UNIVERSITY OF CINCINNATI

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1 UNIVERSITY OF CINCINNATI Date: I,, hereby submit this work as part of the requirements for the degree of: in: It is entitled: This work and its defense approved by: Chair:

2 Extraction of Non-Linear Material Properties of Bio-gels using Atomic Force Microscopy By Sakyasingh Tripathy B.Tech., Indian Institute of Technology-Kanpur India, 2002 Thesis Submitted to Division of Research and Advanced Studies of the University of Cincinnati for the in partial fulfillment of the requirements for the degree of Masters in Science in the Department of Mechanical, Industrial and Nuclear Engineering of the College of Engineering, 2005 Committee Chair Dr. EJ Berger Dr. K Vemaganti Cincinnati, Ohio

3 ABSTRACT There is growing evidence of the importance of mechanical deformations on various facets of cell functioning. This asks for a proper understanding of the cell s characteristics as a mechanical system in different physiological and physical loading conditions. Many researchers use atomic force microscopy (AFM) and the Hertz contact model for elastic material property identification under shallow indentation. For larger indentations, many of the Hertz assumptions are not inherently satisfied and the Hertz model is not directly useful for characterizing nonlinear material properties. We have developed a parameter identification approach for hyperelastic materials property determination from deep indentation data. The approach is chalked out using finite element (FE) reconstructions of the indentation tests. The results suggest that our approach to identifying exponential hyperelastic material parameters based upon large-strain correction to the Hertz model is a viable strategy for interpreting deep indentation AFM data. The extraction method is tested on the experimental AFM data, and the values of the parameter of agarose gels which is our model system are extracted. Viscoelastic and poroelastic material models are used to explain the absence of any time varying characteristics in the AFM data. Analytical solutions for few phenomenological and mechanical models are also presented.

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5 ACKNOWLEDGEMENTS I would like to take this opportunity to express my gratitude to all those involved in the smooth completion of this thesis. This thesis is a result of diligent effort of not only myself but my advisers also. I would take this opportunity to thank Dr. Edward Berger and Dr. Kumar Vemaganti for their invaluable support, guidance and direction. Dr. Hong You has been instrumental in inculcating experimental expertise in me. For his patience and willingness to acknowledge all my naivé questions, I owe him overwhelming debt of gratitude. I gratefully acknowledge the financial support of the National Science Foundation, award number CMS , program manager Dr. Ken Chong. I also wish to thank the faculty at Mechanical Engineering Department of University of Cincinnati for their confidence in my abilities. I would also like to express my gratitude to my colleagues for the many invaluable discussions and constant help. Lastly, I am indebted to my family and friends whose unconditional support and encouragement has kept me motivated and had led to the successful completion of this thesis.

6 Table of Contents List of Tables v List of Figures vi 1 Motivation Cell Biomechanics at a Small Length Scales Current Techniques Atomic Force Microscope Mechanics of indentation using AFM - The Hertz Model Non-Hertzian Effects Material Systems Research Objectives Introduction Atomic Force Microscopy: A tool for indentation studies History of AFM Parts of a Typical AFM i

7 2.1.3 Operating Principle Controller Electronics Applications of AFM Tribology Applications Biological Applications Materials and Methods Experimental Setup Specimen Preparation Confined Compression Apparatus Data Interpretation Elastic Modulus Estimation Non-linear Material Properties Viscoelastic Materials Phenomenological Viscoelastic Models Viscoelastic response in 3-d Poroelastic Models - Continuum Porous Elastic Model and Biphasic Model Hyperelastic Materials Neo-Hookean Material Polynomial Material Model Modified Mooney Rivlin Material ii

8 Exponential Model Finite Element Model for the AFM Contact Problem Finite Element Model for the Confined Compression Problem Material Modeling Linear Viscoelastic Model Continuum Porous Elastic Model Exponential Hyperelastic Model Results and Discussion Phenomenological Viscoelastic Material Model Creep Load Step Load Triangular Load Unconfined Compression in 3 d Confined Compression in 3 d Viscoelastic Finite Element Model Poroelastic Material Model Hyperelastic Material Model Effects of the parameters b 1 and b Representation of the function g Extracting b 1 and b 2 using the logarithmic g Extracting b 1 and b 2 using a polynomial expression for g iii

9 4.7 The Extraction Process Applied to Experimental Data Summary Conclusions and Future Work Conclusions Future Work Experiments Finite Element Analysis References 118 A AFM Terms and Operating Guidelines 119 A.1 Important Terms A.2 Evaluating Noise Level B Useful Abaqus and Matlab codes 123 B.1 Abaqus Input Files B.1.1 UHYPER.for B.2 Matlab Codes B.2.1 Running the FE Simulations using ABAQUS C Boltzmann Superposition Principle 127 iv

10 List of Tables 3.1 The instantaneous and equilibrium modulus of the viscoelastic models The input function for different loading conditions The strain response for Maxwell model The strain response for Kelvin model The strain response for SLS Model The strain response for Zener model The Material parameters Solutions for unconfined compression using elastic dilation and viscoelastic distortion Error in the extracted values of b 1 and b 2 using a polynomial function for g Extracted values of E, b 1 and b 2 from the AFM data v

11 List of Figures 1.1 Hertz contact model Scanning Tunneling Microscope: Working Principle Details of the optical head of the AFM The orientation of the photodiode Various types of tip holders The arrangement of piezos in the scanner Standard cantilevers Working Principle of the AFM AFM setup at the Structural Engineering Laboratory AFM image of 1% agarose gel Experimental setup for confined compression The raw data obtained from the confined compression tests AFM indentation data in f z Force-displacement data extracted from the raw f z plot A rate sensitive material vi

12 3.8 Various phenomenological viscoelastic material models Comparison between various hyperelastic material models The finite element model for AFM indentation The finite element model for the confined compression tests Viscoelastic response to creep load Viscoelastic response to creep load Viscoelastic response to triangular load Viscoelastic response for unconfined compression The variance of Poisson s ratio for viscoelastic models with respect to time Strain response for various models under confined compression Comparison of FE simulations with the experimental data Stress distribution for poroelastic FE model The time steps used in the FE poroelastic model simulations Comparison of FE poroelastic model with the AFM f δ data Extracted time constants for various permeability The Displacement Field of AFM indentation Indentation dependence of local stiffness Softening behavior for lower values of b Hardening behavior for lower values of b Transition from softening to hardening behavior. b Curve Fit to the Indentation Dependent Stiffness vii

13 4.18 The function g for various values of b 1 and R Substrate effects for large indenter radius The function g Vs. b 2 for R = 250nm The function q Vs. b Extracted Values of b 1 and b 2 compared to the actual values:set Extracted Values of b 1 and b 2 compared to the actual values:set Extracted Values of b 1 and b 2 using a polynomial function for g:set Extracted Values of b 1 and b 2 using a polynomial function for g:set f δ behavior of the experimental data in logarithmic scale f δ behavior of the inverse FE problem E-δ behavior of the inverse FE problem Comparison of the local stiffness of the inverse FE solution with the data generated using different contact point in the raw AFM f z file Different load profiles for future AFM experiments Qualitative variance of b 1 and b 2 with the choice of contact point viii

14 Chapter 1 Motivation 1.1 Cell Biomechanics at a Small Length Scales The field of Cellular Biomechanics which studies the responses of cellular systems to different mechanical environment, has grown leaps and bounds owing to the advancement of innovative technologies and smart instruments that can measure physical properties of minute biological systems like the cells. Although, it is yet to reach its full potential as a consistent contributor to the general improvement of human condition, it has one of the brightest future. The primary reason for the slow progress in this field is that biological systems are extremely complicated and also the physical phenomena occur at sub-atomic scale. But with massive innovations in the instrumentation industry, biologists are now able to fathom some of the intricacies involved. Because of the fact that biological systems often have complex geometries and loading conditions, new computational approaches also have to be formulated to solve the problems in an application oriented perspective. Please refer to Humphrey et. 1

15 al. [1] and Zhu et.al. [2] for detailed review on this topic. 1.2 Current Techniques For the proper appreciation of the cell-mechanics, we need to have a clear understanding of its inherent mechanical behavior. Learning about the material properties of the cell is the first and obvious step in that direction. Researchers around the world have used various techniques to measure physical quantities and developed analytical solutions for problems to explain the experimental results. One of the most popular technique to quantify the mechanics of the cells is micropipette aspiration, which is primarily used for measuring the viscoelastic properties [3, 4]. Analytical solutions for the micropipette problem [5] have also been extensively studied. Cell poking [6, 7] is another technique which is used to extract the viscoelastic properties of the cell. With this method the force required to indent cells attached to a glass coverslip is determined and used to find the various properties. Forces on the order of tens of nanonewtons and indentations on the order of one micron can be detected by cell poking method [8]. Chen et. al. developed a magnetic twisting cytometry technique in which mechanical stress is applied on a single cell surface receptor using ligand-coated microbeads, and the cellular mechanical responses are measured [9]. Another approach to the problem is the use of optical tweezers, where small silica beads are used to apply calibrated forces on the cell surface and the cell deformations are monitored [10]. These two bead manipulations techniques (micropipette aspiration and optical tweezers) were compared by Laurent et. al. to find a slight difference in the predicted young s mod- 2

16 ulus while both predicted similar relaxation time constants [11]. Atomic Force Microscope [12, 13] is substantially used in the literature to extract the elastic and low strain mechanical properties of cells. 1.3 Atomic Force Microscope AFM (Chapter 2) is mostly used for imaging of the cells as well as obtaining force indentation data from cells. A plethora of work has been done using AFM to characterize the material properties. Radmacher et. al. were one of the earliest to report the use of AFM indentations to understand the material characteristics of the sample [14]. They used the classical Hertz contact theory [15] for extracting the relative local elastic properties and viscoelastic models (Voigt) to explain the time-dependent characteristics. The hertzian mechanics has been mostly used to explain the contact between the cantilever tip and the sample surface, and also used to extract the elastic modulus of the sample. Domke et. al. used AFM with standard silicon nitride cantilevers to extract the Young s modulus of gelatin films [16]. Using the Hooke s Law for the cantilever deflection and the hertzian contact mechanics to explain the deformation of the sample, they calculated the Young s modulus of the sample (Sec ). They also concluded that for thin films ( 1 µm), it was possible to determine the elastic properties with a thickness down to 50 nm and a Young s modulus of 20 kpa, without any substrate effects. Relative local stiffness of MDCK strain II epithelial cells using a spherical indenter was done by Hassan el. al. [17]. They developed the FIEL (force integration to equal limits) 3

