Simulation supported Hertzian theory calculations on living cell samples

Transcription

1 Simulation supported Hertzian theory calculations on living cell samples Masterarbeit Von Dipl. Ing. Florian Biersack Hochschule München Fakultät 06 Mikro- und Nanotechnologie Erstprüfer: Prof. Dr. Alfred Kersch Zweitprüfer: Prof. Dr. Hauke Clausen-Schaumann

2 Abstract Analysis of Young s modulus from living cell samples is a major concern in biomechanics. In order to determine the Young s modulus, measurements from atomic force microscopy are evaluated with Hertzian analytical theory, which is the major tool for the evaluation in this thesis. The purpose of this thesis is a thorough validation of various analytical formulas from Sneddon s extension of the Hertzian contact theory and furthermore the extension of these formula to include relevant effects. The approach of this thesis is numerical simulation with COMSOL to reproduce the results of Sneddon s analytical formulas. Therefore contact analysis between geometrically different cantilever forms and a living cell sample is done. The cantilever tip will be modeled as a cone, paraboloid and sphere. It turns out that there are deviations between numerical and analytical solution depending on material parameter and finite size effects of the homogeneous sample. These deviations are investigated and the analytical formulas are finally improved. The thesis closes with an outlook to expand the investigation to inhomogeneous samples. 2

3 Contents 1 Introduction Living cell mechanics Mechanical properties of cytoskeleton Atomic force microskopy COMSOL simulation software COMSOL Multiphysics Solid mechanics interface Objectives Contact problem Hertzian theory Sneddon theory Relation between load and penetration for different axisymmetric punch profiles Equation for a punch in spherical form Equation for a punch in paraboloid form Equation for a punch in conical form AFM versus Sneddon s analytical solution Simulation models Model for a spherical form Model for a paraboloid form Model for a conical form Simulation equations Model settings in COMSOL Convergence problem of the contact model Simulation results Simulations for different indenter forms into an infinity elastic half-space Simulations for variable Poisson s ratios Verification of simulation results Simulations for variable sample thicknesses Sphere into variable sample thicknesses Paraboloid form into variable sample thicknesses Conical Indenter for variable sample thicknesses Thick-Thin sample comparison

4 8 Improvement of Sneddon s analytical solution Discussion of the results and conclusions Outlook References List of Figures

5 1 Introduction 1.1 Living cell mechanics Living cells or living organisms are far more complex material systems than engineered materials, such as metals, semiconductors, ceramics and polymers. Properties of living cells are dynamic functions and integrated functions that include remodeling, reproduction, communication, growth, control, metabolism and apoptosis (programmed cell death) (1). Within the past few decades, many studies have established the connections between biological functions of different organs and structure mechanical responses. Examples are heart, bone, cartilage, blood vessel, lung and cardiac and skeletal muscles (2),(3),(4). These studies have led to better understanding of how the biological functions of the body are related to biosolid and biofluid mechanics. In order to understand fundamental mechanisms of biological materials, more systematic studies of deformation, structural dynamics and mechanochemical transduction in living cells and biomolecules are needed. In the last years the development of instruments capable of mechanically probing and manipulating single cells and biomolecules at forces and displacements smaller than 1 piconewton (1 pn = N) and 1 nanometer (1 nm = m) has made large improvements. These advances have provided opportunities for operation of cellular machinery and the interactions between cells, proteins and nucleic acids (5),(6). A variety of functions are performed by the biological cell constitutes, such as, the synthesis, sorting, storage and transport of molecules, the recognition, transmission and transduction of signals, the expression of genetic information and the powering of molecular motors and machines. By continuously altering the structure, living cells can convert energy from one form to another and respond to external environments (7). For example, endothelial cells lining the interior walls of blood vessels alter the expression of stress-sensitive genes in response to shear flow in the blood (8). For the cell body to move forward during cell migration, contractile forces must be generated within the cell (9). Adhesion of cells to extracellular matrix (ECM) through focal adhesion complexes provides 5

6 both signaling and structural functions (10). The mechanical deformation of cells can also be caused by external forces and geometric constraints. A description of these processes constitutes a structural mechanics problem. Size of most biological cells is μm, and they comprise many constituents (Fig. 1). Figure 1: Cell structure (11) The cell is covered by a phospholipid bilayer membrane (Fig. 2). It is reinforced with protein molecules, and the interior of the cell includes a liquid phase (cytosol), a nucleus, the cytoskeleton consisting of networks of microtubules, actin and intermediate filaments, organelles of different sizes and shapes, and other proteins. Figure 2: Cell membrane (11) 6

7 The resistance of single cells to elastic deformation, as quantified by an effective elastic modulus, ranges from 102 to 105 Pa (Fig. 3),(12). The Young s Modulus of living cells is much smaller than that of metals, ceramics and polymers. The deformability of cells is determined largely by the cytoskeleton. The rigidity of the cytoskeletion is influenced by the mechanical and chemical environments including cell cell and cell ECM interactions. Figure 3: Approximate range of values of the elastic modulus of biological cells in comparisons with those of metals, ceramics and polymers (11) 7

