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Time-resolved polarized fluorescence spectroscopy of photoreactive systems in uniaxially organized molecular media
2May 1997
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 269 (1997) 244-256
Time-resolved polarized fluorescence spectroscopy of photoreactive systems in uniaxially organized molecular media J.J. Fisz Institute of Physics, N. Copernicus University, ul. Grudziqdzka 5/7, PL 87-100 Toruri, Poland
Received 23 October 1996; in final form 21 February 1997
Abstract
A description of the time-resolved polarized fluorescence spectroscopy on photoreactive systems incorporated into a uniaxially organized molecular medium is discussed. The treatment assumes many-ground-state species interconverting in the excited states. The ground- and excited-state species undergo different solute-matrix aligning interactions. The rotational motion of the excited-state species is state-dependent. The diffusion model in the state-dependent potential for asymmetric and symmetric rotors is considered. Expressions enabling the computation of polarized fluorescence decays are derived and discussed. The case when the state-to-state kinetics of the systems are controlled by fluorophore-matrix aligning interactions and rotational dynamics of the fluorophores is considered. Some possible experimental arrangements enabling studies of such molecular systems as monolayers, bilayers, membranes as well as solid/air and liquid/air surfaces, are briefly considered. Indicated is a possible application of the theoretical and experimental methods discussed in this work to the time-resolved polarized total internal reflection fluorescence spectroscopy of thin layers of different kinds of molecular assemblies at solid/liquid and liquid/liquid interfaces. © 1997 Elsevier Science B.V. To the memory o f my dear daughter Karolina
1. Introduction
In this work we discuss a description of the time-resolved polarized fluorescence experiments on photoreactive species embedded in uniaxially organized molecular systems. The case considered in this work is depicted in Scheme 1. It is assumed that the system is composed of a greater number of ground-state species interconverting in the excited states. Due to the excitation of the sample at a certain excitation wavelength Aex a greater number of ground-state species can be excited simultaneously (the absorption bands can overlap). The polarized fluorescence monitored at a particular emission wavelength h~m can be contributed to by signals from a greater number o f excited states (the emission bands can overlap). It is assumed that the ground-state species can be excited to the lowest two electronic excited states and that both electronic transition dipole moments have different directions in every species. The state-to-state kinetic relaxation of the system is described by the rates indicated in Scheme 1. The kinetic rate k~ represents a rate for the internal conversion process from the electronic state i** to the i* one (where i = 1 ..... s). The fluorescence is detected from the lowest electronic excited state. It is assumed that the ground-state species undergo different solute-matrix aligning interactions (described by the effective potentials Vg,i(/2)) and that the excited-state solute-matrix interactions are different for different 0009-2614/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PI1 S0009-26 14(97)00270-4
J.J. Fisz/ Cheraical Physics Letters 269 (1997) 244-256
245
species (they are described by the effective potentials Vex,f(O)). The state-dependent species-associated solutematrix interactions lead automatically to different equilibrium angular distributions of the species in all the ground and excited states, probed in the experiment. The rotational dynamics of the excited-state species are state-dependent and are described in terms of the species-associated Green's functions pi* (19-0,010"2, t), which describe the conditional probability of finding a molecule in an excited state i at time t and at an angular orientation ~2 (the Euler angles), which at time t = 0 was oriented at 120. The state-dependence of the rotational motion of the species results, in our treatment, from the state-dependence of the solute-matrix aligning interactions (the rotational dynamics are restricted by the aligning interactions - the rotational motion in a state-dependent potential) and can also result from the state-dependence of the diffusion t e n s o r s ( D (i), D (i) , D~ i) for an asymmetric diffusor or Dli), D~_~ for a symmetric diffusor).
i,
i,
! ,
i ~ z : ~ k , al v ~ \ , i
k24- "..
equilibration Scheme 1 We derive expressions enabling the computation of the polarized decays of the fluorescence for asymmetric and symmetric diffusors rotating in species-associated excited-state potentials, which after convolution with the instrument response function, can be compared with the measured polarized decays. The treatment is further developed to a case when the kinetics of the system are controlled by fluorophore-matrix aligning interactions and by the rotational dynamics of the fluorophores. Such a case can occur when the state-to-state kinetic rates depend on the angular orientation of the fluorophores in an organized environment. The symmetry relations enabling a substantial simplification of the eigenvalues and eigenvectors problem will be discussed. The symmetry relations for the matrix representation of the time-development operator, coupling the state-dependent molecular alignments and rotational dynamics with the state-to-state kinetics, decompose the system of simultaneous differential equations into independent subsystems which can be treated separately. We discuss examples of possible experimental arrangements that can be useful in studies of such uniaxially organized molecular systems as monolayers, bilayers, multibilayers, membranes as well as solid/air and liquid/air surfaces. The advantages of the experimental arrangements discussed are indicated. The experimental arrangements can also be employed in studies of thin layers of different kinds of molecular assemblies at solid/liquid and liquid/liquid interfaces, by means of time-resolved polarized total internal reflection fluorescence (TIRF) spectroscopy. The theory developed in this work together with the discussed experimental arrangements enabling variableangle time-resolved polarized fluorescence measurements, can be useful tools in studies of organized molecular media. The application of photoreactive fluorophores as probes for monitoring the micro-environmental properties of an organized medium can be a source of information on the structural and dynamical properties of the system on a micro-scale (e.g. micro-packing, local mobility (micro-viscosity) and micro-polarity). The problem discussed in this Letter is one of the many cases considered briefly in our previous article [ 1 ] (i.e. ordered systems, solutions and labeled macromolecules), and which is further developed in this work. It is important to mention here the excellent work by Szabo [ 2 ] who considered many aspects of the fluorescence depolarization in uniaxial systems. One of the problems discussed was the many-ground- and many-excited-state
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J.J. Fisz/Chemical Physics Letters 269 (1997) 244-256
problem. However, this problem was treated under the assumption that the rotational dynamics and orientational ordering of the fluorophores were not state-dependent. Also important is the pioneering work by Cross et al. [ 3 ] who formulated a description of fluorescence depolarization experiments with the assumption of state-dependent rotational dynamics in the diffusion model, in the case of solutions. 2. Polarized emission Consider the situation displayed in Scheme 2. An organized (anisotropic) molecular system is excited at t = 0 by a S-pulse of light with polarization ei and polarized emission with polarization ~f is detected at t ~> 0. /2ab,k,~ represents the absorption transition dipole moment of a molecule in the k th ground state and which has been photoselected at t = 0 and at hem to one of the excited states (k* or k**) or to both excited states. It is assumed that the directions of the transition dipole moments /2,,b,kl and /2,,b,k2 are different in the molecule-fixed frame. /2era,( is the emission dipole moment of the same molecule in the ( t h excited state at a later time t. The orientations of Oi and Of in the laboratory-fixed frame XLYLZL are described by the polar angles (0i, ~oi) and (Of, ~of), respectively. The orientations of the absorption and emission dipole moments are defined in the molecule-fixed frame in terms of the polar angles (0~~) , ~o~k'J) ) ( r / = I, 2) and (0~~) , ~o~z~)). The angular orientations of the molecule-fixed frames at t = 0 and t > 0 are defined in the laboratory-fixed frame by two systems of Euler angles Oo = (ao, tgo, Yo) and s'2 = (a, fl, y).
ZL
excitation t--0
enfiuion t>0
-¢÷ \ ~ ) YL
XL Scheme 2
The intensity of the polarized emission for the system shown in Scheme 1 is described by the following expression: 2
l~i~,(t, Aex, Aem) = C(Aex, ~ . e m ) sC,k=-I
K~c(~-em)Z ¢Ok'q(Aex) r/=l
• ex,k,7,,,ex, fg.k(O0) pk~¢(.O0,010, t) Pe(~,~(Aem) dr20 dr2
×
Q
(1)
J2g2o C(Aex, ~-em) includes all excitation and emission wavelength-dependent instrumental factors. K~(~.em ) is the intensity of the fluorescence band of the Cth state at hem. (.Okrt(,~.ex) (~] = 1,2) represents the absorbances of the k* th and k** th absorption bands at Aex. Pex,~,(hex)~') and Pe~m~(Aem), describe the angular dependence of the polarized excitation (absorption) and emission probabilities [4], respectively, fg,k(O0) describe the equilibrium orientational distribution of the solute molecules in the ground state k and are given by the Boltzmann distributions
fg,k(/2) = Ng,k exp(--gg,k($'2)/kBT) ,
(2)
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J.J. FiszlChemical Physics Letters 269 (1997) 244-256
where Ng,k is the normalization constant. Vg,k(/2) are the species-associated ground-state aligning potentials. These potentials can be represented by the following series: oo
Vgk(/2),
= kBT Z'JPq(g'k) Dp,q(J)* (/2) ,
(3)
Jpq
where the expansion coefficients Ujpq • (g,k) describe the species-associated ground-state solute-matrix aligning interactions. The distribution function fg,k(/2"2) can be developed in the basis set of the Wigner rotation matrices [5], i.e. oo
fg,k(/2) = Z 2j877.2 + 1 (Dpq (j) )g,k --pq T)(j ) ,( /2) , Jpq
(4)
where /D (j) /g,r \ • are the orientational order parameters of molecules in the k th ground state, \ pq fg,,(/2) __pq f)(J) ( / 2 ) d/2.
