48, 49]. This mesoscale method using a spherically tipped glass indenter is cost-effective compared to the AFM indentation method although the two methods are based on similar working principles.

One rather unique method for mesoscale indentation is to utilize a water jet to locally deform the top surface of a hydrogel specimen. Chevalier et al. proposed the water jet indentation method in which a pressurized water jet flow indented PAAM gels [50]. Because the method exerted a hydrodynamic pressure on an area of 0.05 mm2, which is approximately equivalent to 220 × 220 μm2, it measured the mesoscale elasticity of the tested gels.

3.5 Magnetic Force-Based Method

In addition to being used as the ball indenter for the indentation methods introduced in Sects. 3.1, 3.2, and 3.3, steel balls can be included in a hydrogel specimen to deform the hydrogel locally. Lin et al. embedded a 0.79-mm-diameter steel ball in DNA-cross-linked PAAM gels and manipulated the ball using a magnetic force to deform the gel locally [51, 52]. Then, the elastic modulus of the hydrogel was measured by relating the linear displacement of the ball which was measured using video microscopy, and the magnetic force exerted to the ball. Considering the diameter of the steel ball used, the method measured the local elastic modulus of the hydrogel on the mesoscale. This magnetic-tweezer-like method has an advantage in that it is a nonintrusive method, but removing the embedded sphere can cause damage of gel samples. Also, it requires a calibration step to evaluate the magnetic force applied to the steel ball.

3.6 Pipette Aspiration Method

The pipette aspiration method pinches a small portion of a hydrogel specimen by applying a negative pressure to the sample with a glass micropipette. Then, the Young’s modulus of the hydrogel is determined by a relationship between the applied pressure and the hydrogel’s aspirated length into the pipette. This relationship depends on the Poisson’s ratio and thickness of the gel specimen, and the relative micropipette wall thickness determined by the inner and outer radii of the used pipette [53, 54]. Recently, Buffinton et al. compared the pipette aspiration method (contact area of ~1 mm2) with the macroscale compression method and the mesoscale indentation method (contact area of ~0.1 mm2) using PAAM gels [54]. They observed differences in the measured elastic moduli among these methods, which appears to be because these methods imposed different loading conditions to the hydrogel.

The pipette aspiration method can also be applied to hydrogel beads. Kleinberger et al. aspirated alginate gel beads using micropipette aspiration and calculated the Young’s modulus of the hydrogel spheres using
$$ E=\Delta p\frac{3R}{2\pi l\varphi}, $$
(5)
where Δp is the pressure difference applied across the gel sphere, φ is the wall function depending on the wall thickness of the used pipette, l is the length of the gel portion aspirated into the pipette, and R is the inner radius of the pipette [55]. Equation (5) requires a thin-walled pipette having a small inner diameter compared to the gel bead diameter.

Related to the pipette aspiration method, the pipette-based method of Wyss et al. is noteworthy [56]. They used tapered micro-capillaries to squeeze whole PAAM gel spheres and determined the elastic modulus of the gel based on a relationship between the deformation of the gel spheres and the applied pressure difference across the gel spheres. Their method was validated with the compression method, and the Young’s modulus values measured in the pipette measurement were a little higher than those measured in the bulk measurement.

4 Microscale Measurements

4.1 AFM Indentation Method

Indentation tests using atomic force microscopy (AFM) have been widely used to measure the elastic modulus of various hydrogels [15, 23, 45, 57, 58, 59, 60, 61, 62]. Typically, a hydrogel specimen is indented in a liquid by the sharp or spherical tip of an AFM probe with a predetermined trigger force and probe speed (Fig. 6a). A force-distance curve is recorded and converted to a force-indentation depth curve, and then the Young’s modulus of the hydrogel is estimated by fitting a proper contact mechanics model (e.g., Hertz model and Sneddon model) against the curve (Fig. 6b). Here, the Sneddon model for spherical indenters is
$$ {\displaystyle \begin{array}{l}F=\frac{E}{2\left(1-{v}^2\right)}\left[\left({a}^2+{r}^2\right)\ln \left(\frac{r+a}{r-a}\right)-2 ar\right],\\ {}\delta =\frac{1}{2}a\ln \left(\frac{r+a}{r-a}\right),\end{array}} $$
(6)
and it is known to be more robust for deeper indentations than the Hertz model, Eq. (4) [63, 64, 65]. In this step, AFM post-processing software such as AtomicJ [66] can be used.
Fig. 6