8 1.2 Mechanical properties of cytoskeleton Actin filaments are helical polymers, 5-10 nm in diameter, with a persistence length ~ 15 μm (Fig.4). Figure 4: Actin filaments (13) They often exist as short, only a few hundred nanometers long, filaments in highest concentration near the surface of cells (14). The filaments are forming an isotropic crosslinked mesh. Also, there are longer filaments (app. 1 μm), which are able to form highly cross-linked aggregates and bundles that are much stronger than individual filaments. It also exists a sort of longer actin filaments (10 μm), which are forming an integral component of the contractile apparatus in muscle cells and stress fibers in non-muscle adherent cells (15). The actin cortex is largely responsible for controlling cell shape and dynamic surface motility. Microtubules are long, cylindrical, approximately 25 nm in diameter filaments, with a persistence length ~ 1 mm (Fig. 5). They are responsible for large scale cellular properties: stabilization of mechanical structures, force transmission across the cytoplasm, oranization of organelles and processes like cell division. 8

9 Figure 5: Microtubules (13) Important differences in the specific molecular interactions give rise to distinct kinetic and mechanical properties of microtubules, because microtubules can span the entire length of a cell, kinesin and dynein. Therefore they have been the subject of many single molecule characterization studies (16). Intermediate filaments are concentrated around cell-cell junctions, but also span the cytoplasm. They have not been as extensively characterized as the other filaments. Therefore not all of their functions are understood. Intermediate filaments are composed of polypeptide rods, arranged in rope-like fibers that are 8-10 nm in diameter (Fig. 6). They are able to resist rupture under large force and can be found in cells, which bear mechanical stress. Figure 6: Intermediate filaments (13) 9

10 Additionally, each polypeptide contains flexible head and tail domains that are variable between types of intermediate filaments. The rods are coupled anti-parallel giving intermediate filaments a non-polar structure. They are preventing directional functionality like that of actin filaments or microtubule and arranged in a staggered fashion, with a much larger contact area between monomers than in actin or microtubules. An explanation for the large stress-bearing capacity of intermediate filaments is this structure. Under large deformations intermediate filaments also exhibit a non-linear increase in tension. This is known as strain-hardening, which could also arise from complex monomer-monomer interactions along their length (17). In axons of neuronal cells intermediate filament (neurofilament) sidechains have been shown to act as disordered entropic springs that repel large particles and other filaments (18). In this way permeability as well as mechanical integrity of the axon is maintained. In smooth muscle, desmin intermediate filaments are distributed throughout the cytoplasm and are thought to support and link the contractile fibers and cytoskeletal attachment nodes (19). The mechanical properties of cytoskeleton show that a living cell sample is an inhomogenous structure. The simulation models of this thesis will use a simplified homogeneous material with small scales. The real living cell, which is a inhomogeneous material must be considered as a model with effective Young s Modulus (chapter 10: Outlook) 10

11 2 Atomic force microskopy Figure 7: Schematic diagram of the atomic force microscope (20) The atomic force microscopy (AFM, Fig. 7) is widely used to characterize the surface of all kinds of materials, conductive as well as insulating samples. Although the most innovative advent of the AFM is its ability to measure locally different physical properties, the AFM offers real 3-dimensional images of the sample surface (21). Figure 8: Cantilever tip (20) 11

12 The AFM uses a cantilever tip (Fig. 8) to measure forces and displacements. A real cantilever geometry is a pyramid with a small tip. Therefore, the cantilever geometry is considered as an axisymmetric approximation with a paraboloid (small indentations: d < R) or a cone (large indentations: d > R). Additionally, in this thesis an indenter in form of a sphere will be considered for small indentations. Typical tip radius is R = nm. In this thesis, a paraboloid with tip radius of 250 nm will be used to discuss the cantilever tip experiments with a large scale. Accuracy in nanometer precision is given with a sub-nano-newton force resolution. In its typical configuration, a cantilever is on top of a piezoelectric actuator. Positioning and scanning of the tip is available over 10s to 100s of microns with subnanometer resolution. The piezo can make a raster scan with the tip across the surface. Forces on the tip that cause deflection of the cantilever can be measured to create an image of the surface. By measuring cantilever deflections, a laser is reflected off the end of the beam into a position sensitive photodiode. The motion of the reflected laser spot, which is measured by the photodiode are the cantilever results. In the most common imaging mode, a feedback electronics loop is closed around the actuator. So, a predetermined deflection is constantly maintained by adjusting the cantilever height, as it scans across topographical features. Primary advantage of the AFM as an imaging tool is, that it does not suffer from the diffraction limitations of optical or scanning electron microscopes. Such wave-based microscopes are generally limited to a resolution of the same order as the imaging wavelength. The application of the AFM, important for this work, is to measure forces as a function of cantilever distance from the surface (22). Again, by measuring the position of a reflected laser beam, the deflection of the cantilever can be plotted as a function of distance to the surface. If the cantilever spring constant is known, this deflection can be translated into a force. Investigation of the force-indentation curve is the main part of this thesis. Such force curves are used to measure intermolecular forces of single macromolecules, such as DNA and proteins (23). If a better force resolution is required, optical tweezers will be used, but at the expense of temporal resolution. Typical force resolution of an AFM is ~100 pn with a 1 khz bandwidth (24). The capabilities of the AFM are tantalizingly close to allowing 12

13 measurement of force interactions around 5pN with unprecedented bandwidth (25), (26), (27). Estimation of Young s modulus from the force-indentation curve requires a model which is fitted to the data. 3 COMSOL simulation software 3.1 COMSOL Multiphysics COMSOL Multiphysics (Fig. 9) is a finite element method (FEM) analysis, solver and simulation software for various physics and engineering applications. It also offers an interface to MATLAB for a large variety of programming, preprocessing and postprocessing possibilities. In addition COMSOL Multiphysics also allows for entering coupled systems of partial differential equations (PDEs). The PDEs can be entered directly or using the so-called weak form. Figure 9: COMSOL Multiphysics (28) 13