(Dpq)g,k (J) = f
(5)
12
Pk--,¢ (12o, 01/2, t) are the Green's functions (propagators) describing the conditional probability of finding an excited molecule at time t and at an angular orientation/2 in state s¢, which was excited to the primary excited state k and with initial angular orientation of /2o. These propagators describe the state-dependent rotational motion of molecules, coupled with the state-to-state kinetic relaxation process. By performing the integration in (1) and employing the orthogonality relations for the Wigner functions [5], one can obtain a formula for polarized emission for many-ground- and many-excited-state systems in an organized medium. In the case of macroscopically ordered uniaxial systems (e.g. membranes, liquid crystals, solid/liquid and liquid/liquid interfaces etc.) the solute-matrix interaction coefficients become uJ.pgqk) = @gqk)6pO, (J) and consequently, \/D (j)pq/g,K \ ' = (Doq)g,k 8p0. The same holds for the excited-state potentials and order parameters. Symmetry properties for the solute-matrix interactions automatically lead to a corresponding simplification of the description of the molecular rotational dynamics. The same problem can be considered by employing group theory methods [6]. Finally, for a uniaxial molecular system (i.e. a rotation of the sample around the axis of molecular alignment by an arbitrary angle, is an invariant) the polarized fluorescence decay component has the form
s I~, #f(t,/~ex, ~-em): C(aex, ~-em) Z
2
[
K~'(~'em)~ 0)k'o(~'ex) Ph(k'()(t) + 2 P2(0i) Sa~tw'~:)(t)
(,k=l
~=1
+2P2(Of) S~k'e)(t) + 4 Z
C2*,p(Oi'~°i) C2,p(Of,~of) 4)~kn'O(t) ,
(6)
p=-2
where
Ph(k'e)(t)=fffg,d~)pk--e(~,Ol/2,
(7)
t) d~d/2,
/2~o S~akm()(t) = Z
fg,k($'~0) D(2)*(/20)0ql Pk~¢(/2°'0l/2't) d/2o d/2 ~z,q,,vA
,WA
]
qi=--2 2
= Z q1=--2
/T~(2)\(') \ ~ 0 q l Ig,k--*~ C2:ql(Oik'~}'@Ak'~)) ,
(8)
J.J. Fisz/Chemical Physics Letters 269 (1997) 244-256
248
fg,k(E20) pk-+~(O0'O[ 12't) DCz)t~~°^~ "u, dE2°
Se{~'¢)(t) = Z
~'21~*,q2~'t'za(()L'E , WE ^(() )
q2=-2 2
= ~
/D(2)\(t) r * r'~(( ) .~(¢) \ Oq2/k---+~"2,q2~.ffE ,WE ) ,
(9)
q2=--2
r.
--Pql
Pq2t JlO) dOo dO
q~,q2=-z'hh0 ,,~(k,7) (kn) . ~ . ,,~() (~:) X C2,ql{ffA '~PA ) t~2,q2{,O'E , ~ E ) 2
=
~
t~(k,7) 'V'VA _(ko) )'-'2,qzt"E r * -,Q(sc) ,wE ,,,(~:) ) .
2 2C v P2 q, ql"'" (, k' ~- -A+' p~q' t2)
(10)
q~,q2=--2
Cj,m(O, ¢) are modified spherical harmonics [5], i.e. Cj,,,(O, ~p) = [ 4 ~ / ( 2 j + 1 )] U2Yj,m(O, ~v), where Yj,m(O, ~v) are the spherical harmonics. In the relations given above Ph (~g) (t) describes polarization-free decay of fluorescence. Sa~k~g) ( t ) / P h (kg) (t) means the order parameter of the absorption dipole moments in the kth ground state (in the absorption band r/ = 1,2) which is detected through the emission from the excited state sc. Se[k'~) ( t ) / P h (kg) (t) is the timedependent order parameter of the emission dipole moments in the ~: th state of the molecules which at t = 0 have been excited from the k th ground state. ~b~'()(t)/Ph~k'¢)(t) represents the correlation (time-dependent mutual angular orientation) between the absorption dipole moment of a molecule excited at t = 0 to k* or k** and the emission dipole moment of the same molecule in the ~rth state at a later time t. n ( 2 ) \ (t) /Ph(k,~) ¢tx The normalized second rank order parameters /\'-'0qj/g,k-~(/--" ~,j are the molecular order parameters in the k th ground state recovered through the polarized emission from state (. (D0(~)~¢/ph(k'¢) (t) are the molecular order parameters of the molecules in state (, that were originally excited to state k. ~VpPqt (k --+ sc, t ) / P h < g) (t) are the molecular correlation functions describing the angular relationship between molecules that at time t = 0 r~(2)\(t) and were excited to state k and at a later time emitted fluorescence from state (. x[O(2)\(t) Oq~/g,k---+(, /\L~Oq2/k---+( ~ V pq~ (k ---+~, t) couple the state-dependent molecular ordering and rotational dynamics with the state-to-state kinetics of the system.
2.1. State-dependent rotational dynamics of the symmetric and asymmetric diffusors in the state-dependent excited-state aligning potentials In order to find explicit forms for the Green's functions pk~((O0,01/'2, t) one can consider dependence of the propagators p__.((/20, 0112, t) = )--]-~-Ipk--+¢(/20, 01/2, t) A k.ex, and which describe tional probability of finding a molecule in the excited state ( at time t, which at t = 0 was excited to primary excited states (the sum over k). Ak,ex = 1 if state k is the primary excited one and A~xx = 0 These propagators obey the equation of motion: ~p~l(OO,01/2, t)=-
Htl)+kl+Zkli i=2
~ p ~ s ( O o , O l O , t) = -
- Z i=l
p~l(OO,0l/'2, t ) - - Z k a p ~ i ( E l 0 , 0 l ~ , t )
the timethe condiany of the otherwise.
,
i=2
ki, P~i( Oo,OI12, t) + t?I~') + ks + Z i=l
k,i p - - , ( O o , OIf2, t) .
(11)
J.J. Fisz/ Chemical Physics Letters 269 (1997) 244-256
249
In the restricted diffusion model the time-development operator i¢?/(() has the form 3
(12) i=1
where the operators Li are the quantum mechanical orbital momentum operators (i = x, y, z), D: ~) are the components of the diffusion tensor and Vexg(12) is the aligning potential in the excited state (. The solution to the equation of motion ( 11 ) has the form [ 1 ] :
=Z [Z :J+' 8¢r2 $
to
to
k=l
jmn
fm'n'
jmn,j, mtn ,t,t.~ t/ Dra, R(('k) (3')", (/~0)
Z
Ak,ex
s
=~_,pk-.#(0o.Olgl, t) Ak.ex •
(13)
k=l
The expansion coefficients RJ~,m,n,(t) appearing in (13) can be calculated from the following procedure. In the matrix equation corresponding to the system of equations ( 11 ), i.e. p(/'2o, 01O, t) = - A p(/9.o, 01/2, t), the Green's functions are replaced by the expansions p-.e(0O,010, t) = ~-]~, (2j + 1)/87r 2 C:~(t) D~J~) *(/2). This leads to a system of coupled differential equations forthecoefficients C:(m~(t), i.e. ~7(t) = - M C(t). A formal solution to this matrix equation is C(t) = R(t) C(0), where R(t) = V exp ( - A t) V -1. V and A = [ .~qq] are the eigenvector and eigenvalue matrices of M, respectively. M is the matrix representation of A in the basis set of Wigner functions• C(0) is a vector of the initial conditions, [C(0)] (:)Jmn= Cjra.(O)'(~) C)~r.~)(0)(~) can be determined from the initial conditions for the Green's function p_.~(Oo,0lO, t) = (3(Oo - / 2 ) A~,ex, and consequently, Cjm. (() (0) = DOn) (/2o) A~,ex. The matrices Iql and R(t) have the following structure:
M =
U 1) K21 K31 "'" Ks1 / Kl2 U (2) K3z "'" Ks2 KI3 K23 U (3) "'" Ks3 , Kls
K2s
//R(l'l)(t)
[R(2,1)(t)
K3s . . . U (s) R(l,2)(t ) R(l,3)(t)
... Rfl,s)(t ) \
R(2,2)(t) R(2,a)(t)
R(2,s)(t)
R(t) = / Rf3'l)(t) Rf3'2)(t) R(3'3)(t)
R(3's)(t)
/
•
:
]
"
\ R f s , l)(t) RCs,2)(t) Rfs,3)(t)
Rfs,s)(t)
J .
(14)
tr(~) s U (~) are submatrices with elements [IJ(~)]p,q = '-" j,,,,,.j, z, n, = t4(~) -j,,,n,j,z,., + (k~ + Y]-i=l,i.,e ki~) (3j,,,..j'm'.', where -
2j'+ I
m"2(O) fD ix#,
D o(j') *:/2) d O ,.,,
(15)
12
The submatrices Kxv are diagonal, i.e. [Kxy]jmn,j,m,n , = - kxy (3jmn,j'm'n', where kxy is the rate for the state-tostate kinetic relaxation from an excited state x to another excited state y. The matrix elements of R (~'k) (t) are R.l~m'nk,~,m,n,(t) = [(V [exp(-Aqq t ) ] V -l)(~'k)]jmn,j,m,n,, where ~: means the state from which the polarized
J.J. Fisz/Chemical Physics Letters 269 (1997) 244-256
250
emission is observed and k denotes one of the primary excited states. The algorithm described here is based on one of the methods of solving the Fokker-Planck equation [8], and which in our problem, has been employed to a system of coupled differential equations for Green's functions [1]. From (13) the Green's functions p,~¢(O0,01/2, t) can be found, and hence, Ph ~*'¢), Sa~k''¢) (t), Se(*'~) (t) and ~b(p*''~) (t) given by ( 7 ) - ( I 0 ) can be replaced by the relations
Ph(k'()(t)= E ~'~ \/D O,n (j')\ •~(~"~) ~O00,jtOn ~,~: ~/g,k, j/n / 2 oo (j~n~ R~o~ok.),,on,(t)2J'V:q~(k,t,j = 0) ) ,...2,q, [-,* tUA :,~(k~7)(krl) Sa~km()(/) = E ,~oA ) , ql=-2
2
(17)
.
s~k'('(t)-- Z q2=--2 ¢b(-k m ( ) (- t - =
(16)
e~(o'k),j/On/(t) \ O,n,/g,k) C2",q2(0 ((), ~
2"
"~
( ~ - " R 2pq2,j'pn' ¢¢'k) (t) J Vpn pql," "" ,,~k,) _~*,),) C 2,q2 * (O(~¢),~p(~¢)) • kZ..~ '~l¢'t= O) ] t-'2.qll, t~A ,q'A q~,q2=-2 \j~n ~ ~
(18)
(19)
2f l/Pqi b t = 0) are the molecular correlation functions at time t = 0 in the excited state k. They can be ,p,, (~,~, expanded in the Clebsch-Gordan series, i.e.
