Rapid Estimation of Fragrance Vapor Pressure Using a Nanostructured Surface–Modified Quartz Crystal Microbalance Sensor

Abstract

Nanostructured oxide coatings play a critical role in determining molecular adsorption and desorption behavior on solid surfaces. In this study, we propose a rapid and simple method to estimate the apparent vapor pressure of fragrance compounds using a quartz crystal microbalance (QCM) sensor modified with a nanostructured silica surface. Here, the term “apparent vapor pressure” refers to the vapor pressure values predicted from the QCM response characteristics, which correlate quantitatively with reference data obtained from conventional thermodynamic calculations. The QCM responses of various fragrances were analyzed in relation to the adsorption–desorption dynamics occurring at the nanostructured interface. We found a quantitative relationship between the sensor responses and the reference vapor pressure values, with a mean absolute percentage error (MAPE) ranging from 19.3% to 220% depending on the compound. This correlation enables rapid evaluation of vapor pressure-related behavior without relying on conventional vapor pressure measurement methods. The results suggest that the surface nanostructure influences the adsorption–desorption balance governed by vapor pressure. This approach provides a practical and efficient means of evaluating the apparent vapor pressure of volatile compounds on nanostructured materials, offering insights into interfacial phenomena relevant to materials science and applied nanosciences.

1. Introduction

Vapor pressure is a crucial parameter for understanding and predicting fragrance behavior in a variety of applications. It reflects the volatility of a fragrance and directly influences its diffusion and persistence [1]. Accurate vapor pressure data enables perfumers to optimize the balance of fragrance components and design formulations with desirable longevity and performance. Additionally, modeling the diffusion of perfume from the skin requires reliable vapor pressure data to predict evaporation and permeation dynamics [2,3,4]. Simplifying the vapor pressure measurements can minimize the consumption of valuable samples and improve the efficiency of fragrance-development workflows.

Several methods have been established to measure the vapor pressure. These include the static method, where vapor-liquid equilibrium is established in a sealed container [5]; the boiling point method, which determines the vapor pressure from boiling temperatures under known pressure [6]; the isoteniscope method, involving equilibrium measurement based on bubble formation [7,8]; and the gas saturation method, which estimates vapor pressure via quantification of vapor in an inert gas stream [8]. Other techniques include differential scanning calorimetry (DSC), which infers vapor pressure from thermal behavior [9], and the triple expansion method, which applies thermodynamic calculations across the expanding chambers [10]. Although these techniques provide accurate results, they typically require large sample volumes, long measurement durations, and device cleaning after each use.

Quartz crystal microbalance (QCM) sensors offer nanogram-level sensitivity for gas detection by measuring mass changes on the crystal surface. As one example of QCM application, Rianjanu et al. used a QCM sensor modified with an electrospun polyvinyl acetate nanofibrous film to measure the vapors of three types of volatile organic compounds of health concern: benzene, toluene, and xylene [11]. They found that the sensor response was affected by the vapor pressure of these compounds, demonstrating the sensitivity of the QCM to the physical vapor properties. When vapor molecules are adsorbed onto the surface of the QCM sensor, the increase in mass causes a corresponding decrease in the oscillation frequency, enabling highly sensitive detection. The degree and speed of adsorption depend on the compatibility between the vapor molecules and the sensor surface. To enhance the detection performance, the QCM surface can be physically or chemically modified using nanomaterials that create functional coatings. These modifications allow the tuning of surface properties, such as polarity and porosity, which directly affect sensor responsiveness to target vapors. In particular, QCM sensors with nanostructured oxide coatings provide a valuable platform to investigate molecular adsorption and desorption processes occurring at nanostructured interfaces.

Various nanomaterials have been used for this purpose. For example, SnO₂ nanowires have been used to detect water vapor [12], mesoporous silica for general volatile organic compounds [13], polyvinyl acetate nanofibers for primary alcohols [14], ZnO nanowires for ammonia [15], and graphene nanosheets for organic solvents [16]. These materials provide high surface areas and tailored pore architectures, which promote efficient adsorption and capillary condensation. This increases the frequency change of the QCM sensor. These mechanisms are particularly important for measuring less volatile compounds by gas sensors.