14 COMSOL Multiphysics contains the following modules for different applications: AC/DC Module, Acoustics Module, CAD Import Module, CFD Module, Chemical Reaction Engineering Module, ECAD Import Module, File Import for CATIA V5, Geomechanics Module, Heat Transfer Module, LiveLink for AutoCAD, LiveLink for Excel, LiveLink for MATLAB, LiveLink for Pro/ENGINEER, LiveLink for Solid Edge, LiveLink for SolidWorks, Material Library, MEMS Module, Microfluidics Module, Nonlinear Structural Materials Module, Optimization Module, Pipe Flow Module, Plasma Module, RF Module, Structural Mechanics Module, Subsurface Flow Module. 3.2 Solid mechanics interface The interface used for this thesis is solid mechanics. It is part of the structural mechanics module, which is available for the following space dimensions: 3D-solid 2D-solid Axisymmetric solid It comprises large deformation (geometrically, nonlinear deformations), shell, beam and contact abilities. Coupled physics are possible. A separate nonlinear structural materials module is newly added to COMSOL 4.3a. Different solver types, such as stationary, timedependent, frequency-domain, eigenmode, linear buckling are included. The axisymmetric variant of the solid mechanics interface uses cylindrical coordinates r, ϕ (phi), and z. In this variant the interface allows force loads only in the r and z directions. 14

15 Figure 10: 2D axisymmetric geometry (29) In the view of 2D axisymmetric geometry, the intersection between the original axially symmetric 3D solid and the half plane is ϕ = 0, r 0. Therefore the axisymmetric projection of the geometry is only in the half plane, r 0 and recovers the original 3D solid by rotating the 2D geometry about the z-axis (Fig. 10), (30). The boundary condition for this geometry is symmetry. In this thesis also fixed constraint and prescribed displacement will be used as boundary conditions. The elastic model will be a linear elastic model. The solved dependent variables for deformations are the displacement field components u, v and w. 15

16 4 Objectives The goal of this thesis is to - Create simulation models for different geometrically forms (sphere, paraboloid, cone) of a cantilever tip to contact a living cell sample - Verification of the numerical results - Simulate the deformations of homogeneous living cell samples in order to obtain the theoretical force-indentation curve with high accuracy - Analyze the limits of the models - Compare the numerical results with the analytical results for Sneddon s analytical solution in an infinity elastic half-space and in variable sample thicknesses - Formulate an improved version of Sneddon s analytical solution - Discuss the reasons of deviations from Sneddon s analytical solution - Discuss the validity of Sneddon s analytical solution 16

17 5 Contact problem The theory of elasticity has a direct application to contact mechanics. The classic Hertzian theory (non-adhesive contact) which was modified by Sneddon is used on contact problems. 5.1 Hertzian theory Geometrical effects on local elastic deformation properties have been considered in 1880 with the Hertzian theory of elastic deformation (31). This theory relates the circular contact area between two spheres with the same radius (Fig. 11). Hertzian contact stress refers to the localized stresses that develop as the two spheres come in contact and deform slightly under the imposed loads. The contact area is supposed to be a plane area. With acting pressure, the contact stress is not constant over the contact area. This amount of deformation is depending on the Young s modulus of the material in contact. The contact stress is given as a function of the normal contact force, the radius of both spheres and the Young s modulus of both bodies. In the theory any surface interactions such as near contact Van der Waals interactions, or contact Adhesive interactions are neglected. In 1965 a modified solution of the Hertzian theory was developed by Ian N. Sneddon (32). 17

18 Figure 11: Hertzian contact between two spheres (33) The generally Hertzian equation for a pressure distribution is shown in (5.1) (5.1) The force F between both spheres and the combined Young s modulus E with ( ) ( ) (5.2) must be known. Other values of this equation are = Poisson s ratio from sphere 1 and sphere 2; = Young s modulus from sphere 1 and sphere 2; = displacement in y-direction; = curvatures of both spheres. (31) 18

19 5.2 Sneddon theory First, an elastic medium that fills an infinitely large half-space (the boundary is an infinite plane) is considered. After applying forces on the free surface, the medium is deformed. The xy-plane is on the circled area of the medium and the filled area corresponds to the positive z-direction (Fig. 12 a). The deformations in the complete half-space can be defined in analytical form (34). Here, only the formula for the displacement in the positive z-direction is needed. Figure 12: a) One point force acting on an elastic half-space; b) A distribution of forces acting on a surface (35) The displacement caused by a force, which is calculated using the following equations: (5.3) (5.4) (5.5) with In particular, one obtains the following displacements of the free surface, which is defined as z = 0: 19

20 (5.6) (5.7) (5.8) with If several forces acting simultaneously (Fig. 12 b), the displacement is a sum of the respective solutions that result from every individual force. For contact problems without friction, the z-component of the displacement (5.8) is a continuous distribution of the normal pressure p(x;y). The displacement of the surface is calculated using (5.9) with (5.10) and Sneddon s assumption of a pressure with a distribution of ( ) (5.11) with the contact radius (5.12) which is exerted on a circle-shaped area. This circle-shaped area with the radius a searches for the vertical displacement of the surface points within the area being acted upon by the pressure. The circle shaped area is caused by indentation d from a sphere of radius R. The resulting vertical displacement is (5.13) The total loading force follows as 20