Atomic force microscopy (AFM) indention for elastic modulus measurement of hydrogels. (a) Schematic of the AFM indentation test using a colloidal probe on a hydrogel sample fixed on a cover glass in a liquid (figure not to scale). Once the spherical tip of the AFM probe is engaged on the gel specimen, the probe is lowered to indent the gel and then retracted. Laser beam enables measuring the deflection of the probe. The indentation force from the tip and the indentation depth on the gel are measured from the spring constant, deflection and vertical displacement of the probe. (b) Exemplary force (F)-indentation depth (zd) curve of AFM indentation (PAAM gel of 4% acrylamide and 0.1% bis-acrylamide) [25]. Here, z and d are the vertical displacement and deflection of the AFM probe, respectively. The Hertz model, Eq. (4), was fitted against the approach part of the curve as shown by the red line, and the Young’s modulus of the gel was measured to be E = 2.54 kPa. Indentation condition: probe speed = 1 μm/s, trigger force = 6 nN, nominal spring constant of AFM probe = 0.06 N/m, and tip radius = 5 μm. (c) Repeatability and precision validation test results of the AFM indentation method [25]. Tested specimens are PAAM gels of 3% acrylamide/0.06% bis-acrylamide and 4% acrylamide/0.1% bis-acrylamide. Compared with the rheometric method, the AFM indentation method showed similar repeatability (changes in the average E values) and better precision (a ratio of the standard deviation to the average value of E) than the rheometric method. Error bar: standard deviation (from about 120 indentation curves for the AFM indentation measurement and from more than three measurements for the rheometric measurement)

For reliable application of the AFM indentation method for hydrogel elasticity measurement, it is critical to know how precise, accurate and repeatable the method is (refer to our summary of previous studies on evaluation of the precision and accuracy of AFM indentation [25] and also Ref. [15, 23, 61]). For this purpose, we recently evaluated the precision, accuracy and repeatability of the AFM indentation method by repeating AFM indentation tests on the same PAAM gels and by comparing the AFM indentation results with rheometric measurements (Fig. 6c) [25]. All AFM indentation tests were conducted in a liquid buffer using MFP-3D-BIO AFM (Asylum Research, Santa Barbara, CA) and probes with a glass bead tip (nominal spring constant = 0.06 N/m; nominal tip diameters = 5 and 12 μm). Force mapping was performed on each hydrogel specimen to obtain multiple force-distance curves.

The precision of the AFM indentation method was evaluated using the relative standard deviation (RSD) of the Young’s modulus (E) values from force mapping (RSDE = σE/mE), assuming that the gel samples were homogeneous. Here, mE and σE are the average and standard deviation of the E values, respectively. The measured RSDE was 1.1–4.6%. Because the RSDE of the rheometric measurement was 2.1–20.7%, the AFM indentation method appeared to have better precision.

For accuracy and repeatability evaluation, we measured the spatially averaged E value of each gel sample at random time intervals. The accuracy of the AFM indentation method was quantified by comparing the \( {m}_{\overline{E}} \) value of the method with that of the rheometric method. Here, \( {m}_{\overline{E}} \) is the average value of the mE values from repeated sets of indentation tests. The \( {m}_{\overline{E}} \) ratio of the AFM indentation method to the rheometric method was 0.67–1.19, which indicates that the two methods showed reasonably good agreement.