17 theory to extract an elasticity map of the cells. When the AFM is used in the relative trigger mode, the force applied to the sample is maintained at a constant value. The FIEL theory uses this idea along with with relative work done at different points to get a map of relative stiffness, which is independent of cantilever spring constant, the cantilever deflection drifts and most importantly the contact point. The choice of contact point in the d-z plot extracted from the AFM is extremely critical [18]. It was also proposed to estimate the contact point as one of the parameters along with the Young s modulus [19]. The force modulation technique in AFM also has been extensively used for imaging viscoelasticity of the cells [20, 21]. To conclude, extensive work has been done using the infinitesimal contact theories to extract elastic properties of the cells and soft tissues and hydro-gels [12, 13, 22]. 1.4 Mechanics of indentation using AFM - The Hertz Model The indentation mechanics is explained using the Hertz contact theory. The assumptions in this theory [15] are 1. The bodies are regarded as an isotropic homogeneous incompressible elastic half space. 2. The contact between the non-confirming surfaces is frictionless, so that only normal pressure is transmitted between them. 3. The strains near the contact area are sufficiently small to lie within the scope of the linear theory of elasticity. 4

18 undeformed surface E, ν F R a δ h x rigid substrate z Figure 1.1: Hertz contact model. F is the normal force; a is the contact area; R is the radius of the indenter; δ is the indentation depth. The Hertz model is definitive when h. Source: [23] 4. It is also assumed that the indenter is rigid, with absolutely no deformation. Fig. 1.1 details the indentation problem. For the Hertzian case, h. While the indenter is assumed to be rigid, the sample s Young s modulus is E and poisson s ratio is ν. Using these assumptions and notations for the axisymmetric indentation problem we have, ( ) 1/3 3F R a = (1.1) 4E δ = a2 R (1.2) where, E = E 1 ν 2. Rearranging the above equations we arrive at the force-indentation (f δ) expression, f = 4 3 E Rδ 3/2 (1.3) 5

19 Since we are considering the sample to be incompressible, we have ν = 0.5 which makes E = 4E 3 and eqn. 1.3 as, f = 16 9 E Rδ 3/2 (1.4) This expression is used in section to extract the Young s modulus from the AFM indentation data Non-Hertzian Effects Most of the assumptions enlisted above for the Hertz model are not exactly satisfied in case of the AFM indentation experiments. The important approximation which is normally violated is that of the thickness. If thin films of sample are used, a hardening behavior is observed because of the hard substrate. Hayes et. al. [24] developed a thin layer correction to the Hertz theory with application to cartilage macroscale indentation tests, while Dimitriadis et al. [18] corrects the contact pressure profile while assuming the Hertzian contact size to be accurate. We will be overcoming this limitation by using thick sample ( 1mm) in comparison with the indenter diameter ( 1µm). The assumption of the linear elasticity is also violated as most of the biomaterials mostly exhibit a high degree of non-linearity. This is one of the areas where this thesis aims to contribute-explaining the AFM indentations using non-hertzian or pseudo-hertzian theories. Few researchers have reported the use of hyperelasticity to explain the indentation tests. Costa et. al. applied finite element method to examine large indentations, relative to the sample thickness and probe dimensions, used conical indenter and nonlinear material properties. 6

20 They considered three different types of incompressible hyperelastic materials to examine the responses: ψ = C 1 (I 1 3) + C 2 (I 2 3) (1.5) ψ = D 1 (I 1 3) + D 2 (I 2 3) 2 (1.6) ψ = B 1 (e B 2(I 1 3) 1) (1.7) where, ψ is the strain energy density; I 1 and I 2 are the Cauchy-Green invariants [25]. The material characteristic equation is independent of the third invariant I 3 because the material is considered incompressible. These three strain energy function characterize the Mooney-Rivlin, simple isotropic polynomial and exponential models, which are discussed in a greater detail later (Chapter 3). They compared these three material models with the infinitesimal behavior and found that E MR = 4(C 1 + C 2 ) (1.8) E poly = 4D 1 (1.9) E exp = 4B 1 B 2 (1.10) were the small strain elastic modulus for the models [26]. Similar strategy was also used by Simha et. al.[27], where they simulated finite element (FE) models of indentations using the hyperelastic material models (Eqns ), maintaining similar infinitesimal behavior (setting the effective modulus identical for all the cases). They used a conical indenter for 7

21 their FE studies which is the idealization of the typical commercial AFM probes. They studied the effect of the different material parameters (specific to the models: B 1, B 2 or C 1, C 2 or D 1, D 2 ) on the material behavior. Considering the dearth of research articles on the use of non-hertzian mechanics to interpret the large-stretch ratio (δ/r 1) AFM indentation data, this thesis s main objective is to contribute to the development of non-hertzian material properties characterization using AFM indentation data. Finite element method is used to examine the large scale deformations using different material models and are compared with the experimental findings. Finally, a methodology to isolate material parameters both linear and non-linear from AFM indentation data, depending on the length scale is formulated. 1.5 Material Systems Two kind of material systems are considered: cells and biogels. Cells like T3T and NRK fibroblasts [12, 28] studied under the AFM are frequently found in the literature. Other kinds of cells like NIH3T3 (mouse fibroblasts) [20] and living MDCK epithelial cells [17] have been also used for AFM imaging and indentation experiments. Another kind of material that is extensively studied using AFM is soft hydrogels like PVA [18]. Here we have used agarose gel for our all of analysis. Agarose is a polysaccharide extracted from marine red algae and consists of β-1,3 linked D-galactose and α-1,4 linked 3,6-anhydro-αL-galactose residues [29]. Agarose is assumed to be chemically and electrically neutral. In the hot solution state, agarose chains are stiff and 8

22 disordered. On cooling below around 40 C, the coils order to form helices that subsequently aggregate into thick bundles [30]. Agarose gels consist of thick bundles of agarose chains and large pores of water. Agarose gel has been commonly used for the cell culture of cartilage and soft tissue. It has also used as a phantom material for material property characterization. Moreover, the modulus of these gels ( 10 50kP a) are in the same range or a pinch stiffer than the soft tissues or cells. Agarose serves as an excellent model system for us to use while working out the analytical/numerical model and the experimental protocols. 1.6 Research Objectives The following are the principal objectives of this thesis To generate a substantial amount of indentation data on agarose gels using the AFM. The topographical images will be obtained using the tapping mode (Sec ), while the force deformations (f δ) generated using contact mode (Sec ). Spherical indenter will be used for generating the f δ data, while the standard V-shaped 200 µm long cantilever with pyramidal tips will be used for imaging. A parameter identification methodology will be developed to extract nonlinear material parameters from AFM indentation data. Exponential hyperelastic material model will be used to explain the large strain behavior. The method will be exercised on numerical data to illustrate its performance across a wide range of material parameters, with the emphasis being on soft materials with Young s modulus on the order of kpa. 9

23 Generate finite element (FE) simulation data simulating the AFM experiments. This data will be compared with the experimental findings. The commercial available software ABAQUS [31] is used for all the FE simulations. An inverse FE problem is solved using the material parameters extracted from the AFM indentation data. This will be a test for the parameter extraction methodology. Viscoelastic and poroelastic material models would be used to predict the extent of time-dependent material characteristics in the AFM indentation experiments. 10

24 Chapter 2 Introduction 2.1 Atomic Force Microscopy: A tool for indentation studies The Atomic Force Microscope (AFM) is one of the more versatile descendant of the Scanning Probe Microscope (SPM) family. Since its invention [32], the AFM has become an increasingly important tool for morphological, physical, and chemical characterization of surfaces at sub-nanometer resolution and is now employed in numerous fields like electronics, biomechanics, the chemical industry etc. The primary difference between the AFM and other classical forms of microscopy is the fact the AFM does the imaging by feeling (mechanical interaction with the surface) rather than seeing. This ability to interact with the sample surface mechanically, adds to the versatility of this imaging technique. Apart from the high quality surface images, the AFM can also be used to obtain qualitative and more 11

25 importantly quantitative information about local forces and interactions at the surface of the sample. Another point that provides the AFM an edge over other microscopes is that it can be used in a fluid environment which permits the dynamic study of living biological specimens History of AFM Development of Scanning Probe Microscope started in the early 80 s after the invention of the Scanning Tunneling Microscope (STM) by Binnig and Rohrer [33]. The STM emerged as a new and revolutionary family of probe microscopes which sense the structure of a surface by some form of interaction (electrical or mechanical) between the surface and the probe. The working principle of the STM was the fact that the tunneling current between a conduction tip and sample is exponentially dependent on their separation (Fig. 2.1). This technique was used for conductive and semiconductive surfaces only. This drawback led to the development of the Atomic Force Microscope (initially called Scanning Force Microscope) [32]. Instead of the tunneling current, the force between the small tip and the surface was used as the physical quantity to be measured. Since sample conductivity was no longer an necessary requirement, the AFM could be used on any kind of material. Initially, the motion of the probe was monitored using a STM tip and measuring the tunneling current. But soon it gave way to the optical lever mechanism (Fig. 2.3). The topography was imaged by maintaining a constant repulsive force between the tip and the indenter. (Section 2.1.3) An alternative was to vibrate the cantilever above the sample and monitor the variations in 12

