Rhodamine 6G 530nm 556nm
Rhodamine B 552nm 580nm
Oxazine 4 615nm 649nm
Oxazine 1 646nm 670nm
All the transition were measured using ethanol as a solvent [14]
Table 2.1 Absorption and fluorescence maxima in ethanol for rhodamine 6G, rhodamine B, oxazine 4 and oxazine 1 in ethanol.
The absorption and emission spectra for low concentrations o f dyes in solutions are characterised by a broad featureless structure compared to ionic and atomic spectra.
There are two main reasons: firstly, in addition to the electronic degrees o f freedom, a
probe molecule possesses quantised rotational and vibrational levels. The rotational levels are very close together, the energy difference between adjacent levels being o f the order o f 0.01-0.00 leV , while vibrational levels are separated by about 0.1 eV. The rotational energy gap is small enough, at ambient temperature, to be described according to Boltzmann statistics. Thus, at any particular wavelength, it is possible to excite a wide range o f ground to excited state rovibronic transitions giving a complex spectrum. The fine structure o f the absorption spectrum can only be resolved by cooling
to w ithin a few degrees Kelvin in a supersonic expansion [15]. Secondly, the collision
rate (lO^^-lO^V^) [16] in solutions does not permit free molecular rotation, leading to a
broadening o f the rovibronic transitions and a loss o f structure in both the absorption and fluorescence spectra.
2.4 Probe Dynamics
2.4.1 Diffusion
Pick [17] described the dynamics o f matter in terms o f a concentration gradient. This process, known as diffusion, is responsible for the transfer o f matter. It arises as the consequence o f a perturbation o f the internal energy o f a system. General examples are the temperature gradient in the case o f heat or the applied electrical potential for
electricity. The flux is proportional to the variation in the number density N(x,t) per unit
distance dx (in one dimension) and is defined as follows:
dN {x,t)
(2.28)
The constant o f proportionality is known as the diffusion coefficient D (units are area
per unit o f time).
The negative sign in Pick’s law comes from the flow variation from higher to lower particle concentration. Application o f the mass conservation into a restricted volume
d N ( x ,t) d { J ^ )
dt dx (2.29)
Combining equations (2.28) and (2.29) and assuming that the diffusion coefficient D is concentration invariant, the equation for concentration time evolution can be written as
d N {x,t) d ^ N {x d ) = D-
dt dx^ (2.30)
The solution o f the parabolic partial differential equation can be achieved using a Fourier spatial transform (assuming initial conditions in which the time-zero mass
concentration is described by a Dirac delta function in the YZ plane N{x,0) = NS(x)
[19])
N (47iDt)
exp
4D t (2.31)
The mean squared distance travelled along the x axis by the diffusing particle (initially
at x=0) can be obtained from the standard average
(2.32)
yielding the Einstein-Smoluchowski equation [2 0]
/ 2\ 1 N “r 2
4D t dx = 2D t (2.33)
Macroscopic diffusion is linked with microscopic processes via Brownian motion [2 1].
Einstein studied this random motion and concluded that it arose from a difference in the
fluctuation pressure due to the asymmetrical collisions o f particles [2 2]. A further
mathematical explanation was put forward by Perrin, applying the principles o f classical mechanics to a fluctuating force F(t). Langevin’s equation is expressed as.
where M is the mass o f a particle moving with velocity v an d ^ is the microscopic
friction constant [23]. Dividing throughout by M and multiplying by jc gives
xx = -g x x -\-x — — (2.3 5)
M
Noting that { x f =2% + Ix x gives
- x ^ - ( x f = - —id + x ^ ^ ^ (2.36)
2 2 M
and using <^xx = \l2 ^ { d / d t)x^ . By the equipartition theorem the average kinetic energy
is written [24], as follows
= (2.37)
where k is Boltzmann’s constant, T the absolute temperature and ( ) denotes a long
time average. Now, if the long time average is imposed in equation (2.36) (^x^^ is
substituted from (2.37), with x and F(t) uncorrelated in time, then
d^ , ( 2 k f \ d ,
which has a solution
In the short-tim e limit, the solution ( t « ^ ') is (x^^ ^ { 2 k T I and the particles
move w ith a thermal velocity ^ 2 k T l M . For long-time limits Brownian
motion becomes the dominant process due to random collisions giving
{ x ^ ) ^ [ 2 k T l M i ) t (2.40)
Combining equations (2.40) and (2.33) an expression is obtained for the diffusion
coefficient (macroscopic magnitude) as a function o f the fnctional force constant (microscopic magnitude)
k j
D = (2.41)
Applying Stokes’ law to the friction coefficient Ç is written as a function o f the viscosity
o f the media r\ and the hydrodynamic radius R (assuming a spherical particle)
Ç=67criR (2.42)
the diffusion coefficient is then,
kgT
2.4.2 Rotational Diffusion o f Spherical Molecules in Isotropic Media
In the long-time diffusion regime, the Brownian motion can be separated into two terms representing the translational and rotational motion independently. In equation (2.31) the translational coefficient was described as an increase in the mean square displacement with time. The extension o f this concept into a three dimensional system,