Currently accepted theories of rotor-solvent interaction center on diffusion. Two derivations of Equation 1 are based on Debye-Stokes-Einstein diffusion and free-volume diffusion, respectively. Förster and Hoffmann [9] provided a rigorous derivation based on classical mechanics and the assumption of Debye-Stokes-Einstein (DSE) diffusion. In their ground-breaking work on triphenylamine dyes, Förster and Hoffmann postulated that each aniline group acts as a nanoscale ellipsoid that obeys the second-order differential equation of rotational motion,

\theta \frac{{d}^{2}\phi}{d{t}^{2}}+\kappa \frac{d\phi}{dt}+\alpha (\phi -{\phi}_{0})=0

(3)

where *φ* is the rotational angle of the aniline group with respect to the ground-state equilibrium position *φ*_{0}, *θ* is the rotational inertia of the aniline group, and *α* reflects the electrostatic force that returns the aniline group to its equilibrium position. The microfriction *κ* is linked to bulk viscosity *η* through the DSE diffusion model,

\kappa =8\pi \text{}{r}^{3}\eta

(4)

where *r* is the effective radius of the aniline group. Under the assumption of strong damping (more precisely, *κ*^{2} ≫ 4*κθ*), a twisted aniline group returns to the ground-state equilibrium in an exponential-decay fashion with a decay constant of *κ*/*α*. Förster and Hoffmann define a function *B*(*φ*) as *B*(*φ*) = *β*(*φ*-*φ*_{0})^{2} that describes the rate of deactivation processes through conformational changes, with *β* being a proportionality constant. With this definition, a differential equation can be found that governs the probability *ρ*(*t*) that the molecule is in the excited state:

\begin{array}{r}-\frac{d\rho (t)}{dt}=\left(\frac{1}{{\tau}_{s}}+B(\phi )\right)\rho (t)\\ =\left(\frac{1}{{\tau}_{s}}+\beta \delta {(1-{e}^{-t/(\kappa /\alpha )})}^{2}\right)\rho (t)\end{array}

(5)

Here, *δ* is the angular difference between the lowest-energy conformations in the excited and ground states and *τ*_{
s
} is the lifetime of the fluorophore in the absence of rotational relaxation events. The quantum yield *ϕ*_{
F
} can be obtained by integrating the excited-state probability:

{\varphi}_{F}=\frac{1}{{\tau}_{0}}{\displaystyle {\int}_{0}^{\infty}\rho}(t)dt

(6)

Although not defined in the original manuscript [9], *τ*_{0} can be assumed to be the natural lifetime, that is, the lifetime of the fluorophore in the absence of any nonradiative deactivation processes as opposed to *τ*_{
s
} , which is the lifetime in the absence of only rotational deactivation processes. According to Förster and Hoffmann, typically *τ*_{
s
} /*τ*_{0} ≈ 0.5 for this class of molecular rotors [9]. To simplify the solution of Equation 5 and its subsequent integration (Equation 6), Förster and Hoffmann examined three special cases. The first case (Equation 7) emerges for low viscosities where the quantum yield reaches a solvent-independent minimum:

{\varphi}_{F,min}=\frac{1}{\beta {\tau}_{0}{\delta}^{2}}

(7)

The second case occurs in solvents of very high viscosity, where radiative relaxation dominates with negligible rotational relaxation, and the quantum yield can be approximated by Equation 8:

{\varphi}_{F,max}=\frac{{\tau}_{s}}{{\tau}_{0}}\left(1-\frac{6{\sigma}^{2}}{{\eta}^{2}}\right)

(8)

In Equation 8, *σ* is a dye-dependent constant that contains all viscosity-independent variables and has units of viscosity:

\sigma ={\left(\frac{{\alpha}^{2}\beta {\delta}^{2}{\tau}_{s}^{3}}{192{\pi}^{2}{r}^{6}}\right)}^{\frac{1}{2}}

(9)

The most important case, the third case, is found for intermediate viscosities *η* ≪ *σ*, when the solution of Equation 6 simplifies to Equation 10:

{\varphi}_{F}=0.893\text{}\text{}\frac{{\tau}_{s}}{{\tau}_{0}}{\left(\frac{\eta}{\sigma}\right)}^{\frac{2}{3}}

(10)

For crystal violet, a triphenylmethane dye, *σ* ≈ 100 Pa s can be found. The remaining constants can be combined into one constant \hat{C}, yielding Equation 11, which is the non-logarithmic form of Equation 1 with an exponent *x* ≡ 2/3 as the result of an integration step:

{\varphi}_{F}=\hat{C}\cdot {\eta}^{2/3}

(11)

Other quantitative treatments of the viscosity-sensitive behavior are based on the premise that the intramolecular reorientation rate *k*_{
or
} depends on rotational diffusivity, more precisely, *k*_{
or
} ∝ *D*. A fluorophore's quantum yield *ϕ*_{
F
} is defined as the radiative relaxation rate *k*_{
R
} relative to the total relaxation rate *k*_{
R
} + *k*_{
NR
} (Equation 12):

