Simulation supported Hertzian theory calculations on living cell

Transcrição

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
Simulation supported Hertzian theory calculations on living cell samples
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
Contents
1
Introduction .................................................................................................................................... 5
1.1
Living cell mechanics................................................................................................................... 5
1.2
Mechanical properties of cytoskeleton ..................................................................................... 8
2
Atomic force microskopy .............................................................................................................. 11
3
COMSOL simulation software ...................................................................................................... 13
3.1
COMSOL Multiphysics............................................................................................................... 13
3.2
Solid mechanics interface ......................................................................................................... 14
4
Objectives...................................................................................................................................... 16
5
Contact problem ........................................................................................................................... 17
5.1
Hertzian theory ......................................................................................................................... 17
5.2
Sneddon theory ........................................................................................................................ 19
5.3
Relation between load and penetration for different axisymmetric punch profiles ............. 21
5.3.1
Equation for a punch in spherical form................................................................................ 21
5.3.2
Equation for a punch in paraboloid form ............................................................................ 22
5.3.3
Equation for a punch in conical form ................................................................................... 23
5.4
6
AFM versus Sneddon´s analytical solution .............................................................................. 24
Simulation models ........................................................................................................................ 25
6.1
Model for a spherical form ....................................................................................................... 25
6.2
Model for a paraboloid form .................................................................................................... 28
6.3
Model for a conical form .......................................................................................................... 30
6.4
Simulation equations ................................................................................................................ 32
6.5
Model settings in COMSOL ....................................................................................................... 34
6.6
Convergence problem of the contact model ........................................................................... 35
7
Simulation results ......................................................................................................................... 36
7.1
Simulations for different indenter forms into an infinity elastic half-space .......................... 36
7.2
Simulations for variable Poisson´s ratios ................................................................................. 40
7.3
Verification of simulation results ............................................................................................. 42
7.4
Simulations for variable sample thicknesses ........................................................................... 47
7.4.1
Sphere into variable sample thicknesses ............................................................................. 48
7.4.2
Paraboloid form into variable sample thicknesses ............................................................. 49
7.4.3
Conical Indenter for variable sample thicknesses ............................................................... 51
7.5
Thick-Thin sample comparison ................................................................................................. 53
3
8
Improvement of Sneddon´s analytical solution .......................................................................... 55
10
Discussion of the results and conclusions................................................................................ 62
11
Outlook...................................................................................................................................... 64
12
References................................................................................................................................. 65