2+j' 2j' vPq' "pn' ( k, t = O) = ( - 1 )P-q'
Z
C(2j'L;-pp)
C(2j'L;-qin')/r~ft)\LIo,n/_ql ]g,k\ ,
(20)
L=I2-j' I where C ( j l j 2 j ; m n ) are the Clebsch-Gordan coefficients [7] and ,:D~L) O,n--ql)g,, k are the ground-state order parameters in state k. The dependence of Ph (kg) (t) on the orientational ordering of the fluorophores in the ground state and the time-dependence of S~akng)(t), in the relations derived, appear when the rates for the state-to-state kinetic relaxation of the fluorophores are controlled by the aligning properties of the fluorophores within an ordered matrix, i.e. these rates are orientation-dependent. This problem will be discussed in the next subsection. If the rates for the state-to-state kinetic relaxation are not orientation-dependent, Ph(kg)(t) and Sa~ g ) (t) simplify to ph{k,f)(t) = R~f ,k) ¢t~: , 0o0,000~
S~akn,f)(t) = "'000,000~ taft ,k) ¢,~ ~l S~kn)
(21)
where
(Doq, )g,k C2.q I
(22)
ql
R(¢,k) r ,~ describes the state-to-state kinetic relaxation alone, over all the excited states involved. Sa~kn) is the 000,000,-: order parameter of the absorption dipole moments in the kth ground state and (D ~z~' are the second rank , 0qt)g,k molecular order parameters in the same state. Taking into account the explicit expressions for Lx D(J')m'n'*' Lv. D(j')m,n,* and L z DO')m n'*' and assuming that the species-associated excited-state aligning potentials are Vex~:(/2), = knrEu~yepxg) ~0pr~J)*(/2) , @
(23)
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251
where , (ex,() describe the species-associated excited-state solute-matrix aligning interactions for a uniaxial molecular system, one obtains the matrix elements of u(£) . ~jmn,j~ nff n~ ••
HJ(~) m?l ff t~ltnt ~ ( [Aj,., (D(x() + D~fi) ) + n '2 D~~)] 6,,n, + (u(+) ~fn' 2f+
~n,n'+2 -'1- Bj,(-) n, ~n,n'-2) (D(x() --
D~,( ) ) ) 6jj, 6tara'
1
+ --2j + 1 Z'ujp(exg) C(Jjj';O - m ' ) Jp x ( a (+) C ( J j ' j ; - p - I
-n')[(D(x ( ) - D~fl)) b (+) Bn,t,+n'+2+ (Dx(~) + D~') ) c (+) 6n,p+n']
+ a (-) C( j j ./.j ; - p + 1 - n r) [(D(x~) + D~fi)) c (-) t3n,p+n' + (D(J) - D(y()) b (-) ¢~n,p+n'--2] + p(p + n') C ( j j r j ; - p - n ' ) D~0 6n,p+n'} 8,nm, ,
(24)
~j,,, = ¼ [ ( j ' ( j ' + 1) - n'(n r 4- 1)) ( j ' ( j ' + 1) - (n' 4- 1)(n' 4- 2))] '/2 , where Aj,n, = ½ [j'(j' + 1) - n'2], R(+) a (+) = ½ [ J ( J + 1) - p ( p 4- 1)] 1/2, b (±) = ½ [ j ( j + 1) - ( p + n ' 4- 1)(p + n ' 4-2)] 1/2 and c (+) =
!2 [J(J + 1) - (p + n' 4- 1)(p + n')] 1/2. For a symmetric diffusor (Dx(~:) = D~~). = D ~ ), D~() = Dlf)) the matrix representation (24) reduces to Hi(') lllll,jl
pill ll I
=l[jt(jt-}-l)--nt2]o(ffL)
+ 2f+2j +--~1 Z
+nt2Ol')l~jj,~mm,~nn
,
u~X,~) C(Jjf;O - m ~)
J
x 2 [a (+) C ( J j ' j ; - 1
- n') c (+) + a (-) C(Jj'j;I - n') c (-) ] D ~ ) 6ram, ~nn',
(25)
and where a (+) = ½ [ J ( J + 1)] 1/2 and c (±) = ½ [ j ( j + 1) - ( n ' + I)] 1/2. The procedure for determining the eigenvector and eigenvalue matrices, V and A, can be radically simplified by decomposing the system of differential equations for the coefficients C(t) into independent subsystems. This is equivalent to a rearrangement of the matrix M into a form with independent blocks for which the eigenvector and eigenvalue matrices can be calculated independently. Such a simplification of the problem comes from the symmetry properties of the aligned phase of the fluorophores and the symmetry of the fluorophores themselves (a point group of symmetry or hydrodynamical symmetry). This information is contained in the form of the matrix representation of the time-development operator/~(~) (see (24) and (25)). In both cases the matrix elements H .jmn,j' (¢) m~n p disappear for m 4: m'. Therefore, the matrix M can be split into independent blocks for different m = m r values. The same concerns the indices n and n t for a symmetric diffusor and a similar procedure can be employed in the case of asymmetric diffusors. The submatrices Kxy are diagonal in the case considered here. When V and A are calculated for the initial values of the adjustable parameters, the time-evolution of ( 16)(19) can be determined, and finally, the polarized fluorescence decays can be constructed and convoluted with the instrument response function. By comparing the convoluted decays with the experimentally detected ones (by employing appropriate computer software) the parameters describing the fluorescence depolarization of systems undergoing the excited-state reactions depicted in Scheme 1, can be recovered.
J.J. Fisz/ Chemical Physics Letters 269 (1997) 244-256
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2.2. The aligning-interactions- and rotational-motion-controlled kinetics of the system We assume here that the rates for the interconversion process over the excited states depend on the angular orientations of the fluorophores. For example, one can consider a photoisomerisation process of flexible fluorophores in a lipid bilayer, leading to a non-emitting photoproduct. The rates for this process will be different for fluorophores allocated in the middle of the bilayer and for those allocated closer towards the polar heads of the lipids (they are closely surrounded by the hydrocarbon chains of the lipids). For this reason, both populations of the fluorophores will have different fluorescence lifetimes. If the fluorescence lifetimes of both populations of the fluorophores are shorter compared to the timescale of exchange between the members of both populations of the fluorophores, the whole system can be treated as being heterogeneous with two distinct sites of different rotational mobility, ordering and photophysics. This case has been clearly pointed out in our earlier work [9] for DPH (1,6-diphenyl-l,3,5-hexatriene) embedded in lipid systems, on the basis of a number of independent steady-state and time-resolved fluorescence experimental data and also on the basis of the data obtained from linear dichroism experiments. The orientation-dependent kinetic rates can be expressed by the following expansions:
ki(12)=kiooo+, ~
ki,Lsr D~L) ( / 2 *) s t
,
(26)
L>O,s,r
where ki,Ls r = ki,LOr 3sO for uniaxial molecular systems, ki,Lsr = ki,LsO t~rO for a symmetric rotor and ki,Lsr = ki.LO0t3sOt~rOin both cases. ki,ooo c a n be identified as an isotropic kinetic rate c o n s t a n t ki,iso, which corresponds to all the kinetic rate constants considered in previous sections of this work. Let us consider a symmetric rotor in a uniaxial medium, for which the angle-dependent kinetic rate can be approximated in the following way:
ki(fl) = ki.lJ cos2 fl + ki,± sin 2 fl = ½ ( ki, ll + 2 ki.±) + 2 (ki, ii _ ki,±) D ~ ) ( / 2 ) ,
(27)
where ki, II and ki,m are the parallel and perpendicular components (with respect to the axis of uniaxial ordering of the fluorophores) of the angle-dependent rate ki(fl), and where D ~ ) ( / ' 2 ) = D~)(afly) = P2(cosfl). By comparing (26) with (27) one obtains ki,o00 = ki,iso = ½ (ki, l[ + 2 ki,±) •
(28)
We assume here that all state-to-state kinetic rates (excluding the rates for radiative processes) can be angledependent ones, in general. One has to modify the submatrices IJ ~¢) and Kxy in (15), by replacing the matrix elements ki t~jnm,j,m,n, by the corresponding ones [ki]jmn,j,
mtn, _
2J'+8~r 2 1 f D~mJ2(O) ki(12) D 'j'm,,,,*(/2).
dO.
(29)
a ki corresponds to k¢ appearing in IJ <¢) and to kp~ appearing in IJ and Kxy. The matrix elements [ki]jnm,j'm'n' for both representations of the angular dependence of the state-to-state kinetic rates, i.e. (26) and (27), are given by the following expressions:
2jr+ 1 [ ki] jmn.j'm'n' = ki,ooo~jmn,j'm'n' + 2j+"----'~ ~ ki, Lrs C ( L j ' j ; - s - m ) L>O,s,r
C ( Lj'j; --r--nt)t~m,s+m, ~n,r+n' , (30)
[ki]jmn,j'm'n' = g(ki, +2ki,±) tSjmn,j'm'n' + 2(ki, ll-ki,±) 22f-+- ~l C '~2 'Jl "j ; v- - m t,) C(2j'j;O-n')~mm, t3nn, (31) respectively. Consequently, the submatrices Kxy are not diagonal in the case discussed here.
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253
When the angular dependence of the rates for the state-to-state kinetic relaxation is assumed, the polarizationfree term Ph(k'¢)(t) (16) depends on the ground-state molecular ordering and also on the state-dependent rotational dynamics restricted by the species-associated excited-state solute-matrix aligning interactions. The time-dependence of S~akn'¢)(t) in (17) appears for the same reasons. This results from the fact that the photochemical channels along which the excited states interconvert are described by the angle-dependent rates, i.e. there are preferred molecular angular orientations at which these reaction channels are more effective.
3. Examples of particular experimental arrangements for the studies of macroscopically ordered uniaxial molecular systems For macroscopically ordered uniaxial systems formula (6) can be further simplified. For the molecular systems considered in this work, the polarized fluorescence experiments (steady-state or time-resolved) must be invariant with respect to the interchange of angles ~oi and ~of i.e. the transformation C~,p(Oi,~oi) C~,p(Oi,~of) and C~,p(Of,~of) ~ C2,p(Of,~oi) must be an invariant of (6). This can be generalized by the requirement that (6) must be invariant with respect to a simultaneous reflection of the unit vectors ~i and ef through any vertical plane (containing the main axis of molecular alignment of the sample). Introducing Ep (0i, ~Pi,0f, ~of) = C~*p (Oi, ~i ) C2,p(0f, ~f) and taking into account that ~b(pk°'¢) * (t) = ~b(_~p'¢) (t) and Ep (0i, ~Pi,0f, ~pf) = E_p (0i, ~Pi,0f, ~pf), the last term in (6) can be rewritten in the following form: 2
4 ~
C2"p ( 0i, ~oi) C2,p( Of, ~f) ~b(pk't/'so)( t ) = 4 Re E0 ( 0i, ~i, Of, ~0f) Re ~b~0k'1'¢) (t)
p=-2
+ 8 Re El (0i, ~Pi,0f, ~pf) Re ~blk'1'¢)(t) + 8 Re E2 (0i, g'i, 0f, q~f) Re ~b~k'7'¢)(t) - 8 Im E1 (0i, q~i,0f, ~pf) Im ~bl~'~:) (t) - 8 Im E2(0i, q~i,0f, ~of) Im ~b~k'7'¢) ( t ) .