{\varphi}_{F}=\frac{{k}_{R}}{{k}_{R}+{k}_{NR}}\approx \frac{{k}_{R}}{{k}_{or}}

(12)

The approximation is valid for molecular rotors, because the intramolecular reorientation rate *k*_{
or
} is the dominant nonradiative relaxation pathway, and *k*_{
or
} ≫ *k*_{
R
} . Furthermore, *k*_{
R
} is a viscosity-independent dye constant. DSE diffusion stipulates that the rotational diffusion constant *D* is inversely proportional to viscosity (Equation 13):

D=\frac{1}{6V\text{}s\text{}g}\text{}\text{}\frac{{k}_{{\rm B}}T}{\eta}

(13)

Here, *V* is the effective volume of the molecule, *s* reflects a boundary condition (*s* = 1 for a stick condition and *s* < 1 for a slip condition), *g* is a shape factor, *k*_{
B
} is Boltzmann's constant, and *T* is the temperature. Vogel and Rettig [60] define a driving force *F* , which is the force constant of the harmonic twist potential in triphenylmethane dyes, such that

{k}_{or}=\frac{2F}{{\zeta}_{S}}

(14)

where *ζ*_{
S
} is the Stokes friction coefficient, defined as *ζ*_{
S
} = 6*V η*, implicitly setting *s* = 1 and *g* = 1. Vogel and Rettig now argue that the product of viscosity and rotational reorientation rate would be constant under the DSE theory (Equation 15) [60].

{k}_{or}\eta =\frac{F}{3V}

(15)

Experimental evidence invalidates Equation 15, because the product *k*_{
or
}*η* increases strongly with decreasing temperature. To explain the deviation from DSE theory, Vogel and Rettig use the microfriction model introduced by Gierer and Wirtz [61]. These authors extend DSE theory by accounting for the finite thickness of molecular layers that surround the fluorophore. By solving the equation of rotational motion for a spherical molecule of radius *r*_{
M
} surrounded by finite layers of spherical solvent molecules with a radius *r*_{
S
} , Gierer and Wirtz obtain a corrected microfriction coefficient *ζ*_{
Micro
} that is related to the DSE macrofriction coefficient *ζ*_{
Macro
} through Equation 16 [61]:

{\zeta}_{Micro}={\zeta}_{Macro}\cdot {\left(\frac{6{r}_{S}}{{r}_{M}}+\frac{1}{{\left(1+{r}_{S}/{r}_{M}\right)}^{3}}\right)}^{-1}

(16)

Vogel and Rettig interpret this result as a superposition of Stokes diffusional freedom 1/*ζ*_{
S
} and free-volume diffusional freedom 1/*ζ*_{
FV
} (Equation 17), where diffusion is facilitated by void spaces between solvent and solute.

\frac{1}{{\zeta}_{Micro}}=\frac{1}{{\zeta}_{S}}+\frac{1}{{\zeta}_{FV}}

(17)

Viscosity decreases with increasing temperature. A commonly-used model is the Arrhenius function

\eta ={\eta}_{0}\cdot exp\left(\frac{{E}_{A}}{{k}_{B}T}\right)

(18)

where *η*_{0} is a material constant, *E*_{
A
} is an apparent activation energy, *k*_{
B
} is Boltzmann's constant, and *T* is the absolute temperature. Since viscosity is assumed to be proportional to the friction coefficient *ζ*, the apparent microviscosity *η*_{
Micro
} of the solvent, which is reported by the molecular rotor, would be smaller than the DSE macroviscosity. More specifically, the apparent microviscosity can be described as the superposition of two Arrhenius terms (Equation 19),

\begin{array}{c}{\eta}_{Micro}=a\cdot exp\left(-\frac{{E}_{\eta}}{{k}_{B}T}\right)\\ +b\cdot exp\left(-\frac{{E}_{FV}}{{k}_{B}T}\right)\end{array}

(19)

where *a* and *b* are related to the material constant *η*_{0}, and *E*_{
η
} and *E*_{
FV
} are the apparent activation energies for DSE macroviscosity and the free-volume viscosity term, respectively. After some arithmetic manipulation, Vogel and Rettig arrive at an extension of Equation 15, namely, *k*_{
or
}*η* = *A* + *B η*^{x} , where *A* corresponds to the term *F*/3*V* in Equation 15 and *x* reflects the relative contribution from Stokes and free-volume diffusion and is not identical to the exponent *x* in Equation 1. With *A* and *B* being experimental constants, this model was found to represent a good fit of experimental data [60]. To obtain an equation similar to Equation 11, *k*_{
or
} can be substituted in Equation 12, leading to Equation 20, which is an alternative model to Equation 1:

{\varphi}_{F}=\frac{{k}_{R}\text{}\eta}{A+B\text{}{\eta}^{x}}

(20)