13
List of Figures ............................................................................................................................ 68
4
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
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 1–100 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
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
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
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
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
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
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 = 10 – 250 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
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
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
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
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
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
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;
sphere 2; = displacement in y-direction;
= Young´s modulus from sphere 1 and
= curvatures of both spheres. (31)
18
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
(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
∫
(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.
5.3.1 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
(
).
(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).
5.3.2 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
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)
5.3.3 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):
(5.22)
Pressure distribution has the assumed form
23
(
√( )
)
(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
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
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
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
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
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 0.5.
29
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 7.4.3 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
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
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
0.499 is used in COMSOL, because it delivers similar results (Fig. 22).
32
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
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 = 0.499. 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 = 0.499 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
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
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. 23 – 28 show the solutions.
36
-
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
-
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
-
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
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.
29 - 31 show the numerical results, obtained for a Poisson´s ratio range ν = 0 – 0.499.
40
-
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=0 – 0.499
-
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=0 – 0.499
41
-
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=0 – 0.499
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 =
0.499. Poisson´s ratio´s between ν = 0 – 0.2 are relatively stable (max. deviation: 5%). The
deviation starts growing with ν = 0.3.
7.3 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. 32 –
37 show new solutions, which are the verifications of the simulation results.
42
-
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
-
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
-
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
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 = 0.499.
If we consider this new insight, a material with low Poisson´s ratio between 0 – 0.2 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
0.27 – 0.30
glass
0.18 – 0.3
foam
0.10 - 0.40
cork
~0.00
46
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 7.4.1 – 7.4.3. 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 = 0.499 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
7.4.1 Sphere into variable sample thicknesses
-
Poisson´s ratio ν = 0.499:
35
30
E [kPa]
25
20
Simulation Value
15
E (Fit)
10
5
0
0
10
20
30
40
Sample Thickness [µm]
Figure 38: Sphere into variable sample thicknesseses and Poisson´s ratio ν = 0.499
Poisson´s ratio ν = 0:
70
60
50
E [kPa]
-
40
Simulation Value
30
E (Fit)
20
10
0
0
10
20
30
40
Figure 39: Sphere into variable sample thicknesseses and Poisson´s ratio ν = 0
48
The first analytical solution with a Poisson´s ratio ν = 0.499 (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.
7.4.2 Paraboloid form into variable sample thicknesses
35
30
25
E [kPa]
-
20
Simulation Value
15
E (Fit)
10
5
0
0
10
20
30
40
Figure 40: Paraboloid form into variable sample thicknesseses and Poisson´s ratio ν = 0.499
49
-
45
40
35
E [kPa]
30
25
20
Simulation Value
15
E (Fit)
10
5
0
0
10
20
30
40
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 = 0.499 (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
