Lennard-Jones Model of Frequency-Selective Barochromism and

10596

J. Phys. Chem. B 2004, 108, 10596-10606

Lennard-Jones Model of Frequency-Selective Barochromism and Thermochromism of Spectral Holes in Glasses†,‡ Indrek Renge* Institute of Physics, UniVersity of Tartu, Riia 142, EE51014 Tartu, Estonia ReceiVed: March 15, 2004; In Final Form: April 29, 2004

Pressure (P) and temperature (T) effects on (quasi)homogeneous optical spectra of dyes in solvent glasses and polymers were investigated by hole burning. The frequency-dependent P- and T-induced shift and broadening of zero-phonon holes burned in the inhomogeneous spectra are rationalized using two-body LennardJones potentials. The difference between the excited-state potential (U*) and the ground-state potential (Ug) yields the absolute vacuum-to-matrix shift as a function of the intermolecular coordinate. The P- and T-shifts can be described in terms of differences between the derivatives U*′ and Ug′ and the ratios of second derivatives U*′′ and Ug′′, respectively. The P-shift increases and the T-shift decreases as the optical transition energy increases. The Lennard-Jones model reproduces this opposite frequency dependence very well, by assuming that, in the excited state, the potential well minimum is displaced to shorter distances (σ* < σg). The force constant of a 6-12 potential vanishes at a distance exceeding the equilibrium value only by 11%. This feature explains qualitatively the phonon mode softening in the presence of a free volume in glasses or defect solids.

1. Introduction Sharp, purely electronic transitions have been studied at low temperature (T) for many organic and inorganic guest-host combinations in the glassy or crystalline state. Narrow zerophonon lines (ZPLs) can be recorded as spectral holes burned in the inhomogeneous spectra1,2 or even in fluorescence emitted by individual molecules.3 Narrow optical features are of great interest, because of their ultimate sensitivity, with respect to the local structure, matrix dynamics, and external perturbations. The position of a ZPL and its shift under pressure are dependent on the structure of the nearest environment of the probe molecule. The phonon wing, the Debye-Waller factor, and the temperature-induced shift and broadening of ZPLs reflect the phonon spectrum and local vibrational dynamics of the system. The relative simplicity of spectroscopic measurements can be in sharp contrast with difficulties in the unambiguous interpretation of spectral behavior in disordered solids, even on a qualitative level. Detailed mechanisms of the solvent shift and the electron-phonon coupling (EPC) processes constitute the two facets of a major problem. Small has noted that the chargetransfer character of a transition would inevitably increase the strength of linear EPC (the Huang-Rhys factor).4 The ZPLs are barely detectable, if the dipole moment change of a chromophore exceeds 5 D.5 On the other hand, the solutesolvent dipole-dipole interaction provides a very efficient mechanism of inhomogeneous broadening.6 The relationship between the polarity and EPC enables one to predict the low-T phononless transition probabilities on the basis of solvent shifts and bandwidths at room temperature.6 The absolute solvent shift was defined with high precision in the work of Sesselmann et al.7 as a difference between the †

Part of the special issue “Gerald Small Festschrift”. This work was supported by the Estonian Science Foundation (Grant No. 4511). This paper is dedicated to Professor Gerald J. Small on the occasion of his 65th birthday. * Author to whom correspondence should be addressed. Telephone: +3727-304800. FAX: +3727-383033. E-mail address: [email protected]. ‡

0-0 origin in a cold supersonic expansion and the spectral line (or hole) position in a solid and, since then, has become a key parameter.8 A pronounced dependence of the Stark effect in spectral holes on the burning frequency was observed for polymers.9,10 The frequency dependence of hole shifts induced by hydrostatic pressure (P) (denoted as a color effect)11 was thoroughly characterized in the following papers12-14 (for a review, see ref 13). Recently, a remarkable sensitivity of temperature (T) shifts of spectral holes to the burning position has been reported.15 Further color effects on pressure-induced broadening,11,16 quasi-homogeneous hole width,9 or thermal broadening15 have been noticed. Relationships between the solvent shift and the inhomogeneous bandwidth have been analyzed with statistical physics methods, and the theory was compared to measurements in polymers9,17-19 and rare gas solids.20,21 The present theoretical understanding of frequency-dependent electric field effects,9,10 P-effects,17-19 and T-effects15 is clearly not complete enough for an adequate description of hole behavior. Potential energy hypersurfaces are needed for conversion spectroscopic energies to structural parameters, such as intermolecular distance r. The very fact of hole broadening by a pressure change reveals that, besides the universal dispersive solvent shift, which is proportional to r-6, at least one additional power dependence must contribute.17 The frequency dependence of the P-shifts is variable for chromophores and matrixes of different polarities, indicating that several solvent shift mechanisms are involved.12-14 The Lennard-Jones (L-J) potential explicitly incorporates attractive and repulsive interactions. Different versions of L-J potentials have been used for the calculation of thermodynamic functions of liquids,22 the modeling of low-T properties of glasses,23 and the treatment of inhomogeneous broadening and P-induced phenomena.17,18,24 In the present work, the color effects on spectral holes will be rationalized in terms of a couple of two-body L-J potentials: one for the ground state and the other for the excited state. Glasses are not in a state of

10.1021/jp0488564 CCC: $27.50 © 2004 American Chemical Society Published on Web 06/09/2004