26 Controller Electronics Feedback Mecanism X,Y Scanner Z AmpMeter ~ I ~ Ve cd Tip d Sample Figure 2.1: Scanning Tunneling Microscope: Working Principle amplitude (Section 2.1.3) to improve the imaging quality. The invention of the AFM has been a revelation to the scientific community and with newer perfected micro cantilevers, it has become the order of the day for surface scientists Parts of a Typical AFM An AFM consists of the following basic parts. The Optical Head. This small rectangular block is the heart of the AFM. It is attached to an X-Y stage, both are kinematically coupled with the scanner via three contact points. A pair of springs on either side of it secures it to the scanner. A laser is embedded on the top, while the photo diode is on the top left. Fig 2.2 shows the details of the head with its 13

27 Photodiode adjust Laser Y-axis adjust Laser X-axis adjust KEY 1. Laser 2. Mirror 3. Cantilever 4. Tilt Mirror 5. Photodetector Head X-axis stage adjust Head Y-axis stage adjust Figure 2.2: Details of the optical head of the AFM [34]. various knobs to adjust the position of the laser, mirror and photodiode. The Photo Diode (Part 5 of Fig. 2.2), which measures the laser reflection off the cantilever, is divided into four quads, and combined voltages of the all of them form the SUM signal. The amplified differential signal between the two top and two bottom elements provides the measure of the cantilever deflection (VERT). This differential signal is either used directly (Contact Mode - Section 2.1.3) or fed to an RMS amplifier (Tapping Mode - Section 2.1.3) for measuring the normal force exerted by the cantilever on the sample. On the other hand, the difference between the two left and two right signal (HOR.) is used in the Lateral Force Microscopy (Other Modes - Section 2.1.3). Fig. 2.3 shows a schematic representation of the diodes. The Tip Holder 14

28 LFM Laser Photodiode Segments B AFM Mirror A Cantilever C D Figure 2.3: The orientation of the photodiode. Contact and TappingMode EFM Force Modulation Top View Bottom View Figure 2.4: Various types of tip holders. [34] 15

29 Figure 2.4 shows few standard tipholders. A typical tip holder has a piezo which is used to vibrate the cantilever in Tapping Mode. The choice of tipholders (Standard, Fluid Cell, Force Modulation tipholder) depends on the nature of the sample and also the applied force range. The fluid cell is capable of dynamically maintaining a physiological environment, which is extremely useful for biological applications. The force modulation tipholder has a large piezo bimorph for applying larger forces on harder samples. The Scanner The piezoactuators provide the scanning mechanism to the AFM. The piezos employed are usually cylindrical tubes of different dimensions, and the inside electrode is grounded and the outer electrode segmented into four quadrants. Figure 2.5 shows the electrical configuration of the piezo-electric actuators inside the scanner. The various types of scanners are characterized depending on the scan size, vertical range, resolution and geometry of the piezos. Probes The probes are the most critical of all the components. They are usually manufactured having a couple of cantilevers on either side. The selection of the cantilever is extremely important and is governed by the type of sample and the range of the applied force. The cantilevers are usually classified either according to the type of the attached tip: sharp tipped, tipless, spherical beaded tip or according to the stiffness of the cantilever. Mostly, the cantilevers are made of silicon or silicon nitride and the back surface is 16

30 + - Z ~ Z X X ~ + - Y ~ + X - Y Y Figure 2.5: The arrangement of piezos in the scanner [34] painted with a reflective coating. (usually gold or chromium). They usually have two shapes: rectangular and triangular or V - shaped (Fig. 2.6). But recently the urge to expand the capabilities of AFM has led to a lot of innovations of the tips, like chemically altered tips [35], carbon nanotube cantilevers [36], thermocouple incorporated tips [34] etc Operating Principle The heart of the AFM is the cantilever with a nanofabricated tip attached. This tip deflects when it interacts with the sample surface. This deflection is used to generate the topography of the surface. The optical lever method is the most popular mode of measuring the deflections of the the cantilever. A laser beam is focused on the cantilever and the deviation 17

31 Probe Cantilever Tip A:Silicon Tips used in Tapping Mode B:Silicon Nitride Tips used in Contact Mode Figure 2.6: Standard cantilevers: (A) Rectangular tips (B) Triangular tips. [37] of the reflection on the photodiode gives the estimate of the deflection (Fig. 2.7-B). The modern-day AFM is equipped to be used in various operating modes, like the Contact Mode, Tapping Mode, Lateral Force Microscopy, Force Modulation etc. [34]. The contact and the tapping modes are our primary interests and are explained in detailed below. The Contact Mode As the name suggest there is explicit contact of the cantilever and the sample during the process. The central theme of this operation mode is to apply a very small but constant force to the sample. If the tip approaches any feature on the surface, the force exerted (and thus deflection) changes, and the tip has to be moved in the z-direction by the scanner to maintain the original force. A map of this z-displacements gives us the topography of the sample. (Fig. 2.7-C) 18

32 The Tapping Mode In this mode the tip is vibrated at its resonant frequency just above the surface. When it encounters a surface feature, the interaction changes the amplitude of the vibrations. The RMS value of the amplitude is monitored and the tip moved in the z-direction to maintain the amplitude. As in the contact mode, these z-displacements reproduce the topography. (Fig. 2.7-B) Other Modes Apart from the above two modes and the STM, the AFM can be customized to be used in many scanning techniques, like phase imaging, magnetic force microscopy, electric force microscopy, surface potential microscopy, force modulation and lateral force microscopy (LFM). In LFM, the cantilever scans laterally (perpendicular to their length) and are torqued by the frictional force. The relative map of the lateral forces encountered along the surface yields a map of high and low friction sites. Force modulation produces a map of relative elasticity of the surface. It measures the local elasticity by oscillating a cantilever such that the tip indents slightly into the sample. Refer to the AFM manufacturer s manuals [34] for the detailed description of the various mode of operation Controller Electronics A digital signal processor is used to convert the computer commands to electrical pulses, which in turn controls the tip-sample interaction. The reflection from the cantilever hitting on the photo-diode is processed differently in different operation modes. In Lateral Force 19

33 (A-B) or (C-D) Photodiode Signal Photodiode Mirror Laser Tip Tip Holder Z-Piezo Sample Deflection Voltage Setpoint Voltage Difference A/D Converter Computer D/A Converter A. The Working of a typical AFM RMS Voltage Oscillating Tip Sample Direct Voltage Tip in Contact Sample B. Tapping Mode C. Contact Mode Figure 2.7: Working Principle of the AFM. Fig A shows the general principle and the control mechanism. Refer to the Fig. 2.3 for the details of (A-B) and (C-D) signal. This signal strength is compared to the setpoint voltage and the difference governs the amount of motion in the z-piezo. Note that the sample is stationary for the whole process. This z-displacements over tip in x y space renders the topography of the sample. Fig. B shows the tapping mode, where the RMS voltage is monitored to measure the amplitude of oscillating tip, and compared with the setpoint amplitude. In contact mode as shown Fig. C the direct (A-B) signal is used for comparison. LFM uses similar technique as the contact mode, but instead of (A-B) signal the (C-D) signal gives the torque experienced by the tip, which gives a surface frictional map. 20

34 Microscopy, the difference between the two vertical sections (C and D, refer to Fig. 2.3) is used while the two horizontal sections (A and B) is used in contact mode or taping mode AFM. When the tip is moved over the surface of the sample, the voltage difference gives the height information of the surface. The signal is filtered and amplified to get a clean and clear topographical map of the sample. A feedback loop is used to maintain a constant interacting force (Contact mode AFM and LFM) or a constant RMS amplitude (Tapping mode AFM). The Feedback Mechanism The use of an feedback mechanism is imperative for optimized control of the interactions between the different sample and the tip. The mechanism uses three kind of gains: proportional, integral and lookahead. The tip scans over the surface, the tip deflection is monitored. At every measurement, it is compared with the set deflection voltage (setpoint voltage) and depending on the difference, the z-scanner is moved. The proportional gain controls the rate at which this movement occurs. For example if the proportional gain is set to 2, then the rate at which the scanner moves is twice than when the gain is set as 1. The cumulative error accumulated over time is rectified using the integral gain. Its effect is to reduce the error (difference between the actual deflection and the set deflection) by integrating the errors over a longer period of time. This smooths out the short-term and fluctuating effects of proportional gain. The mechanism is more sensitive to the integral gain than proportional gain. When a prior knowledge of the topography is available, the lookahead can be used to keep the tip-deflections within proper limits so that the other gains can perform better by keeping the error to the minimum. Before starting the imaging, the setpoint is provided. 21

35 In Contact Mode, this amount defines the amount of the cantilever deflection. So higher the setpoint, higher is the applied force. But in Tapping Mode, the setpoint is determined by the RMS amplitude of the oscillation tip. So when the setpoint is decreased, the amplitude decreases, but the applied force increases. Please refer to Appendix A for detailed description of commonly used terms. 2.2 Applications of AFM AFM is one of the most widely used techniques for imaging at nano-scales. It is vastly popular with scientists because it provides information both qualitatively and quantitatively. Recent advances in nanofabrication of tips have further helped AFM expand into very specific applications Tribology Applications Imaging at sub-atomic level of non conducting materials led to the AFM branch out of the parent STM. Apart from its imaging qualifications, the AFM has also being extensively used to study contact mechanics at a nanometer scale [38]. The frictional forces at nanometerscales can be extracted using the Lateral force mode in the AFM. This enables scientist to study atomic scale stick-slip behavior [39]. Surface hardness and characterization is another important application of AFM. It has also being used to determine the thickness of thin films [40]. Force modulation is another feature available with the AFM to study the relative elasticity of the sample [14]. More Recently, AFM has been used for macromolecular crys- 22