7.4.3 Conical Indenter for variable sample thicknesses
-
35
30
E [kPa]
25
20
Simulation Value
15
E (Fit)
10
5
0
0
10
20
30
40
Figure 42: Conical form into variable sample thicknesseses and Poisson´s ratio ν = 0.499
40
35
30
25
E [kPa]
-
20
Simulation Value
15
E (Fit)
10
5
0
0
10
20
30
40
Figure 43: Conical form into variable sample thicknesseses and Poisson´s ratio ν = 0
51
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. 42 – 43). 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
7.5 Thick-Thin sample comparison
-
Spherical indenter:
45
Young´s Modulus [kPa]
40
35
30
25
20
Thin Sample
15
Thick Sample
10
5
0
0
0,5
1
1,5
2
2,5
Force [nN]
Figure 44: Thick (20 µm) – Thin (0.5 µm) comparison for a spherical indenter
Paraboloid form indenter:
40
35
Young´s Modulus [kPa]
-
30
25
20
Thin Sample
15
Thick Sample
10
5
0
0
0,5
1
1,5
Force [nN]
Figure 45: Thick (20 µm) – Thin (0.5 µm) comparison for a paraboloid form indenter
53
-
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. 44 – 45). 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
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= 0 – 0.5. 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 = 0 – 0.5
The best fit is obtained for a polynomial equation of the form:
(8.1)
55
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 7.4.2 for a paraboloid are repeated with the
extended equation. Improved solutions with this extension are shown in Fig. 47, Fig. 48 and
Fig. 49.
56
-
Paraboloid form indenter with Poisson´s ratio ν = 0.499
Figure 47: New Numerical and analytical solution for a paraboloid form (R=250 nm) into a thick sample (D = 30 µm) with
ν = 0.499
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 = 0.499
57
-
Paraboloid form into variable sample thicknesses
40
35
30
E [kPa]
25
20
Simulation Value
15
E (Fit)
10
5
0
0
10
20
30
40
Figure 49: New calculations of a paraboloid form into variable sample thicknesseses and Poisson´s ratio ν = 0.499
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 = 0 – 0.5
58
This extension has less coefficient values but slightly lower accuracy.
(8.4)
with the coefficient value
1.0855
which results in
(
)
(8.5)
Equation 8.5 still contains a correction coefficient A = 1.0855. 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
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 = 0.499.
60
1,6
1,4
Force [nN]
1,2
1
0,8
Sphere (v = 0.499)
Paraboloid (v = 0.499)
0,6
0,4
0,2
0
0
50
100
150
Indentation [nm]
200
250
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
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
solution are inevitably transferred as errors to the derived values. To estimate the form
effects of the cantilever, three different models for the indenter were considered: sphere,
paraboloid and a cone. The different types of cantilever tips have lead to the same analytical
and numerical results for an infinity elastic half-space, if the Poisson´s ratio is ν~0.0.
Unfortunately the created finite element model (FEM) for a cone indenter was difficult to
solve in COMSOL such as further conclusions could not be drawn.
Another point from the analysis of the simulation results is the significant dependence of the
results on sample thickness. If the sample thickness comes close on the order of the applied
indentation, the cantilever strongly feels the underlying petri-dish while indenting. With
Sneddon´s analytical solution, the sample will always appear to be a lot stiffer than it actually
is. If the Poisson´s ratio of the sample is ν = 0, the overestimated Young´s modulus can reach
193 % in case of a sphere, while a paraboloid reaches 91 %! This means, an extension of
Sneddon´s formula with a correction factor for thin samples must be implemented.
The effect of radial forces can simply be added to Sneddon´s formula with a correction factor
including Poisson´s ratio. Two different correction factors for Sneddon´s formula are
proposed, valid for Poisson´s ratio´s ν = 0 – 0.5.
It seems that a correction coefficient A = 1.0855 is still necessary to obtain a perfect fit to the
numerical solution for a sample thickness of 30µm. Numerical results for the thickness
dependence are available for a thickness correction factor.
The neglection of radial forces in Sneddon´s analytical solution appears in his assumption
about the pressure distribution. The comparison between Sneddon´s assumption and the
numerical pressure distribution shows a mismatch for ν = 0.499.
Finally, the influence of different cantilever tip geometry (sphere, paraboloid) with same tip
radius was analyzed to see a growing deviation in the force-indentation curve by a ratio
(indentation/tip radius) > 20%.
With the improvements of Sneddon´s formula, a clear separation of effects from neglected
radial forces and finite sample thickness has been done.
63
11 Outlook
The accuracy of Sneddon´s analytical solution for a homogenous half-space was discussed in
this thesis. But living cell samples are complex structures. In a next step, the influence of