(32)
This expression can be projected into an invariant form with respect to the above-mentioned symmetry by making use of the projection operator ¢9[ ] = ½ (E+o'o) [ ]. E is the identity symmetry operation and o'v is the reflection symmetry operation with respect to the XZ vertical plane (which means an invariance with respect to a reflection by an arbitrary vertical plane, due to the axial symmetry of the system). Taking into account that O"t, : C 2,p ( 0, (p) = ( - 1 )P C2.-p ( 0, ~p), one finds immediately that 0 : Re Ep ( 0i, ~i, 0f, ~f ) = Re Ep ( 0i, (Pi, 0f, ~f) whereas 0 : Imgp(Oi,~i,Of, q~f) -- 0 . This result means that the invariant form of (32) does not contain the last two terms. Taking into account that Re~b(pkn'¢) (t) = ½ (~b(pk'7,O (t) + ~b(__kg'~) (t)) and the explicit forms of C2,p(0, ~) [5], the formula (6) becomes 2 /,~,~r(t, Aex,~.em) ---C(Aex,,~em) ~ K{(Aem) EOAkr/(Aex) ~,k=-I r/=l x [ph(k'~:) (t) + 2 P2(Oi)
s~k~'~)(t) + 2 P2(Of) S~k'() (t) W 4 P2(Oi) P2(Of) ~b(0kn'~¢)(t)
+ 3 sin(20i) sin(2Of) COS(~ i -- 6Of) 1 (¢~Ikr/,~")(t) + ~b(__k~"~:)(t)) + 3 sin2(Oi) sin2(Of) cos2(q~i--~Of) 1 (t~k'q,~:)(t) + ~b(__k~'~) (t))] .
(33)
It is important to stress here that formula (33) describes the polarized fluorescence decays for uniaxial systems independently if the uniaxial distribution of the fluorophores within the sample has polar or apolar character i.e. whether the symmetry of the uniaxial system is C ~ ) or D ~ ) (both symmetry groups have been introduced in Ref. [61). The fact that the correlation functions ~(p~'~)(t) appear in the combination !2 (~b(p~'~) (t) + ~b(~ '~)_ (t)) means that the molecular correlation functions 22vPql(k.pq2 " ~ ~' t) also appear in
254
J.J. Fisz/Chemical Physics Letters 269 (1997) 2 4 4 - 2 5 6
similar combinations, i.e. 1 (22vpq, (k --+ ~:, t) -'1.-22 V p %ql ( k --* ~:, t)), in (10). Taking into account the symmetry property of the Clebsch-Gordan coefficients, C(jlj2L; mn) = ( - 1 ) j~+j2-L C ( j l j 2 L ; - m - n ) , the odd rank order parameters (L = 1,3) disappear in the sum of ½(22VpPqql(k ~ ~:,t = 0) +22 V-pq,(k --Pq2 " " -"+ ( , t = 0)) developed in the Clebsch-Gordan series, because they appear with opposed signs in both constituent correlation functions. Consequently, the odd rank order parameters do not appear in (33) independently of whether the uniaxial distribution of the fluorophores within the matrix has polar or apolar character. For example, this applies to the fluorophores adsorbed at an interface or a surface as well as to polar fluorophores aligned by an external electric field, where the aligning interactions have evidently polar character. This problem also relates to the simplest case i.e. to a one-excited-state problem, and which corresponds to typical polarized fluorescence experiments on membranes or liquid crystals. It is important to stress that the symmetry discussed means that the distribution of the molecules and dynamical evolution of the system on the both sides of a plane defined by the unit vector ~i and the axis of the molecular alignment, are indistinguishable. In Fig. 1 are shown some examples of possible experimental arrangements for the steady-state and timeresolved polarized fluorescence measurements on macroscopically ordered uniaxial systems. They are: (A) macroscopically ordered systems deposited on the surface of a semi-cylinder (e.g. Langmuir-Blodgett (LB) films, a phospholipid bilayer, a multibilayer system and also a solid/air or liquid/air surface); (B) macroscopically ordered systems between two semi-cylinders (e.g. multibilayers, polymers) and (C) a macroscopically ordered system between two glass or quartz plates (e.g. multibilayers, polymers) placed within a cylindrical cuvette filled with an immersion oil. The experimental arrangements shown in Fig. 1 enable steady-state or time-resolved excitation and/or detection angle-dependent polarized fluorescence experiments. Using semicylinders [case (A) and (B)] or a cylindrical cuvette filled with an immersion oil [case (C)] eliminates many complications that can occur due to the multireflection of the exciting light and the emitted fluorescence. Also a real angular resolution of the experiments, within the samples, is high compared to the same experiments on samples ordered between two glass or quartz plates and placed in air. The experimental arrangements indicated in Fig. 1 essentially reduce, or eliminate, the refraction of the exciting light and the emitted fluorescence at the sample/air boundary. The advantage of the variable-angle steady-state or time-resolved experiments is such that by changing the direction of excitation of the sample and/or the direction of detection of the polarized fluorescence, one can change in a controlled way the contribution of all six terms Ph (kg) (t), S(ak~'~) (t), S~ek'~) (t), ~b0~k~g:)(t), Re &~k'Tg)(t) and Re &~k,7,¢)(t), in relation (33). This procedure provides much more experimental data, which can result in more reliable information on the parameters (state-dependent solute-matrix interaction coefficients, order parameters, rates of rotational dynamics, direction of absorption and emission dipole moments
cuzi,s~ on
, (excitation) Ix ~ p ly V ~
(A)
J~
organized systems
excitation (emission)
iZ Y
e~mi~ion Izy i
(B)~
Ix
excitation
.... i
:
X
"
emission
Izy :
(c)~~3 Ix
excitation
~..... ' ', ~
;
Fig. !. Some examples of the experimental arrangements for the time-resolved fluorescence depolarization studies of organized systems.
J.J. Fisz/ Chemical Physics Letters 269 (1997) 244-256
255
as well as the rates for the state-to-state kinetics of the system under study) recovered from the global analysis of the experimental data available. The experimental arrangements shown in Fig. 1, and case (A), in particular, can be employed in studies of thin layers at solid/liquid and liquid/liquid interfaces. When the excitation angle of the sample is adjusted so that a total internal reflection of the exciting light takes place, the molecular system is excited by the evanescent field. The penetration depth of the sample depends on the excitation angle (within the total internal reflection angular regime), the refractive indices of the two media forming the interface and on the excitation wavelength [ 10-12]. The relations derived in this work can also be employed to these kinds of experiments, allowing the study of photoreactions at the interfaces. In this case the species-associated ground- and excited-state aligning potentials and also the species-associated rotational dynamics of the fluorophores must be distinguished from the corresponding bulk parameters (which can be determined from experiments at an excitation angle outside the total internal reflection angular regime). For experiments with a variable-depth penetration, these quantities have to be described in terms of discrete or continuous distributions as a function of the depth penetration. Furthermore, the penetration depth-dependent concentration profiles of the species have to be considered, in a more general case. Finally, the modified expression for the polarized fluorescence decay has to be integrated over the whole area penetrated by the evanescent field. This problem will be the subject of an independent discussion.
4. Discussion
Formula (33), describing the polarized fluorescence intensity and applying to all the cases shown in Fig. 1, represents a S-pulse excitation fluorescence decay. For the purpose of the analysis of the experimental data, formula (33) must be convoluted with the instrument response function on the exciting laser pulse. Expressions enabling the calculation of the convolution integrals for the scatter method and the method of monoexponential reference compounds, for fluorescence depolarization experiments on systems undergoing excited-state reactions, are given in Ref. [ 13]. From a theoretical point of view, the global analysis of the excitation and emission multiwavelength variable-angle polarized fluorescence decay surfaces should enable the determination of all parameters describing the polarized fluorescence decay of the system undergoing the reaction depicted in Scheme 1. These parameters are: (a) coefficients U (g'k) and u~ x'~) describing the ground- and excited-state aligning interactions; (b) the components of the state-dependent'~"~diffusion tensors Dt(); (c) the state-to-state kinetic relaxation rate constants (or the parameters describing the angular dependence of the kinetic rates, if they are angle-dependent); (d) the angles describing the orientations of the absorption and emission transition dipole moments for all the electronic states involved and (e) the spectral contributions of all the absorption and emission bands involved at the excitation and emission wavelengths at which the polarized fluorescence decays have been collected. The problem discussed in this work concerns a more general case in which a greater number of ground and excited states is assumed. One has to realize, however, that in practice the number of ground and excited states for which the experimental data can be analyzed successfully, will be limited. Further investigations based on the analysis of computer generated synthetic polarized fluorescence decays for different particular cases (e.g. different number of electronic states involved, different shapes of the state-dependent aligning potentials), are needed to explore this problem in more detail. Such studies are necessary in order to predict all limitations of the applicability of the formalism discussed in this work to real experimental cases. Furthermore, such studies seem to be the only way of working out an appropriate numerical algorithm necessary for analyses of the experimental data and to predict the optimum experimental conditions ensuring a high precision of the experimental data analysis and the reliability of the recovered parameters. A systematic study of the problem mentioned here will be presented elsewhere.
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J.J. Fisz/Chemical Phystcs Letters 269 (1997) 244-256
Acknowledgement This work was supported in part by the Polish Committee for Scientific Research (KBN), within Project 2.P303.089.04. References [1] [2] [3] [4] [5] [6] [7] [8 ] [9] [10] [11] [ 12]
J.J. Fisz, Chem. Phys. 181 (1994) 417. A. Szabo, J. Chem. Phys. 81 (1984) 150. A.J. Cross, D.W. Waldeck and G.R. Fleming, J. Chem. Phys. 78 (1983) 6455. J.J. Flsz, Chem. Phys. Lett. 262 (1996) 507, D.M. Brink and G.R. Satchler, Angular momentum (Oxford Univ. Press, Oxford, 1968). J.J. Fisz, Chem. Phys. 114 (1987) 165~ M.E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957). H. Risken, The Fokker-Planck equation. Methods of solutions and applications (Springer, Berlin, 1984). J.J. Flsz, Chem. Phys, 132 (1989) 315. N.J. Harrick, Internal reflection spectroscopy (Wiley/Interscience, New York, 1967). D. Axelrod, T.P. Burghard and N.L. Thompson, Annu. Rev. Biophys. Bioeng. 13 (1984) 247. H. Masuhara, EC. De Schryver, N. Kitamum and N. Tamai, Microchemistry. Spectroscopy and chemistry in small domains (NorthHolland, Amsterdam, 1994). [13] J.J. Fisz, Chem. Phys. Lett. 262 (1996) 495.