Barochromism and Thermochromism of Spectral Holes thermodynamic equilibrium. The structural disorder in glasses means that most particles, including the impurity molecules added by doping, cannot occupy the potential energy minimum. Our approach may be distinguished as a method of frozen potentials, because the relaxation along the L-J coordinate is neglected. Except for the minimum, the same energy would correspond to two different coordinate values, and, therefore, the intermolecular interaction energies are degenerate in the ground state (S0) and the excited state (S1). Such degeneracy may appear to be a major obstacle in probing the local structure and dynamics of glasses by ZPLs. Fortunately, the frequencyselective behavior of spectral holes shows that the degeneracy in S1-S0 transition energies seems to be lifted because of the displacement of potential minima of the S0 and S1 states. As a result, a unique correspondence between the spectroscopic energy and the configurational coordinate r can be established. Therefore, loose sites that contain a free volume in the vicinity of the chromophore would suffer smaller solvent shifts and can be assigned to the high-frequency edge of the band, whereas tightly packed or appressed centers absorb on the red edge. The first part of the paper describes the properties of onedimensional 6-12 potentials, representing an effective guesthost dimer in the ground state and the excited state. Relationships will be established between the derivatives of L-J functions on one hand, and the P-shift coefficients, local phonon frequencies, and the quadratic electron phonon coupling constants on the other. In the second part, the predictions of the L-J model will be brought in connection with experimental observables, such as inhomogeneous broadening and frequency-selective line shifts induced by pressure and temperature. The relevance of the L-J approach, with respect to temperature and pressure broadening and linear EPC, is discussed in short. The magnitudes of potential well depths and equilibrium distances will be estimated. 2. Experimental Section Polycyclic hydrocarbons, phthalocyanine, solvents, lowdensity polyethylene (PEld, with a density of 0.915 g/cm3 and a melting temperature of 115 °C), and polystyrene (PS, with a molecular weight of ca. 280 000) were purchased from Aldrich. Commercial Plexiglas was used as a high-molecular-weight poly(methyl methacrylate) (PMMA). Chlorin,25 tetra-tert-butylporphyrazine (t-Bu-TAP),26 a squarene dye,12,13 and benzo[k]fluoranthene derivatives27 were available from our previous work. The 0.1-0.5-mm-thick polymer films were cast from a solution and freed from the solvent by heating under vacuum as described in ref 26, where the details of measurement can also be found. In short, the samples were fixed in a continuous flow optical cryostat (Oxford, model CF1204), and the temperature was maintained within 1 °C with an Oxford model ITC-4 temperature controller. A dye laser (Lambda Physik, model LPD 3002E) that was pumped with an excimer laser (Lambda Physik, model LPX 100) was applied for hole burning. Holes were explored in transmission by scanning the same dye laser with attenuated energy. A two-channel Molectron JD2000 Joulemeter Ratiometer was used for recording, with the sensitive model J3S-10 (109 V/J) probe in the sample channel, and the less-sensitive probe (model J3-09) (103 V/J) was used for measuring the laser pulse energy. The (quasi-)homogeneous hole widths were corrected for laser line width by subtracting its double value, 5 GHz, from the measured width. Pressure studies were performed in a stainless-steel cell 20 mm in diameter, supplied with two sapphire windows. The sample volume of 2 mm × 4 mm × 4 mm was filled with gaseous helium to 200

J. Phys. Chem. B, Vol. 108, No. 29, 2004 10597

Figure 1. Lennard-Jones (L-J) potential U (eq 1, ) σ ) 1) and the first two derivatives U′ and U′′, equal to the force (-F) and the force constant (k), respectively.

bar and cooled to 5 K, and then the holes were burned. The pressure was released step by step, and the holes were recorded under isothermal conditions (for details, see ref 12). Standard computer programs xvgr for Unix and Origin were used for the data analysis. 3. The Lennard-Jones Model 3.1. Solvent Shift and Pressure Shift. Intermolecular potential energy (U) is often characterized by a Lennard-Jones (L-J) type of function that contains the potential well depth () and the equilibrium distance (σ) as parameters:

U)

[(σr )

12

-2

(σr ) ] 6

(1)

The power coefficients are somewhat arbitrary; in particular, the repulsive term r-12 may be replaced by r-9 (see ref 24) or an exponential law.23 The coordinate r may describe either interparticle distance in a dimer or an average distance between the solute and several solvent molecules in a cluster. The derivative of eq 1 yields the force vanishing at the equilibrium distance r ) σ:

-F )

σ 12 σ6 dU ) U′ ) -12r-1 dr r r

[( ) ( ) ]

(2)

The second derivative of eq 1 is equal to the force constant k:

k)

σ 12 σ6 d 2U ) U′′ ) 12r-2 13 -7 2 r r dr

[ ()

( )]

(3)

The functions (eqs 1-3) are depicted in Figure 1 for ) σ ) 1. In the absence of external perturbations, the system resides in the bottom of the potential well at T ) 0, performs anharmonic vibrations if the value of kBT lies between 0 and , and dissociates if the value of kBT exceeds . The equilibrium guest-host distance may not be accessible in the solid, because of the constraints imposed by lattice forces in the case of crystals or the loss of molecular mobility below the glass-transition temperature (Tg) in the case of glasses. For the same reason, the oscillation along the L-J coordinate may be impossible in the solid state. Evidently, in the condensed phase, a single solute-solvent distance should be replaced by a general or effective coordinate. The well depths and the minimum position σ will be marked with indices g and an asterisk (/), to denote the ground-state

10598 J. Phys. Chem. B, Vol. 108, No. 29, 2004

Renge

TABLE 1: Properties of Electronic Transitions, Expressed in Terms of Lennard-Jones Parameters property

symbol

potential energy in the ground state potential energy in the excited state transition energy solvent shift same at band maximum pressure shift coefficient same at band maximum slope of dν/dP vs ∆ν same at band maximum vanishing point of the pressure shift force constant QEPC constanta a

abbreviation

expression

Ug

g[(σg/r)12 - 2(σg/r)6]

