stojkovic - Experimental Particle Physics Department
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Transcript stojkovic - Experimental Particle Physics Department
University of Ljubljana
Faculty of Mathematics and Physics
Biljana Stojković
Mentor: Prof. Dr Igor Poberaj
Ljubljana, December 4th, 2012
Outline
Introduction
Microrheology
Optical tweezers
Passive Microrheology
Active Microrheology
Rheology of bacterial network
Future work
Microrheology Rheology
Rheology is the study of the deformation and flow of a
material in response to applied force.
solid
polymers
bacteria
DNA
materials
properties
gels
foams
fluid
V
I
S
K
O
E
L
A
S
T
I
C
Applying oscillatory shear strain:
𝜸 𝒕 = 𝜸𝟎 𝒔𝒊𝒏(𝝎𝒕)
Resultant shear stress: 𝜎 𝑡 = 𝜎0 𝑠𝑖𝑛(𝜔𝑡 + 𝛿)
𝜎 𝑡 = 𝛾0 𝐺 ′ 𝜔 sin 𝜔𝑡 + 𝐺 ′′ 𝜔 cos(𝜔𝑡)
𝐺 ∗ 𝜔 = 𝐺 ′ 𝜔 + 𝑖𝐺 ′′ 𝜔
𝐺′′
tan 𝛿 =
𝐺′
𝐺′′
𝜂=
𝜔
Microrheology is
“rheology on the micrometer length scale”
» Microscopic probe particles
» Locally measure viscoelastic parameters
» Study of heterogeneous environments
» Requires less than 10 microliters of sample
» Biological samples – limited amount of material
» Important for fundamental reaserch and in industrial applycations
Current techniques can be divided into two main categories:
•
active methods that involve probe manipulation
•
passive methods that rely on thermal fluctuations of the probe
Technique in microrheology
Optical tweezers technique
Tightly focused laser beam
Dielectric particles with
surrounding medium
higher
refraction
index
that
Wavelength of the laser size of the object being trapped
Maximum force strenght is in the range of 0.1-100 pN
Powerful laser beam (power on sample 10 − 100 mW)
Microscope objective with high numerical aperture
(𝑁𝐴 = 𝑛 sin 𝜃 ≥ 1)
of
How we could describe the trapping of dielectric bead?
R<<λ, point dipol
λ R
𝑭𝒕𝒓𝒂𝒑 = −𝒌𝒙
R>>λ, ray optics
Optical tweezers set-up
Force calibration
• Bead is held in stationary trap
• Equation of motion:
𝑑2 𝑥
𝑑𝑥
𝑚 2 = −𝑘𝑥 − 𝛾
+ 𝐹(𝑡)
𝑑𝑡
𝑑𝑡
• Power Spectral Density (PSD):
𝑋(𝑓)
2
𝑘𝐵 𝑇
= 2 2
𝛾𝜋 (𝑓𝑐 + 𝑓 2 )
𝒌
𝒇𝒄 =
𝟐𝝅𝜸
𝑿(𝟎)
𝟐
𝟒𝜸𝒌𝑩 𝑻
=
𝒌𝟐
Force calibration
• Boltzman statistic
• In the equilibrium, the probability density of
the 1D particle position:
𝜌 𝑥, 𝑦 = 𝐶
𝑈(𝑥)
−𝑘 𝑇
𝑒 𝐵
• Trap potential can be obtained from
normalization histogram of trapped particle
postition as:
𝜌(𝑥)
𝑈 𝑥 = −𝑘𝐵 𝑇 ln
𝐶
• Fit parabola with:
𝑦 = 𝑎𝑥 2 + 𝑏
𝒌𝒙 = 𝒂/𝒌𝑩 𝑻
Passive microrheology
• Brownian motion
• Two ways for determination shear modulus:
𝑟 2 (τ) = [𝑟 𝑡 + τ − 𝑟 𝑡 ]2
1.