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 269 (1997) 244-256
Time-resolved polarized fluorescence spectroscopy of photoreactive systems in uniaxially organized molecular media J.J. Fisz Institute of Physics, N. Copernicus University, ul. Grudziqdzka 5/7, PL 87-100 Toruri, Poland
Received 23 October 1996; in final form 21 February 1997
Abstract
A description of the time-resolved polarized fluorescence spectroscopy on photoreactive systems incorporated into a uniaxially organized molecular medium is discussed. The treatment assumes many-ground-state species interconverting in the excited states. The ground- and excited-state species undergo different solute-matrix aligning interactions. The rotational motion of the excited-state species is state-dependent. The diffusion model in the state-dependent potential for asymmetric and symmetric rotors is considered. Expressions enabling the computation of polarized fluorescence decays are derived and discussed. The case when the state-to-state kinetics of the systems are controlled by fluorophore-matrix aligning interactions and rotational dynamics of the fluorophores is considered. Some possible experimental arrangements enabling studies of such molecular systems as monolayers, bilayers, membranes as well as solid/air and liquid/air surfaces, are briefly considered. Indicated is a possible application of the theoretical and experimental methods discussed in this work to the time-resolved polarized total internal reflection fluorescence spectroscopy of thin layers of different kinds of molecular assemblies at solid/liquid and liquid/liquid interfaces. © 1997 Elsevier Science B.V. To the memory o f my dear daughter Karolina
1. Introduction
In this work we discuss a description of the time-resolved polarized fluorescence experiments on photoreactive species embedded in uniaxially organized molecular systems. The case considered in this work is depicted in Scheme 1. It is assumed that the system is composed of a greater number of ground-state species interconverting in the excited states. Due to the excitation of the sample at a certain excitation wavelength Aex a greater number of ground-state species can be excited simultaneously (the absorption bands can overlap). The polarized fluorescence monitored at a particular emission wavelength h~m can be contributed to by signals from a greater number o f excited states (the emission bands can overlap). It is assumed that the ground-state species can be excited to the lowest two electronic excited states and that both electronic transition dipole moments have different directions in every species. The state-to-state kinetic relaxation of the system is described by the rates indicated in Scheme 1. The kinetic rate k~ represents a rate for the internal conversion process from the electronic state i** to the i* one (where i = 1 ..... s). The fluorescence is detected from the lowest electronic excited state. It is assumed that the ground-state species undergo different solute-matrix aligning interactions (described by the effective potentials Vg,i(/2)) and that the excited-state solute-matrix interactions are different for different 0009-2614/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PI1 S0009-26 14(97)00270-4
J.J. Fisz/ Cheraical Physics Letters 269 (1997) 244-256
245
species (they are described by the effective potentials Vex,f(O)). The state-dependent species-associated solutematrix interactions lead automatically to different equilibrium angular distributions of the species in all the ground and excited states, probed in the experiment. The rotational dynamics of the excited-state species are state-dependent and are described in terms of the species-associated Green's functions pi* (19-0,010"2, t), which describe the conditional probability of finding a molecule in an excited state i at time t and at an angular orientation ~2 (the Euler angles), which at time t = 0 was oriented at 120. The state-dependence of the rotational motion of the species results, in our treatment, from the state-dependence of the solute-matrix aligning interactions (the rotational dynamics are restricted by the aligning interactions - the rotational motion in a state-dependent potential) and can also result from the state-dependence of the diffusion t e n s o r s ( D (i), D (i) , D~ i) for an asymmetric diffusor or Dli), D~_~ for a symmetric diffusor).
i,
i,
! ,
i ~ z : ~ k , al v ~ \ , i
k24- "..
equilibration Scheme 1 We derive expressions enabling the computation of the polarized decays of the fluorescence for asymmetric and symmetric diffusors rotating in species-associated excited-state potentials, which after convolution with the instrument response function, can be compared with the measured polarized decays. The treatment is further developed to a case when the kinetics of the system are controlled by fluorophore-matrix aligning interactions and by the rotational dynamics of the fluorophores. Such a case can occur when the state-to-state kinetic rates depend on the angular orientation of the fluorophores in an organized environment. The symmetry relations enabling a substantial simplification of the eigenvalues and eigenvectors problem will be discussed. The symmetry relations for the matrix representation of the time-development operator, coupling the state-dependent molecular alignments and rotational dynamics with the state-to-state kinetics, decompose the system of simultaneous differential equations into independent subsystems which can be treated separately. We discuss examples of possible experimental arrangements that can be useful in studies of such uniaxially organized molecular systems as monolayers, bilayers, multibilayers, membranes as well as solid/air and liquid/air surfaces. The advantages of the experimental arrangements discussed are indicated. The experimental arrangements can also be employed in studies of thin layers of different kinds of molecular assemblies at solid/liquid and liquid/liquid interfaces, by means of time-resolved polarized total internal reflection fluorescence (TIRF) spectroscopy. The theory developed in this work together with the discussed experimental arrangements enabling variableangle time-resolved polarized fluorescence measurements, can be useful tools in studies of organized molecular media. The application of photoreactive fluorophores as probes for monitoring the micro-environmental properties of an organized medium can be a source of information on the structural and dynamical properties of the system on a micro-scale (e.g. micro-packing, local mobility (micro-viscosity) and micro-polarity). The problem discussed in this Letter is one of the many cases considered briefly in our previous article [ 1 ] (i.e. ordered systems, solutions and labeled macromolecules), and which is further developed in this work. It is important to mention here the excellent work by Szabo [ 2 ] who considered many aspects of the fluorescence depolarization in uniaxial systems. One of the problems discussed was the many-ground- and many-excited-state
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J.J. Fisz/Chemical Physics Letters 269 (1997) 244-256
problem. However, this problem was treated under the assumption that the rotational dynamics and orientational ordering of the fluorophores were not state-dependent. Also important is the pioneering work by Cross et al. [ 3 ] who formulated a description of fluorescence depolarization experiments with the assumption of state-dependent rotational dynamics in the diffusion model, in the case of solutions. 2. Polarized emission Consider the situation displayed in Scheme 2. An organized (anisotropic) molecular system is excited at t = 0 by a S-pulse of light with polarization ei and polarized emission with polarization ~f is detected at t ~> 0. /2ab,k,~ represents the absorption transition dipole moment of a molecule in the k th ground state and which has been photoselected at t = 0 and at hem to one of the excited states (k* or k**) or to both excited states. It is assumed that the directions of the transition dipole moments /2,,b,kl and /2,,b,k2 are different in the molecule-fixed frame. /2era,( is the emission dipole moment of the same molecule in the ( t h excited state at a later time t. The orientations of Oi and Of in the laboratory-fixed frame XLYLZL are described by the polar angles (0i, ~oi) and (Of, ~of), respectively. The orientations of the absorption and emission dipole moments are defined in the molecule-fixed frame in terms of the polar angles (0~~) , ~o~k'J) ) ( r / = I, 2) and (0~~) , ~o~z~)). The angular orientations of the molecule-fixed frames at t = 0 and t > 0 are defined in the laboratory-fixed frame by two systems of Euler angles Oo = (ao, tgo, Yo) and s'2 = (a, fl, y).
ZL
excitation t--0
enfiuion t>0
-¢÷ \ ~ ) YL
XL Scheme 2
The intensity of the polarized emission for the system shown in Scheme 1 is described by the following expression: 2
l~i~,(t, Aex, Aem) = C(Aex, ~ . e m ) sC,k=-I
K~c(~-em)Z ¢Ok'q(Aex) r/=l
• ex,k,7,,,ex, fg.k(O0) pk~¢(.O0,010, t) Pe(~,~(Aem) dr20 dr2
×
Q
(1)
J2g2o C(Aex, ~-em) includes all excitation and emission wavelength-dependent instrumental factors. K~(~.em ) is the intensity of the fluorescence band of the Cth state at hem. (.Okrt(,~.ex) (~] = 1,2) represents the absorbances of the k* th and k** th absorption bands at Aex. Pex,~,(hex)~') and Pe~m~(Aem), describe the angular dependence of the polarized excitation (absorption) and emission probabilities [4], respectively, fg,k(O0) describe the equilibrium orientational distribution of the solute molecules in the ground state k and are given by the Boltzmann distributions
fg,k(/2) = Ng,k exp(--gg,k($'2)/kBT) ,
(2)
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J.J. FiszlChemical Physics Letters 269 (1997) 244-256
where Ng,k is the normalization constant. Vg,k(/2) are the species-associated ground-state aligning potentials. These potentials can be represented by the following series: oo
Vgk(/2),
= kBT Z'JPq(g'k) Dp,q(J)* (/2) ,
(3)
Jpq
where the expansion coefficients Ujpq • (g,k) describe the species-associated ground-state solute-matrix aligning interactions. The distribution function fg,k(/2"2) can be developed in the basis set of the Wigner rotation matrices [5], i.e. oo
fg,k(/2) = Z 2j877.2 + 1 (Dpq (j) )g,k --pq T)(j ) ,( /2) , Jpq
(4)
where /D (j) /g,r \ • are the orientational order parameters of molecules in the k th ground state, \ pq fg,,(/2) __pq f)(J) ( / 2 ) d/2.