21 (5.14) The displacement of the surface inside and outside of the area under pressure is shown in Fig. 13. Figure 13: Surface displacement resulting from a pressure distribution (35) 5.3 Relation between load and penetration for different axisymmetric punch profiles According to literature, Sneddon developed three formulas to describe the contact between a sphere, paraboloid, cone and an elastic half-space (32). Sneddon s method describes the contact between an axisymmetric tip and an infinite elastic half-space Equation for a punch in spherical form According to Sneddon s equation for a sphere, the applied load is given by: [ ( ) ] (5.15) with the reduced Young s Modulus 21

22 ( ). (5.16) The equation for the indentation d follows ( ). (5.17) According to equation (5.17), the indentation d depends only on geometrical properties, such as the contact radius a, and the sphere radius R. The applied load F is a function of contact radius a (5.12), sphere radius R, and mechanical properties of the elastic half-space, represented by Young s modulus E and Poisson s ratio v. With a known indentation d, the contact radius a from equation (5.17) can be calculated and be inserted into equation (5.15) Equation for a punch in paraboloid form Figure 14: Punch in paraboloid form (35) The contact between a paraboloid and an elastic half-space is shown in Fig. 14. The displacement of the points on the surface in the contact area between an even surface and a paraboloid with the radius R is equal to (5.18) To cause the displacement in (5.18), following relationship is shown in (5.19) (5.19) 22

23 Therefore, variables a and b must fulfill the following requirements:,. It follows for the contact radius (5.12) and for the maximum pressure ( ) (5.20) with the reduced Young s modulus from equation (5.10). Substituting from equation (5.12) and equation (5.20) into equation (5.14) results in obtaining a normal force of (5.21) Equation for a punch in conical form Figure 15: Punch in conical form (35) The contact between a conical indenter and an elastic half-space is shown in Fig. 15. When acting into an elastic half-space with a conical indenter, the indentation depth and the contact radius are given through the following relationship (32): Pressure distribution has the assumed form (5.22) 23

24 ( ( ) ) (5.23) and the total force is calculated as (5.24) 5.4 AFM versus Sneddon s analytical solution In case of using AFM for a very hard sample, movement in z-direction of the piezo (while in contact) will lead to a deflection d of the cantilever, which is identical to z. This means: d = z for hard samples, which can not be penetrated by a cantilever. If a soft sample is used, the cantilever furthermore can penetrate the sample. The same movement in z-direction will lead to a smaller deflection of the cantilever as a result of the elastic indentation δ. This means: d = z - δ. The loading force F is proportional to the deflection d of the cantilever and the force constant k of the cantilever. It can be written as (5.25) In a force-indentation curve the deflection as a function of the z movement by the piezo will be measured. Equation (5.23) can be used to calculate the curve from measured data set. When considering the indenting cone, Sneddon assumed an infinity sharp tip. In reality, the tip has a finite tip radius (R = 10 nm 250 nm), which is often difficult to measure. The tip radius was observed and discussed and there have been some suggested methods of receiving the tip radius of the indenter (36). Therefore in reality for small indentations (d; indentation < D; sample thickness) Sneddon s analytical solution for a paraboloid is used to 24

25 analyze the curves (5.21), while higher indentations are analyzed with the equation for a cone (5.24). 6 Simulation models A cantilever was considered on a plane elastic living cell sample and modelled as a twodimensional axisymmetric structure to simplify the mesh generation and reduce the computing time. This approach is common for the analysis of symmetrical structures. Cantilever and cell block are elastic, homogeneous, and isotropic. Cantilever and sample constitute a contact problem. 6.1 Model for a spherical form First, a model for a sphere and rectangle living cell sample in contact is shown in Fig. 16. This model describes Sneddon s method for a spherical punch into an infinity elastic half-space. It is the simplest contact geometry for COMSOL with the lowest calculation time. Fig. 16 also shows mesh and contact pair of the 2D-axisymmetric model. The dashed line on the left boundaries is the symmetry axis. 25

26 Figure 16: Simulation model for a sphere with mesh and contact pair In Fig. 17 the simulation result for a elastic deformation with spherical indenter is shown in 3D-solid by rotating the 2D-geometry about the z-axis. 26

27 Figure 17: Simulation result for elastic deformation (including mesh deformation) with spherical indentation. Mesh quality is shown in a color range between red (high) and blue (low). The scale dimensions are in micrometers. Sphere radius is 1 micrometer and the sample has a length of 5 micrometer with a variable thickness between 0.5 micrometer and 30 micrometer. As long as the mesh quality in deformation is higher than 0.3, simulation results are veliable. A sphere shows the best mesh quality. 27

28 6.2 Model for a paraboloid form A model for a paraboloid indenter in contact with a rectangle living cell sample has been set up. This model contains Sneddon s method for a paraboloid punch into an infinity elastic half-space. Fig. 18 shows mesh and contact pair of the model. Figure 18: Simulation model for a paraboloid with mesh and contact pair In Fig. 19 the simulation result for a elastic deformation with paraboloid indenter is shown in 3D-solid by rotating the 2D-geometry about the z-axis. 28

29 Figure 19: Simulation result for elastic deformation (including mesh deformation) with paraboloid indentation. Mesh quality is shown in a color range between red (high) and blue (low). The paraboloid has a tip radius of 250 nanometer and the sample has a length of 5 micrometer with a variable thickness between 0.5 micrometer and 30 micrometer. The mesh quality is still good with a lowest value of