The repeatability of the AFM indentation method was quantified by using the RSD of the mE values (\( {\mathrm{RSD}}_{\overline{E}}={\sigma}_{\overline{E}}/{m}_{\overline{E}} \)). Here, \( {\sigma}_{\overline{E}} \) is the standard deviation of the mE values. Because the \( {\mathrm{RSD}}_{\overline{E}} \) was measured to be 2.6–6.0%, we found that the AFM indentation method was repeatable for the examined PAAM gel samples. To sum up, we have confirmed that the AFM indentation method is repeatable with good precision when conducted carefully.

It needs to be emphasized that the quality of the AFM indentation tests is dependent on experimental conditions and data analysis methods. Experimental conditions include in situ calibration of the AFM probe spring constant, the geometry and size of the AFM probe tips, sample fixation, and indentation parameters such as probe speed and trigger force. For instance, spherical tips are known to be better for consistent elasticity measurements than sharp tips [42, 65, 67, 68]. Also, it is important to choose a proper contact mechanics model depending on the tip geometry of the AFM probes and the thickness, properties and deformation of the hydrogel samples [60, 69, 70]. Although the AFM indentation method is a versatile tool for probing the mechanical properties of hydrogels, the method is low throughput because of its time consumption, and it can be challenging to determine the contact point between the probe tip and the sample because of their adhesive interactions [71].

4.2 Magnetic Force-Based Method

Similar to the mesoscale method of applying a magnetic force to a sphere embedded in a hydrogel (see Sect. 3.5), Chippada et al. embedded nickel needles with a dimension of 10 × 1 × 1 μm3 in PAAM gels and applied a magnetic force or torque to the needles using a magnetic manipulator [22, 72]. The two-dimensional translations and rotations of the needles were measured using video microscopy, and the forces and torques on the needles were calculated using finite element analysis (FEA) simulations. Therefore, they could measure the local elastic modulus of the PAAM gels on a microscopic scale. It needs to be noted that in this method, the bulk elastic property of hydrogels can be affected by the inclusion of solid microneedles.

4.3 MEMS-Based Method

Microelectromechanical systems (MEMS) technology can be used to measure the elastic modulus of tiny hydrogels. One example is force-feedback MEMS micro-grippers. These grippers could compress micrometer-sized alginate gel beads of 15–25 μm in diameter and thus measure the Young’s modulus of the gel spheres [73]. Another application of MEMS is to measure the elastic property of hydrogels using MEMS resonant sensors [74]. These MEMS-based methods have an advantage in that they can measure the microscopic elasticity of hydrogels in similar ways to the macroscopic methods, but their adaptation may be limited by the fact that the methods require resources and expertise for MEMS fabrication.

5 Conclusion

The elasticity of hydrogels refers to their capability to deform instantly, responding to a mechanical loading, and then to restore upon removal of the loading. This elastic behavior of hydrogels is described by their Young’s modulus, and this elastic modulus is a criterion of selecting a hydrogel material depending on the need. Currently various experimental methods are available for the measurement of hydrogels’ elastic modulus, and this review chapter has categorized them into three different scales: macroscale (1–10 mm order scale), mesoscale (sub-mm order scale), and microscale (10–100 μm order scale). The macroscale methods summarized in Sect. 2 are appropriate for measuring the bulk elastic modulus of homogeneous hydrogel materials, but these methods have drawbacks in that they require separately prepared hydrogel specimens of relatively large volumes, and they cannot measure the elastic modulus distribution of hydrogels. In contrast, the microscale methods reviewed in Sect. 4 are advantageous in that they can probe the local elastic modulus at various points on an inhomogeneous hydrogel specimen of relatively small volume, and hydrogel specimens can be kept hydrated easily in these methods. However, these microscale methods usually require complicated equipment of high cost. The mesoscale methods introduced in Sect. 3 fill the spatial scale between the macroscale methods and the microscale methods with reasonable requirements. Regardless of the pros and cons of these methods, however, it needs to be kept in mind that one should choose an appropriate elastic modulus measurement method depending on the application, dimension and expected property of the hydrogel, and that it is ideal to measure the elastic modulus of the hydrogels under conditions that are as similar as possible to the in situ condition of their application.