(Dpq)g,k (J) = f
(5)
12
Pk--,¢ (12o, 01/2, t) are the Green's functions (propagators) describing the conditional probability of finding an excited molecule at time t and at an angular orientation/2 in state s¢, which was excited to the primary excited state k and with initial angular orientation of /2o. These propagators describe the state-dependent rotational motion of molecules, coupled with the state-to-state kinetic relaxation process. By performing the integration in (1) and employing the orthogonality relations for the Wigner functions [5], one can obtain a formula for polarized emission for many-ground- and many-excited-state systems in an organized medium. In the case of macroscopically ordered uniaxial systems (e.g. membranes, liquid crystals, solid/liquid and liquid/liquid interfaces etc.) the solute-matrix interaction coefficients become uJ.pgqk) = @gqk)6pO, (J) and consequently, \/D (j)pq/g,K \ ' = (Doq)g,k 8p0. The same holds for the excited-state potentials and order parameters. Symmetry properties for the solute-matrix interactions automatically lead to a corresponding simplification of the description of the molecular rotational dynamics. The same problem can be considered by employing group theory methods [6]. Finally, for a uniaxial molecular system (i.e. a rotation of the sample around the axis of molecular alignment by an arbitrary angle, is an invariant) the polarized fluorescence decay component has the form
s I~, #f(t,/~ex, ~-em): C(aex, ~-em) Z
2
[
K~'(~'em)~ 0)k'o(~'ex) Ph(k'()(t) + 2 P2(0i) Sa~tw'~:)(t)
(,k=l
~=1
+2P2(Of) S~k'e)(t) + 4 Z
C2*,p(Oi'~°i) C2,p(Of,~of) 4)~kn'O(t) ,
(6)
p=-2
where
Ph(k'e)(t)=fffg,d~)pk--e(~,Ol/2,
(7)
t) d~d/2,
/2~o S~akm()(t) = Z
fg,k($'~0) D(2)*(/20)0ql Pk~¢(/2°'0l/2't) d/2o d/2 ~z,q,,vA
,WA
]
qi=--2 2
= Z q1=--2
/T~(2)\(') \ ~ 0 q l Ig,k--*~ C2:ql(Oik'~}'@Ak'~)) ,
(8)
J.J. Fisz/Chemical Physics Letters 269 (1997) 244-256
248
fg,k(E20) pk-+~(O0'O[ 12't) DCz)t~~°^~ "u, dE2°
Se{~'¢)(t) = Z
~'21~*,q2~'t'za(()L'E , WE ^(() )
q2=-2 2
= ~
/D(2)\(t) r * r'~(( ) .~(¢) \ Oq2/k---+~"2,q2~.ffE ,WE ) ,
(9)
q2=--2
r.
--Pql
Pq2t JlO) dOo dO
q~,q2=-z'hh0 ,,~(k,7) (kn) . ~ . ,,~() (~:) X C2,ql{ffA '~PA ) t~2,q2{,O'E , ~ E ) 2
=
~
t~(k,7) 'V'VA _(ko) )'-'2,qzt"E r * -,Q(sc) ,wE ,,,(~:) ) .
2 2C v P2 q, ql"'" (, k' ~- -A+' p~q' t2)
(10)
q~,q2=--2
Cj,m(O, ¢) are modified spherical harmonics [5], i.e. Cj,,,(O, ~p) = [ 4 ~ / ( 2 j + 1 )] U2Yj,m(O, ~v), where Yj,m(O, ~v) are the spherical harmonics. In the relations given above Ph (~g) (t) describes polarization-free decay of fluorescence. Sa~k~g) ( t ) / P h (kg) (t) means the order parameter of the absorption dipole moments in the kth ground state (in the absorption band r/ = 1,2) which is detected through the emission from the excited state sc. Se[k'~) ( t ) / P h (kg) (t) is the timedependent order parameter of the emission dipole moments in the ~: th state of the molecules which at t = 0 have been excited from the k th ground state. ~b~'()(t)/Ph~k'¢)(t) represents the correlation (time-dependent mutual angular orientation) between the absorption dipole moment of a molecule excited at t = 0 to k* or k** and the emission dipole moment of the same molecule in the ~rth state at a later time t. n ( 2 ) \ (t) /Ph(k,~) ¢tx The normalized second rank order parameters /\'-'0qj/g,k-~(/--" ~,j are the molecular order parameters in the k th ground state recovered through the polarized emission from state (. (D0(~)~¢/ph(k'¢) (t) are the molecular order parameters of the molecules in state (, that were originally excited to state k. ~VpPqt (k --+ sc, t ) / P h < g) (t) are the molecular correlation functions describing the angular relationship between molecules that at time t = 0 r~(2)\(t) and were excited to state k and at a later time emitted fluorescence from state (. x[O(2)\(t) Oq~/g,k---+(, /\L~Oq2/k---+( ~ V pq~ (k ---+~, t) couple the state-dependent molecular ordering and rotational dynamics with the state-to-state kinetics of the system.
2.1. State-dependent rotational dynamics of the symmetric and asymmetric diffusors in the state-dependent excited-state aligning potentials In order to find explicit forms for the Green's functions pk~((O0,01/'2, t) one can consider dependence of the propagators p__.((/20, 0112, t) = )--]-~-Ipk--+¢(/20, 01/2, t) A k.ex, and which describe tional probability of finding a molecule in the excited state ( at time t, which at t = 0 was excited to primary excited states (the sum over k). Ak,ex = 1 if state k is the primary excited one and A~xx = 0 These propagators obey the equation of motion: ~p~l(OO,01/2, t)=-
Htl)+kl+Zkli i=2
~ p ~ s ( O o , O l O , t) = -
- Z i=l
p~l(OO,0l/'2, t ) - - Z k a p ~ i ( E l 0 , 0 l ~ , t )
the timethe condiany of the otherwise.
,
i=2
ki, P~i( Oo,OI12, t) + t?I~') + ks + Z i=l
k,i p - - , ( O o , OIf2, t) .
(11)
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In the restricted diffusion model the time-development operator i¢?/(() has the form 3
(12) i=1
where the operators Li are the quantum mechanical orbital momentum operators (i = x, y, z), D: ~) are the components of the diffusion tensor and Vexg(12) is the aligning potential in the excited state (. The solution to the equation of motion ( 11 ) has the form [ 1 ] :
=Z [Z :J+' 8¢r2 $
to
to
k=l
jmn
fm'n'
jmn,j, mtn ,t,t.~ t/ Dra, R(('k) (3')", (/~0)
Z
Ak,ex
s
=~_,pk-.#(0o.Olgl, t) Ak.ex •
(13)
k=l
The expansion coefficients RJ~,m,n,(t) appearing in (13) can be calculated from the following procedure. In the matrix equation corresponding to the system of equations ( 11 ), i.e. p(/'2o, 01O, t) = - A p(/9.o, 01/2, t), the Green's functions are replaced by the expansions p-.e(0O,010, t) = ~-]~, (2j + 1)/87r 2 C:~(t) D~J~) *(/2). This leads to a system of coupled differential equations forthecoefficients C:(m~(t), i.e. ~7(t) = - M C(t). A formal solution to this matrix equation is C(t) = R(t) C(0), where R(t) = V exp ( - A t) V -1. V and A = [ .~qq] are the eigenvector and eigenvalue matrices of M, respectively. M is the matrix representation of A in the basis set of Wigner functions• C(0) is a vector of the initial conditions, [C(0)] (:)Jmn= Cjra.(O)'(~) C)~r.~)(0)(~) can be determined from the initial conditions for the Green's function p_.~(Oo,0lO, t) = (3(Oo - / 2 ) A~,ex, and consequently, Cjm. (() (0) = DOn) (/2o) A~,ex. The matrices Iql and R(t) have the following structure:
M =
U 1) K21 K31 "'" Ks1 / Kl2 U (2) K3z "'" Ks2 KI3 K23 U (3) "'" Ks3 , Kls
K2s
//R(l'l)(t)
[R(2,1)(t)
K3s . . . U (s) R(l,2)(t ) R(l,3)(t)
... Rfl,s)(t ) \
R(2,2)(t) R(2,a)(t)
R(2,s)(t)
R(t) = / Rf3'l)(t) Rf3'2)(t) R(3'3)(t)
R(3's)(t)
/
•
:
]
"
\ R f s , l)(t) RCs,2)(t) Rfs,3)(t)
Rfs,s)(t)
J .
(14)
tr(~) s U (~) are submatrices with elements [IJ(~)]p,q = '-" j,,,,,.j, z, n, = t4(~) -j,,,n,j,z,., + (k~ + Y]-i=l,i.,e ki~) (3j,,,..j'm'.', where -
2j'+ I
m"2(O) fD ix#,
D o(j') *:/2) d O ,.,,
(15)
12
The submatrices Kxv are diagonal, i.e. [Kxy]jmn,j,m,n , = - kxy (3jmn,j'm'n', where kxy is the rate for the state-tostate kinetic relaxation from an excited state x to another excited state y. The matrix elements of R (~'k) (t) are R.l~m'nk,~,m,n,(t) = [(V [exp(-Aqq t ) ] V -l)(~'k)]jmn,j,m,n,, where ~: means the state from which the polarized
J.J. Fisz/Chemical Physics Letters 269 (1997) 244-256
250
emission is observed and k denotes one of the primary excited states. The algorithm described here is based on one of the methods of solving the Fokker-Planck equation [8], and which in our problem, has been employed to a system of coupled differential equations for Green's functions [1]. From (13) the Green's functions p,~¢(O0,01/2, t) can be found, and hence, Ph ~*'¢), Sa~k''¢) (t), Se(*'~) (t) and ~b(p*''~) (t) given by ( 7 ) - ( I 0 ) can be replaced by the relations
Ph(k'()(t)= E ~'~ \/D O,n (j')\ •~(~"~) ~O00,jtOn ~,~: ~/g,k, j/n / 2 oo (j~n~ R~o~ok.),,on,(t)2J'V:q~(k,t,j = 0) ) ,...2,q, [-,* tUA :,~(k~7)(krl) Sa~km()(/) = E ,~oA ) , ql=-2
2
(17)
.
s~k'('(t)-- Z q2=--2 ¢b(-k m ( ) (- t - =
(16)
e~(o'k),j/On/(t) \ O,n,/g,k) C2",q2(0 ((), ~
2"
"~
( ~ - " R 2pq2,j'pn' ¢¢'k) (t) J Vpn pql," "" ,,~k,) _~*,),) C 2,q2 * (O(~¢),~p(~¢)) • kZ..~ '~l¢'t= O) ] t-'2.qll, t~A ,q'A q~,q2=-2 \j~n ~ ~
(18)
(19)
2f l/Pqi b t = 0) are the molecular correlation functions at time t = 0 in the excited state k. They can be ,p,, (~,~, expanded in the Clebsch-Gordan series, i.e.