Although previous studies have shown that QCM sensor responses may be influenced by vapor pressure, systematic investigations across a wide range of fragrance types are lacking. Moreover, many earlier studies relied primarily on endpoint responses, such as signal saturation values, rather than analyzing the dynamic behavior of the response curve. By extracting time-resolved features, such as the response time, recovery time, and slope characteristics, a more nuanced understanding of the interaction between vapor molecules and the sensor surface can be achieved. Shiba et al. used a surface stress sensor (MSS) coated with nanoparticles to identify various liquid vapors, including water, tea, alcohol, and ethanol solutions [17]. Using a sensor array, they analyzed time-response curves and extracted multiple features, demonstrating that alcohol concentration could be estimated using a predictive formula. This underscores the value of analyzing not only the saturation levels but also the shape of the transient response curves, even when using a single sensor, to enrich vapor analysis.

Other than QCM, there is an impedance-type sensor modified with a silica nanoparticle thin film as a quick and easy vapor measurement method [18]. In this method, droplets of the target vapor are capillary-condensed in the voids of the nanoparticle thin film and are measured electrically. In the past, it was reported that there is a correlation between the sensor response and the vapor pressure for polar substances. However, because droplets are measured electrically, their application to substances with low electrical conductivities is difficult. This is due to the fact that volatile organic compounds with low polarity exhibit small changes in electrical conductivity, making it difficult to obtain sufficient response in impedance-type sensors [19].

Here, we achieved rapid and simple measurements of fragrance vapors using a QCM sensor modified with a silica nanoparticle thin film. This method utilizes capillary condensation in nanoscale pores to provide short-term vapor measurements. A particularly noteworthy point is that the time response waveform of the sensor changes significantly depending on the apparent vapor pressure of the fragrance. The apparent vapor pressure, estimated from QCM responses, represents the apparent vapor pressure of each fragrance. We then extracted the characteristic features from the response waveforms of each fragrance and clarified the quantitative relationship between these features and apparent vapor pressure. Using a prediction model constructed through cross-validation, we propose a method for estimating the apparent vapor pressure from the dynamic response of fragrance vapors. Finally, we evaluated the practicality and specificity of this method by comparing its performance with that of existing representative vapor pressure measurement methods. This study suggests that the nanostructured surface morphology of the silica nanoparticle film may influence the adsorption–desorption dynamics of fragrance molecules, thereby affecting the apparent vapor pressure estimated from the QCM responses and suggesting a possible connection between interfacial adsorption behavior and molecular volatility.

2. Theory

A QCM sensor modified with a thin film of silica nanoparticles on its surface was used to capture fragrance vapors, based on the following principle (Figure 1). The QCM sensor used in this study consisted of a gold-coated quartz substrate and a silicon nanoparticle thin film layered on top (Figure 1a). The time-dependent changes in the frequency variation obtained from the QCM sensor and the corresponding diffusion process of the fragrant vapor are illustrated in a schematic diagram (Figure 1b). Nanoscale pores formed between the particles within the silica nanoparticle thin film. When the QCM sensor is exposed to fragrance vapors, the vapors gradually penetrate the nanoscale pores within the thin film. As this process progresses, capillary condensation (a phenomenon in which highly adsorbent gases condense and liquefy at low vapor pressure in microscopic pores) occurs, causing a change in the frequency response of the sensor [20]. As condensation progresses, the frequency response of the sensor reaches a peak and eventually transitions to a metastable state. When the exposure to fragrance vapor is complete and the sensor is re-exposed to ambient air, fragrance molecules evaporate from the nanoscale voids into the atmosphere, causing the frequency response of the sensor to change accordingly. The relationship between the vapor pressure and vapor diffusion processes can be expressed as follows: according to Fick’s law and the gas equation, molecular flux within the pores is expressed as J = −D∇c = −D(∇p/RT), where D is the diffusion coefficient, ∇c is the vapor concentration gradient, ∇p is the vapor pressure gradient, R is the gas constant, and T is temperature. The molecular flux was proportional to the pressure gradient. According to this relationship, the higher the vapor pressure, the faster the diffusion into the membrane. A QCM sensor detects changes in the frequency of vibrations caused by the adhesion of substances to the electrode surface, with the frequency decreasing proportionally to the mass of the adhered substance. The relationship between the frequency change obtained from the QCM sensor and mass is given by Sauerbrey’s equation [21]:

$$\delta m={\displaystyle \frac{S\sqrt{\rho \mu }}{2N{F}^2}}\left(-\delta F\right)$$