U*

*[(σ*/r)12 - 2(σ*/r)6] + ν00

ν ∆ν ∆νm dν/dP dνm/dP a

U* - Ug U* - Ug - ν00 -1/3βTr∆U′ (dν/dP)′/∆U′

am ∆ν0(P)

∆νm - (dνm/dP)/am

kg or k* W

Ug′′or U*′′ U*′′/Ug′′ - 1

remarks g, potential well depth; σg, equilibrium distance *, σ*, same as above; ν00, vacuum frequency

*[(σ*/r)12 - 2(σ*/r)6] - g[(σg/r)12 - 2(σg/r)6] + ν00 *[(σ*/r)12 - 2(σ*/r)6] - g[(σg/r)12 - 2(σg/r)6] *[(σ*/σg)12 - 2(σ*/σg)6] + g 4βT{*[(σ*/r)12 - (σ*/r)6] - g[(σg/r)12 - (σg/r)6]} 4βT*[(σ*/σg)12 - (σ*/σg)6] 2βT{*[2(σ*/r)12 - (σ*/r)6] - g[2(σg/r)12 - (σg/r)6]}/ {*[(σ*/r)12 - (σ*/r)6] - g[(σg/r)12 - (σg/r)6]} 2βT[2(σ*/σg)12 - (σ*/σg)6 - g/*]/[(σ*/σg)12 - (σ*/σg)6] ∆νm - 2*[(σ*/σg)12 - (σ*/σg)6]2/[2(σ*/σg)12 -(σ*/σg)6 g/*] 12r-2[13(σ/r)12 - 7(σ/r)6] {*[13(σ*/r)12 - 7(σ*/r)6]/g[13(σg/r)12 - 7(σg/r)6]} - 1

r ) σg βT, volume compressibility r ) σg r ) σg r ) σg ) g or *, σ ) σg or σ* U*′′/Ug′′ ) k*/kg ) ω*2/ω2g

Quadratic electron-phonon coupling constant.

Figure 2. Zero-phonon transitions between the L-J energies of the ground state (Ug; upper thick line) and the excited state (U* - ν00, where ν00 is the transition energy in a free guest molecule; dashed line). The length of the arrows shows solvent shifts at the maximum and on the edges of an inhomogeneous band. Several harmonic potentials u were calculated from the second derivatives as u ) 1/2kr2.

and excited-state potentials, respectively (Table 1). The excitedstate potential is upshifted by the transition energy in a vacuum ν00:

U* ) *

[(σ*r )

12

-2

(σ*r ) ] + ν 6

0 0

(4)

The absolute solvent shift of a transition ∆ν (in energy units) is a difference between the potential functions of the S1 and S0 states, minus the transition energy in a vacuum:

∆ν ) ν - ν00 ) U* - Ug - ν00

(5)

Figure 2 illustrates L-J potentials for the S0 (g ) σg ) 1) and S1 states (* ) 1.2, σ* ) 0.97). The excited-state function is downshifted by ν00; therefore, the difference between the curves is equal to the solvent shift, ∆ν (see eq 5). The relative depth of the potential wells (*/g) can be estimated as a ratio of polarizabilities of the respective states (R*/Rg), according to the London expression for dispersion energy.28 The ratio */g ) 1.2 corresponds to a dispersive solvent shift of approximately -1000 cm-1 per unit Lorentz-Lorenz parameter, or an absolute solvent shift of -300 cm-1 in common polymers.28 Obviously, the largest pool of centers contributing to the peak maximum of the inhomogeneous band has an r value that is similar to σg.

Figure 3. Solvent shifts as a function of the configurational coordinate r for transitions between the ground-state potential Ug (g ) σg ) 1) and the excited-state potentials U* (* ) 1.2) with various equilibrium positions σ*.

The average ground-state energy fluctuation is ∼kBTg above g, which is indicated by a horizontal bar in Figure 2. On the other hand, the variation range of r is ∼1 ( 0.03, in terms of relative distances, based on the fact that glasses contain ∼10% free volume. The arrows denote zero-phonon transitions at the maximum and on both edges of an inhomogeneous absorption contour. If the equilibrium distance in the excited state is shorter than σg, as exemplified in Figure 2 for σ*/σg ) 0.97, a unique dependence between the spectroscopic energy and the intermolecular distance should exist. The analysis of experimental data will show that this assumption seems to be close to reality. The absence of degeneracy is favorable from the standpoint of probing local properties by means of optical transition energies. The solvent shift ∆ν is calculated as a function of r (eq 5) and displayed in Figure 3 for our model system with various σ* (see Table 1 for a general expression):

∆ν ) 1.2

[(σ*r )

12

-2

(σ*r ) ] - [(1r) 6

12

-2

(1r) ] 6

(6)

If σ* ) σg, the largest (negative) solvent shift occurs at the equilibrium distance and amounts to -(* - g). In the case of σ* * σg, the maximum solvent shift increases quickly, in particular, when σ* < σg. However, in the vicinity of σg, the solvent shift is relatively insensitive, with respect to σ*. The correlation between the transition frequency and the coordinate becomes unique, when the minimum in ∆ν is displaced outside

Barochromism and Thermochromism of Spectral Holes


Figure 4. (a) Solvent shifts of band maxima separated to repulsive (eq 9, solid ascending curves) and attractive components (eq 10, dashed descending curves) for the red-shifted (* ) 1.2) and blue-shifted (* ) 0.8) transitions, plotted as a function of relative displacement of potential minima σ*/σg. In a realistic interval of σ*/σg between (*/ g)1/6 and (*/g)1/12, both interactions produce shifts of the same sign. (b) Slope of the plot of pressure shift (P-shift) coefficients (dν/dP) versus the solvent shift ∆ν in the region of the band maximum am (eq 15, dashed line) and the frequency at which the pressure-induced shift vanishes (∆ν0(P); from eq 16, solid line). If the σ*/σg scales as (*/ g)1/6 or (*/g)1/12, the slopes are equal either to 4βT or 2βT (βT is the volume compressibility; here, βT ) 1) and the P-shift vanishes at the vacuum frequency of the chromophore ν00.

the actual range of r (for example, 1 ( 0.03). This occurs already at the mismatch in relative σg and σ* values of (g/*)1/12 (Figure 4a; for further details, see Results and Discussion, Part 1).

