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 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. 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 is not zero. So the intensity is not zero! The intensity of light

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 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 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 In one dimension, the force on a dielectric particle is 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 wowo w o 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) Microscope objective  Ambient refractive index, n

Gaussian beam profile: quadratic approximation (trap stiffness) For small displacements from the central position we can approximate the Gaussian using a quadratic intensity profile Laser Intensity Intensity gradient

The intensity profile of a laser beam is given by the equation Where w o is the radius of the beam and I o is the intensity at its centre. Verify that for small displacements (r<<w o ) 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

Measuring forces with optical tweezers 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

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 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 F trap F applied x

Problem II A 100nm radius polymer particle having a relative permittivity of  p =5 is suspended in water  w =80, n w =1.33  A 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 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 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