(1)

where δm is the mass change, S is the electrode area, ρ is the density of the quartz crystal, μ is the shear stress of the quartz crystal, N is the overtone order, and F is the nominal frequency. Based on this relationship, the measured frequency change values were converted to the amount of condensed fragrance vapor.

3. Materials and Methods

3.1. Sensor Fabrication

Silica nanoparticle thin films were used as gas-capturing layers. The fabrication process of the QCM sensor modified with silica nanoparticle thin films was as follows. A non-porous silica nanoparticle dispersion solution (Sicastar, particle diameter: 50 nm, particle concentration: 25 mg/mL, Micromod Partikeltechnologie GmbH, Rostock, Germany) was diluted 10-fold with ethanol as the solvent and filtered through a filter (Whatman Mini-UniPrep, Cytiva, Waltham, MA, USA; pore size: 0.45 μm) to remove aggregates. This solution was dispensed onto one side of a quartz crystal (resonant frequency: 20 MHz, quartz crystal diameter: 8 mm, TAMADEVICE Co., Ltd., Kawasaki, Japan) at 20 μL and spin-coated at 1000 rpm for 2 min. The samples were then dried on a hot plate at 50 °C. The particle size of 50 nm was selected based on previous work by Kano et al. [18], who investigated capillary condensation behavior in non-sintered silica nanoparticle films of different particle diameters. Their study showed that films prepared from 10 nm particles exhibited slower adsorption and desorption due to finer pore radii, whereas 50 nm particles provided a more balanced response and recovery behavior. In view of these findings, the 50 nm silica nanoparticles were chosen in this study as a suitable size to achieve stable and rapid vapor responses in the QCM measurements. Figure 2b shows the SEM image of a spin-coated nanoparticle film, which was obtained by the authors using a scanning electron microscope (S-4800, Hitachi High-Tech, Tokyo, Japan). The film was spin-coated on a gold-sputtered silicon substrate using the same spin-coating conditions as those used for the QCM sensor fabrication, and is therefore representative of the coating morphology on the sensor surface.

3.2. Measurement System

The headspace vapor was measured by inserting the fabricated QCM sensor into a vial containing the sample (Figure 3a). The fabricated QCM sensor was connected to a control PC using a simple portable QCM measuring device (THQ-100P-SW type, TAMADEVICE Co., Ltd., Japan). A glass vial (No. 7, capacity: 50 mL, Maruem Corporation, Osaka, Japan) was filled with 200 μL of the fragrance. This amount was determined as the minimum volume sufficient to saturate the vial headspace while maintaining a stable vapor concentration; smaller volumes resulted in unstable responses. The sensor was then fixed to a stand. To suppress the influence of the surrounding airflow, the entire setup was installed inside a draft chamber that was shut down. The sampling interval was set to 1 s. Operations such as starting/stopping measurements and setting the sampling interval were performed using the QCMeasure software (version 1.11, TAMADEVICE Co., Ltd., Japan). The measurement results were output as data for the number of samples (time), frequency, and frequency change value from the start of the measurement. To explain the changes in the time-response waveform due to differences in the vapor pressure of the fragrance, a typical response image is shown in Figure 3b.