Figure 5. Dependence of (a) P-shift coefficients dν/dP (eq 13) and (b) quadratic electron-phonon coupling (QEPC) constant W (eq 19) on the solvent shifts for transitions between the ground-state potential Ug (g ) σg ) 1) and the excited-state potentials U* (* ) 1.2) for different values of σ*. At very small relative displacements of potential minima (σ*/σg ≈ 1), the degeneracy in transition energies would lead to baric or thermal hole splitting.

The pressure shift of an energy level can be calculated from the derivative of the potential (eq 2), using linear isothermal compressibility, which is defined as RT ) -(dr/dP)T/r:

dU σ 12 σ6 ) -RTrU′ ) RTrF ) 12RT dP r r

[( ) ( ) ]

(11)

The P-shift of transition energy between the levels can be expressed as

[

]

d(U* - Ug) dν ) -RTr ) -RTr∆U′ dP dr

(12)

In terms of bulk compressibility βT (βT ) 3RT), the P-shift coefficient dν/dP for our model system is equal to (see Table 1 for full expression):

{ [( ) ( ) ] [( ) ( ) ]}

σ* 12 σ* 6 1 12 16 dν ) 4βT 1.2 dP r r r r

(13)

In Figure 5a, the P-shift coefficients dν/dP are plotted versus the solvent shift ∆ν for different equilibrium distances of the excited state σ*. The pressure shift would vanish at the band maximum, if σ* ) σg. When σ* happens to be very similar to σg, the transition energies are degenerate and a splitting of spectral holes should occur. For σ*/σg < 0.985, an approximately linear color effect is predicted over the band, in accordance with observations. The slope of the plot of dν/dP versus the solvent shift or the hole position, denoted by a, is calculated as a ratio of the derivatives of dν/dP and ∆ν:


Renge

d(dν/dP)/dr ) d(∆ν)/dr 2βT{*[2(σ*/r)12 - (σ*/r)6] - g[2(σg/r)12 - (σg/r)6]}

Bose factors reduce to a linear and a quadratic dependence, respectively:

a)

*[(σ*/r) - (σ*/r) ] - g[(σg/r) - (σg/r) ] 12

6

12

6

(14)

In the vicinity of the band maximum (r ) σg), one obtains (Figure 4b)

am )

2βT[2(σ*/σg)12 - (σ*/σg)6 - g/*] (σ*/σg) - (σ*/σg) 12

6

(15)

Equations 14 and 15 can be simplified, if the ratio of potential well depths (g/*) is equal to either (σ*/σg)12 or (σ*/σg).6 In the former case, the first and third members cancel in the numerator and denominator of eq 14 and a slope of a ) 2βT is obtained. When g/* ) (σ*/σg)6, the second and fourth members disappear, leaving a slope of a ) 4βT (see Figure 4b). In many less-polar systems, the experimental a values do indeed lie between 2βT and 4βT.12,13 The extrapolated frequency ν0(P) where the P-shift would vanish can be determined from the solvent shift (∆νm), the P-shift coefficient at the band maximum (dνm/dP), and the slope (am) (for full expression, see Table 1):

∆ν0(P) ) ν0(P) - ν00 ) ∆νm -

dνm/dP am

(16)

A correct extrapolation to the 0-0 origin in a vacuum (∆ν0(P) ) 0) is obtained from the P-shift measurement in a solid matrix, if the relative well depths g/* and equilibrium distances are equal either to (σ*/σg)12 or (σ*/σg)6 (see Figure 4b). 3.2. Temperature Shift and Broadening. The line broadening and shift at temperatures above 0 K are of dynamic origin, in contrast to inhomogeneous broadening, which is a consequence of static, microscopic solvent shifts. Broadening is dependent on the dynamic modulation of electronic levels by matrix fluctuations, whereas the shift arises as a result of a phonon frequency change that accompanies optical excitation. In the case of weak interaction of an electronic transition with a single harmonic pseudo-local mode, the nonperturbative quadratic electron phonon coupling (QEPC) theory provides the following expressions for the line shift (∆ν(T)) and broadening (∆Γ) (in frequency units):29,30

∆ν(T) ) ∆Γ )

kBWT 2h

k2BW2τgT2 4h2

(20) (21)

The QEPC constant W characterizes the difference in curvatures of potential energy surfaces for a (pseudo-)local vibration between the ground state and the excited electronic state of the chromophore. The second derivative d2U/dr2 ≡ U′′ yields the force constant k (eq 3, Figure 1), which, in turn, would determine the harmonic frequency as ω ) (2π)-1(k/µ)1/2 (where µ is the reduced mass). Note that negative force constants correspond to imaginary frequencies and have no direct physical meaning. Figure 5b displays the QEPC constant W, calculated as U*′′/Ug′′ - 1, versus the solvent shift. If there is no shift of potential minima, W would be dependent only on the relative depth of the potential wells, as (*/g) - 1, and, in our model, this produces a thermal blue shift (W ) 0.2 > 0). At σ*/σg values close to unity, a degeneracy remains in the QEPC constants that could result in hole splitting. When σ*/σg e 0.98, the thermal shift will be bathochromic in the entire range of solvent shifts accessible to hole burning. Moreover, a decrease of negative T-shifts with decreasing frequency is evident, in agreement with experiment (see below). Harmonic potentials (u) can be constructed as a function of r (Figure 2):