Notes

Acknowledgments

We acknowledge supports from the Nebraska Tobacco Settlement Biomedical Research Development Fund through (1) Bioengineering for Human Health Grant of the University of Nebraska-Lincoln (UNL) and the University of Nebraska Medical Center (UNMC) and (2) Biomedical Research Seed Grant of UNL. AFM measurements were performed at the NanoEngineering Research Core Facility of UNL, which is partially funded from Nebraska Research Initiative Funds.

References

  1. 1.
    Anseth KS, Bowman CN, Brannon-Peppas L (1996) Mechanical properties of hydrogels and their experimental determination. Biomaterials 17:1647–1657CrossRefPubMedGoogle Scholar
  2. 2.
    Ahearne M, Yang Y, Liu K-K (2008) Mechanical characterization of hydrogels for tissue engineering application. Topics in tissue Engineering 4:1–16Google Scholar
  3. 3.
    Oyen ML (2014) Mechanical characterization of hydrogel materials. Int Mater Rev 59:44–59CrossRefGoogle Scholar
  4. 4.
    Vedadghavami A, Minooei F, Mohammadi MH, Khetani S, Rezaei A, Mashayekhan S, Sanati-Nezhad A (2017) Manufacturing of hydrogel biomaterials with controlled mechanical properties for tissue engineering applications. Acta Biomater 62:42–63CrossRefPubMedGoogle Scholar
  5. 5.
    Schramm-Baxter J, Katrencik J, Mitragotri S (2004) Jet injection into polyacrylamide gels: investigation of jet injection mechanics. J Biomech 37:1181–1188CrossRefPubMedGoogle Scholar
  6. 6.
    Buckley CT, Thorpe SD, O’Brien FJ, Robinson AJ, Kelly DJ (2009) The effect of concentration, thermal history and cell seeding density on the initial mechanical properties of agarose hydrogels. J Mech Behav Biomed Mater 2:512–521CrossRefPubMedGoogle Scholar
  7. 7.
    Delaine-Smitha RM, Burneya S, Balkwillb FR, Knight MM (2016) Experimental validation of a flat punch indentation methodology calibrated against unconfined compression tests for determination of soft tissue biomechanics. J Mech Behav Biomed Mater 60:401–415CrossRefGoogle Scholar
  8. 8.
    Lee KY, Mooney DJ (2012) Alginate: properties and biomedical applications. Prog Polym Sci 37:106–126CrossRefPubMedPubMedCentralGoogle Scholar
  9. 9.
    Boontheekul T, Kong H-J, Mooney DJ (2005) Controlling alginate gel degradation utilizing partial oxidation and bimodal molecular weight distribution. Biomaterials 26:2455–2465CrossRefPubMedGoogle Scholar
  10. 10.
    Ouwerx C, Velings N, Mestdagh MM, Axelos MAV (1998) Physico-chemical properties and rheology of alginate gel beads formed with various divalent cations. Polym Gels Netw 6:393–408CrossRefGoogle Scholar
  11. 11.
    Chan E-S, Lim T-K, Voo W-P, Pogaku R, Tey BT, Zhang Z (2011) Effect of formulation of alginate beads on their mechanical behavior and stiffness. Particuology 9:228–234CrossRefGoogle Scholar
  12. 12.
    Normand V, Lootens DL, Amici E, Plucknett KP, Aymard P (2000) New insight into agarose gel mechanical properties. Biomacromolecules 1:730–738CrossRefPubMedGoogle Scholar
  13. 13.
    Brujan E-A, Nahen K, Schmidt P, Vogel A (2001) Dynamics of laser-induced cavitation bubbles near an elastic boundary. J Fluid Mech 433:251–281CrossRefGoogle Scholar
  14. 14.