36 tallography [41] and even for nanoscale lithography [42]. Considering the versatility of the applications of AFM, it is the ideal microscope for a surface scientist Biological Applications Recently, the AFM has gained popularity in the field of life science mostly cell and molecular biology not only because of the fact that it can image at sub-nano scale, but also for the fact that it offers several advantages over conventional microscopic techniques used for studying biological materials. Firstly, most biological molecules and cells can be imaged directly with AFM, requiring little sample pre-treatment. But more importantly, AFM can be used in a fluid environment, making it capable of imaging biological molecules and cells in their physiological environments. Although AFM started as an imaging tool for various cells like endothelial cells [43] and glial cells [44] and even erythrocytes [45] at resolution of 30 nm or less, it is being actively used to gather quantitative information about the interactive forces at a cellular level. The probing technique is often used for the estimation the local elastic behavior [18, 22, 28] and viscous properties [20] and microrheology [46] of living cells like cardiac, skeletal muscles and vascular endothelium [47]. Apart from being actively used to map the strain distribution of cells [48], AFM has also being widely used to identify sub-surface features of cells like cardiac myocytes [49] and carcinoma cells [50]. Dvorak el. al. even used AFM to track reproducible kinetic changes in the localized stiffness of vertebrate cells [51]. Imaging of DNA [52], and its surface study [53] is another important application of the AFM. AFM 23

37 has also been used for nanomanupulation of the DNA, Hansma el. al. excised nm portion from plasmid DNA using AFM [54]. Today, AFM is extensively used for the study of the dynamic behavior of biological system in real time at a sub-atomic scale. Extensive work has be done to estimate the local elastic behavior [18, 22, 28] and viscous properties [20] of cells or biogels and microrheology [46] of living cells like cardiac, skeletal muscles and vascular endothelium [47]. Apart from being actively used to map the strain distribution of cells [48], AFM has also being widely used to identify sub-surface features of cells like cardiac myocytes [49] and carcinoma cells [50]. 24

38 Chapter 3 Materials and Methods 3.1 Experimental Setup The AFM (Fig. 3.1) used for most of the experimental data was located at the Structural Engineering Laboratory, Civil Engineering Department at University of Virginia. It is a Multimode Nanoscope IV (Veeco Metrology, Santa Barbara, CA) with a PicoForce scanner. The PicoForce (PF) scanner incorporates a closed-loop Z-axis with a 20 µm vertical range and an XY scan size of > 40 µm in each direction. Unless otherwise stated, the indentation data in obtained using the PF scanner. All imaging was done using the standard 200 µm long silicon nitride V-shaped cantilevers (Veeco Metrology, Santa Barbara, CA) (Fig. 2.6) with pyramidal tips. The nominal stiffness of these tips was 0.06 N/m and the resonant frequency was about 8 khz (±2 khz) in fluid. But for indentation experiments, spherical beaded cantilevers were used. Usually, borosilicate glass spheres of either 1 µm or 5 µm diameter are attached to tip less cantilevers to obtain the 25

39 Objective Tip Holder PF Scanner Multimode Figure 3.1: AFM setup at the Structural Engineering Laboratory. 26

40 beaded tips. These tips were commercially obtained from Novascan Technologies, Inc. The thermal tuning system to calculate the cantilever stiffness which is incorporated into the PF scanner is used to determine the cantilever stiffness. The deflection signal over a time period is collected using the Contact Mode at thermal equilibrium while the cantilever is suspended away from any solid surface. Brownian motion of the surrounding molecules impact random impulses to the cantilever. This function obtained in time domain is Fourier transformed to obtain the Power Spectral Density (PSD) in the frequency domain. The area under the resonant peak in the spectrum gives the power, which is compared with the potential energy of a harmonic oscillating system to obtain the cantilever spring constant. A detail derivation can be found in [55]. The Multimode PicoForce automates this thermal tune calculations. The sampling was carried out over a time period of ten seconds to obtain the PSD with 25 Hz frequency resolution [34]. This thermal tuning method was invariably used to obtain the stiffness value in experimental conditions. If this feature in not available in a particular system, then methods listed in the literature, like comparing with a reference cantilever of known spring constant [56] can be used to obtain the cantilever stiffness. Images were acquired either in air or in de-ionized water (MilliQ Ultra pure or Fisher DIUF) using the TappingMode (Section 2.1.3), while the indentations were performed using the ContactMode (Section 2.1.3). The scan rate was adjusted to get an optimal image. The deflection sensitivity was calibrated with the flat mica substrate. 27

41 3.2 Specimen Preparation The agarose (Invitrogen Life Technologies, Carlsbad, CA.) was dissolved in de-ionized ultra pure water (MilliQ Ultra pure or Fisher DIUF) by heating at 200 C for roughly ten minutes. It was cooled at room temperature for five minutes before slowing pouring it into a container to solidify. Bubbles, if any were pushed away to the side using a disposable tip. This was set aside for about 30 minutes to one hour in a closed container to solidify. Thin ( 1 mm) and small strips ( 1 cm 1 cm) were cut from the gel brick and glued to a hard substrate before any experimentation. Gels of different concentration were prepared by adding different amounts of agarose. For example, a 1% gel sample was prepared by adding 1 gm of Agarose in 100 ml of water. 3.3 Confined Compression Apparatus We also used macroscale tests involving confined compression. This experiments were undertaken to study the creep behavior of the agarose gel. The experimental setup for confined compression experiments is shown in Fig It consists of a flat-ended cylindrical indenter, which is loaded by dead weights. The load is measured by a load cell having a linear variable differential transformer (LVDT) is used for measuring displacements. The total weight of the indenter assembly is around 50 grams, which is the minimum creep load that can be applied. Cylinders of 3 mm deep and 3 mm diameter was cut and placed into the well of similar dimensions. From this data, the stress versus stretch data was extracted assuming a uniform pressure distribution over the surface. Determining the deformation at time t = 0+ from 28

42 Figure 3.2: AFM image of 1% agarose gel. The image was obtained using the standard 200µm V-shaped cantilevers with pyramidal tips (NP-20, Veeco Instruments) in the contact mode. 29

43 Figure 3.3: Experimental setup for confined compression. the experimental data presents a serious problem because it is very difficult to determine the instant of time when the creep force is applied as seen in Fig We set the contact time, when the force was more than 75% of the final load (which was 50 gms). That point was reset as t = 0. For agarose gels, whose Young s modulus being around 20 kpa, the force applied is too large for the response to be completely elastic. 30

44 10 0 Confined compression raw data Force (gms) Time Figure 3.4: The raw data obtained from the confined compression tests. 3.4 Data Interpretation An in-house MATLAB code was used to extract force-displacement data from the raw AFM data (Fig. 3.6) after choosing a contact point from the AFM f-z data (Fig. 3.5). For a detailed description see [19, 57]. This force-displacement data was used to extract the elastic modulus as well as the hyperelastic material parameters as described below Elastic Modulus Estimation The Hertz model [15] is used to extract the elastic properties. As discussed earlier, these kind of materials are considered incompressible, hence a Poission ratio of 0.5 is assumed throughout the discussion. The f δ data (Fig. 3.6) from the AFM is compared with the Hertzian response: F HERT Z = 16 REδ 3/2 9 (3.1) 31

45 0 f (pn) Chosen Contact Point (z 0,f 0 ) z (nm) 0 Figure 3.5: Raw AFM indentation data in f z. A spherical indenter of 1µm diameter was used to obtain the data. The contact point chosen was marked in red. If this point is (z 0, f 0 ), we calculate the force as (f i f 0 ) and δ as (z i z 0 ). The noise in the non-contact region of the data is about tens of piconewtons. 32

46 AFM data Hertz fit f (pn) δ Figure 3.6: Force-displacement data extracted from the raw f z plot. The data is curve fit with the Hertzian equation to obtain the elastic modulus. 33

47 where R is the radius of the indenter and E is the Young s modulus. The function fminsearch of the commercially available software Matlab [58] is used to minimize the error between the AFM f δ data, F AF M (orf) and the Hertzian response, F HERT Z (Eqn. 3.1) and extract the Young s modulus E of the agarose gel samples. The function fminsearch uses the Nelder- Mead simplex search method ([59, 58]). This is a direct search method that does not use numerical or analytic gradients. The square of the error (Eqn. 3.2) is minimized to extract the Young s modulus. e f = n (FAF i M FHERT i Z) 2 (3.2) i=0 where n= the number of data points Non-linear Material Properties Both time-dependent and time-independent material properties are considered in this study. Hyperelastic material models will be used to explain the large scale deformations while porous-elasticity and viscoelasticity models will be used to explain the time-dependent properties. Viscoelastic models are only used for the confined compression experiments while hyperelasticity and poroelasticity are applied to the AFM indentation experiments Viscoelastic Materials An elastic solid returns to its original configuration after removal of the deforming force, restoring all the energy. On the other hand a viscous liquid has no definite shape and flows irreversibly under the application of a force. The mechanical properties of gels lie in between 34

48 these two extremes. We can add some fluid characteristics to our elastic models to obtain time-dependent response from the system. From Newton s Law of linear viscosity [60] using the usual index notation we have σ xy = η ɛ xy t (3.3) where η is the viscosity of the fluid, ɛ xy is the shear strain and σ xy is the shear stress. Hooke s law for a linear elastic material states, σ xy = Gɛ xy (3.4) where G is the shear modulus. A possible formulation of linear viscoelastic behavior would be just adding the elastic part (time-independent) with the inelastic part (viscous or timedependent): σ xy = Gɛ xy + η ɛ xy t (3.5) This is a simple viscoelastic model called the Kelvin model, which is discussed later in a greater detail. There are basically two modes of depicting linear viscoelasticity: creep and stress relaxation. One-dimensional creep behavior is observed when a constant load σ = σ 0 H(t) (3.6) where, H(t) is the heavyside function. 1 if t a 0, H(t a) = 0 if t a < 0, 35

