The polynomial-exponent model that we propose in this manuscript is motivated by the limitations of the classical models conventionally considered for calibrating photoluminescence sensors when they are applied to measurements in the frequency domain. In this section, we analyze in detail the classical calibration models and their limitations when applied in phase-resolved photoluminescence.

#### 2.2. Limitations of the Stern–Volmer Model

The Stern–Volmer model provides a very good qualitative description of a wide range of photoluminescent sensors. The accuracy provided by this model is acceptable for some sensing phases, but it is insufficient for most of them, due to deviations of the photoluminescence from the first order linear model [

1,

3,

4,

11]. These deviations are associated with the presence of different excitation and/or deactivation mechanisms, the distribution of the quencher in the volume of the luminescent material, biased measurements of the modulation-factor or phase-shift due to instrumental configuration, etc. Taking into account the mathematical formulation of the model, two different situations should be considered: (a) the relationship between the quencher concentration and the non-radiative deactivation constant is not linear (i.e., Equation (

2) is insufficient); or (b) the order of the differential equation is higher than one.

From the mathematical perspective, both situations are completely different, since in the first case, the photoluminescence sensor is a first order system (i.e., with a monoexponential evolution of the luminescent response, with a well defined lifetime, and providing identical lifetime estimations from modulation-factor or from phase-shift and at whatever modulation frequency), while in the second case there is not a well-defined luminescence lifetime, the estimated values depend on the estimation procedure (so they are referred to as “apparent lifetimes”) and therefore the calibration parameters depend on the sensor configuration (modulation frequency, whether the apparent lifetime is estimated either from the modulation-factor or from the phase-shift, etc.).

For the first situation (first order differential equation but nonlinear relationship between quencher concentration and deactivation constant), the Lehrer model is an extension of the Stern–Volmer model [

1,

9]:

where a proportion

x of the luminescent material is assumed to be affected by the quencher while the rest,

$1-x$, is not affected. Compared with the Stern–Volmer model, the 3-parameter Lehrer model increases the flexibility in the description of the

${\tau}_{q}\left(C\right)$ curves, providing a better calibration for many photoluminescence sensors.

The Demas model can also be considered an extension of the Stern–Volmer model [

1,

7]:

and represents two independent quenching contributions, each one with a different Stern–Volmer constant. The 4-parameter Demas model increases the flexibility with respect to the Stern–Volmer and Lehrer models and provides more accurate calibrations. Additionally, the Demas model is scalable (and can be extended to more than two sites by including more terms):

where

${\tau}_{0,n}$ is the contribution of the

n-th site to the lifetime at null quencher concentration.

For the second situation (higher order differential equation), a multi-exponential photoluminescence model is proposed [

1,

22,

23], providing the following frequency response:

where the index

n represents each mono-exponential process contributing to the multi-exponential response; for each process

n,

${M}_{0,n}$ is the modulation factor at low frequency and null concentration,

${\tau}_{0,n}$ is the lifetime at null concentration, and the relation between the concentration and the lifetime for each mono-exponential process is described by a Stern–Volmer equation [

1,

24]:

with its corresponding Stern–Volmer constant

${k}_{n}$. In this model, each mono-exponential process requires three parameters to be calibrated (

${M}_{0,n}$,

${\tau}_{0,n}$, and

${k}_{n}$). By transforming the frequency response into the corresponding differential equation, it can be demonstrated that an

N-process model is equivalent to a

N-order differential equation.

Interestingly, the Demas multi-site first order model and the multi-exponential model provide an equivalent equation for low frequencies. Effectively, if

$\omega {\tau}_{q,n}\ll 1$ for all the processes, the dependence with frequency disappears, and Equation (

15) can be rewritten as:

and, since the modulation-factor is proportional to the lifetime at low frequency (according to Equation (

11)), both models are identical at low frequency.