2+j' 2j' vPq' "pn' ( k, t = O) = ( - 1 )P-q'
Z
C(2j'L;-pp)
C(2j'L;-qin')/r~ft)\LIo,n/_ql ]g,k\ ,
(20)
L=I2-j' I where C ( j l j 2 j ; m n ) are the Clebsch-Gordan coefficients [7] and ,:D~L) O,n--ql)g,, k are the ground-state order parameters in state k. The dependence of Ph (kg) (t) on the orientational ordering of the fluorophores in the ground state and the time-dependence of S~akng)(t), in the relations derived, appear when the rates for the state-to-state kinetic relaxation of the fluorophores are controlled by the aligning properties of the fluorophores within an ordered matrix, i.e. these rates are orientation-dependent. This problem will be discussed in the next subsection. If the rates for the state-to-state kinetic relaxation are not orientation-dependent, Ph(kg)(t) and Sa~ g ) (t) simplify to ph{k,f)(t) = R~f ,k) ¢t~: , 0o0,000~
S~akn,f)(t) = "'000,000~ taft ,k) ¢,~ ~l S~kn)
(21)
where
(Doq, )g,k C2.q I
(22)
ql
R(¢,k) r ,~ describes the state-to-state kinetic relaxation alone, over all the excited states involved. Sa~kn) is the 000,000,-: order parameter of the absorption dipole moments in the kth ground state and (D ~z~' are the second rank , 0qt)g,k molecular order parameters in the same state. Taking into account the explicit expressions for Lx D(J')m'n'*' Lv. D(j')m,n,* and L z DO')m n'*' and assuming that the species-associated excited-state aligning potentials are Vex~:(/2), = knrEu~yepxg) ~0pr~J)*(/2) , @
(23)
J.J. Fisz/Chemical Physics Letters 269 (1997) 244-256
251
where , (ex,() describe the species-associated excited-state solute-matrix aligning interactions for a uniaxial molecular system, one obtains the matrix elements of u(£) . ~jmn,j~ nff n~ ••
HJ(~) m?l ff t~ltnt ~ ( [Aj,., (D(x() + D~fi) ) + n '2 D~~)] 6,,n, + (u(+) ~fn' 2f+
~n,n'+2 -'1- Bj,(-) n, ~n,n'-2) (D(x() --
D~,( ) ) ) 6jj, 6tara'
1
+ --2j + 1 Z'ujp(exg) C(Jjj';O - m ' ) Jp x ( a (+) C ( J j ' j ; - p - I
-n')[(D(x ( ) - D~fl)) b (+) Bn,t,+n'+2+ (Dx(~) + D~') ) c (+) 6n,p+n']
+ a (-) C( j j ./.j ; - p + 1 - n r) [(D(x~) + D~fi)) c (-) t3n,p+n' + (D(J) - D(y()) b (-) ¢~n,p+n'--2] + p(p + n') C ( j j r j ; - p - n ' ) D~0 6n,p+n'} 8,nm, ,
(24)
~j,,, = ¼ [ ( j ' ( j ' + 1) - n'(n r 4- 1)) ( j ' ( j ' + 1) - (n' 4- 1)(n' 4- 2))] '/2 , where Aj,n, = ½ [j'(j' + 1) - n'2], R(+) a (+) = ½ [ J ( J + 1) - p ( p 4- 1)] 1/2, b (±) = ½ [ j ( j + 1) - ( p + n ' 4- 1)(p + n ' 4-2)] 1/2 and c (+) =
!2 [J(J + 1) - (p + n' 4- 1)(p + n')] 1/2. For a symmetric diffusor (Dx(~:) = D~~). = D ~ ), D~() = Dlf)) the matrix representation (24) reduces to Hi(') lllll,jl
pill ll I
=l[jt(jt-}-l)--nt2]o(ffL)
+ 2f+2j +--~1 Z
+nt2Ol')l~jj,~mm,~nn
,
u~X,~) C(Jjf;O - m ~)
J
x 2 [a (+) C ( J j ' j ; - 1
- n') c (+) + a (-) C(Jj'j;I - n') c (-) ] D ~ ) 6ram, ~nn',
(25)
and where a (+) = ½ [ J ( J + 1)] 1/2 and c (±) = ½ [ j ( j + 1) - ( n ' + I)] 1/2. The procedure for determining the eigenvector and eigenvalue matrices, V and A, can be radically simplified by decomposing the system of differential equations for the coefficients C(t) into independent subsystems. This is equivalent to a rearrangement of the matrix M into a form with independent blocks for which the eigenvector and eigenvalue matrices can be calculated independently. Such a simplification of the problem comes from the symmetry properties of the aligned phase of the fluorophores and the symmetry of the fluorophores themselves (a point group of symmetry or hydrodynamical symmetry). This information is contained in the form of the matrix representation of the time-development operator/~(~) (see (24) and (25)). In both cases the matrix elements H .jmn,j' (¢) m~n p disappear for m 4: m'. Therefore, the matrix M can be split into independent blocks for different m = m r values. The same concerns the indices n and n t for a symmetric diffusor and a similar procedure can be employed in the case of asymmetric diffusors. The submatrices Kxy are diagonal in the case considered here. When V and A are calculated for the initial values of the adjustable parameters, the time-evolution of ( 16)(19) can be determined, and finally, the polarized fluorescence decays can be constructed and convoluted with the instrument response function. By comparing the convoluted decays with the experimentally detected ones (by employing appropriate computer software) the parameters describing the fluorescence depolarization of systems undergoing the excited-state reactions depicted in Scheme 1, can be recovered.
J.J. Fisz/ Chemical Physics Letters 269 (1997) 244-256
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2.2. The aligning-interactions- and rotational-motion-controlled kinetics of the system We assume here that the rates for the interconversion process over the excited states depend on the angular orientations of the fluorophores. For example, one can consider a photoisomerisation process of flexible fluorophores in a lipid bilayer, leading to a non-emitting photoproduct. The rates for this process will be different for fluorophores allocated in the middle of the bilayer and for those allocated closer towards the polar heads of the lipids (they are closely surrounded by the hydrocarbon chains of the lipids). For this reason, both populations of the fluorophores will have different fluorescence lifetimes. If the fluorescence lifetimes of both populations of the fluorophores are shorter compared to the timescale of exchange between the members of both populations of the fluorophores, the whole system can be treated as being heterogeneous with two distinct sites of different rotational mobility, ordering and photophysics. This case has been clearly pointed out in our earlier work [9] for DPH (1,6-diphenyl-l,3,5-hexatriene) embedded in lipid systems, on the basis of a number of independent steady-state and time-resolved fluorescence experimental data and also on the basis of the data obtained from linear dichroism experiments. The orientation-dependent kinetic rates can be expressed by the following expansions:
ki(12)=kiooo+, ~
ki,Lsr D~L) ( / 2 *) s t
,
(26)
L>O,s,r
where ki,Ls r = ki,LOr 3sO for uniaxial molecular systems, ki,Lsr = ki,LsO t~rO for a symmetric rotor and ki,Lsr = ki.LO0t3sOt~rOin both cases. ki,ooo c a n be identified as an isotropic kinetic rate c o n s t a n t ki,iso, which corresponds to all the kinetic rate constants considered in previous sections of this work. Let us consider a symmetric rotor in a uniaxial medium, for which the angle-dependent kinetic rate can be approximated in the following way:
ki(fl) = ki.lJ cos2 fl + ki,± sin 2 fl = ½ ( ki, ll + 2 ki.±) + 2 (ki, ii _ ki,±) D ~ ) ( / 2 ) ,
(27)
where ki, II and ki,m are the parallel and perpendicular components (with respect to the axis of uniaxial ordering of the fluorophores) of the angle-dependent rate ki(fl), and where D ~ ) ( / ' 2 ) = D~)(afly) = P2(cosfl). By comparing (26) with (27) one obtains ki,o00 = ki,iso = ½ (ki, l[ + 2 ki,±) •
(28)
We assume here that all state-to-state kinetic rates (excluding the rates for radiative processes) can be angledependent ones, in general. One has to modify the submatrices IJ ~¢) and Kxy in (15), by replacing the matrix elements ki t~jnm,j,m,n, by the corresponding ones [ki]jmn,j,
mtn, _
2J'+8~r 2 1 f D~mJ2(O) ki(12) D 'j'm,,,,*(/2).
dO.
(29)
a ki corresponds to k¢ appearing in IJ <¢) and to kp~ appearing in IJ and Kxy. The matrix elements [ki]jnm,j'm'n' for both representations of the angular dependence of the state-to-state kinetic rates, i.e. (26) and (27), are given by the following expressions:
2jr+ 1 [ ki] jmn.j'm'n' = ki,ooo~jmn,j'm'n' + 2j+"----'~ ~ ki, Lrs C ( L j ' j ; - s - m ) L>O,s,r
C ( Lj'j; --r--nt)t~m,s+m, ~n,r+n' , (30)
[ki]jmn,j'm'n' = g(ki, +2ki,±) tSjmn,j'm'n' + 2(ki, ll-ki,±) 22f-+- ~l C '~2 'Jl "j ; v- - m t,) C(2j'j;O-n')~mm, t3nn, (31) respectively. Consequently, the submatrices Kxy are not diagonal in the case discussed here.
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When the angular dependence of the rates for the state-to-state kinetic relaxation is assumed, the polarizationfree term Ph(k'¢)(t) (16) depends on the ground-state molecular ordering and also on the state-dependent rotational dynamics restricted by the species-associated excited-state solute-matrix aligning interactions. The time-dependence of S~akn'¢)(t) in (17) appears for the same reasons. This results from the fact that the photochemical channels along which the excited states interconvert are described by the angle-dependent rates, i.e. there are preferred molecular angular orientations at which these reaction channels are more effective.
3. Examples of particular experimental arrangements for the studies of macroscopically ordered uniaxial molecular systems For macroscopically ordered uniaxial systems formula (6) can be further simplified. For the molecular systems considered in this work, the polarized fluorescence experiments (steady-state or time-resolved) must be invariant with respect to the interchange of angles ~oi and ~of i.e. the transformation C~,p(Oi,~oi) C~,p(Oi,~of) and C~,p(Of,~of) ~ C2,p(Of,~oi) must be an invariant of (6). This can be generalized by the requirement that (6) must be invariant with respect to a simultaneous reflection of the unit vectors ~i and ef through any vertical plane (containing the main axis of molecular alignment of the sample). Introducing Ep (0i, ~Pi,0f, ~of) = C~*p (Oi, ~i ) C2,p(0f, ~f) and taking into account that ~b(pk°'¢) * (t) = ~b(_~p'¢) (t) and Ep (0i, ~Pi,0f, ~pf) = E_p (0i, ~Pi,0f, ~pf), the last term in (6) can be rewritten in the following form: 2
4 ~
C2"p ( 0i, ~oi) C2,p( Of, ~f) ~b(pk't/'so)( t ) = 4 Re E0 ( 0i, ~i, Of, ~0f) Re ~b~0k'1'¢) (t)
p=-2
+ 8 Re El (0i, ~Pi,0f, ~pf) Re ~blk'1'¢)(t) + 8 Re E2 (0i, g'i, 0f, q~f) Re ~b~k'7'¢)(t) - 8 Im E1 (0i, q~i,0f, ~pf) Im ~bl~'~:) (t) - 8 Im E2(0i, q~i,0f, ~of) Im ~b~k'7'¢) ( t ) .