49 Applied strain Time Stress Response Time Figure 3.7: Time-dependent response of a rate sensitive material: (a) load profile (b) stress profile. The values are arbitrarily scaled for comparison. is applied to the sample. After an instantaneous deformation, the sample slowly deforms to reach the final state of deformation. The strain response to the constant load σ 0 can be written as ɛ(t) = σ 0 J(t) (3.7) where J(t) is called the creep compliance, which is only a function of time. It describes the stress-strain behavior for linear viscoelastic materials. Conversely, a decay in stress is observed when a constant strain (ɛ = ɛ 0 H(t)) is applied. And likewise we can define a stress relaxation modulus G(t), G(t) = σ(t) ɛ 0 (3.8) Obviously, the load response of these materials depends on the loading profile (Fig. 3.7) rather than just the final and initial values. Such a material is said to be rate sensitive. And as we are assuming linear viscoelasticity, the Boltzmann Superposition Principle is used 36

50 E 1 E 1 η 1 η 1 a) Maxwell Model b) Kelvin Model E 1 η 1 E 1 E 2 E 2 η 1 c) Zener Model d) SLS Model Figure 3.8: Various phenomenological viscoelastic material models. to calculate the final deformation by the addition of contributions from each loading step. A detailed description is provided in Appendix-C Phenomenological Viscoelastic Models Phenomenological material models are discrete-element models built from various combinations of time-independent components like springs and time-dependent components like dashpots. Though they do not represent the structural or microscopic components within the materials, they represent the overall responses to external load profile (stress or strain histories). Different material properties can be obtained using various combinations of springs and dashpots. Four such phenomenological models are studied here. 37

51 The Maxwell Model This is the simplest model one can conceive where a dash-pot and a spring are serially connected (Fig. 3.8a). The respective strains of the spring and the dashpot can be simply added to obtain the total strain. So, the governing differential equation is, ɛ = σ E 1 + σ η 1 (3.9) Kelvin or Voigt Model Another simple model consists of a single spring and a single dashpot, but this time they are connected in parallel (Fig. 3.8-b). In this case, the stresses can be added to get the total stress. So the governing equation is σ = E 1 ɛ + η 1 ɛ (3.10) Standard Linear Solid Model We can increase the complexity of the model by adding a spring element to our model. This spring is in series with a Kelvin unit (Fig. 3.8-d). The differential equation of this model is, Zener Model σ + E 1 + E 2 η 2 σ = E 1 ɛ + E 1E 2 η 2 ɛ (3.11) If we add a spring in parallel to a Maxwell model we arrive at the Zener model (Fig. 3.8c), which has one parameter more than the Maxwell model. Its governing equation is σ + σ τ 1 = (E 1 + E 2 ) ɛ + E 1 E 2 ɛ η 1 (3.12) 38

52 Model Instantaneous Modulus Equilibrium Modulus Maxwell E 1 - Kelvin - E 1 SLS E 1 E 1 E 2 E 1 + E 2 Zener E 1 + E 2 E 2 Table 3.1: The instantaneous and equilibrium modulus of the viscoelastic models. where τ 1 = η 1 /E 1. The equilibrium stiffness is calculated by making t or in other words removing the time derivative terms. For Maxwell model the equilibrium stiffness does not make any sense because of the damper is connected serially to the load. But for Kelvin it would be just E. Similarly, the instantaneous modulus is defined at t = 0+. Table 3.1 enlists the instantaneous and equilibrium moduli. If we look carefully at the differential equations for all four models, we can generalize them as [61], p 0 σ + p 1 σ + p 2 σ = q 0 ɛ + q 1 ɛ + q 2 ɛ (3.13) or, which can be simplified as m 0 p i d i σ dt i = n 0 q j d j ɛ dt j (3.14) Pσ = Qɛ (3.15) 39

53 where the operators P and Q are P = m d i n p i dt, Q = i 0 0 q j d j dt j (3.16) Viscoelastic response in 3-d The above discussion is only applicable to uniaxial application of load. Now, we consider the general case of all the three dimensions for analysis. The stress tensor can be written as the sum of the hydrostatic part and the deviatoric part. [σ] = [σ s ] + [σ d ] (3.17) where σ s is the hydrostatic stress component and the σ d is the deviatoric component. The deviatoric part can also be broken down into 5 planar components like [σ deviatoric ] = 0 σ xy 0 σ xy σ yz 0 σ yz 0 (3.18) σ xz σ xz σxx d 0 0 σxx d σzz d σzz d (3.19) As the hydrostatic part of the stress only produces dilation and no distortion, we can relate the hydrostatic stresses with the similar hydrostatic component of the strains using the general viscoelastic eqn. (3.15). P σ s = Q ɛ s (3.20) 40

54 Following the same argument we can relate the individual deviatoric components with each other P σ d = Q ɛ d. (3.21) Unconfined Compression In unconfined compression, the stress is uni-axial but the sample is free to deform radially. So we have σxx s = σyy s = σzz s = 1 3 σ 0 (3.22) σ d xx = 2 3 σ 0, σ d yy = σ d zz = 1 3 σ 0 (3.23) and ɛ s xx = ɛ s yy = ɛ s zz = 2 3 (ɛ xx + 2ɛ yy ) (3.24) ɛ d xx = 2 3 (ɛ xx ɛ yy ), ɛ d yy = ɛ d zz = 1 3 (ɛ xx ɛ yy ) (3.25) Now using these quantities in the respective general Eqns. (3.20) (3.21),we have P σ 0 = Q ɛ xx + 2Q ɛ yy (3.26) P σ 0 = Q ɛ xx Q ɛ yy (3.27) and after some manipulation we have the equations as (P Q + 2Q P )σ 0 = 3Q Q ɛ xx (3.28) (P Q Q P )σ 0 = 3Q Q ɛ yy (3.29) 41

55 It can be recalled from eqn. (3.16) that P and Q are the operators on the stress and strain respectively (Eqn. 3.15). These operators depend on the type of model chosen for the particular response. For example if we consider the hydrostatic response to be completely elastic, we have P = 1, Q = 3K ( as σ s xx = 3Kɛ s xx) (3.30) where K is the bulk modulus. And, if we consider the distortion to follow any viscoelastic model like the Maxwell model we have from Eqn. (3.9) P = (1 + η E d dt ); Q = η d dt (3.31) We can use any of the other three models also for the distortion part. Confined Compression In this case deformation is only allowed in one direction and restricted in the other two. So we have σ xx = p ; σ yy = σ zz (3.32) ɛ yy = ɛ zz = 0 (3.33) We have all the three components of stresses σ xx, σ yy and σ zz contributing toward the strain ɛ xx. Using Eqns. (3.20) and (3.21) for all the components and summing up we find, (P Q + 2Q P )σ xx + 2(P Q Q P )σ yy = 3Q Q ɛ xx (3.34) (2P Q + Q P )σ yy = (P Q Q P )σ xx (ɛ yy = 0) (3.35) 42

56 The dilation is considered to be elastic and the distortion to be viscoelastic. So we have P = 1, Q = 3K (3.36) P = p 1 + p 2 d dt, Q = q 1 + q 2 d dt (3.37) where p 1, p 2, q 1, q 2 can be specified for the selected viscoelastic model. These equations are solved with different boundary conditions in Chapter 4. The phenomenological viscoelastic models present a relatively simpler approach to study the timedependence response of a material. We used these models to explain the confined compression experiments. While finite element analysis generated data is compared with both the confined and unconfined compression. The goal of these comparisons is to comment on the time-dependence of the gels and thus being able to suggest its role in the AFM indentation experiments Poroelastic Models - Continuum Porous Elastic Model and Biphasic Model Poroelasticity is a continuum theory for the analysis of a porous medium consisting of an elastic matrix with interconnected fluid-saturated pores. The basic phenomenon underlining the poroelastic behavior is the coupling between the solid and the fluid phases. A change in applied pressure produces a change in the fluid pressure or fluid mass and when the fluid pressure or the fluid mass changes it produces a volume change in the pores. The magnitude of these couplings depends on the compressibility of the framework, the fluid and the solid phase. The ratio of the volume of its solid and fluid phases ( porosity ) also determines the 43

57 characteristics of the porous material. Additionally, The nature of the fluid also plays a role; if the fluid is viscous the behavior of the system becomes more time dependent. A nonuniform fluid pressure distribution leads to time-dependent fluid flow according to the Darcy s law [60]. The time dependence of the pore pressure produces time dependence of the poroelastic stress and strains, which in turn couple back to the pore pressure field. The basic phenomenological model for porous elastic materials was first proposed by Biot [62]. Though initially developed for applications in hydrology and geomechanics, the concepts of poroelastic materials are now used in a multitude of applications like petroleum engineering, soft tissue mechanics, etc [60]. There are many similarities in mechanical properties and behavior between geomechanical and certain biological structures. The pores in biological structures like the bone or soft tissues like articular cartilage are very important for the functionality of the tissues. Poroelastic theories have been extensively employed in the study of articular cartilage and other soft tissues [63]. Finite element models have been widely used to extract solutions and because of the complicated nature of the problem different models have been offered. Various techniques like uniaxial compression test [64], indentation tests [65, 66], joint contact analysis [67] have been used to study the poroelastic behavior of biological systems. The biphasic model [68] which utilizes the mixture theory is more comprehensively used for studying the rheological behavior of biological systems. The poroelastic theory is different from the biphasic theory in that the former specifies a continuous distribution of pores in the solid matrix whereas the latter specifies a continuous distribution of solid and fluid phases 44

58 [69]. Though this difference is obvious, it has been shown [70] using the poroelastic approach in ABAQUS [31] gives identical results to those using the biphasic theory [66]. Definition of related terms: Pore pressure is defined as the pressure of the fluid occupying the pore space. Void ratio is the ratio of the volume of each phase and is equal to V f /V s where V f is the volume fraction of the fluid phase and V s is the volume fraction of the solid phase. Permeability is the ease with which the fluid passes though the solid matrix. It gives a relationship between the volume flow rate per unit area of the wetting fluid though a porous medium and the gradient of the effective fluid pressure [31]. Mathematically there are various ways of representing it. One of the earliest and more widely used definition is that of Darcy s. He deduced an empirical relationship for one dimensional flow q z = k dh dz (3.38) where q z, the specific discharge, is the volume of fluid crossing a unit area per unit time; h, the head, the height above a reference datum attained by a column of water in equilibrium with a point in the column and k, the permeability, is the constant of proportionality. This defines the permeability of the fully saturated medium which is a function of the void ratio and temperature. Some authors refer to this definition of permeability (units of LT 1 ) as hydraulic conductivity of the porous medium. 45