However, the differences between both models become significant for phase-resolved photoluminescence sensors using higher modulation frequencies (or for multifrequency photoluminescence). Experimental results suggest that deviation from the simple Stern–Volmer model is usually better explained with a multiexponential model than with a multi-site quenching of the lifetime, since the apparent lifetimes estimated at different frequencies, either from modulation-factor or phase-shift, provide inconsistent estimations [

23,

29,

30,

31,

32], while a multiexponential model provides a globally consistent description of the photoluminescence response at different modulation frequencies [

23].

Even though the multiexponential model is preferable for a global description of the photoluminescence multifrequency response, the sensor calibration is usually performed with the Demas model and using apparent lifetimes (either from modulation-factor or phase-shift based and estimated at a single frequency) because the combination of the different measured parameters in a single analyte determination is not very common and it is a relatively recent proposal [

23,

29,

30]. In addition, the Demas model is expected to provide a reasonable calibration when it is applied to the apparent lifetimes because at moderate frequencies the relation between

$m(C,j\omega )$ and

${\tau}_{m}\left(C\right)$, or between

$\varphi (C,j\omega )$ and

${\tau}_{\varphi}\left(C\right)$, is close to linear, and usually very high modulation frequencies are not applied in phase-resolve photoluminescence (because the emission amplitude is low at high modulation frequencies, and the response would be too noisy for an accurate analyte determination) [

23].

#### 2.3. Limitations of the Demas Model

According to the previous discussion, even though the Demas multi-site description is not appropriate for a multiexponential photoluminescence sensor, at very low frequencies, it is perfectly consistent (since it is mathematically equivalent to the multiexponential model). On the other hand, if the deviation from the Stern–Volmer model is moderate, or at moderate frequencies (relative to the involved lifetimes), the Demas model provides a reasonable calibration, usually with accuracy significantly improved with respect to the Stern–Volmer model (because the inconsistency can be compensated by the model flexibility). However, the inconsistency of the Demas model limits its accuracy in multiexponential photoluminescence sensors excited with modulated light sources.

Additionally, the instrumental bias (affecting the modulation-factor and the phase-shift measurements) and its propagation to the apparent lifetimes also contribute to limiting the accuracy of the Demas-based calibration. An accurate estimation of the modulation-factor and phase-shift requires the acquisition of the excitation and emission signals

${x}_{exc}\left(t\right)$ and

${x}_{em}\left(t\right)$ and the subsequent digital signal processing [

1,

27,

28]. The illumination is usually provided by an electrical signal which feeds a driver circuit connected to the illumination source, and the excitation signal is acquired by sampling the electrical signal used for the illumination system. The acquisition of the excitation signal is not difficult since this signal is usually provided by a signal generator, with large amplitude and high signal-to-noise ratio. However, the acquisition of the emission signal in a photoluminescence sensor is usually affected by several problems. The photoluminescence response is converted into an electrical signal by a photodetector (for example, a photodiode, a phototransistor, or a photomultiplier tube), and the electrical signal has to be amplified and sampled. Depending on the intensity of the photoluminescence response and the required sensitivity, large gains are usually required in the transduction, and usually the transductors and the amplification circuitry cause linear and nonlinear distortions (group delay, frequency response associated with the instrument including gain and phase-shift distortions, nonlinear distortion due to the transducer response, etc.). A significant bias in the estimation of the phase-shift is very common in photoluminescence sensors (associated with group delay in the photodetector and frequency response of the photodetector and preamplifier). The main source of bias in the amplitude measurements would be associated with the polarization of the electronic and optoelectronic components, but in a linear system it does not affect frequency based measurements (only at null frequency). Therefore, the bias in the modulation-factor is expected to be moderate and would be associated with nonlinear effects in the photodetector.

These effects can be estimated, calibrated, and compensated in order to remove them from the frequency response of the photoluminescence sensor and therefore to obtain a more consistent calibration of the sensor using a Demas or a multiexponential model. However, an accurate calibration and compensation of the instrumentation effects for the operation conditions (or a join calibration of both the instrumentation and the photoluminescent subsystems) are not easy [

1,

24,

27,

28].