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