3.3. Measurement Procedure

The fragrance vapor was measured according to the following procedure (Figure 4). First, 200 μL of fragrance was sealed in a vial and left to stand for 5 min. Next, the sensor was placed in a vial and exposed for 5 min to allow the sensor surface to acclimate to the fragrance vapor prior to the measurement. The sensor was then removed and left to stand in the air for 5 min. Next, for the measurement, the sensor was placed in a vial and exposed to fragrance vapor. Because the time required to reach the maximum change varies depending on the fragrance, the exposure time was adjusted accordingly (specifically, preliminary measurements were conducted, and based on the results, the exposure time was set to 30, 60, or 90 s). After the measurement, the sensor was removed and left to rest in the air for 5 min. This measurement and the air resting process were repeated 12 times. The exposure time was set to 30 s for high-volatility compounds (e.g., limonene, α-terpinene, isobutyl isobutyrate, isoamyl butyrate, ethyl hexanoate, p-cymene, benzaldehyde, hexyl acetate, hexyl alcohol, benzyl acetate), 60 s for medium-volatility compounds (e.g., linalyl acetate, ethyl benzoate, undecanal), and 90 s for low-volatility compounds (e.g., phenethyl alcohol).

3.4. Analysis

In this study, three characteristic quantities representing the response characteristics of each fragrance were defined from the time-response waveforms of the sensor. Specifically, these are the sensor response time t₉₀, which represents the vapor adsorption process; the sensor recovery time t₁₀, which represents the desorption process of condensed vapor; and Δ, where represents the slope of the response overshoot. These features were calculated based on the representative points on the response waveform (Figure 5). Accordingly, a is the initial value at the start of the exposure, b is the peak value, c is the value at the end of the exposure, and d is the frequency change value when the response decreases and stabilizes. In addition, e represents the point where the frequency change value reaches 90% of the adsorption process between A and B (e = 0.9(b − a)), and f represents the point where the frequency change value decreases to 10% of the desorption process between C and D (f = 0.1(d − c)). Each feature was calculated as follows: t₉₀ = t_e − t_a (time until 90% of the value increased), t₁₀ = t_f − t_c (time until 90% of the value decreased), and Δ = (b − c)/(t_b − t_c) (slope of the graph after the peak).

To investigate whether the relationship between the features obtained in this study and the vapor pressure can be applied to estimate the vapor pressure of unknown samples, cross-validation was performed. Leave-one-group-out cross-validation (LOGO-CV) was used for cross-validation. LOGO-CV divides data into groups and performs verification, which prevents information leakage within the same group [22]. In this study, each fragrance was defined as one group, and the generalization performance of the regression model using the least squares method (LSM) was evaluated using this method. Using LOGO-CV, it is possible to ensure that the same fragrance data is not duplicated in the test and training data. The implementation environment consisted of Python (v3.13.1), scikit-learn (v1.6.0), numpy (v2.2.0), and pandas (v2.2.3).

The evaluation metrics used were the root mean square error (RMSE) and mean absolute percentage error (MAPE). These metrics are widely used to evaluate the accuracy of prediction models and are considered particularly effective when dealing with data of different scales [23]. The RMSE and MAPE are defined as follows:

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2}$$

(2)

$$\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}={\displaystyle \frac{100}{N}}{\sum }_{i=1}^N\left|{\displaystyle \frac{y_i-{\widehat{y}}_i}{{\widehat{y}}_i}}\right|$$

(3)

where N: Number of test data, ${\widehat{y}}_i$: Reference value of the target quantity of the i-th test data, y_i: Predicted value of the target quantity of the i-th test data. The scores were calculated for each combination of the test and training data, and the average values were used as the final scores. The predicted values were calculated using the regression equations obtained by regression analysis for each fragrance (Table S1). Statistical analysis was performed by calculating the mean and standard deviation from 12 repeated measurements for each fragrance. RMSE and MAPE were used as quantitative indicators of prediction accuracy.

The detailed analysis procedure is as follows. For each type of fragrance, 12 sets of measurement data were collected, and data that were clearly abnormal owing to operational errors were excluded. Nine sets of data were randomly selected. To ensure reproducibility, the random number generator was fixed to “random_state = 0.” The selected data were divided into 14 groups (14 types of fragrances were used) according to the fragrance type. Each group was used as test data one at a time, and the remaining data were used as training data to create 14 combinations. A model was constructed using the training data, and the predicted values y were calculated for the test data. The constructed model follows the regression equations (Equations (4)–(10)) below, depending on the number of features used. For one feature:

$$y_{1,\left(x_1\right)}={\beta }_0+{\beta }_1{x}_1$$

(4)

$$y_{1,\left(x_2\right)}={\beta }_0+{\beta }_2{x}_2$$

(5)

$$y_{1,\left(x_3\right)}={\beta }_0+{\beta }_3{x}_3$$

(6)

