Lecture 8: Measurement of Nanoscale forces II
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
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
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
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
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
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
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)
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
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!
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
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
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
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
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
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
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.
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!
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
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