    Roeder BA, Kokini K, Sturgis JE, Robinson JP, Voytik-Harbin SL (2002) Tensile mechanical properties of three-dimensional type I collagen extracellular matrices with varied microstructure. J Biomech Eng 124:214–222CrossRefPubMedGoogle Scholar
  15. 15.
    Engler A, Bacakova L, Newman C, Hategan A, Griffin M, Discher D (2004) Substrate compliance versus ligand density in cell on gel responses. Biophys J 86:617–628CrossRefPubMedPubMedCentralGoogle Scholar
  16. 16.
    Peyton SR, Putnam AJ (2005) Extracellular matrix rigidity governs smooth muscle cell motility in a biphasic fashion. J Cell Physiol 204:198–209CrossRefPubMedGoogle Scholar
  17. 17.
    Sun J-Y, Zhao X, Illeperuma WRK, Chaudhuri O, Oh KH, Mooney DJ, Vlassak JJ, Suo Z (2012) Highly stretchable and tough hydrogels. Nature 489:133–136CrossRefPubMedPubMedCentralGoogle Scholar
  18. 18.
    Gross W, Kress H (2017) Simultaneous measurement of the Young’s modulus and the Poisson ratio of thin elastic layers. Soft Matter 13:1048–1055CrossRefPubMedGoogle Scholar
  19. 19.
    Wong JY, Velasco A, Rajagopalan P, Pham Q (2003) Directed movement of vascular smooth muscle cells on gradient-compliant hydrogels. Langmuir 19:1908–1913CrossRefGoogle Scholar
  20. 20.
    Kong HJ, Wong E, Mooney DJ (2003) Independent control of rigidity and toughness of polymeric hydrogels. Macromolecules 36:4582–4588CrossRefGoogle Scholar
  21. 21.
    Janmey PA, Georges PC, Hvidt S (2007) Basic rheology for biologists. In: Wang Y-L, Discher DE (eds) Cell mechanics. Academic, San Diego, pp 3–27Google Scholar
  22. 22.
    Chippada U, Yurke B, Langrana NA (2010) Simultaneous determination of Young’s modulus, shear modulus, and Poisson’s ratio of soft hydrogels. J Mater Res 25:545–555CrossRefGoogle Scholar
  23. 23.
    Hammond NA, Kamm RD (2013) Mechanical characterization of self-assembling peptide hydrogels by microindentation. J Biomed Mater Res B 101:981–990CrossRefGoogle Scholar
  24. 24.
    Lee D, Rahman MM, Zhou Y, Ryu S (2015) Three-dimensional confocal microscopy indentation method for hydrogel elasticity measurement. Langmuir 31:9684–9693CrossRefPubMedGoogle Scholar
  25. 25.
    Lee D, Ryu S (2017) A validation study of the repeatability and accuracy of atomic force microscopy indentation using polyacrylamide gels and colloidal probes. J Biomech Eng 139:044502CrossRefGoogle Scholar
  26. 26.
    Kandow CE, Georges PC, Janmey PA, Beningo KA (2007) Polyacrylamide hydrogels for cell mechanics: steps toward optimization and alternative uses. In: Wang Y-L, Discher DE (eds) Cell mechanics. Academic, San Diego, pp 29–46CrossRefGoogle Scholar
  27. 27.
    Tse JR, Engler AJ (2010) Preparation of hydrogel substrates with tunable mechanical properties. Curr Protoc Cell Biol 16:10–16Google Scholar
  28. 28.
    Ross KA, Scanlon MG (1999) Analysis of the elastic modulus of agar gel by indentation. J Texture Stud 30:17–27CrossRefGoogle Scholar
  29. 29.
    Peng X, Huang J, Qin L, Xiong C, Fang J (2009) A method to determine Young’s modulus of soft gels for cell adhesion. Acta Mech Sin 25:565–570CrossRefGoogle Scholar
  30. 30.
    Lo C-M, Wang H-B, Dembo M, Wang Y-L (2000) Cell movement is guided by the rigidity of the substrate. Biophys J 79:144–152CrossRefPubMedPubMedCentralGoogle Scholar
  31. 31.
    Munevar S, Wang Y-L, Dembo M (2001) Traction force microscopy of migrating normal and H-ras transformed 3T3 fibroblasts. Biophys J 80:1744–1757CrossRefPubMedPubMedCentralGoogle Scholar