1 u ) kr2 2

(22)

At the equilibrium distance, k ) 72*/σ2g and increases rapidly as r becomes shorter. It is easy to see that, at a distance r ) (13/7)1/6σ ) 1.109σ, the force constant is reduced to zero. A soft, zero-frequency mode can appear, if the equilibrium distance is exceeded only by 10%. Consequently, intermolecular potentials with inflection points predict mode softening in the presence of free volume in disordered solids. Realistic interaction potentials provide a simple connection between the empty space and universal anomalies in thermal, acoustical, and optical properties of disordered solids. 4. Results and Discussion

1 ∆ν(T) ) Wωgn(ωg) 2

(17)

1 ∆Γ ) W2ωg2τgn(ωg)[n(ωg) + 1] 4

(18)

where W is the QEPC constant, ωg the phonon frequency (if the impurity is in the ground state), τg the lifetime of the phonon level, and n(ωg) the Bose factor (n(ωg) ) {exp[hωg/(kBT)] 1}-1). The QEPC constant is dependent on the harmonic frequencies of a (pseudo-)local vibration in the S0 and S1 state (ω*):

W)

( )

ω* 2 -1 ωg

(19)

At the low-temperature limit (kBT , hωg), one obtains, from eqs 17 and 18, that both shift and broadening obey the same Arrhenius law [∼exp[-hωg/(kBT)]. At high T (kBT . hωg), the

4.1. Solvent Shift. A wide distribution of solvent shifts for individual chromophores manifests itself as a huge inhomogeneous width of optical spectra in glasses. The band maximum corresponds to a major fraction of centers with intermolecular coordinates r close to σg. The shift of band maximum ∆νm is dependent mainly on the difference in potential well depths (* - g), rather than on the relative horizontal displacement of minima σ*/σg (see Figures 2 and 3). However, in the case of exact coincidence of σ* and σg, an unrealistic band shape with a sharply rising peak on the red edge would follow from the two-particle model. The role of repulsive interactions has been a matter of discussion in the literature about solvatochromism.31-33 Partitioning of the solvent shift to repulsive and attractive contributions is straightforward in the L-J model (eqs 7 and 8). In redshifted spectra (* > g), the repulsive shift ∆νrep would be positive and the attractive shift ∆νattr would be negative, if σ*/ σg is larger than (g/*)1/12 (see Figure 4a). Surprisingly, the



TABLE 2: Solvent Shifts of Band Maxima, Inhomogeneous Bandwidths, and Pressure Shift Coefficients of Spectral Holes at 6 K, for Various Dopantsa matrixb PMMA ethanol PEld PS PC PMMA ethanol isopentane 3-MePe 1,3-DMCH squalane Tol-Benz PEld

∆νm (cm-1)

Γih (cm-1)

a (10-5 bar-1)

∆ν0(P) (cm-1)

2.25 ( 0.17 3.34 ( 0.12 4.38 ( 0.12

80 ( 17 -27 ( 4 -45 ( 4

Chlorin, p ) -659 ( 20 cm-1, ν00 ) 15912(j) cm-1 230 -0.121 136 -0.076 84 -0.173 209

-97 -101 -177 -177 -50 -87 -97 -156 -166 -183 -184 -186 -190

dνm/dP (GHz/bar)

Tetra-tert-butylporphyrazine, p ) -988 ( 39 cm-1, ν00 ) 16326 ( 8(s) cm-1 286 -0.164 2.04 ( 0.13 328 -0.263 2.42 ( 0.17 188 -0.202 3.08 ( 0.11 102 -0.221 4.47 ( 0.1 112 -0.241 4.59 ( 0.11 140 -0.269 4.79 ( 0.11 170 -0.289 3.84 ( 0.20 278 155 -0.279 3.8 ( 0.1

212 ( 15 287 ( 27 119 ( 9 8(4 9(4 3(5 61 ( 15 62 ( 10

-407, -445c -441, -443d -453

3,4-Benzopyrene, p ) -1567 ( 60 118, ?c 146, 198d 246

-687

Phthalocyanine, p ) -2240 ( 59 cm-1,e ν00 ) 15132(j) cm-1 103 -0.451 3.93 ( 0.3

-303 ( 30

PMMA ethanol

-721 -739

Squarene Dye, p ) -2617 ( 84 cm-1, ν00 ) 16353 ( 20(s) cm-1 354 -0.325 2.92 ( 0.12 204 -0.268 3.67 ( 0.19

-357 ( 17 -502 ( 16

ethanolf

Cyclohexylbenzo[k]fluoranthene, p ) -3016 ( 81 cm-1,g ν00 ) 25282 cm-1 -840 283 -0.57 3.48 ( 0.12

-289 ( 20

ethanol

Cyclopropylbenzo[k]fluoranthene, p ) -3016 ( 81 cm-1,g ν00 ) 24744 cm-1 -840 226 -0.67 3.99 ( 0.23

-292 ( 34

ethanol 3-MePe PMMA

9,10-Dichloroanthracene, p ) -4176 ( 81 cm-1,h ν00 ) 25950(j) cm-1 -1209, -1228i 218, 240i -0.875 4.75 ( 0.23 -1216, -1253d 171, 210d -0.90 5.57 ( 0.10 -1328 362 -0.855 5.9 ( 0.5