30 6.3 Model for a conical form A model for a conical indenter in contact with a rectangle cell sample is the most difficult geometry for COMSOL and has the highest calculation time. In chapter the difficulties with this model in COMSOL will be discussed. This model contains Sneddon s method for a conical punch into an elastic half-space. Fig. 20 shows mesh and contact pair of the model. Figure 20: Simulation model for a cone with mesh and contact pair In Fig. 21 the simulation result for a elastic deformation with conical indenter is shown in 3Dsolid by rotating the 2D geometry about the z-axis. 30

31 Figure 21: Simulation result for elastic deformation (including mesh deformation) with conical indentation. Mesh quality is shown in a color range between red (high) and blue (low). The cone (α = 60 ) has no tip radius and the sample has a length of 5 micrometer with a variable thickness between 0.5 micrometer and 30 micrometer. With maximum indentation, the mesh quality comes close to 0.3, where the results become unreliable. 31

32 6.4 Simulation equations - Contact algorithm COMSOL s contact algorithm is an optimal convergence for Newton-Raphson iteration. If the gap distance between the slave and master boundaries at a given equilibrium iteration is becoming negative, (master boundary is indenting the slave boundary), the user defined normal penalty factor is expressed with Lagrange multipliers for contact pressure (37). (6.1) (6.2) The distance between two existing nodes on master and slave boundaries is the gap g, while defines the contact stiffness. - Hyperelastic material model A hyperelastic material model is often used for rubber material, which has similar properties like a living cell sample. But in this thesis a linear elastic material with a Poisson s ratio of is used in COMSOL, because it delivers similar results (Fig. 22). 32

33 Figure 22: Hyperelastic model and linear elastic model results in COMSOL - Linear elastic material model The total strain tensor is written in terms of the displacement gradient. (6.3) The Duhamel-Hooke s law relates the stress tensor or the strain tensor and temperature (6.4) C is the fourth order elasticity tensor, which stands for the double-dot tensor product (or double contraction). (29) The stress and strain is calculated as. (6.5) 33

34 6.5 Model settings in COMSOL The solid mechanics module with stationary solver is used for this 2D-axisymmetric problem. In definitions, a contact pair was created between the contact boundaries of indenter and cell sample in order to treat the contact problem. Contact friction is not activated. The different geometries from chapter 6.1 chapter 6.3 are finalized by the method form an assembly. Material proberties from both domains have a difference in stiffness. The indenter domain is always considered as a stiff material (Structural Steel) while the living cell domain is considered as a soft material (Linear elastic material model) with a Young s modulus of 20 kpa and a Poisson s ratio ν = In reality, a living cell sample is not homogeneous. But in this model, it is necessary to simulate a basic infinity elastic halfspace. Living cell samples are assumed to be perfect incompressible materials. It has a Poisson s ratio of 0.5, but COMSOL has problems with this value and doesn t converge. Therefore a close Poisson s ratio ν = is set for the sample. Other material parameters like density are set according to the default value of COMSOL. In addition, the properties were assumed to be independent of temperatures. Boundary conditions are set as follows: The contact boundary on the indenter is set as the master boundary and the contact boundary on the living cell is set as the slave boundary. Boundary conditions for the living cell domain is set as free in the x direction. So, while indentation the living cell can move in x-direction. The bottom of the living cell domain is fixed which means perfect adhesion with the Petri dish. Prescribed displacement is used for the whole indenter domain in z-direction. The maximum displacement and steps are set as parameters (10 nanometer steps, max.: 200 nanometer indentation). An accurate contact pressure can be achieved by refining the mesh around the contact zone. So, the mesh at the contact area is finer than in the bottom. In addition the moving mesh module is activated for large deformations and follows the deformations of the cantilever. COMSOL suggests that there should be at least 10 nodes along the slave contact boundary. The slave boundary should always have a finer mesh than the master boundary. In the spherical model and paraboloid model triangular elements were used and in the conical model distributed elements were used in contact area. The structural 34

35 mechanics computations use the assumption that the material is linear elastic and take geometric nonlinearities into account. 6.6 Convergence problem of the contact model Analysis is performed on the indentation between cantilver and cell sample without friction to help the solver to converge and find a solution. If the mesh becomes finer, a smaller parametric step should be chosen. The disadvantage of this method is the additional computional time for the solver to find a solution. Thus a sufficiently fine mesh size is needed in order to obtain accurate results. For larger indentation steps, a solution is calculated without reaching the tolerance criterion and the accuracy of the results can be doubted. In the contact pair parameters, the penalty factor can help the solver to start the convergence or reduce the convergence time. This is very important, if a convergence problem occurs. But for complicated geometry designs (like conical cantilever form), this estimation can be very difficult. If the solver fails to find a solution, it can be very difficult to set improved parameters because the error file gives not enough information. 35

36 7 Simulation results The simulation models, which are described in chapter 6 were made for three different cantilever geometry. With the simulation results, the analytical solutions and numerical solutions of characteristic force-indentation curves plots will be discussed for all these cantilever models. 7.1 Simulations for different indenter forms into an infinity elastic halfspace According to Sneddon theory his equations are valid for a punch in an infinite elastic halfspace (chapter 5.2). Therefore first simulations and analytical solutions are calculated for a very large simulation domain. The sample thickness is 30 micrometer by a maximum indentation of 200 nanometer, which means a ratio of 0.67 %. Therefore, sample thickness can be considered as a nearly infinite elastic half-space. An additional log x-axis log y-axis scale plot was set up for a better visualization of deviations. Fig show the solutions. 36