59 Generally speaking, for a fluid velocity v w permeability is defined as k = k s 1 + β v w v w k (3.39) where k s is the dependence of permeability on saturation of the wetting liquid and β is a velocity coefficient which may be a function of the void ratio Hyperelastic Materials Elastic materials for which the work done by the stresses in independent of the load path are said to be hyperelastic materials. For these materials, the work done is depends on the initial and final configurations. As a consequence of the path-independent behavior, a stored strain energy function or an elastic potential can be formulated as the work done by the stresses from the initial to the final configuration [25], S = 2 ψ(c) C = w(e) E (3.40) where, gradient), and ψ = the stored energy potential, S = second Piola-Kirchhoff stress tensor, C = the right Cauchy-Green deformation tensor (C = F T F, F is the deformation E(Green strain) = 1 (C - I) 2 The path-independent behavior of this material model can be further illustrated by the fact that the second Piola-Kirchhoff stress and the Green strain are work conjugates, which 46

60 gives: 1 2 E2 E 1 S : E = w(e 2 ) w(e 1 ) (3.41) C2 C 1 S : C = ψ(c 2 ) ψ(c 1 ) (3.42) We consider the material to be isotropic, so the relationship between ψ and C is independent of the material axes. Hence the potential for an isotropic hyperelastic material can be expressed as the function of the principal invariants (I 1, I 2, I 3 ) of the right Cauchy-Green deformation tensor, C : ψ(c) = ψ(i 1, I 2, I 3 ) (3.43) where, I 1 (C) = trace(c) = C ii I 2 (C) = 1 2 [(trace(c))2 trace(c 2 )] = 1[(C 2 ii) 2 C ij C ji ] I 3 (C) = det(c) Various material model can be formed using different function for the stain energy function Neo-Hookean Material As the name suggests, this kind of material model is an extension of the isotropic linear model (Hooke s Law) to large deformations. The stain energy function for such a material is described as: ψ = µ 0 2 (I 1 3) µ 0 ln J + λ 0 2 (ln J)2 (3.44) 47

61 where λ 0 and µ 0 are the Lamé constants of the linear theory, J = det(f), and I 1 is the first invariant of C Polynomial Material Model Given isotropy and the additive decomposition of volumetric and deviatoric strain energy contribution, the potential can be written [31] as ψ = f(i 1 3, I 2 3) + g(j 1) (3.45) which can be expanded without any loss of generality as n ψ = c ij (I 1 3) i (I 2 3) j + i+j=1 n i=1 1 d i (J 1) 2i (3.46) This is the polynomial representation of the strain energy function. For an incompressible material we know J = det(f) = 1, or I 3 = J 2 = 1 so we have ψ = ψ(i 1, I 2 ). So the second term in the above equation (Eqn. 3.46) vanishes. A simple second order polynomial strain energy function for incompressible hyperelastic material can be written as ψ = c 1 (I 1 3) + c 2 (I 1 3) 2 (3.47) Modified Mooney Rivlin Material The Mooney-Rivlin model is an incompressible and isotropic hyperelastic constitutive model for large deformation for elastomers. It is a modified and simplistic version of the polynomial 48

62 parameter Mooney Rivlin Model Polynomial Model Exponential Model Elastic Response 1.0 Normalized In-place stress Stretch Ratio Figure 3.9: Stress versus stretch response of the material models under plane strain uniaxial tension and compression. The values of the respective parameters are picked so as to obtain a Young s modulus [26, 27] of 20 kpa for all the models. The stress values are arbitrarily scaled for comparison. 49

63 material model. ψ = c 1 (I 1 3) + c 2 (I 2 3) (3.48) This model was shown to be extremely compliant with the experimental results for large deformation. The Mooney-Rivlin model is an example of a Neo-Hookean material Exponential Model The strain energy is represented as an exponential function in this incompressible and isotropic model, ψ = b 1 (e b 2(I 1 3) 1) (3.49) This model is primarily used to describe various biomaterials. The parameter b 1 is the elastic or linear contributor while b 2 governs the the non-linearity in the model. The response of the three models: the simple polynomial, 2-parameter Mooney-Rivlin and exponential models are compared. The response is shown in Fig As we would expect the exponential is the most non-linear of the three while the 2-parameter Mooney-Rivlin model is closer to the elastic response. This suggests the use of exponential model if we expect the material to be behave non-linearly even for low stretch ratios like A detailed comparison between the three models can be found in [26]. 50

64 Indenter Radius Rigid Spherical Indenter. Frictionless Contact. Axisymmetric Boundary Condition units Positive Displacements units The dimentions are not to scale. Figure 3.10: The finite element model used to study the AFM indentations. Source: ABAQUS CAE. 51

65 3.5 Finite Element Model for the AFM Contact Problem Commercially available software ABAQUS [31] was used for all the finite element (FE) simulations. The FE models used to study the AFM indentation problem are examined here. Figure 3.10 shows the basic FE model in detail. The spherical indenter was assumed to be analytically rigid while different material models like incompressible hyperelasticity and porous-elastic were used for the sample (Agarose gel). Axisymmetric boundary conditions were imposed on the left edge. The contact between the spherical indenter and the gel was considered to be frictionless. The mesh was biased towards the contact area. The model dimensions were 2000 nm in the radial direction and 4000 nm through the thickness. The boundary conditions in the case of the porous-elastic model were slightly modified to accommodate the fluid flow. Details specific to the material model are discussed in the following sections. All post-processing FE simulation data like to generate f δ response, local stiffness etc. was done using MATLAB [58]. 3.6 Finite Element Model for the Confined Compression Problem Figure 3.11 shows the details for the FE model used for the confined compression. While the model was x-axisymmetric, the bottom surface was constrained in the y-direction. To simulate the confined compression environment, the outer surface nodes had their x displacement 52

66 Pressure Axisymmetric Boundary Condition 3000 units x-displacement constrained y-displacement constrained 1500 units The dimentions are not to scale. Figure 3.11: The finite element model for the confined compression tests. Source: ABAQUS CAE. degree of freedom restricted. The radial dimension was 1.5 mm while it was 3 mm deep. A dead load of 50 gm was simulated on the top surface. 3.7 Material Modeling Three different material models are used for the finite element analysis. The viscoelastic model was used to explain the confined compression tests. The porous-elastic material model was used to characterize the time-dependent properties of the AFM indentations, and the hyperelastic material model was used to explain the softening and hardening of the materials. 53

67 3.7.1 Linear Viscoelastic Model The finite element viscoelastic material model defines the stress response (τ(t)) when a time varying shear strain, γ(t) is applied to the material, as t τ(t) = G 0 g R (t t ) γ(t )dt (3.50) 0 where g R is the normalized shear relaxation modulus and G 0 is the instantaneous shear modulus. And similarly, for volumetric response t p(t) = K 0 k R (t t ) ɛ vol (t )dt (3.51) 0 where p is the hydrostatic pressure, K 0 is the instantaneous elastic bulk modulus, k R (t) is the dimensionless bulk relaxation modulus, and ɛ vol is the volume strain [31]. The viscoelastic material model is usually defined by a Prony series expansion of the dimensionless relaxation modulus. g R = 1 N g i (1 e t/τ i ) (3.52) i=1 where N, g i and τ i are materials parameters. In ABAQUS, the user can either directly give these Prony parameters or can supply test data (creep or shear relaxation). The software then finds a best fit curve for the data using Eqn.( 3.52) with a given N and calculates the unknown parameters. We provided the confined compression data to the model. An elastic modulus of 84.5 kpa was provided and the Poisson s ratio was set to be This was calculated from the final equilibrium position of the compression test. As we have only strain in one dimension ɛ ax, we calculated the 54

68 volumetric strains ɛ vol = (1 2ν)ɛ ax Continuum Porous Elastic Model In these analyses, various build-in porous elements are used. These elements are based on the Forchheimer s law for determining the flow velocities and pore pressure [31]. Built-in axisymmetric elements like CAX4P (4-node bilinear displacement and pore pressure) and CAX8P (8-node biquadratic displacement and bilinear pore pressure) are used for this model. The porous-elastic material model in ABAQUS requires the fully saturated permeability (at each void ratio for non-linear permeability), the initial void ratio, the elastic constants (Young s modulus and Possion s ratio). The extracted values of Young s modulus from the AFM indentation data were used. As the sample sits on a rigid impregnable substrate, zero pore pressure boundary conditions were applied to the bottom surface. For the top surface, while the displacement degree of freedoms (dof) were free, the pore pressure for all the nodes in contact with the indenter had zero pore pressure, while the rest were free. (A trial run was performed with all the top nodes free with respect to the pore pressure dof, and the contact nodes were picked. In the second run, all the contact nodes had the pore pressure dof restricted and new set of contact points determined. After a few iterations, the converged set of contact nodes were restricted finally.) In the first step the displacement to the bottom edge was applied linearly over time, while in the second step, this displacement was maintained to monitor the creep 55

69 effect of the model Exponential Hyperelastic Model The strain energy potential for an exponential model is represented by Eqn. (3.49). This is picked over other potential functions because of its appropriateness to be used for biological soft tissues to explain large scale deformations. A lot of research papers [27, 26] have advocated the use of the exponential model for soft tissues. This model is implemented in ABAQUS as a user-defined subroutine UHYPER [31] (Appendix B). A range of < > Pa for b 1 and < > for b 2 is examined. The radius of the indenter was also varied. A range of R ={125,250,500,1000}(nm) was explored. For each point in this parameter space, we allowed the maximum indentation δ max to be equal to the indenter radius R. Later when the parameters are extracted from the AFM indentation data, they are used for the inverse FE problem. The results obtained from these analyses and the comparisons are presented in the next chapter. 56