collagen fibres should be analyzed. Collagen fibres are thin (D = 80 nm – 120 nm) and often
appear in a mesh structure, which has a higher Young´s Modulus than the rest of the cell
sample. It is interesting to analyze how accurate the Young´s modulus of a single collagen
fibre can be estimated. The best solution would be a modification of Sneddon´s formula,
which could estimate an effective Young´s Modulus of the collagen fibres and the softer cell
sample. To obtain qualitative solutions, a 3D-model in COMSOL is necessary. In such a way,
different collagen structure configurations can be analyzed. A first example for such a model
is shown in (Fig. 53).
Figure 53: 3D-model of an indenting AFM tip into a living cell sample with collagen fibres
64
12 References
(1) Alberts, B. et al. Molecular Biology of the Cell 4th edn (Garland, New York, 2002)
(2) Fung, Y. C. Biomechanics: Mechanical Proberties of Living Tissues 2nd edn (Springer,
New York, 1993)
(3) Fung, Y. C. Biomechanics: Motion, Flow, Stress and Growth (Springer, New York,
1990)
(4) Fung, Y. C. Biomechanics: Circulation 2nd edn (Springer, New York, 1997)
(5) Bustamante, C. Bryant, Z. & Smith, S. B. Ten years of tension: single-molecule DANN
mechanics. Nature 421, 423 – 427 (2003)
(6) Leckband, D. Measuring the forces that control protein interactions. Annu. Rev.
Biophys. Biomol. Struct. 29, 1 – 26 (2000)
(7) Chen, C. S. Mrksich, M., Huan, S., Whitesides, G. M. & Ingber, D. E. Geometric control
of cell life and death. Science 276, 1425 – 1428 (1997)
(8) Mc Cormick, S. M. et al. DANN microarray reveals changes in gene expression of
shear stressed human umbilical vein endothelial cells. Proc. Natl Acad Sci. USA 98,
8955 – 8960 (2001)
(9) Stossel, T. P. On the crawling of animal cells. Science 260, 1086 – 1094 (1993)
(10) Ingber, D. E. Mechanical signaling and the cellular response to extracellular matrix
in angiogenesis and cardiovascular physiology. Circ. Res. 91, 877 – 887 (2002)
(11) S. Suresh, (2003) Cell and molecular mechanics of biological materials, Nature
Materials Vol. 2, Nature Publishing Group, p. 716
(12) Hochmuth, R. M. Micropipette aspiration of living cells. J. Biomech. 33, 15 – 22
(2000)
(13) Michael E. Manwaring, Jennifer F. Walsh, Patrick A. Tresco, (2004) Contact guidance
induced organization of extracellular matrix, Biomaterials, Vol. 25, Elsevier, pp. 3631
- 3638
(14) Stossel, T. P., J. Condeelis, L. Cooley, J. H. Hartwig, A. Noegel, M. Schleicher, S. S.
Shapiro, Filamins as integrators of cell mechanics and signalling, Nat. Rev. Mol. Cell
Biol. 2, 138 – 145 (2001)
65
(15) Huxley, H., J. Hanson, Changes in the cross-striations of muscle during contraction
and stretch and their structural interpretation, Nature 173, 973 – 976 (1954)
(16) Jonathon Howard: Mechanics of Motor Proteins and the Cytoskeleton, Sinauer,
2001
(17) Janmey, P. A., S. Hvidt, F. George, J. Lamb, and T. Stossel. (1990). Nature, 347:95
(18) KUMAR, M.S. and OWENS, G.K. (2003). Combinatorial control of smooth
musclespecific gene expression. Arterioscler Thromb Vasc Biol 23: 737-47
(19) Draeger, A., Amos, W. B., Ikebe, M. and Small, J. V. (1990). The cytoskeletal and
contractile apparatus of smooth muscle: contraction bands and segmentation of the
contractile elements. J. Cell Biol. 111, 2463-2473
(20) J. J. van der Loo, J. Jacot, P.H.M. Bovendeerd, (2008) The Development in Cardiac
Stiffness in Embryonic, Neonatal and Adult Mice Evaluated with Atomic Force
Microscopy, Internship, p. 2
(21) Cuenot, S., Fretigny, C., Demoustier-Champagne, S. & Nysten B., (2003),
Measurement of elastic modulus of nanotubes by resonant contact atomic force
microscopy, J. Appl. Phys. 93, 5650
(22) D. Rugar, H. J. Mamin, P. Guethner, S. E. Lambert, J. E. Stern, I. McFadyen, and T.
Yogi, "Magnetic force microscopy: general principles and application to longitudinal
recording media," J. Appl. Phys., vol. 68, no. 3, pp. 1169-1183, 1990.
(23) Florin, E.-L.; Moy, V. T.; Gaub, H. E., Science (1994), 264, 415 – 417
(24) S. P. Jarvis, U. Duerig, M. A. Lantz, H. Yamada, and H. Tokumoto, "Feedback
stabilized force-sensors: a gateway to the direct measurement of interaction
potentials," Applied Physics A (Materials), vol. 66, pp. S211-S213, 1998.
(25) Radmacher, M.; Fritz, M.; Kacher, C. M.; Cleveland, J. P.; Hansma, P. K. Biophys. J.
(1996), 70, 556 – 567
(26) Hofmann, U. G.; Rotsch, C.; Parak, W. J.; Radmacher, M. J. Struct. Biol. (1997), 119,
84 – 91
(27) G. Stemme, "Resonant silicon sensors," Journal of Micromechanics and
Microengineering, Vol. 1, pp. 113-125, (1991)
(28) COMSOL, http://www.comsol.de/press/news/article/624/
(29) COMSOL Multiphysics User´s Guide, (2010), COMSOl 4.1, p. 504
66
(30) COMSOL Multiphysics User´s Guide, (2010), COMSOl 4.1
(31) H. Hertz, Über die Berührung fester elastischer Körper, Macmillan, London, 1896, p.
156
(32) I. N. Sneddon, The relation between load and penetration in the asisymmetric
Boussinesq problem for a punch of arbitrary profile, Int. J. Engng Sci. Vol. 3, pp. 47 –
57, Pergamon Press, 1965
(33) Beichuan Yan, Richard A. Regueiro, Stein Sture, (2010) Three-dimensional ellipsoidal