-602 ( 33 -710 ( 35 -675 ( 46

3-MePe ethanol PMMA PEld

, ) 25265(j) cm -0.3 -0.256 -0.235

cm-1

ν00

-1

6.77 ( 0.13 6.19 ( 0.41 4.05 ( 0.6

-288 ( 20 -299 ( 11 -276 ( 35

a

The parameter p represents the slope of the plot of absorption band maxima in n-alkanes versus Lorentz-Lorenz function at room temperature; is the transition wavenumber in a cold supersonic jet (j) (see refs 38-41) or extrapolated from solvent shifts (s); ∆νm is the vacuum-to-matrix shift of band maximum; Γih is the double value of the half-widths at half-maximum for the long-wavelength slope of the band; dνm/dP is the P-shift coefficient at the band maximum; a is the slope of the frequency dependence of the P-shift coefficient; and ∆ν0(P) is the P-shift vanishing point minus ν00. b Legend for the matrixes is as follows: PMMA, poly(methyl methacrylate); PEld, low-density polyethylene; PS, polystyrene; 3-MePe, 3-methylpentane; PC, propylene carbonate; 1,3-DMCH, 1,3-dimethylcyclohexane; and Tol-Benz, toluene-benzene (2:1 v/v) glass. c Cyclooctatetraene (1.5%, v/v) was added to enable hole burning. d tert-Butyl perbenzoate (5%) was added to enhance hole burning. e For fluorescence. f Cycloheptatriene (10%, v/v) was added to enhance hole burning. g For benzo[k]fluoranthene. h For anthracene. i Cyclooctatetraene (1.5%) and triethylamine (10%) were added to enhance hole burning.

ν00

repulsive shift vanishes at σ*/σg ) (g/*)1/12 and becomes bathochromic at larger displacements. At even greater relative shifts of potential minima, when σ*/σg < (g/*)1/6, the attractive interaction with a r-6 dependence would cause a blue shift. Therefore, for σ*/σg lying between (g/*)1/6 and (g/*)1/12, both “interactions” produce red shifts. Moreover, if the ratio σ*/σg lies in this interval, the potential functions Ug and U* - ν00 do not cross and the solvent shift cannot be reversed by tuning r, for example, by applying high pressure. All π-π* and n-π* transitions in centrosymmetric chromophors known to us possess negative shifts. Positive (blue) shifts of transitions localized on atoms or σ bonds have been rationalized in terms of repulsive strain in the host cavity.31,33,34 The equilibrium distance in the excited state σ* is assumed to be larger than σg, and probably g > *. It follows from eqs 7 and 8 that, in this case for “realistic” σ*/σg between (g/*)1/12 and (g/*)1/6, both interactions would cause blue shifts (see Figure 4a). The L-J potential is applicable to particles devoid of electrical moments. This criterion is satisfied ideally for rare gas atoms, and perhaps less perfectly for aliphatic hydrocarbons (alkanes, polyethylene) as host matrixes. By contrast, the guest molecules,

such as aromatic hydrocarbons, carry appreciable quadrupole moments that can show strong variation in different electronic states.35 Absorption or fluorescence band maxima in liquid n-alkanes at room temperature exhibit almost-perfect linear correlation with the Lorentz-Lorenz function φ(n2) (φ(n2) ) (n2 - 1)/(n2 + 2), where n is the refractive index of the solvent):13,28,36

νm ) ν00 + pφ(n2)

(23)

Despite the limited range of φ(n2) from 0.22 (n-pentane) to 0.26 (n-hexadecane), the plot extrapolates nicely to a vacuum frequency ν00.36 The slope p can vary between -102 cm-1 for the S1 bands in porphyrins and -104 cm-1 for the strongest transitions in polyarenes, polyenes, and fullerenes (Table 2).8,13,28,37 According to London and Lorentz-Lorenz formulas, p is proportional to the difference of polarizabilty densities between the ground state and the excited state. A linear plot of p, multiplied by molecular weight Mw for a cavity volume, and the polarizability difference ∆R between the ground state and the excited state (∆R ) R* - Rg), obtained from the Stark effect measurements was built for polyarenes:28


∆R ) -(0.4 ( 1.5) - [(18.2 ( 1.4) × 10-6]pMw

Renge

(24)

The relative depth of the potential wells */g is approximately equal to R*/Rg or 1 + 1.8 × 10-4p, because the polarizabilities Rg (in Å3 units) of common C, N, and O atoms in organic compounds amount to ∼0.1Mw.42 The band maxima of large molecules in solid rare gases obey eq 23 perfectly well, with slopes p very similar to that for liquid n-alkanes.8 Thus, the dispersive contribution to the solvent shift can be estimated solely on the basis of the parameter p and macroscopic refractive index of the environment. In the absence of polar and conformation effects, the frequency spread in glasses could be expressed in terms of local polarizability or density fluctuations. 4.2. Inhomogeneous Broadening. The inhomogeneous bandwidth (Γih) is characterized by a variation of solvent shifts, exemplified by the left and right arrows in Figure 2. The peak shift ∆νm, represented by the central transition in Figure 2, and Γih both are dependent on the difference * - g and should, therefore, exhibit a strong mutual correlation. The high sensitivity of Γih to the relative displacement of potential minima σ*/ σg can be deduced from Figure 3, by comparing the solvent shifts for a given spread of r (for example, from 0.97 to 1.03). The width Γih is also dependent on the glass temperature of the matrix (Tg), as shown by the height (kBTg above g) and span of the horizontal level built on the ground-state potential curve in Figure 2. Experimentally, bandwidths have been plotted versus the solvent shifts of peak maxima in solvent and polymer glasses,6,13,43 and an ∼1:10 ratio for Γih and |∆νm| was found for less-polar systems. Moreover, a permanent contribution to the width of ∼70-100 cm-1 was assigned to either multipolar (or bond dipolar) or repulsive interactions. The former explanation seems preferable, because, in the L-J model, the repulsive and dispersive interactions are (anti)correlated and should vanish together. A survey of data in Table 2 confirms that R-bands of polyarenes are narrower (118 cm-1 for 3,4-benzopyrene in 3-methylpentane) than strong p-bands (171 cm-1 for 9,10dichloroanthracene in the same glass). The widths can be compared with ∆νm ) -407 and -1216 cm-1, respectively. Inhomogeneous broadening is considerably larger in polymers that have a Tg value above that of ethanol or 3-methylpentane by 100 K (PEld) to 200 K (PMMA).44,45 A comparison of nonpolar and polar media, such as 3-methylpentane versus ethanol or PEld versus PMMA exposes a very strong dependence of Γih on matrix polarity. A more detailed discussion of the relationships between Γih and ∆νm in solvent and polymer glasses of different polarity can be found in refs 6, 13, and 43. Inhomogeneous broadening can be treated in terms of the relative sizes of guest and host particles, the number density of host molecules, and the shifts produced by each solvent unit using statistical methods.9,17-21 Because of its generality, the stochastic, microscopic theory has a great potential. However, the formal approach can be misleading, if the complex solvent shift mechanisms are not taken into account properly. Several inconsistencies of the earlier work will be mentioned shortly. Inhomogeneous broadening caused by a small number of big solvent molecules, each producing a large solvent shift, would evidently exceed the broadening, because of a large number of small host particles capable of a more uniform packing.17,18 Phthalocyanine has a much larger Γih value in polystyrene (276 cm-1) than in PE (116 cm-1 from earlier works,18,19 103 cm-1 in this work; see Table 2), apparently confirming the idea.