For two features:

$$y_{2,\left(x_1,x_2\right)}={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2$$

(7)

$$y_{2,\left(x_1,x_3\right)}={\beta }_0+{\beta }_1{x}_1+{\beta }_3{x}_3$$

(8)

$$y_{2,\left(x_2,x_3\right)}={\beta }_0+{\beta }_2{x}_2+{\beta }_3{x}_3$$

(9)

For three features:

$$y_{3,\left(x_1,x_2,x_3\right)}={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3$$

(10)

Here, each symbol is defined as follows: x₁: Feature 1 (log₁₀|Δ|), x₂: Feature 2 (log₁₀t₉₀), x₃: Feature 3 (log₁₀t₁₀), y_n: Predicted value (n indicates the number of features used), β₀: intercept, β_n: regression coefficient corresponding to feature x_n (n = 1, 2, 3). The subscript (x_i, x_j) indicates the combination of features used in the model construction (e.g., (x₁, x₂), (x₁, x₃), (x₂, x₃), etc.). This procedure was repeated such that all data groups were used as the test data. The RMSE was calculated for each fragrance, and the mean RMSE was calculated by dividing the total RMSE of all fragrances by the number of fragrances 14 for each combination of features. The combinations of features were set to the following seven combinations, each containing at least one of the three features (log₁₀|Δ|, log₁₀t₉₀, and log₁₀t₁₀). (a) log₁₀|Δ|, (b) log₁₀t₉₀, (c) log₁₀t₁₀, (d) log₁₀|Δ|, log₁₀t₉₀, (e) log₁₀|Δ|, log₁₀t₁₀, (f) log₁₀t₉₀, log₁₀t₁₀, (g) log₁₀|Δ|, log₁₀t₉₀, log₁₀t₁₀. In addition, for comparison with existing methods, MAPE was calculated for the combination with the best RMSE score. Note that the data used in this analysis refer specifically to the features extracted from the measurement data (described in Section 4.3).

3.5. Fragrances

Fourteen commercially available fragrances were used in the study (

Table 1. Fragrances used in the measurements. The reference vapor pressure at the measurement temperature was corrected using the molar vaporization enthalpy and boiling point values from the literature. The molar enthalpy of vaporization and boiling point at standard pressure were obtained from the NIST Chemistry WebBook (Standard Reference Data, SRD 69) and ChemSpider (Royal Society of Chemistry).

Name	Scent	Molecular Formula	Molecular Weight	Molar Enthalpy of Vaporization [kJ/mol]	Boiling Point at Standard Pressure [K]	Reference Vapor Pressure at the Temperature During Measurement Pvap [Pa]
α-Terpinene	Wood	C10H16	136.23	39.4	447.2	490.34 at 24.37 °C
Isobutyl Isobutyrate	Tropical fruit	C8H16O2	144.21	48.5	421	320.21 at 24.27 °C
Isoamyl Butyrate	Fruity	C9H18O2	158.24	47.4	451	137.43 at 23.1 °C
Ethyl Hexanoate	Fruity	C8H16O2	144.21	50.6	441.2	131.25 at 24.55 °C
Limonene	Citrus	C10H16	136.23	49.2	450	128.18 at 25.37 °C
p-Cymene	Fresh citrus	C10H14	134.22	49.2	450	114.81 at 23.72 °C
Benzaldehyde	Bitter almond	C7H6O	106.12	48.0	452	111.28 at 21.6 °C
Hexyl Acetate	Sweet fruity	C8H16O2	144.21	51.9	443	100.70 at 24.03 °C
Hexyl Alcohol	Leaves	C6H14O	102.17	61.0	430	47.06 at 23.44 °C
Benzyl Acetate	Jasmine	C9H10O2	150.17	55.5	479.2	20.34 at 24.27 °C
Linalyl Acetate	Bergamot	C12H20O2	196.29	57.8	493.2	9.09 at 23.76 °C
Ethyl Benzoate	Fruity	C9H10O2	150.17	61.1	485	6.55 at 23.18 °C
Phenethyl Alcohol	Rose	C8H10O	122.16	69.0	491.4	1.44 at 22.72 °C
Undecanal	Citrus	C11H22O	170.3	64.6	545.15	0.64 at 23.19 °C

