Monomer-Excimer Equilibrium
The scheme is:
M  M
1
k DM

 km
kMD
M
1
D
 kD
2M
k
Where k m and D represent the sum of the rate constants of all intrinsic
processes (including radiative) causing the decay of monomer and excimer. The primary
1

excited state, M , is generated by the absorption of a photon by the ground state
species, M.
Then, at the end o a brief excitation pulse,
d [ 1M  ]/ dt  kMD [ 1D]  (kM  kDM [ M ])[ 1M  ]
 kMD [ 1D ]  X [ 1M  ]
and,
d[ 1D ]/ dt  kDM [ 1M  ][M ]  (kD  kMD )[ 1D ]
 kDM [ 1M  ][ M ]  Y [ 1D ]
(1) and (2) are a pair of coupled differential equations the solution of which yields
the time dependence of the monomer and the excimer.
The pair of equations can be solved using Laplace transform or secular equation
methods, whence:
[ 1M  ] [ 1M  ]0 {( 2 X )exp( 1t )  ( X   1)exp( 2t )}/{ 2  1}
and
[ 1D ]  kDM [ 1M  ]0 [ M ]{exp( 1t )  exp( 2t )}/{ 2   1}
since the monomer fluorescence response function can be defined as
iM (t )  k FM [ 1M  ] /[ 1M  ]0
and the corresponding excimer fluorescence response function is
iD (t )  k FD [ 1D ] /[ 1M  ]0
then
iM (t )  kFM ( 2 X ){exp( 1t )  A.exp( 2t )}/{ 2  1}
and
iD (t )  kFDkDM [M ]{exp( 1t )  exp( 2t)}/{ 2  1}
where
A= ( X  1 ) /(2  X ) , k FM and
and the excimer respectively, and
kFD
are the radiative rate constants for the monomer
1
1,2  [( X  Y )  {(Y  X )2  4 kDM kMD [M ]} 2 ]/ 2
The  quantities are the multipliers of time in the exponential terms. They have
dimensions of reciprocal time and are composite rate constants.
Note that equation (5)-and (3)-involves the sum of exponential terms, whereas equation
(6)-and (4)-involves a difference. This is what is expected by inspecting the experimental
time profiles, as in figure 1.
The Dynamic Equilibrium Condition
In the above scheme, suppose that k DM , k MD  k M , k D , i.e. the rate of establishment of
the equilibrium is much more rapid than the intrinsic decays of the excited states. Thus
M  and 1D are in equilibrium at all times during the decay event.
Then  2   1 and exp (2t )  exp(1t )
1
Also,
iM (t )  B.kFM .kMD. exp(t ) and iD (t )  B. k FD.k DM [M }exp{t )
where B= 1/(kMD  kDM [M ])
Thus the two excited states have a common decay lifetime, 1/  , and it can be shown that
  mkM  dk D
1

where m and d are the mole fractions of M and
1
D in the equilibrium mixture.
From measurements of  as a function of [M], kDM and kMD can be extracted. From
these we can evaluate the equilibrium constant for excimer formation:
keq  k DM / kMD
and by measuring the temperature dependence of keq , thermodynamic parameters for
the excimer formation can be obtained:
d (ln Keq ) / dT  H 0 / RT 2
and
G0   RT ln Keq  H 0  T S 0