(32)
This expression can be projected into an invariant form with respect to the above-mentioned symmetry by making use of the projection operator ¢9[ ] = ½ (E+o'o) [ ]. E is the identity symmetry operation and o'v is the reflection symmetry operation with respect to the XZ vertical plane (which means an invariance with respect to a reflection by an arbitrary vertical plane, due to the axial symmetry of the system). Taking into account that O"t, : C 2,p ( 0, (p) = ( - 1 )P C2.-p ( 0, ~p), one finds immediately that 0 : Re Ep ( 0i, ~i, 0f, ~f ) = Re Ep ( 0i, (Pi, 0f, ~f) whereas 0 : Imgp(Oi,~i,Of, q~f) -- 0 . This result means that the invariant form of (32) does not contain the last two terms. Taking into account that Re~b(pkn'¢) (t) = ½ (~b(pk'7,O (t) + ~b(__kg'~) (t)) and the explicit forms of C2,p(0, ~) [5], the formula (6) becomes 2 /,~,~r(t, Aex,~.em) ---C(Aex,,~em) ~ K{(Aem) EOAkr/(Aex) ~,k=-I r/=l x [ph(k'~:) (t) + 2 P2(Oi)
s~k~'~)(t) + 2 P2(Of) S~k'() (t) W 4 P2(Oi) P2(Of) ~b(0kn'~¢)(t)
+ 3 sin(20i) sin(2Of) COS(~ i -- 6Of) 1 (¢~Ikr/,~")(t) + ~b(__k~"~:)(t)) + 3 sin2(Oi) sin2(Of) cos2(q~i--~Of) 1 (t~k'q,~:)(t) + ~b(__k~'~) (t))] .
(33)
It is important to stress here that formula (33) describes the polarized fluorescence decays for uniaxial systems independently if the uniaxial distribution of the fluorophores within the sample has polar or apolar character i.e. whether the symmetry of the uniaxial system is C ~ ) or D ~ ) (both symmetry groups have been introduced in Ref. [61). The fact that the correlation functions ~(p~'~)(t) appear in the combination !2 (~b(p~'~) (t) + ~b(~ '~)_ (t)) means that the molecular correlation functions 22vPql(k.pq2 " ~ ~' t) also appear in
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J.J. Fisz/Chemical Physics Letters 269 (1997) 2 4 4 - 2 5 6
similar combinations, i.e. 1 (22vpq, (k --+ ~:, t) -'1.-22 V p %ql ( k --* ~:, t)), in (10). Taking into account the symmetry property of the Clebsch-Gordan coefficients, C(jlj2L; mn) = ( - 1 ) j~+j2-L C ( j l j 2 L ; - m - n ) , the odd rank order parameters (L = 1,3) disappear in the sum of ½(22VpPqql(k ~ ~:,t = 0) +22 V-pq,(k --Pq2 " " -"+ ( , t = 0)) developed in the Clebsch-Gordan series, because they appear with opposed signs in both constituent correlation functions. Consequently, the odd rank order parameters do not appear in (33) independently of whether the uniaxial distribution of the fluorophores within the matrix has polar or apolar character. For example, this applies to the fluorophores adsorbed at an interface or a surface as well as to polar fluorophores aligned by an external electric field, where the aligning interactions have evidently polar character. This problem also relates to the simplest case i.e. to a one-excited-state problem, and which corresponds to typical polarized fluorescence experiments on membranes or liquid crystals. It is important to stress that the symmetry discussed means that the distribution of the molecules and dynamical evolution of the system on the both sides of a plane defined by the unit vector ~i and the axis of the molecular alignment, are indistinguishable. In Fig. 1 are shown some examples of possible experimental arrangements for the steady-state and timeresolved polarized fluorescence measurements on macroscopically ordered uniaxial systems. They are: (A) macroscopically ordered systems deposited on the surface of a semi-cylinder (e.g. Langmuir-Blodgett (LB) films, a phospholipid bilayer, a multibilayer system and also a solid/air or liquid/air surface); (B) macroscopically ordered systems between two semi-cylinders (e.g. multibilayers, polymers) and (C) a macroscopically ordered system between two glass or quartz plates (e.g. multibilayers, polymers) placed within a cylindrical cuvette filled with an immersion oil. The experimental arrangements shown in Fig. 1 enable steady-state or time-resolved excitation and/or detection angle-dependent polarized fluorescence experiments. Using semicylinders [case (A) and (B)] or a cylindrical cuvette filled with an immersion oil [case (C)] eliminates many complications that can occur due to the multireflection of the exciting light and the emitted fluorescence. Also a real angular resolution of the experiments, within the samples, is high compared to the same experiments on samples ordered between two glass or quartz plates and placed in air. The experimental arrangements indicated in Fig. 1 essentially reduce, or eliminate, the refraction of the exciting light and the emitted fluorescence at the sample/air boundary. The advantage of the variable-angle steady-state or time-resolved experiments is such that by changing the direction of excitation of the sample and/or the direction of detection of the polarized fluorescence, one can change in a controlled way the contribution of all six terms Ph (kg) (t), S(ak~'~) (t), S~ek'~) (t), ~b0~k~g:)(t), Re &~k'Tg)(t) and Re &~k,7,¢)(t), in relation (33). This procedure provides much more experimental data, which can result in more reliable information on the parameters (state-dependent solute-matrix interaction coefficients, order parameters, rates of rotational dynamics, direction of absorption and emission dipole moments
cuzi,s~ on
, (excitation) Ix ~ p ly V ~
(A)
J~
organized systems
excitation (emission)
iZ Y
e~mi~ion Izy i
(B)~
Ix
excitation
.... i
:
X
"
emission
Izy :
(c)~~3 Ix
excitation
~..... ' ', ~
;
Fig. !. Some examples of the experimental arrangements for the time-resolved fluorescence depolarization studies of organized systems.
J.J. Fisz/ Chemical Physics Letters 269 (1997) 244-256
255
as well as the rates for the state-to-state kinetics of the system under study) recovered from the global analysis of the experimental data available. The experimental arrangements shown in Fig. 1, and case (A), in particular, can be employed in studies of thin layers at solid/liquid and liquid/liquid interfaces. When the excitation angle of the sample is adjusted so that a total internal reflection of the exciting light takes place, the molecular system is excited by the evanescent field. The penetration depth of the sample depends on the excitation angle (within the total internal reflection angular regime), the refractive indices of the two media forming the interface and on the excitation wavelength [ 10-12]. The relations derived in this work can also be employed to these kinds of experiments, allowing the study of photoreactions at the interfaces. In this case the species-associated ground- and excited-state aligning potentials and also the species-associated rotational dynamics of the fluorophores must be distinguished from the corresponding bulk parameters (which can be determined from experiments at an excitation angle outside the total internal reflection angular regime). For experiments with a variable-depth penetration, these quantities have to be described in terms of discrete or continuous distributions as a function of the depth penetration. Furthermore, the penetration depth-dependent concentration profiles of the species have to be considered, in a more general case. Finally, the modified expression for the polarized fluorescence decay has to be integrated over the whole area penetrated by the evanescent field. This problem will be the subject of an independent discussion.
4. Discussion
Formula (33), describing the polarized fluorescence intensity and applying to all the cases shown in Fig. 1, represents a S-pulse excitation fluorescence decay. For the purpose of the analysis of the experimental data, formula (33) must be convoluted with the instrument response function on the exciting laser pulse. Expressions enabling the calculation of the convolution integrals for the scatter method and the method of monoexponential reference compounds, for fluorescence depolarization experiments on systems undergoing excited-state reactions, are given in Ref. [ 13]. From a theoretical point of view, the global analysis of the excitation and emission multiwavelength variable-angle polarized fluorescence decay surfaces should enable the determination of all parameters describing the polarized fluorescence decay of the system undergoing the reaction depicted in Scheme 1. These parameters are: (a) coefficients U (g'k) and u~ x'~) describing the ground- and excited-state aligning interactions; (b) the components of the state-dependent'~"~diffusion tensors Dt(); (c) the state-to-state kinetic relaxation rate constants (or the parameters describing the angular dependence of the kinetic rates, if they are angle-dependent); (d) the angles describing the orientations of the absorption and emission transition dipole moments for all the electronic states involved and (e) the spectral contributions of all the absorption and emission bands involved at the excitation and emission wavelengths at which the polarized fluorescence decays have been collected. The problem discussed in this work concerns a more general case in which a greater number of ground and excited states is assumed. One has to realize, however, that in practice the number of ground and excited states for which the experimental data can be analyzed successfully, will be limited. Further investigations based on the analysis of computer generated synthetic polarized fluorescence decays for different particular cases (e.g. different number of electronic states involved, different shapes of the state-dependent aligning potentials), are needed to explore this problem in more detail. Such studies are necessary in order to predict all limitations of the applicability of the formalism discussed in this work to real experimental cases. Furthermore, such studies seem to be the only way of working out an appropriate numerical algorithm necessary for analyses of the experimental data and to predict the optimum experimental conditions ensuring a high precision of the experimental data analysis and the reliability of the recovered parameters. A systematic study of the problem mentioned here will be presented elsewhere.
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Acknowledgement This work was supported in part by the Polish Committee for Scientific Research (KBN), within Project 2.P303.089.04. References [1] [2] [3] [4] [5] [6] [7] [8 ] [9] [10] [11] [ 12]
J.J. Fisz, Chem. Phys. 181 (1994) 417. A. Szabo, J. Chem. Phys. 81 (1984) 150. A.J. Cross, D.W. Waldeck and G.R. Fleming, J. Chem. Phys. 78 (1983) 6455. J.J. Flsz, Chem. Phys. Lett. 262 (1996) 507, D.M. Brink and G.R. Satchler, Angular momentum (Oxford Univ. Press, Oxford, 1968). J.J. Fisz, Chem. Phys. 114 (1987) 165~ M.E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957). H. Risken, The Fokker-Planck equation. Methods of solutions and applications (Springer, Berlin, 1984). J.J. Flsz, Chem. Phys, 132 (1989) 315. N.J. Harrick, Internal reflection spectroscopy (Wiley/Interscience, New York, 1967). D. Axelrod, T.P. Burghard and N.L. Thompson, Annu. Rev. Biophys. Bioeng. 13 (1984) 247. H. Masuhara, EC. De Schryver, N. Kitamum and N. Tamai, Microchemistry. Spectroscopy and chemistry in small domains (NorthHolland, Amsterdam, 1994). [13] J.J. Fisz, Chem. Phys. Lett. 262 (1996) 495.