However, it is well-known that aromatic solvents are pseudopolar, as a result of quadrupole moments and polarizability anisotropy. The spectra in a glassy toluene-benzene mixture are always considerably broader than in aliphatic hosts of similar granulation, e.g. 1,3-dimethylcyclohexane (278 and 140 cm-1 for t-Bu-TAP). The broadening in aromatic glasses has obviously no relationship with a comparatively large size of benzene molecules or phenyl fragments. In a similar system (octaethylporphine in polystyrene), the Stark effect on spectral holes shows a characteristic enhancement with decreasing frequency.9 The conclusion that inhomogeneous broadening (Γih ) 152 cm-1) is dominated by dispersion interaction is very difficult to reconcile with negligible refractive index dependence of the S1-S0 band of octaethylporphine in liquid n-alkanes (p ) -128 ( 19 cm-1).46 The polarizability difference (∆R ) 20 ( 4 Å3) obtained by the authors,9 who were obviously unaware of solvent shifts in porphyrins, by far exceeds the solvatochromic value (2.8 ( 2 Å3).46 Free-base tetrapyrroles are almost ideally suited as hole burning probes, because of intrinsic photochemistry and weak EPC. Unfortunately, although formally devoid of dipole moment, porphyrins are characterized by rather complicated polarity-dependent spectroscopy.12-14,46 Finally, the bandwidth analysis of phthalocyanine cocondensed with rare gases has led to a conclusion that pigment is residing in the low-density interstitial regions between the nanocrystalline grains of the host matrix.20,21 The plot of band maxima (from Table 1 in ref 21) versus φ(n2) for crystalline argon, krypton, and xenon at 5 K8 is perfectly linear (correlation coefficient is r ) 0.99999). The slope p (-1982 ( 8 cm-1) resembles that for phthalocyanine fluorescence in n-alkanes (-2240 ( 59 cm-1),25 and the extrapolation to a solvent-free state yields a frequency (15 124 ( 2 cm-1) that is very close to ν00 in a supersonic jet (15 132 cm-1).40 In xenon (φ(n2) ) 0.295),8 and polyethylene (φ(n2) ) 0.329),45,47 the solvent shifts are similar (-592 and -687 cm-1, respectively). Therefore, the magnitude of dispersive shifts suggests that pigment molecules are placed in the surroundings with local polarizability (and density) that is comparable to that for bulk rare gas crystals, rather than in a special environment of reduced density. 4.3. Pressure Shift. One can tune the configurational coordinate r with external pressure, but only toward reduction, as negative P or stretching seems to be hard to realize on a three-dimensional sample. In our concept, the L-J “master” potentials remain unaltered upon application of pressure P, and the spectral shift is associated with the change of r. One can speak of frozen potentials for this reason, and also because vibrational relaxation along the L-J coordinate is being neglected. This approach differs from the model advanced by Drickamer48 and Curie,49 presuming displacements of harmonic potentials in both energy and distance with P. In addition, the L-J model includes anharmonicity in a natural way. Harmonic frequencies can be calculated from the second derivatives of master curves (eq 3, Figure 2) for any r fixed either in a host cavity or tuned to by applying external P. The P-shift of transition energy can be expressed in terms of a difference between the derivatives U*′ and Ug′ (eq 12). For a better comparison with experiment in Figure 5a, the P-shift coefficients dν/dP are plotted versus the solvent shift ∆ν, rather than r. Because the bandwidth is ∼10% of the dispersive shift,6,43 the shift -0.20 ( 0.01 in arbitrary units corresponds to the upper half of the inhomogeneous site distribution function and the full scale from -0.25 to -0.15 covers the frequency spread accessible to hole burning in a favorable case. An almost-



Figure 7. Absorption spectra of t-Bu-TAP in solvent glasses and dependence of pressure-shift coefficients (dν/dP) ((O) isopentane and (0) propylene carbonate) and broadening coefficients (dΓ/dP), taken with a minus sign ((b) isopentane and (9) propylene carbonate) on absolute solvent shift at 6 K.