  32. 32.
    Reinhart-King CA, Dembo M, Hammer DA (2003) Endothelial cell traction forces on RGD-derivatized polyacrylamide substrata. Langmuir 19:1573–1579CrossRefGoogle Scholar
  33. 33.
    Damljanovic V, Lagerholm BC, Jacobson K (2005) Bulk and micropatterned conjugation of extracellular matrix proteins to characterized polyacrylamide substrates for cell mechanotransduction assays. Biotechniques 39:847–851CrossRefPubMedGoogle Scholar
  34. 34.
    Frey MT, Engler A, Discher DE, Lee J, Wang YL (2007) Microscopic methods for measuring the elasticity of gel substrates for cell culture: microspheres, microindenters, and atomic force microscopy. In: Wang Y-L, Discher DE (eds) Cell mechanics. Academic, San Diego, pp 47–65CrossRefGoogle Scholar
  35. 35.
    Long R, Hall MS, Wu M, Hui C-Y (2011) Effects of gel thickness on microscopic indentation measurements of gel modulus. Biophys J 101:643–650CrossRefPubMedPubMedCentralGoogle Scholar
  36. 36.
    D’Sa DJ, de Juan Pardo EM, de las Rivas Astiz R, Sen S, Kumar S (2009) High-throughput indentational elasticity measurements of hydrogel extracellular matrix substrates. Appl Phys Lett 95:063701CrossRefPubMedCentralGoogle Scholar
  37. 37.
    Peng X, Huang J, Deng H, Xiong C, Fang J (2011) A multi-sphere indentation method to determine Young’s modulus of soft polymeric materials based on the Johnson–Kendall–Roberts contact model. Meas Sci Technol 22:027003CrossRefGoogle Scholar
  38. 38.
    Kuznetsova TG, Starodubtseva MN, Yegorenkov NI, Chizhik SA, Zhdanov RI (2007) Atomic force microscopy probing of cell elasticity. Micron 38:824–833CrossRefPubMedGoogle Scholar
  39. 39.
    Yoffe EH (1984) Modified Hertz theory for spherical indentation. Philos Mag A 50:813–828CrossRefGoogle Scholar
  40. 40.
    Mahaffy RE, Shih CK, MacKintosh FC, Käs J (2000) Scanning probe-based frequency-dependent microrheology of polymer gels and biological cells. Phys Rev Lett 85:880–883CrossRefPubMedGoogle Scholar
  41. 41.
    Mahaffy RE, Park S, Gerde E, Käs J, Shih CK (2004) Quantitative analysis of the viscoelastic properties of thin regions of fibroblasts using atomic force microscopy. Biophys J 86:1777–1793CrossRefPubMedPubMedCentralGoogle Scholar
  42. 42.
    Dimitriadis EK, Horkay F, Maresca J, Kachar B, Chadwick RS (2002) Determination of elastic moduli of thin layers of soft material using the atomic force microscope. Biophys J 82:2798–2810CrossRefPubMedPubMedCentralGoogle Scholar
  43. 43.
    Yang Y, Bagnaninchi PO, Ahearne M, Wang RK, Liu K-K (2007) A novel optical coherence tomography-based micro-indentation technique for mechanical characterization of hydrogels. J R Soc Interface 4:1169–1173CrossRefPubMedPubMedCentralGoogle Scholar
  44. 44.
    Lee SJ, Sun J, Flint JJ, Guo S, Xie HK, King MA, Sarntinoranont M (2011) Optically based-indentation technique for acute rat brain tissue slices and thin biomaterials. J Biomed Mater Res B 97:84–95CrossRefGoogle Scholar
  45. 45.
    Jacot JG, Dianis S, Schnall J, Wong JY (2006) A simple microindentation technique for mapping the microscale compliance of soft hydrated materials and tissues. J Biomed Mater Res A 79:485–494CrossRefPubMedGoogle Scholar
  46. 46.