37 - Spherical Indenter in infinite elastic half-space: Figure 23: Numerical and analytical solution for a sphere (R=1 µm) into a thick sample (D = 30 µm) Figure 24: log x log y scale plot for a sphere (R=1 µm) into a thick sample (D = 30 µm) 37

38 - Paraboloid form indenter in infinity elastic half-space: Figure 25: Numerical and analytical solution for a paraboloid form (R=250 nm) into a thick sample (D = 30 µm) Figure 26: log x log y scale plot for a paraboloid form (R=250 nm) into a thick sample (D = 30 µm) 38

39 - Conical form indenter in infinity elastic half-space: Figure 27: Numerical and analytical solution for a conical form (α=60 ) into a thick sample (D = 30 µm) Figure 28: log x log y scale plot for a conical form (α=60 ) into a thick sample (D = 30 µm) 39

40 The numerical solution shows a mismatch between numerical and analytical solution on all three cantilever models. In a first view it seems the Young s modulus is underestimated. The deviation can reach a maximum of 23 %. The reason could be that Sneddon s formula only considers vertical forces. Radial forces are maybe neglected. Friction was always not activated in the simulation. Therefore, this can t be the reason of mismatch. In order to proof the theory with neglected radial forces, chapter 7.2 describes a new model configuration for COMSOL. 7.2 Simulations for variable Poisson s ratios In chapter 7.1 a mismatch between numerical solution and analytical solution is shown, which could be caused by Sneddon s neglection of radial forces. To reduce the radial forces in the simulation model, there is a possibility in a reduced Poisson s ratio. The Poisson effect describes when a material is compressed in one direction, and it usually tends to expand in the other two directions prependicular to the direction of compression. It is the negative ratio of transverse to axial strain (38). Thus a reduced Poisson s ratio forces the applied load more and more only into z-direction. The simulations are calculated for all three geometrical forms of the cantilever with variable Poisson s ratios, a constant Young s modulus of 20 kpa and a sample thickness of 30 µm. Fig show the numerical results, obtained for a Poisson s ratio range ν =

41 - Spherical indenter with variable Poisson s ratios: Figure 29: Numerical solutions for a sphere (R=1 µm) into a thick sample (D = 30 µm) with different Poisson s ratios ν= Paraboloid form indenter with variable Poisson s ratios: Figure 30: Numerical solutions for a paraboloid form (R=250 nm) into a thick sample (D = 30 µm) with different Poisson s ratios ν=

42 - Conical form indenter with different Poisson s ratios: Figure 31: Numerical solutions for a conical form (α=60 ) into a thick sample (D = 30 µm) with different Poisson s ratios ν= The numerical results are depending on Poisson s ratio as also observed for Sneddon s analytical solution. Comparing the numerical results with the analytical solution in chapter 7.1, the Young s modulus is underestimated by the Sneddon assumption. In all three geometrically forms, the maximum deviation occurs for a Poisson s ratio ν = Poisson s ratio s between ν = are relatively stable (max. deviation: 5%). The deviation starts growing with ν = Verification of simulation results By setting Poisson s ratio ν = 0 for the Linear elastic material model of the cell sample, the simulation has a zero radial displacement. In this case, the analytical solution matches perfect with the numerical solution for all three geometrical forms of the cantilever. Fig show new solutions, which are the verifications of the simulation results. 42

43 - Spherical indenter with Poisson s ratio ν = 0: Figure 32: Numerical and analytical solution for a sphere (R=1 µm) into a thick sample (D = 30 µm) with Poisson s ratio ν=0 Figure 33: log x log y scale plot for a sphere (R=1 µm) into a thick sample (D = 30 µm) with Poisson s ratio ν=0 43

44 - Paraboloid form indenter with Poisson s ratio ν = 0: Figure 34: Numerical and analytical solution for a paraboloid form (R=250 nm) into a thick sample (D = 30 µm) with Poisson s ratio ν=0 Figure 35: log x log y scale plot for a paraboloid form (R=250 nm) into a thick sample (D = 30 µm) with Poisson s ratio ν=0 44

45 - Conical form indenter with Poisson s ratio ν = 0: Figure 36: Numerical and analytical solution for a conical form (α=60 ) into a thick sample (D = 30 µm) with Poisson s ratio ν=0 Figure 37: log x log y scale plot for a conical form (α=60 ) into a thick sample (D = 30 µm) with Poisson s ratio ν=0 45

46 In view of these results, Sneddon s formula only considers vertical forces. Normally, a linear isotropic material subjected only to compressive forces. The deformation of a material in the direction of one axis will produce a deformation of the material along the other axis in three dimensions. But here it is not fulfilled. The radial forces are neglected and this is the main reason for a mismatch between numerical solution and analytical solution for a Poisson s ratio ν = If we consider this new insight, a material with low Poisson s ratio between could be well described by the analytical model. A view into a table of different materials (Table 1) with their Poisson s ratio shows that the perfect material for Sneddon s analytical solution should be like cork. Table 1: Different material with Poisson s ratio (39) Material Poisson s ratio rubber ~0.50 gold 0.42 magnesium 0.35 steel glass foam cork ~