Free volume was also recognized as an important determinant of a molecular rotor's quantum yield by Loutfy and coworkers [62–64]. The treatment by Loutfy *et al*. is based on Doolittle's [65] empirical relationship between viscosity and free volume, Equation 21,

\eta =A\cdot exp\left(B\frac{{v}_{o}}{{v}_{f}}\right)

(21)

where *A* and *B* are empirical, solvent-dependent constants with *B* ≈ 1, *v*_{
o
} is the occupied volume, and *v*_{
f
} is the free volume. The free volume is the temperature-dependent factor, and for glass-forming liquids, the free volume reaches a minimum at the glass transition temperature [66]. The ratio *v*_{
f
} /*v*_{
o
} is the relative free space for a liquid and becomes very small, typically 0.025, at the glass transition of many alcohols. Loutfy and Arnold [63] provide experimental evidence that the quantum yield of a fluorophore follows a relationship analogous to Equation 21,

{\varphi}_{F}=\frac{{k}_{R}}{{k}_{NR,0}}\cdot exp\left(x\frac{{v}_{o}}{{v}_{f}}\right)

(22)

where *k*_{NR,0}is interpreted as an intrinsic, fluorophore-dependent constant, and *x* is the slope found in plots of *logϕ*_{
F
} over {v}_{f}^{-1}. Equation 22 allows to express the rotational relaxation rate as a function of free volume, namely, *k*_{
or
} = *k*_{NR,0}*exp*(-*x v*_{
v
} /*v*_{
f
} ). Contrary to the assumptions by Vogel and Rettig and by Förster and Hoffmann, Loutfy and Arnold found 15 a power-law relationship between viscosity and diffusional reorientation rate. Equation 21 can be used to replace the free-volume term by viscosity, which yields Equation 23:

{\varphi}_{F}=\frac{{k}_{R}}{{k}_{NR,0}}{\left(\frac{\eta}{A}\right)}^{x}=C\cdot {\eta}^{x}

(23)

By combining the dye- and solvent-dependent constants *k*_{
R
} , *k*_{N R,0}, and *A*^{x} into one constant *C*, Equation 1 readily emerges. It is noteworthy that the rigorous derivation by Förster and Hoffmann - under the assumption of rotational friction according to the DSE model - and the more empirical derivation by Loutfy and Arnold - under the assumption of a power-law microfriction behavior - lead to the same relationship between quantum yield and bulk viscosity. Contrary to the Förster- Hoffmann derivation, however, the exponent *x* in Equation 23 can vary with the solvent and the molecular rotor molecule.

In practice, each of the models has limited applicability. A comparison of the model by Vogel and Rettig to the model by Loutfy *et al*. is shown in Figure 3. Each data point represents solvent viscosity and measured intensity of the molecular rotor DCVJ at a concentration of 5 *μ* M. Intensity is proportional to quantum yield, and an additional proportionality constant needs to be introduced in Equations 1 and 20 to reflect concentration and instrument constants. It can be seen that Equation 1 describes the data from polar solvents well in accordance with the literature [15, 34, 40, 54]. Equation 20 describes the data almost equally well, although Equation 1 is statistically the preferred model (F-test, *P* = 0.89). The model in Equation 20 tends to underestimate the viscosity at very low and very high solvent viscosities. In fact, when *A* ≪ *B*, Equation 20 takes up the form of Equation 1. The curve fit in Figure 3 provided *A ≈ B*/10. However, this ratio is not in agreement with values reported by Vogel and Rettig, where *A* and *B* are in the same order of magnitude. In this example, high viscosities were achieved with a viscosity gradient of mixtures of glycerol and a low-viscosity alcohol, such as ethylene glycol or methanol. This method is commonly used in the literature [33, 40, 54, 56, 63, 67] to achieve large-scale variations in viscosity with relatively small variations in polarity.

Deviations from the model can be seen in several instances. Water with its very high polarity reduces the barrier to the TICT state [50] and causes an anomalous low fluorescence. Polar aprotic solvents, such as dimethylsulfoxide, dimethylformamide, and acetone show a higher DCVJ intensity than predicted by the models, and nonpolar solvents (methylene chloride, benzene and toluene) have an even higher intensity, because nonpolar solvents stabilize the LE state. It can be seen that at low viscosities, other effects than microfriction dominate. Law [15] has reported that the chain length of short-chain 1-alkanols has a very small effect on the quantum yield, which would corroborate the low viscosity case presented by Förster and Hoffmann (Equation 7). According to Law [15], long-chain 1-alkanols also deviate from the models (light blue dotted line in Figure 3), because the alkane chain becomes the main determinant of intramolecular rotation, and the viscosity of alkanes is known to be much lower than that of the corresponding 1-alkanols.

In summary, rotational diffusivity is the most important determinant of intramolecular rotation rate and therefore a molecular rotor's quantum yield. However, when the intramolecular rotation rate becomes very high in solvents of low viscosity, additional effects, such as hydrogen bond formation, excimer formation, and polar-polar interaction are no longer negligible and cause significant deviations from established models that describe the relationship between quantum yield and viscosity.