    Frey MT, Wang Y-L (2009) A photo-modulatable material for probing cellular responses to substrate rigidity. Soft Matter 5:1918–1924CrossRefPubMedPubMedCentralGoogle Scholar
  47. 47.
    Beekmans SV, Iannuzzi D (2016) Characterizing tissue stiffness at the tip of a rigid needle using an opto-mechanical force sensor. Biomed Microdevices 18:15CrossRefPubMedPubMedCentralGoogle Scholar
  48. 48.
    Saxena T, Gilbert JL, Hasenwinkel JM (2009) A versatile mesoindentation system to evaluate the micromechanical properties of soft, hydrated substrates on a cellular scale. J Biomed Mater Res A 90:1206–1217CrossRefPubMedGoogle Scholar
  49. 49.
    Levental I, Levental KR, Klein EA, Assoian R, Miller RT, Wells RG, Janmey PA (2010) A simple indentation device for measuring micrometer-scale tissue stiffness. J Phys Condens Matter 22:194120CrossRefPubMedPubMedCentralGoogle Scholar
  50. 50.
    Chevalier NR, Dantan P, Gazquez E, Cornelissen AJM, Fleury V (2016) Water jet indentation for local elasticity measurements of soft materials. Eur Phys J E 39:10CrossRefPubMedGoogle Scholar
  51. 51.
    Lin DC, Yurke B, Langrana NA (2004) Mechanical properties of a reversible, DNA-crosslinked polyacrylamide hydrogel. J Biomech Eng 126:104–110CrossRefPubMedGoogle Scholar
  52. 52.
    Lin DC, Yurke B, Langrana NA (2005) Inducing reversible stiffness changes in DNA-crosslinked gels. J Mater Res 20:1456–1464CrossRefGoogle Scholar
  53. 53.
    Boudou T, Ohayon J, Picart C, Tracqui P (2006) An extended relationship for the characterization of Young’s modulus and Poisson’s ratio of tunable polyacrylamide gels. Biorheology 43:721–728PubMedGoogle Scholar
  54. 54.
    Buffinton CM, Tong KJ, Blaho RA, Buffinton EM, Ebenstein DM (2015) Comparison of mechanical testing methods for biomaterials: pipette aspiration, nanoindentation, and macroscale testing. J Mech Behav Biomed Mater 51:367–379CrossRefPubMedGoogle Scholar
  55. 55.
    Kleinberger RM, Burke NAA, Dalnoki-Veress K, Stöver HDH (2013) Systematic study of alginate-based microcapsules by micropipette aspiration and confocal fluorescence microscopy. Mater Sci Eng C 33:4295–4304CrossRefGoogle Scholar
  56. 56.
    Wyss HM, Franke T, Mele E, Weitz DA (2010) Capillary micromechanics: measuring the elasticity of microscopic soft objects. Soft Matter 6:4550–4555CrossRefGoogle Scholar
  57. 57.
    Engler AJ, Richert L, Wong JY, Picart C, Discher DE (2004) Surface probe measurements of the elasticity of sectioned tissue, thin gels and polyelectrolyte multilayer films: correlations between surface substrate stiffness and cell adhesion. Surf Sci 570:142–154CrossRefGoogle Scholar
  58. 58.
    Radmacher M (2007) Studying the mechanics of cellular processes by atomic force microscopy. In: Wang Y-L, Discher DE (eds) Cell mechanics. Academic, San Diego, pp 347–372CrossRefGoogle Scholar
  59. 59.
    Engler AJ, Rehfeldt F, Sen S, Discher DE (2007) Microtissue elasticity: measurements by atomic force microscopy and its influence on cell differentiation. In: Wang Y-L, Discher DE (eds) Cell mechanics. Academic, San Diego, pp 521–545CrossRefGoogle Scholar
  60. 60.
    Markert CD, Guo X, Skardal A, Wang Z, Bharadwaj S, Zhang Y, Bonin K, Guthold M (2013) Characterizing the micro-scale elastic modulus of hydrogels for use in regenerative medicine. J Mech Behav Biomed Mater 27:115–127CrossRefPubMedGoogle Scholar