Figure 6. Absorption spectra of t-Bu-TAP and phthalocyanine (Pc) in low-density polyethylene at 7 K and the dependence of hole properties on absolute solvent shift: (O) P-shift coefficients (dν/dP), (b) pressure broadening coefficients (dΓ/dP), (/) thermal shift (∆ν(T)), and (0) thermal broadening (∆Γ) between 7 K and 35 K.

linear dependence of dν/dP vs ∆ν is predicted for ∆ν ≈ 0.2, if the ratio of equilibrium distances σ*/σg is 0). A linear Stark shift in a solvent cavity field or a reaction field is characterized by enormous inhomogeneous broadening Γih, comparable to the shift ∆νm.5,6 The local electric field strength scales linearly with compression (∼RT) that would correspond to a as small as 1/3βT.12 For t-BuTAP doped in nonpolar hosts, the plots of dν/dP extrapolate closely to ν00, whereas in polar PC, PMMA, and ethanol, ν0(P) exceeds ν00. Consequently, the tetra-azaporphine chromophore is moderately (pseudo-)polar, as confirmed also by roomtemperature solvatochromism (see Figure 3 in ref 12). 4.4. Pressure Broadening. Deep holes burned with a considerable light dose retain a Lorentzian shape, despite


Renge

broadening. With changing helium gas pressure, the hole shape is gradually rendered Gaussian. The Gaussian component can be extracted using a standard fitting to a Voigt profile, while keeping the initial Lorentz width fixed. In relative terms, the color effect on broadening coefficient dΓ/dP (filled symbols in Figures 6-8) is much less pronounced than that for the shift coefficient. The frequency dependence of dΓ/dP is approximately linear for Pc/PEld, t-Bu-TAP/isopentane, and 3,4-BP/ 3-MePe. In other systems, the broadening increases on the red edge of the band, shows little change in the middle, and may again slightly increase on the high-frequency edge (note that, in Figures 7 and 8, the sign of dΓ/dP is inverted). The pressure broadening is caused by lifting the accidental degeneracy in the S1-S0 transition energy. It has been emphasized that a solvent shift mechanism characterized by a singlepower dependence cannot produce broadening, provided the compression is isotropic.17 Our two-particle L-J model does not explain pressure broadening, because of a fixed link between r and ∆ν. If σ* ) σg, the red-shifted centers have r values close to σg, whereas both the appressed (r < σg) and the loosely packed (r > σg) impurity molecules will contribute simultaneously to the high-frequency edge of the band. Consequently, a hole splitting that increases with frequency should occur as a result of such degeneracy. On the average, this effect may manifest itself as an enhancement of pressure broadening on a blue edge of the band. In contrast, stochastic theory of inhomogeneous broadening developed beyond the Gaussian approximation predicts a stronger pressure broadening for the red-shifted centers.19 Contradicting model considerations may reflect reality to some extent, because the broadening is stronger on the band edges for t-Bu-TAP/PC and 9,10-ClA/3-MePe. The absence of a color effect for phthalocyanine in PEld (Figure 6) could result from a compensation effect between different mechanisms. Both the inhomogeneous width Γih and the pressure broadening are strongly affected by matrix polarity. For t-Bu-TAP, both parameters increase by a factor of 3 in highly polar propylene carbonate, as compared to isopentane glass (Figure 7). The inhomogeneous broadening and pressure broadening seem to be closely related. 4.5. Temperature Shift. The observed thermal line shift (∆νobs) consists of a density-dependent solvent shift and a pure thermal, phonon-induced contribution.7,15,51 The shift caused by thermal expansion can be estimated from P-shift coefficients dν/dP, isobaric thermal expansion coefficient RP (RP ) (dV/ dT)P/V), and the isothermal volume compressibility βT of the matrix. A pure thermal shift ∆ν(T) is obtained after subtraction of the density-dependent component:7,14,51

∆ν(T) ) ∆νobs -

( )( )∫ dν 1 dP βT

T

0

RP dT

(25)

The thermal shift of dynamic origin has a characteristic, approximately linear, dependence on burning frequency.15 For several dye-polymer combinations, the phonon-induced line shift reaches the largest negative magnitudes on the highfrequency absorption edge (Figure 6). On the opposite (red) edge of the band, the shift is less negative, or even negligible, as in the case of octaethylporphine in PMMA and chlorin in PEld.15 The color effect on ∆ν(T) seems to be universal and relatively insensitive, with respect to the band shift ∆νm and the polarity of the matrix (PEld or PMMA). Very remarkably, the frequency dependencies of thermal and baric shifts are opposite: with increasing frequency, the P-shift increases and the T-shift diminishes (the negative shift in-

Figure 9. Temperature-dependent widths of holes burned at different positions over the absorption band of chlorin in polystyrene matrix. Quasi-homogeneous width (Γqh) is obtained for shallow holes burned and measured at the same temperature (Tb ) Tm) in the indicated wavelength range ((0.5 nm). Broadening ∆Γ was measured for deep holes hole burned at low temperature (Tb ) 7 K), and ∆Γc was obtained after passing thermal cycles to Tc (Tb ) Tm ) 7 K). The temperature dependence of Γqh at the band maximum (denoted by an asterisk, /) was fitted to a power law of T1.67. Inset shows the absorption spectrum of chlorin in polystyrene at 8 K.

creases). Predominately negative values and the opposite color effects of both phenomena are nicely reproduced in terms of the L-J model, by assuming that the ratio of equilibrium distances σ*/σg is equal or less than (g/*)1/12 (0.985 in Figure 5). The identical (negative) sign of both the solvent and the T-shift seems to be hard to understand, bearing in mind that a negative solvent shift presumes a stabilization of the exited state. A deeper potential well corresponds to a higher vibrational frequency in the excited state that, in turn, would produce a blue shift (eq 19, the thick horizontal line in the case of σ* ) 1 in Figure 5b). The model of displaced L-J potentials offers an elegant solution to this paradox. 4.6. Temperature Broadening. Both dynamic and static properties of glasses can be addressed by measuring hole widths, depending on the procedure. Shallow holes (

Lennard-Jones Model of Frequency-Selective Barochromism and

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