70 Chapter 4 Results and Discussion 4.1 Phenomenological Viscoelastic Material Model The differential equations discussed in Section for various phenomenological viscoelastic models are solved using Hereditary Integrals (Appendix C). Three different kind of loading conditions are considered: creep, step and triangular load. The loading functions are presented in table Creep Load In this section we study the behavior of all the above models under creep load σ = σ 0 H(t), where H(t) is the heavyside function. Figure 4.1 shows the creep response of the governing equation under the creep load of the different models. The solutions are presented in tables 4.2, 4.3, 4.4 and 4.5. The parameters for all the models (Table 4.6) were extracted by using a least square fit of the analytical solution of the creep loading to the confined compression 57

71 Load Creep: Loading function σ 0 H(t) Step Load: σ 0 H(t) σ 0 H(t T 2 ) Triangular Load: 2σ 0 t T H(t) 4σ 0 T (t T 2 )H(t T 2 ) Table 4.1: The input function for different loading conditions. experimental data Step Load In case of step loading, after applying a constant load for T/2, the load is relaxed. Figure 4.2 shows the responses of different models to step load and tables 4.2, 4.3, 4.4, 4.5 list the analytical solutions to the equations. The parameters obtained from curve fitting of the confined compression tests are used. In all the above models we can see that they all reach the same final deformation, but all have different strain rates and also instantaneous deformation. There is no instantaneous deformation observed in the Kelvin model because of the damper (Fig. 3.8) directly attached to the load Triangular Load A stress with a triangular profile is used in this case to analyze the strain responses. Tables 4.2, 4.3, 4.4, 4.5 show all the solutions for the strain response. The responses can be seen in 58

72 Zener Model Kelvin Model SLS Model Expt. Values Strain σ σ 0 H(0) 0.1 t Time, t (sec) Figure 4.1: Viscoelastic response to creep load. 59

73 Zener Model Kelvin Model SLS Model Strain σ t Time, t (secs) Figure 4.2: Viscoelastic response to creep load. 60

74 Zener Model Kelvin Model SLS Model 0.5 Strain t Time, t (secs) Figure 4.3: Viscoelastic response to triangular load. fig Its seen that the responses are similar for all the three cases. This is because the load increment is too fast for creep to set it. So the time dependence is quite small and a first order approximation will suffice for this loading, which is the typical loading profile of AFM indentation experiment. The parameters (Table 4.6) extracted from the experimental confined compression data are used. From table 4.6 we observe that the time constants are nearly similar for all the cases which we expect. We also observe that the modulus Maxwell and Kelvin are comparable ( ) E1 E 2 and are also equal to the effective instantaneous modulus for SLS and Zener E 1 + E 2 (E 2 ). 61

75 Input Creep: Step Load: Triangular Load: Maxwell Model [ ] 1 E (1 + t ) σ τ 0 [ ] σ(t) E σ 0 t 2 T η H(t) 2σ 0 T η (t2 T t + T 2 ) 4 Table 4.2: The strain response for Maxwell model. Input Creep Step Load Triangular Load σ 0 E 2σ [ 0 t τ(1 e t τ ) T E Kelvin Model [ ] 1 t (1 + e τ ) σ 0 E [(1 e t τ 1 )H(t) (1 e (t 1 T/2 ] H(t) 4σ 0 T E ] τ 1 )H(t T ) 2 [ ] (t T ) τ(1 (t T 2 ) 2 e τ ) H(t T ) 2 Table 4.3: The strain response for Kelvin model. Input Creep Step Load Triangular Load SLS Model [ ] (1 e t τ 2 ) σ 0 H(t) E 1 E 2 σ(t) + σ ] 0 [(1 e t τ 2 )H(t) (1 e (t 1 T/2) τ E 1 E 2 )H(t T ) 2 σ(t) + 2σ [ ] 0 t τ 2 (1 e t τ 2 ) H(t) E 1 T E 2 4σ 0 T E 2 [ (t T 2 ) τ 2(1 e (t T 2 ) τ 2 ) ] H(t T ) 2 Table 4.4: The strain response for SLS Model. 62

76 Input Creep Step Load Triangular Load Zener Model [ 1 1 E ] 1 e t τ σ 0 E 2 E 1 + E 2 σ(t) σ 0 E ] 1 + [(1 e t τ 1 )H(t) (1 e (t 1 T/2) τ E 1 + E 2 E 2 (E 1 + E 2 ) 1 )H(t T ) 2 σ(t) 2σ 0 E [ ] 1 + t τ(1 e t τ ) H(t) E 1 + E 2 T E 2 (E 1 + E 2 ) [ ] 4σ 0 E 1 (t T 2 T E 2 (E 1 + E 2 ) ) τ(1 (t T 2 ) e τ ) H(t T ) 2 Table 4.5: The strain response for Zener model. Models Maxwell Kelvin SLS Zener E 1 τ 1 E 1 τ 1 E 2 E 1 τ 1 E 1 E 2 τ 1 Parameters 86400Pa sec 86485Pa sec Pa Pa sec Pa 85717Pa sec Table 4.6: The Material parameters. Refer to Fig. 3.8 for the description of the parameters. These parameters are obtained by curve fitting the respective analytical expression to the confined compression test data. 63

77 4.2 Unconfined Compression in 3 d In Section we had arrived at the differential equations for the general unconfined compression loading. (P Q + 2Q P )σ 0 = 3Q Q ɛ xx (4.1) (P Q Q P )σ 0 = 3Q Q ɛ yy. (4.2) where, P = 1, Q = 3K, (4.3) for elastic dilation. If we choose the Maxwell model for the deviatoric part we have P = (1 + η E d dt ); Q = η d dt (4.4) Identically, we can choose any one of the other three viscoelastic models. These equations for all the four models were developed and solved. Table 4.7 presents the solutions for all the models. The values of ɛ xx and ɛ yy are plotted for all the 4 cases (Fig. 4.4). We also notice that at t = 0 + we have both ɛ xx and ɛ yy are positive. The reason of this is that as we have considered only elastic response for dilation so the model tries to expand in all direction initially. But later when time increases the creep slowly sets in. It can be observed that the SLS and the Zener models predict identical strains. This is because if both the solutions are compared at t >> 0 and with the material parameters (Es and νs) substituted, both of them will be identical. Both of the models are 3-parameter models, hence this similarity. as both of the models are 3-parameter approximations. 64

78 Maxwell Model ɛ xx = [ 1 9K + 2 3E + 2t 3η ]σ 0 ɛ yy = [ 1 9K 1 3E t 3η ]σ 0 ɛ xx = 1 9K [(1 + 6K E Kelvin Model t )(1 e τ ) + e t τ ]σ 0 ɛ yy = 1 [(1 3K 9K E SLS Model t )(1 e τ ) + e t τ ]σ 0 ɛ xx = 1 6K [(1 + 9K E 1 )e t τ ɛ yy = 1 3K [(1 9K E 1 )e t τ +(1 + 6K( 1 E E 2 ))(1 e t τ )]σ 0 +(1 3K( 1 E E 2 ))(1 e t τ )]σ 0 ɛ xx = 1 9K [(1 + Zener Model 6K E 1 +E 2 )e t τ ɛ yy = 1 [(1 9K 3K E 1 +E 2 )e t τ +(1 + 6K E 2 )(1 e t τ )]σ 0 +(1 3K E 2 )(1 e t τ )]σ 0 Table 4.7: Solutions for unconfined compression using elastic dilation and viscoelastic distortion. 65

79 5 x Strain Time, t (secs) Figure 4.4: Viscoelastic response for unconfined compression Poisson s ratio Maxwell Kelvin SLS Zener Time, t (secs) Figure 4.5: The variance of Poisson s ratio for viscoelastic models with respect to time. 66

80 The plot of the ratio ɛ xx ɛ yy (Fig. 4.5) with time is insightful. From the figure it is obvious that a concept of a Poisson s ratio (ν = ɛ xx ɛ yy ) is not very meaningful for viscoelastic materials. We can clearly see that the Poisson s ratio increase from a negative value (other than Maxwell Model) to reach to a final value of near zero but positive, which we can roughly say the time independent Poisson s ratio. This number has been used in the finite element viscoelastic model. 4.3 Confined Compression in 3 d From Section we have the differential equations for confined compression as, (P Q + 2Q P )σ xx + 2(P Q Q P )σ yy = 3Q Q ɛ xx (4.5) (2P Q + Q P )σ yy = (P Q Q P )σ xx (ɛ yy = 0) (4.6) where, P = 1, Q = 3K, (4.7) for elastic dilation and P = p 1 + p 2 d dt, Q = q 1 + q 2 d dt (4.8) is the general expression for any viscoelastic model. Using Laplace transformation [71] to solve the equation we get ɛ xx = p[ p 1α λ (1 e λt ) + p 2 αe λt ] (4.9) 67

81 Kelvin SLS Zener Expt Strain Time, t (secs) Figure 4.6: Strain response for various models under confined compression. where, α = 3 2q 2 + 3Kp 2 λ = 2q 1 + 3Kp 1 2q 2 + 3Kp 2. Using the values of p 1, p 2, q 1, q 2 for a specific model, we can arrive at the exact solution for that model. Figure 4.6 shows the response of all the model compared with the experimental data. The material parameters (Table 4.6) extracted from the confined compression experiments by curve fitting the analytical expression for creep loading is used. We see that the SLS model is closest to the actual experimental data. The error in the time constant of the SLS model is about 12%. 4.4 Viscoelastic Finite Element Model The finite element modeling is done as explained in Section ABAQUS was provided with the SLS response and also the experimental data to calculate the Prony parameters 68