  61. 61.
    Li C, Allen J, Alliston T, Pruitt LA (2011) The use of polyacrylamide gels for mechanical calibration of cartilage – a combined nanoindentation and unconfined compression study. J Mech Behav Biomed Mater 4:1540–1547CrossRefPubMedPubMedCentralGoogle Scholar
  62. 62.
    Ondeck MG, Engler AJ (2016) Mechanical characterization of a dynamic and tunable methacrylated hyaluronic acid hydrogel. J Biomech Eng 138:021003CrossRefPubMedGoogle Scholar
  63. 63.
    Sneddon IN (1965) The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int J Eng Sci 3:47–57CrossRefGoogle Scholar
  64. 64.
    Heuberger M, Dietler G, Schlapbach L (1995) Mapping the local Young’s modulus by analysis of the elastic deformations occurring in atomic force microscopy. Nanotechnology 6:12–23CrossRefGoogle Scholar
  65. 65.
    Puricelli L, Galluzzi M, Schulte C, Podestà A, Milani P (2015) Nanomechanical and topographical imaging of living cells by atomic force microscopy with colloidal probes. Rev Sci Instrum 86:033705CrossRefPubMedGoogle Scholar
  66. 66.
    Hermanowicz P, Sarna M, Burda K, Gabryś H (2014) AtomicJ: an open source software for analysis of force curves. Rev Sci Instrum 85:063703CrossRefPubMedGoogle Scholar
  67. 67.
    Carl P, Schillers H (2008) Elasticity measurement of living cells with an atomic force microscope: data acquisition and processing. Pflugers Arch 457:551–559CrossRefPubMedGoogle Scholar
  68. 68.
    Harris AR, Charras GT (2011) Experimental validation of atomic force microscopy-based cell elasticity measurements. Nanotechnology 22:345102CrossRefPubMedGoogle Scholar
  69. 69.
    Lin DC, Horkay F (2008) Nanomechanics of polymer gels and biological tissues: a critical review of analytical approaches in the Hertzian regime and beyond. Soft Matter 4:669–682CrossRefGoogle Scholar
  70. 70.
    McKee CT, Last JA, Russell P, Murphy CJ (2011) Indentation versus tensile measurements of Young’s modulus of soft biological tissues. Tissue Eng B 17:155–164CrossRefGoogle Scholar
  71. 71.
    Lin DC, Dimitriadis EK, Horkay F (2007) Robust strategies for automated AFM force curve analysis-I. Non-adhesive indentation of soft, inhomogeneous materials. J Biomech Eng 129:430–440CrossRefPubMedGoogle Scholar
  72. 72.
    Chippada U, Yurke B, Georges PC, Langrana NA (2008) A nonintrusive method of measuring the local mechanical properties of soft hydrogels using magnetic microneedles. J Biomech Eng 31:021014Google Scholar
  73. 73.
    Kim K, Liu X, Zhang Y, Cheng J, Wu XY, Sun Y (2009) Elastic and viscoelastic characterization of microcapsules for drug delivery using a force-feedback MEMS microgripper. Biomed Microdevices 11:421–427CrossRefPubMedGoogle Scholar
  74. 74.
    Corbin EA, Millet LJ, Pikul JH, Johnson CL, Georgiadis JG, King WP, Bashir R (2013) Micromechanical properties of hydrogels measured with MEMS resonant sensors. Biomed Microdevices 15:311–319CrossRefPubMedGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical and Materials EngineeringLincolnUSA
  2. 2.Nebraska Center for Materials and Nanoscience, University of Nebraska-LincolnLincolnUSA

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