47 7.4 Simulations for variable sample thicknesses After verification of the simulation results for an infinite elastic half-space in chapter 7.3, the dependence of the extracted Young s modulus for variable sample thicknesses will be analyzed in chapter In many experiments the sample is not an infinite elastic half-space, therefore the influence from thinner samples on Sneddon s analytical solution must be investigated. Thus analytical solutions were calculated for all three cantilever models with a sample thickness range between 0.5 µm 30 µm. The influence of thin samples to Sneddon s analytical solution is already a known problem (40). In addition, the force-indentation curves were calculated for a Poisson s ratio ν = and ν = 0. The method was done in following steps. Force-indentation curves were numerical calculated for all three cantilever models. Then, the data were fitted to the analytical model with equation (5.15; sphere), equation (5.21; paraboloid) and equation (5.24; cone) to extract the Young s Modulus E. Poisson s ratio ν was assumed to be known. 47

48 E [kpa] E [kpa] Simulation supported Hertzian theory calculations on living cell samples Sphere into variable sample thicknesses - Poisson s ratio ν = 0.499: Simulation Value E (Fit) Sample Thickness [µm] Figure 38: Sphere into variable sample thicknesseses and Poisson s ratio ν = Poisson s ratio ν = 0: Simulation Value E (Fit) Sample Thickness [µm] Figure 39: Sphere into variable sample thicknesseses and Poisson s ratio ν = 0 48

49 E [kpa] Simulation supported Hertzian theory calculations on living cell samples The first analytical solution with a Poisson s ratio ν = (Fig. 38) shows an underestimated Young s modulus for thick samples and an overestimated Young s modulus of 50 % for thin samples. If the Poisson s ratio of simulation is set to ν = 0 (Fig. 39), the analytical solution fulfills Sneddon s analytical solution, but the overestimated Young s modulus for thin samples reaches 193 %. It is another proof that Sneddon s formula neglects radial forces, which causes a deviation from the simulation value of the Young s modulus. Thin samples are causing an overestimated Young s modulus, because the hardening influence of the underlying petri-dish is growing with decreasing sample thickness Paraboloid form into variable sample thicknesses - Poisson s ratio ν = 0.499: Simulation Value E (Fit) Sample Thickness [µm] Figure 40: Paraboloid form into variable sample thicknesseses and Poisson s ratio ν =

50 E [kpa] Simulation supported Hertzian theory calculations on living cell samples - Poisson s ratio ν = 0: Simulation Value E (Fit) Sample Thickness [µm] Figure 41: Paraboloid form into variable sample thicknesseses and Poisson s ratio ν = 0 It is the similar result than in the previous chapter. The analytical solution with a Poisson s ratio ν = (Fig. 40) shows an underestimated Young s modulus for thick samples, which can be corrected by setting the simulations Poisson s ratio ν = 0 (Fig. 41). Again, thin samples are causing an overestimated Young s modulus (deviation for ν = 0.499: 49 %; deviation for ν = 0: 91 %) as a consequence of the rising influence of the stiff petri-dish to the measurement. 50

51 E [kpa] E [kpa] Simulation supported Hertzian theory calculations on living cell samples Conical Indenter for variable sample thicknesses - Poisson s ratio ν = 0.499: Simulation Value E (Fit) Sample Thickness [µm] Figure 42: Conical form into variable sample thicknesseses and Poisson s ratio ν = Poisson s ratio ν = 0: Simulation Value E (Fit) Sample Thickness [µm] Figure 43: Conical form into variable sample thicknesseses and Poisson s ratio ν = 0 51

52 Unfortunately, COMSOL has many problems with non-curved structures as a contact pair. For sharp cantilever tips (α < 45 ) and a sample with Poisson s ratio ν = 0, the solver doesn t converge. Therefore, a wide cone (α = 60 ) was used for the simulations, and the solver could converge. Coincidentally, in chapter 7.3 a cone simulation has shown similar manner like a sphere and a paraboloid in the force-indentation curve. But calculations for variable sample thicknesses show large deviations from the numerical solution (Fig ). A first attempt to solve this problem was to set up a finer mesh in contact area, followed by an activated adaptive mesh refinement. Another attempt was a new configuration for the solver. The stationary solver type was changed from MUMPS to PARADISO and finally SPOOLES. Also the relative tolerance of the study steps was increased. As a last step, the segregated solver s maximum number of iterations and the scaling of the dependent variables were increased. However, the numerical result could not be improved. There seems to be an algorithmic problem in the COMSOL solver. The solver fails to find a solution, if the initial gap between cantilever and deformable surfaces is too high. The maximum error seems to appear for a conical indenter. 52

53 Young s Modulus [kpa] Young s Modulus [kpa] Simulation supported Hertzian theory calculations on living cell samples 7.5 Thick-Thin sample comparison - Spherical indenter: ,5 1 1,5 2 2,5 Force [nn] Thin Sample Thick Sample Figure 44: Thick (20 µm) Thin (0.5 µm) comparison for a spherical indenter - Paraboloid form indenter: Thin Sample Thick Sample ,5 1 1,5 Force [nn] Figure 45: Thick (20 µm) Thin (0.5 µm) comparison for a paraboloid form indenter 53