2. Linear response theory:
𝑥 𝜔 = 𝛼(𝜔)𝐹(𝜔)
𝛼 𝜔 = 𝛼 ′ 𝜔 + 𝑖𝛼 ′′ 𝜔
𝛼
′′
𝜔 𝑆(𝜔)
𝜔 =
4𝑘𝐵 𝑇
2
′
𝛼 𝜔 =
𝜋
𝐺∗
∞
∞
cos (𝜔𝑡)
0
1
𝑓 =
6𝜋𝛼𝑎
0
𝛼 ′′ 𝜔′ sin(𝜔′ 𝑡)𝑑𝜔′ 𝑑𝑡
Active microrheology
One-particle active
Oscillations of trap:
𝑥𝑡 𝑡 = 𝐴 sin 𝜔𝑡
The response of the bead is:
The equation of motion:
𝑥 𝑡 = 𝐷 𝜔 sin 𝜔𝑡 − 𝛿 𝜔
𝑘 + 𝑘𝑚 𝑥 𝑡 + 6𝜋𝜂𝑎𝑥 𝑡 = 𝑘𝐴𝑠𝑖𝑛(𝜔𝑡)
The viscoelastic moduli are calculated as:
𝐺′
𝑘𝑚
𝑘 cos 𝛿(𝜔)
𝜔 =
=
−1
6𝜋𝑎 6𝜋𝑎 𝑑(𝜔)
𝑘 sin 𝛿(𝜔)
𝐺′′ 𝜔 = 𝜔𝜂(𝜔) =
6𝜋𝑎 𝑑(𝜔)
Active microrheology
Two-particle active
The displacements od the probe particle:
𝑥
𝑦
2
2
1
𝜔 = 𝐴𝐼𝐼 (𝜔)𝐹𝑥 (𝜔)
1
𝜔 = 𝐴⊥ (𝜔)𝐹𝑦 (𝜔)
The same displacements can be also expressed directly as:
𝑥
2
𝜔 =𝛼
𝜔 𝐹𝑥
2
𝜔 −𝑘
2
𝑥
2
+ 𝛼𝐼𝐼 𝐹𝑥
1
𝜔 −𝑘
1
𝑥
1
𝑦
2
𝜔 = 𝛼 (2) (𝜔) 𝐹𝑦
2
𝜔 −𝑘
2
𝑦
2
+ 𝛼⊥ 𝐹𝑦
1
𝜔 −𝑘
1
𝑦
1
2
Active microrheology
Mutual response functions:
𝟏
𝜶𝑰𝑰 =
𝟒𝝅𝑹𝑮(𝝎)
𝟏
𝜶⊥ =
𝟖𝝅𝑹𝑮(𝝎)
Single particle response functions:
𝜶
𝟏
=𝜶
𝟐
𝟏
=
𝟔𝝅𝒂𝑮(𝝎)
Complex viscoelastic modulus:
𝐺 𝜔 =
𝐺′
𝜔
+ 𝑖𝐺 ′′
𝑘𝑚
𝜔 =
+ 𝑖𝜔𝜂
6𝜋𝑎
Rheology of bacteria network
Bacteria – single cell organisms
Different modes:
• Free floating mode
• Formation of biofilms
Biofilms
Free-floating organisms attach to a surface
Colonies of bacteria embedded in an extracellular matrix (EPS)
EPS consist of:
• Polymers and proteins
• accompanied with nucleic acids and lipids
EPS:
Protect microorganisms from hostile enviroment
Support cells with nutrients
Allow comunication between cells
Biofilm development
Stationary phase
Death phase
Log phase
Lag phase
Complexity of biofilm arises:
• Spatial heterogeneities in extracellular chemical concentration;
• Regulation of water content of the biofilm by controling the
composition of EPS matrix;
• Spatial
heterogeneities
on
gene expression
heterogeneities in polymer and surfactant production
creates
The production and assembly of cells, polymer, cross-links
and surfactants result in a structure that is heterogeneous and
dynamic.
Why is this study important
• Biofilm mechanics is important for survival in some enviroments
• Well-known viscoelasticity of bioflims can provide insight into the
mechanics of biofilms
• Quantitative measure of the “strength” of a biofilm could be useful for:
• Development of drugs for inhibition of biofilm growth
• In identifying drug targets
• Characterizing the effect of specific molecular changes of biofilms.
Future work
We will use optical tweezers to study viscoelastic
properties of different biological samples;
We want to understand fundamentally how
the
viscoelasticity changes on different lenght scales on different
frequencies;
The methods will be first tested on water;
The final testground will be viscoelastic characterization
of bacterial biofilms at different stages of biofilm evolution.
References
• Annu. Rev. Biophys. Biomol. Struct. 1994. 23.’247-85
• Annu. Rev. Condens. Matter Phys. 2010.1:301-322.
• Natan Osterman, Study of viscoelastic properties, interparticle potentials and self
ordering in soft matter with magneto-optical tweezers, Doctoral thesis, University
Ljubljana, 2009.
• Natan Osterman, TweezPal – Optical tweezers analysis and calibration software,
Computer Physics Communications 181 (2010) 1911–1916
• Oscar Björnham, A study of bacterial adhesion on a single – cell level by means of
force measuring optical tweezers and simulations, Department of Applied Physics
and Electronics, Umeå University, Sweden 2009
• Mark C. Williams, Optical Tweezers: Measuring Piconewton Forces, Northeastern
University