Precision optical force measurements require well-calibrated optical tweezers. The forces acting on an optically trapped particle are approximate to a Hookean spring with zero rest-length at the centre of the trap. Brownian motion causes small move- ments of the trapped particle about the centre of the trap. There are three main methods which have been used for calibration which I will now discuss.
The simplest method for calibration is based on the equipartition theorem. The energy of the trapped bead is equal to 1
1. MANIPULATION AND PORATION OF MICROSCOPIC PARTICLES USING LIGHT
is 12khx2i where k is the spring constant and hx2i is the variance in the motion of the bead. Equating these two energies yields an expression for the trap stiffness:
k= hx
2i
kBT
(1.5) This method does not require a high-bandwidth detector of trap position and so it is possible to implement with a fast CCD camera. Whilst it is simple to implement, an independent temperature measurement must be made and the assumption is made that this remains constant for the duration of the trapping event. Additionally, any drift in the system will affect the calculated variance of particle position.
A more rigorous method based on a particle’s Brownian motion is to use a quadrant photodiode (QPD) to detect the bead position and then generate a power spectrum. The power spectrum for a trapped bead is [47]:
SXX(f) =
kBT
π2β(f2 0 +f2)
(1.6) where β is the hydrodynamic drag coefficient (β = 6πηa for Stokes drag coeffi- cient on a bead of radiusain a medium of viscosityη) andf0 is the rolloff frequency.
The trap stiffness can then be calculated using the following [48]:
k= 2πβf0 (1.7)
This method is generally considered the gold standard for trap stiffness measure- ments. A key requirement is the detector must have a high enough bandwidth in excess of the rolloff frequency to avoid aliasing effects. Trapped particles typically have a rolloff frequency of the order of a few hundred Hz or a few kHz [49], so high- bandwidth detectors like QPDs (placed at the back focal plane) are often used. It is
1. MANIPULATION AND PORATION OF MICROSCOPIC PARTICLES USING LIGHT
also important to calibrate the QPD response. The Nature Protocols paper by Lee et al provides a step-by-step guide on building and calibrating an optical tweezer with a QPD [49].
The third method to calibrate an optical trap is based on viscous drag and the particle’s escape velocity. If we consider a particle being dragged through a medium at an increasing velocity the drag force will depend on the Stokes drag force. If the bead is close to a boundary such as the glass coverslip, a correction factor (Faxen’s law) should be applied to give:
Fdrag = 6πηav 1− 9 16( a h) + 1 8( a h)3 − 45 256( a h)4− 1 16( a h)5 (1.8)
where v is the velocityh is the distance from the glass coverslip.
By experimentally finding the velocity at which the optically trapped particle falls out of the trap, the escape force (i.e. the trapping force) can be equated to the viscous drag force:
Fdrag ≈Fescape (1.9)
Hence the Q-value can be obtained:
Q= Fescapec
nP ≈
Fdragc
nP (1.10)
Another permutation to obtain the escape velocity is to use a stationary particle and a fluid flow. The drag force method is less commonly used as it requires either a motorised stage travelling at a known velocity or a well-controlled fluid flow. Section 2.2 will discuss the properties of microfluidic flows in greater detail.
1. MANIPULATION AND PORATION OF MICROSCOPIC PARTICLES USING LIGHT
1.6.1
Comparison of calibration methods
The three methods to calibrate optical traps vary in terms of complexity and the information obtained about the system. Both the equipartition-theorem method and the power-spectrum method rely on observing thermal fluctuations around the centre of the optical trap. Force measurements are often conducted 50 nm to 100 nm from the trap centre and thermal fluctuation calibrations do not give accurate in- formation on the shape of the trapping potential at large displacements from the trap centre [50]. The drag method provides the best way to measure the linearity of the optical trap, provided the viscous drag force can be accurately calculated (Equation 1.8). However, this requires a motorised stage which may not always be possible. Additional benefits of the equipartition-theorem method it is independent of the viscosity of the surrounding medium and does not require the size of the trapped object to be measured.
The power-spectrum method can provide additional information to diagnose problems with an optical trap by observing deviations away from the ideal Lorentzian shape. A sharp rise in the power-spectral density at low frequency is an indication of instrumental drift whilst electrical noise is visible as delta functions [50]. The high bandwidth of the QPD makes it the preferred tool to measure particle position in order to accurately determine the roll-off frequency of a power-spectrum. If the bandwidth of the detector is too low this is observed in the power-spectrum as a decrease of Sxx (Equation 1.6) faster than f−2. In Chapter 3 the system does not
contain a QPD nor motorised stage so the equipartition-theorem method was used for trap stiffness measurements.
1. MANIPULATION AND PORATION OF MICROSCOPIC PARTICLES USING LIGHT