82 (Sec ). Figure 4.7 shows the FE responses of both compared with the raw experimental data. It is observed that the FE models predict a time constant about 20 seconds but curve fitting on the experimental values, a time constant of about 60 secs is obtained. The FE model using experimental creep compliances have a lower instantaneous deformation than the experimental data. This is observed because the exact starting point of application of the load is not accurately known. The comparisons are not completely satisfying, as the dead load ( 50 gm) applied in the experiment (and hence in the FE simulations also) is in excess for these kind of materials (Young s modulus 20 kpa). From both analytical and FE viscoelastic studies we found that the force applied is excessive. Hence we do not have more accurate comparison with the experimental data. With such high load we are bound to see plastic deformations in the sample. So the plastic time constant of the gels is around 60 seconds. Although we were not able to predict the time constant of the gel with supreme confidence, we can certainly comment that the time constant is much larger than the frequency of the AFM indentation, 0.1 seconds. Thus we can be assured that the the AFM indentation experiments will not reflect any creep or time-dependent characteristics. 4.5 Poroelastic Material Model The AFM indentation experiments were simulated using finite element models using the poroelastic material model as explained in section Figure 4.8 shows the stress distribution of the poroelastic FE model near the contact region. The reaction force vs. indentation depth (f δ) obtained from the AFM experiments were compared with the results of the 69

83 Strain FE using Expt Compliance FE using SLS Compliance Experimental Values Time, t (secs) Figure 4.7: Comparison of FE simulations with the experimental data. Figure 4.8: A snapshot of the stress distribution near the contact region for poroelastic FE model. A permeability of 10 5 is used for this simulation. This distribution is at the end of the second step. 70

84 400 Displacement BC of the bottom surface ~ Time, t (secs) Figure 4.9: The time steps used in the FE poroelastic model simulations. first step of the simulations. Refer to Chapter 3 and Fig. 4.9 for the time steps used for the simulations. The AFM data and the poroelastic data are compared in the Fig The forces were arbitrarily normalized for qualitative comparison, as the exact values of the poroelastic material parameters for the material are not known. As we can clearly see, the poroelastic model fairly captures the AFM indentation behavior. The AFM has the minimum frequency of operation of 10 Hz. By using a 0.1 second time period for the first step we have incorporated this into the FE model. In the second step of the analysis we apply a constant load (Fig. 4.9) for a longer time (depending on the permeability). This will give us a picture of the time constant for the poroelastic behavior. Now we evaluate the time constants for different but possible values of permeability of 1% agarose gels. The permeability (hydraulic conductivity LT 1 ) generally lies between the range of to [72] for 1% agarose gels. The time constant is the 71

85 10 1 Normalized indentation force (nn) FE model porous elastic AFM indentation data Indentation depth, δ (nm) Figure 4.10: Comparison of FE poroelastic model with the AFM f δ data. The force is arbitrarily normalized for comparison. The radius of the indenter was 500 nm. δ max was nearly 300nm. We see a slight softening behavior at larger indentation depth. 72

86 τ(sec) Permeability Figure 4.11: Extracted time constants for various permeability. time interval required for a system to change a specified fraction from one state or condition ( to another. Mathematically, its the time taken to reach 1 1 ) (approx. 63%) of the final e value, in this case the final reaction force. It is evaluted from the second step where a constant displacement is applied to the bottom surface. From Fig we can clearly see that within the range of permeability the material is expected to possess, the time constant is on an average 30 seconds which is well over 0.1 seconds-the time period for a single indentation during the AFM experiments. This is a satisfactory proof of that fact that AFM experiment operate too fast for poroelastic effects to start, hence we do not expect any time-variance in the f δ behavior of the AFM indentation experiments. 73

87 f Rigid Indenter δ max a Figure 4.12: The displacement field near the contact area. The dotted line is the undeformed configuration while the solid line is the final state; δ max R a is the contact radius. 4.6 Hyperelastic Material Model = 1; f is the indentation force and The exponential model discussed in section is used for the material modeling. Quasistatic solutions for the prescribed incremental axial displacements were obtained. Fig shows a typical deformed displacement field. The resulting indentation force (f) is the axial component of the reaction force on the indenter and δ is the axial displacement of the indenter relative to the undeformed surface. These values are extracted from the FE model to get the f δ behavior of the model. The following parameter range was explored: b 1 = {1, 5, 10, 50, 100} 10 9 (Pa), b 2 = {0.1, 0.25, 0.5, 0.75, 1, 2.5, 5, 7.5, 10} and R = {125, 250, 500, 1000} (nm) (Refer to Sec ). The force-indentation data for this parameter space is gathered and systematically studied. 74

88 7.4 x Local stiffness, ζ (Pa) Indentation depth, δ (m) x 10 Figure 4.13: Indentation dependence of local stiffness. R = 250nm; b 1 = P a; b 2 = To find the correction to the Hertzian behavior for a spherical indenter for hyperelastic material, we propose an indentation-dependent stiffness, ζ(δ): F (δ) = C ζ(δ) δ 3 2 (4.10) where, C = 16 R, R being the radius of the indenter. 9 In other words, we are explaining hyperelastic material as an elastic material with an indentation-dependent stiffness. The f δ data extracted from the FE analysis was used to compute this indentation-dependent stiffness using a local Hertzian fit. For a given point δ k, the local stiffness ζ(δ k ) was calculated through a local Hertzian fit (Eqn. 3.1) over three neighboring points, δ k 1, δ k and δ k+1. Figure 4.13 shows a representative plot of the the extracted values of ζ for a data set (R = 250nm; b 1 = P a; b 2 = 5). It is clear from the fig that the local stiffness, ζ is not constant with indentation depth, as we would expect if the response were completely elastic. 75

89 Indentation force, f (nn) Hyperelastic (b 1 =10;b 2 =0.1) Hertz (E=6b 1 b 2 ) Indentation depth, δ (nm) Figure 4.14: Softening behavior observed for R = 500nm; b 1 = P a; b 2 = Effects of the parameters b 1 and b 2 It was observed that for small values of b 2 the material tends to show a softening behavior. The force-indentation curve lies below the Hertzian curve with same Young s modulus (6b 1 b 2 ). This can be clearly seen in Fig But beyond a critical value of b 2, the material starts to harden (Fig. 4.15). So we can clearly observe an transformation (Fig. 4.16) from softening to hardening behavior. This was found for all values of b 1 in our parameter space. For this we can safely conclude that the softening/hardening behavior is associated with the parameter b 2 of the model. Based on these observations and numerical experiments, the following one term correction form of the deflection-dependent stiffness seems reasonable for the case of 76

90 Indentation force, f (nn) Hyperelastic (b 1 =10; b 2 =10) Hertz E= (6b 1 b 2 ) Indentation depth, δ (nm) Figure 4.15: Hardening behavior observed for R = 500nm; b 1 = P a; b 2 = 10. spherical indenter. ζ(δ) = E 0 + m(b 1, b 2 ) δ R, (4.11) where, E 0 = 6b 1 b 2 is the Young s modulus of the material. Note that the above expression is a first order expression. Now from the Fig it is clear that the dependence of E on δ is not linear, and one can arguably find a better fit using a higher-order polynomial fit. The overall goal however was to get a correction that is simple to use yet sufficiently accurate. If necessary, additional higher terms can be taken into account later on. On further analysis of the data with respect to the parameter b 1, it was found that the slope function m is linearly dependent on b 1. So we can rewrite the Eqn. (4.11) as, ζ(δ) = E 0 + g(b 2 )b 1 δ R (4.12) 77

91 ζ 6b 1 b Relative Local Stiffness, b 2 =10 Elastic b 2 =0.25 Increasing b x 10 7 Indentation depth, δ (nm) Figure 4.16: Transition from softening to hardening behavior. R = 250nm; b 1 = P a. 4.4 x Indentation Dependent Stiffness, ζ (Pa) c m FE data Curve Fit Straight Line x 10 7 Indentation depth, δ (nm) Figure 4.17: A straight line is curve fitted to the indentation dependent modulus data. The intercept c and the slope m are then compared to 6b 1 b 2 and g( )b 1 respectively. 78

92 g(b 2 ) R=125 nm,b =1e9 Pa 1 R=125 nm,b =5e9 Pa 1 R=125 nm,b =10e9 Pa 1 R=125 nm,b =50e9 Pa 1 R=125 nm,b 1 =100e9 Pa R=250 nm,b 1 =1e9 Pa R=250 nm,b =5e9 Pa 1 R=250 nm,b =10e9 Pa 1 R=250 nm,b 1 =50e9 Pa R=250 nm,b =100e9 Pa 1 R=500 nm,b =1e9 Pa 1 R=500 nm,b 1 =5e9 Pa R=500 nm,b 1 =10e9 Pa R=500 nm,b =50e9 Pa 1 R=500 nm,b =100e9 Pa b 2 Figure 4.18: The function g for various values of b 1 and R. The function g in the above equation is extracted from a number of numerical experiments (Fig. 4.18). All the results are for a fixed domain size of 2000 nm in the radial direction and 4000 nm through thickness. For a given inventor size, the data-points coincide for all the values of b 1 considered. When the size of the indenter is increased, however, there is a small increase in this function g. This is believed to be due to the finiteness of the domain, which only comes into play as the size of the indenter is increased. In fact, experiments with the larger indenter sizes showed the stress profile reaching the boundary of the finite element domain. (Fig. 4.19) The accuracy with which g is determined from the FE experiments certainly depends on the mesh characteristics. Using a finer mesh or a different element type 79

93 R=1000nm 4000 units Substrate effects 2000 units Figure 4.19: Substrate Effects for large indenter radius. might lead to a more accurate description of the function g. But, if our hypothesis is true, then this step need be performed only once. The function g is negative for small values of b 2, and then rapidly increases for higher values of b 2. The transition from softening to hardening appears somewhere between 1 and 2.5. To determine this point more accurately, further FE experiments were carried out for this range of b 2. These showed that, with R = 250 nm, b 1 = Pa, the change in sign occurs around b 2 = The data from these experiments are shown in Fig To verify that the radius R and the parameter b 1 have no role to play for this critical value of b 2, a few more FE simulations were carried out with different values of R and b 1. Very small variations was observed. Please refer to Tables for the details. 80