54 - Conical form indenter: In chapter 7.4.3, problems with conical form indenter were already discussed. Because of explained problems, a veliable simulation was not possible. Thick samples give always a value close to the analytical solution to the defined Young s modulus (20kPa) from the simulation results (Fig ). However, for thin samples and high loading forces, the calculated Young s modulus is rising to >20kPa. The effect was already discussed in chapter 7.4. Only for very small forces, the calculated Young s modulus approached for the value of the thick film within an accuracy of 60%. If the loading forces would approach zero, the calculated Young s modulus might reach the value of the thick film. 54

55 8 Improvement of Sneddon s analytical solution This section presents a correction factor to Sneddon s analytical solution for a paraboloid to improve the validy from ν = 0 to all values of ν for a nearly infinite elastic half-space. Sneddon s analytical solution for a sphere is not used in AFM-applications. Therefore only Sneddon s formula for the paraboloid form indenter will be extended, because simulations for a conical form were not valid. The extension will correct the changing properties with variable Poisson s ratios. First, a polynomial correction value is fitted depending on variable Poisson s ratios. The range of Poisson s ratio s is between ν= The numerical results are used from a 30µm thick sample. The fit calculations are plotted in Fig. 46. Figure 46: Polynomial fitting curve of correction values depending on Poisson s ratios ν = The best fit is obtained for a polynomial equation of the form: (8.1) 55

56 with the coefficient values 004. The extension will be implemented in Sneddon s formula for a paraboloid: (8.2) which results in (8.3) with. The calculations from chapter 7.1 and chapter for a paraboloid are repeated with the extended equation. Improved solutions with this extension are shown in Fig. 47, Fig. 48 and Fig

57 - Paraboloid form indenter with Poisson s ratio ν = Figure 47: New Numerical and analytical solution for a paraboloid form (R=250 nm) into a thick sample (D = 30 µm) with ν = Figure 48: New log x log y scale plot for a paraboloid form (R=250 nm) into a thick sample (D = 30 µm) with ν =

58 E [kpa] Simulation supported Hertzian theory calculations on living cell samples - Paraboloid form into variable sample thicknesses Simulation Value E (Fit) Sample Thickness [µm] Figure 49: New calculations of a paraboloid form into variable sample thicknesseses and Poisson s ratio ν = Alternatively to the polynomial correction factor, fit function (8.4) with only one coefficient value can be implemented in Sneddon s formula. Figure 50: Fitting curve of correction value depending on Poisson s ratios ν =

59 This extension has less coefficient values but slightly lower accuracy. (8.4) with the coefficient value which results in ( ) (8.5) Equation 8.5 still contains a correction coefficient A = This is attributed to the fact that the numerical simulation has been done with a sample thickness of 30 µm. The factor A is assumed to be A = 1 for an infinite half-space sample. 59

60 9 Additional results Figure 51: Pressure distribution from COMSOL and Sneddon s analytical assumption The neglection of radial forces in Sneddon s analytical solution appears in his assumption about the pressure distribution, which is described by equation (5.11) for a paraboloid form indenter. This assumption can be compared to the numerical result of the pressure distribution. Fig. 51 shows incorrectness of this assumption for ν =

61 Force [nn] Simulation supported Hertzian theory calculations on living cell samples 1,6 1,4 1,2 1 0,8 0,6 Sphere (v = 0.499) Paraboloid (v = 0.499) 0,4 0, Indentation [nm] Figure 52: Geometrical form effect from sphere and paraboloid A final analysis of the influence from different cantilever tip forms is shown in Fig. 52. The numerical solutions were calculated for a sphere and paraboloid with the same radius R = 250 nanometer, which is close to maximum indentation in order to see the highest deviation. For a ratio of 20% between indentation and tip radius, the force-indentation curves of both tip forms are equal. For a ratio > 20% a growing deviation appears, caused by the geometrically differences. This means, for a very sharp cantilever with spherical tip geometry, deviations in the analytical solution will occur. 61

62 10 Discussion of the results and conclusions Sneddon s analytical solution is a simple way for solving a class of indentation problems in elastic isotropic infinite half-spaces. It assumes a frictionless, non adhesive, normal indentation. The analytical solutions are an extension of Hertzian theory to derive the forceindentation relationship for different geometrically forms, such as a spherical-, a paraboloid form-, and a conical indenter. The analytical solutions are derived under the following three assumptions. 1. An infinity half-space is required 2. An ideal geometry for the indenter (perfect sphere, perfect paraboloid, perfect cone) with known parameters for his equation are required 3. A linear elastic and incompressible material is required In addition, there is further implicit assumption that Sneddon s analytical solution neglects radial forces. This means, his method only describes forces in z-direction. A simulation model with an infinite elastic half-space should proof it, because according to Sneddon s assumption infinite thickness yields the best analytical solution. A nearly infinite elastic halfspace was created by a 30µm thick homogeneous sample in COMSOL. This sample was penetrated by a cantilever tip. The indentation was 200nm, which means 0.67 % of the sample thickness, which is enough to ensure that most of the indentation energy is stored in the sample without influence of the bottom. After observing an underestimated Young s modulus (max. 23 %) for this sample a new model configuration was necessary to proof the assumption of neglected radial forces. The simulations with variable Poisson s ratios have indicated that radial forces must be included in order to obtain valid results for the Young s modulus from force-indentation curves. This means, an extension with correction factor must be implemented into Sneddon s formula. The requirements on the validity of the analytical solutions are limitations for applications to many experimental situations. For example, a conical indenter of infinitely sharpness is impossible to fulfill in real experiments. In experiments with an AFM the cantilever will always have a radius on the tip. These deviations from requirements of Sneddon s analytical 62