discrete element modeling of granular materials and its coupling with finite element
facets, Engineering Computations, Vol. 27 Iss: 4, pp. 519 - 550
(34) L. D. Landau, E. M. Lifschitz, Theory of elasticity, Theoretical Physics, Vol. 7,
Butterworth-Heinemann (1999)
(35) Valentin L. Popov, Contact Mechanics and Friction, Springer, (2010), pp. 56 – 64
(36) Shih, C. W., Yang, M., Li, J. C. M., (1991), Effect of tip radius on nanoindentation, J.
Mater, Res. 6, 2623 – 2628
(37) A. Faraji, (2005), Elastic and Elastoplastic Contact Analysis using Boundary Elements
and Mathematical Programming, Topics in Engineering, Vol. 45
(38) Lekhnitskii, SG., 1963, Theory of elasticity of an anisotropic elastic body, Holden-Day
Inc.
(39) Boresi, A. P, Schmidt, R. J. and Sidebottom, O. M., (1993), Advanced Mechanics of
Materials, Wiley.
(40) Domke, J., Radmacher, M., (1998), Measuring the Elastic Properties of Thin Polymer
Films with the Atomic Force Microscope, Langmuir, 14, 3320 – 3325
67
13 List of Figures
Figure 1: Cell structure (11) ..................................................................................................................... 6
Figure 2: Cell membrane (11) .................................................................................................................. 6
Figure 3: Approximate range of values of the elastic modulus of biological cells in comparisons with
those of metals, ceramics and polymers (11) ......................................................................................... 7
Figure 4: Actin filaments (13) .................................................................................................................. 8
Figure 5: Microtubules (13) ..................................................................................................................... 9
Figure 6: Intermediate filaments (13) ..................................................................................................... 9
Figure 7: Schematic diagram of the atomic force microscope (20) ...................................................... 11
Figure 8: Cantilever tip (20) ................................................................................................................... 11
Figure 9: COMSOL Multiphysics (28) ..................................................................................................... 13
Figure 10: 2D axisymmetric geometry (29) ........................................................................................... 15
Figure 11: Hertzian contact between two spheres (33) ........................................................................ 18
Figure 12: a) One point force acting on an elastic half-space; b) A distribution of forces acting on a
surface (29) ............................................................................................................................................ 19
Figure 13: Surface displacement resulting from a pressure distribution (29) ...................................... 21
Figure 14: Punch in paraboloid form (29) ............................................................................................. 22
Figure 15: Punch in conical form (29).................................................................................................... 23
Figure 16: Simulation model for a sphere with mesh and contact pair ................................................ 26
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). ................... 27
Figure 18: Simulation model for a paraboloid with mesh and contact pair ......................................... 28
Figure 19: Simulation result for elastic deformation (including mesh deformation) with paraboloid
Figure 20: Simulation model for a cone with mesh and contact pair ................................................... 30
Figure 21: Simulation result for elastic deformation (including mesh deformation) with conical
Figure 22: Hyperelastic model and linear elastic model results in COMSOL ........................................ 33
Figure 23: Numerical and analytical solution for a sphere (R=1 µm) into a thick sample (D = 30 µm) 37
Figure 24: log x – log y scale plot for a sphere (R=1 µm) into a thick sample (D = 30 µm) ................... 37
Figure 25: Numerical and analytical solution for a paraboloid form (R=250 nm) into a thick sample (D
= 30 µm) ................................................................................................................................................ 38
Figure 26: log x – log y scale plot for a paraboloid form (R=250 nm) into a thick sample (D = 30 µm) 38
Figure 27: Numerical and analytical solution for a conical form (α=60°) into a thick sample (D = 30
µm) ........................................................................................................................................................ 39
Figure 28: log x – log y scale plot for a conical form (α=60°) into a thick sample (D = 30 µm) ............. 39
