lecture 8: measurement of nanoscale forces ii. what did we cover in the last lecture? the spring...
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Lecture 8: Measurement of Nanoscale forces II
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What did we cover in the last lecture?
The spring constant of an AFM cantilever is determined by its material properties and its physical dimensions
3
3
l
EIk
kzF F
Atomic force microscopes can be used to measure forces
A split photodiode arrangement is used to detect deflections and to measure forces with >10pN precision
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In this lecture…
Optical tweezers
The wave nature of light
Optical trapping
A point dipole in a field revisited
Using lasers to trap particles
Applications of optical trapping
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Optical tweezers
Optical tweezers provide a way of manipulating nano or micron scale objects and measuring very small forces.
Particles are usually trapped by focussing visible light (usually a laser)
Objects can be manipulated by moving the laser beam
Particle
Visible light
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The wave nature of light
Light is an electromagnetic wave.
As it propagates through vacuum or a medium it carries energy.
)sin( kxtEE o
Energy is stored in electric and magnetic fields that oscillate in a direction perpendicular to the direction of travel (more on this in Classical Fields)
Electric field travelling in positive x direction
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The time averaged intensity of light, I, can be related to the magnitude of the electric field, E, by the relationship
in Wm-2
Where c is the speed of light (ms-1) and n is the refractive index of the medium in which it propagates
Note: For a sinusoidal oscillating field, the time average value of <E2> is not zero. So the intensity is not zero!
The intensity of light
2
2E
ncI o
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Optical trapping
Small dielectric particles experience a force when illuminated with visible light.
If a dielectric particle is placed in an intensity gradient a force is exerted on the particle such that it will move to a region of higher intensity
Light Beam
Intensity gradient
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A point dipole in a field
2
~~~. EEEU
Suppose we treat the small particle as a collection of small dipoles
The energy associated with placing the particle in an electric field E is
Where p is the electric dipole moment (see lecture 3)
For a particle of polarisability, , we obtain the energy as
~~.EpU
We can use this to calculate the force on the particle (see OHP)
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The force on a dipole in an intensity gradient
Generalising to 3 dimensions we have
~~~~~~
2k
dz
dIj
dy
dIi
dx
dI
nck
dz
dUj
dy
dUi
dx
dUF
o
In one dimension, the force on a dielectric particle is
~~
2i
dx
dI
nci
dx
dUF
o
So the force always acts along the direction of increasing intensity gradient. Particles move to regions of higher intensity
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Creating a gradient in intensity
The beam profile of a laser is not uniform - it has a Gaussian profile in the radial direction – trapping in x-y plane only
Laser Beam
2
2
2
22
2exp
2
)(exp
oo
oo
w
rI
w
yxII
wo
wo is a measure of the beam width
However, there is a problem! We only get trapping in x-y plane – no gradient in z direction!
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Optical Trapping in 3D
We can use a microscope objective to create a 3D trap!
The changing area of the beam near the focus means that the intensity of the light decreases on either side of the focal point
This traps the particle in z direction also!
The maximum gradient in intensity can be obtained by using a lens with a high numerical aperture (NA)
sinnNA
Microscope objective
Ambient refractive index, n
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Gaussian beam profile: quadratic approximation (trap stiffness)
21 bxII o
For small displacements from the central position we can approximate the Gaussian using a quadratic intensity profile
bxIdx
dIo2
Laser Intensity
Intensity gradient
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The intensity profile of a laser beam is given by the equation
Where wo is the radius of the beam and Io is the intensity at its centre.
Verify that for small displacements (r<<wo) the intensity profile can be approximated by a quadratic form. Hence show that the force exerted on a small particle at the centre of the beam can be expressed in the form F=-kr
Derive an expression for k.
Problem 1
2
2
2exp
oo w
rII
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Measuring forces with optical tweezers
kxF
As we saw in the previous problem, for small displacements, x, the force on a particle in a focussed laser beam is given by
Where k is the trap stiffness and is given by
So if we can measure the displacement we can determine the force
2oo
o
nwc
Ik
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Detecting deflections: Quadrant photodiode
The split photodiode arrangement used in AFM can also be used to detect deflections in optical traps. However a quadrant photodiode is used to detect deflections in both the x and y directions
Quadrant photodiode
(4 photodiodes)
Shadowing (large particles) Fringe pattern (small particles)
Laser beam
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Trap calibration
2~
x
Tkk B
If we want to calibrate a trap we can measure the displacement of a particle under the influence of known forces
Or we can measure the rms displacement of a trapped particle under the influence of thermal motion
FtrapFapplied
x x
Fk applied
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Problem II
A 100nm radius polymer particle having a relative permittivity ofp=5 is suspended in water w=80, nw=1.33A 10W laser beam with a Gaussian profile is focussed down on to the particle such that it creates an optical trap with an effective radius of 1 m.
Calculate the displacement of the trapped particle if a force of 1pN is applied to it.
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Multiple traps: time sharing
http://www.nat.vu.nl/~joost/tetris/
If traps are switched on and off faster than particles can diffuse away, the same trap can be moved around and used to trap many objects
A game of Tetris with glass beads!
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Force Measurements
It is also possible to perform more serious measurements on e.g. single molecules
C. Bustamante et al., Nature, 421(23), 423 (2003)
An optical trap and a fine micropipette can be used to measure the forces exerted by individual DNA molecules
Small polymer beads are tethered to the ends of the molecule
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Summary of key concepts
Particles in a laser beam will experience a force that acts to pull them to regions of higher intensity
~~
2i
dx
dI
nci
dx
dUF
o
Lasers have a Gaussian beam profile which naturally lends itself to trapping of particles
Trapping in 3 dimensions can be achieved by focussing the laser using a microscope objective
Trapping forces on nanoscale particles can be measured with pN precision