Figure 29: Numerical solutions for a sphere (R=1 µm) into a thick sample (D = 30 µm) with different
Poisson´s ratios
ν=0 – 0.499 ........................................................................................................... 41
Figure 30: Numerical solutions for a paraboloid form (R=250 nm) into a thick sample (D = 30 µm) with
different Poisson´s ratios ν=0 – 0.499 ................................................................................................... 41
68
Figure 31: Numerical solutions for a conical form (α=60°) into a thick sample (D = 30 µm) with
different Poisson´s ratios ν=0 – 0.499 ................................................................................................... 42
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 ........................................................................................................................ 43
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
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 ......................................................................................................... 44
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
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 ................................................................................................................. 45
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
Figure 38: Sphere into variable sample thicknesseses and Poisson´s ratio ν = 0.499........................... 48
Figure 39: Sphere into variable sample thicknesseses and Poisson´s ratio ν = 0 .................................. 48
Figure 40: Paraboloid form into variable sample thicknesseses and Poisson´s ratio ν = 0.499 ............ 49
Figure 41: Paraboloid form into variable sample thicknesseses and Poisson´s ratio ν = 0 ................... 50
Figure 42: Conical form into variable sample thicknesseses and Poisson´s ratio ν = 0.499 ................. 51
Figure 43: Conical form into variable sample thicknesseses and Poisson´s ratio ν = 0......................... 51
Figure 44: Thick (20 µm) – Thin (0.5 µm) comparison for a spherical indenter .................................... 53
Figure 45: Thick (20 µm) – Thin (0.5 µm) comparison for a paraboloid form indenter ........................ 53
Figure 46: Polynomial fitting curve of correction values depending on Poisson´s ratios ν = 0 – 0.5 .... 55
Figure 47: New Numerical and analytical solution for a paraboloid form (R=250 nm) into a thick
sample (D = 30 µm) with ν = 0.499 ........................................................................................................ 57
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 = 0.499................................................................................................................................. 57
Figure 49: New calculations of a paraboloid form into variable sample thicknesseses and Poisson´s
ratio ν = 0.499........................................................................................................................................ 58
Figure 50: Fitting curve of correction value depending on Poisson´s ratios ν = 0 – 0.5........................ 58
Figure 51: Pressure distribution from COMSOL and Sneddon´s analytical assumption ....................... 60
Figure 52: Geometrical form effect from sphere and paraboloid......................................................... 61
Figure 53: 3D-model of an indenting AFM tip into a living cell sample with collagen fibres ................ 64
69
Acknowledgement
I would like to thank Prof. Dr. rer. nat. Alfred Kersch who guided me thorough this project
with patience and gave me advice anytime. I am also grateful to Prof. Dr. rer. nat. Hauke
Clausen-Schaumann who offered me the project and for supporting the idea of this thesis.
Thanks to the lab team of Prof. Hauke Clausen-Schaumann for “Nanoanalytics and
Biophysics” and special thanks to Carina Prein who supported me in understanding the AFM
technique. Very great thanks also to my family for supporting me all the time.
70
Erklärung
Gemäß § 13 Abs. 4 RaPo
Name: Florian Biersack
Geboren am 27.02.1982 in Penzberg
Matrikelnummer: 22987402
Ich erkläre hiermit eidesstaatlich, dass ich die vorliegende Arbeit selbständig angefertigt
habe. Die aus fremden Quellen direkt oder indirekt übernommenen Gedanken sind als
solche kenntlich gemacht. Diese Arbeit wurde bisher keiner anderen Prüfungsbehörde
vorgelegt.
_________________________
______________________
Ort, Datum
Unterschrift
71

Simulation supported Hertzian theory calculations on living cell

Transcrição

Documentos relacionados

Can Bt maize change the spatial distribution of predator Cycloneda

here - Gecad

Page 1 of 4 IDENTIFICADOR: 6682d4df-c11c-4076-9189

Supporting Computer Tools for Learning/Teaching Heat Transfer

FREE AND OPEN-SOURCE SIMULATION SOFTWARE “URURAU

Bayesian methods for air pollution data

July 10-11, 2012

Physics-based Real Time Laparoscopic Electrosurgery

Development of an Interface to a Spacecraft Simulator

experimental and numerical simulation of the backward