). The selected fragrances cover a wide range of volatility and polarity, enabling evaluation of both polar and nonpolar compounds with different molecular weights. Because the sensing principle relies on capillary condensation and interfacial adsorption on a silica nanoparticle surface, the developed device is expected to recognize moderately volatile compounds more effectively than extremely low-volatility substances. Benzaldehyde was purchased from FUJIFILM Wako Pure Chemical Corporation (Osaka, Japan), undecanal from Tokyo Chemical Industry (Tokyo, Japan), and the others from Sigma-Aldrich (St. Louis, MO, USA). In selecting the fragrances, those that did not cause olfactory discomfort were selected to facilitate the experiment. Additionally, to calculate the vapor pressure at any temperature during the measurement, the following equation, which approximates the Clausius–Clapeyron equation, was used:

$$P_{\mathrm{v}\mathrm{a}\mathrm{p}}=P_0\mathrm{e}\mathrm{x}\mathrm{p}\left[{\displaystyle \frac{{∆}_{\mathrm{v}\mathrm{a}\mathrm{p}}{H}_{\mathrm{m}}}{R}}\left({\displaystyle \frac{1}{T_0}}-{\displaystyle \frac{1}{T}}\right)\right]$$

(11)

where Δ_vapH_m is the molar evaporation enthalpy, R is the gas constant, P₀ is the standard pressure, T₀ is the boiling point at standard pressure, T is an arbitrary temperature, and P_vap is the saturated vapor pressure at T. Under the following conditions, the temperature must be sufficiently below the critical temperature, the saturated vapor pressure must be sufficiently low, and the temperature difference from the boiling point must be small. This allowed the volume change due to evaporation to be approximated as the volume change of vapor and the volume of vapor to be approximated as the volume of an ideal gas. The temperature and pressure dependences of the molar evaporation enthalpy were ignored. In this study, the temperature was measured at 5-min intervals during 12 consecutive measurements for each sample, and the vapor pressure was calculated based on the average temperature.

The vapor pressure values at the measurement temperature were calculated from the Clausius–Clapeyron equation using the literature data of boiling point and enthalpy of vaporization, and are referred to as the reference vapor pressure in this study. The term apparent vapor pressure is used hereafter to indicate the vapor pressure values predicted from the QCM response data using the regression model described in Section 4.4.

4. Results and Discussions

4.1. Sensor Responsiveness

To determine whether each fragrance exhibited unique responsiveness, we plotted time-dependent changes in the frequency change values for each fragrance (Figure 6). The frequency change values were normalized by dividing them by the maximum frequency change value. This normalized frequency change value is referred to as Δf_norm. Depending on the fragrance, the behavior of Δf_norm increased from 0 to its maximum value, and the slope (rate of decrease) after the peak differed. These results indicate that there are differences in the evaporation speed from the pores of the nanoparticle thin film and the condensation speed in the pores, depending on the fragrance. Thus, it was confirmed that this sensor can obtain different responses depending on the fragrance.

4.2. Measurement Reproducibility

To investigate the reproducibility of the sensor, more than 10 consecutive measurements were performed for each fragrance. The vertical axis was normalized in the same manner, as shown in Figure 6. For example, the plots for limonene showed good agreement (Figure 7). Similarly, the plots for all fragrances used showed good agreement (Figure S1). These results confirm that the sensor responded reproducibly to the fragrances used. All consecutive measurements were performed within two hours. During the measurements, the temperature and humidity did not change significantly, and the measurements were stable. These results suggest that the proposed measurement system can be used to stably measure fragrances with good reproducibility in typical indoor environments. Additional response data for all fragrances are provided in the Supplementary Materials (Figures S1–S4).

4.3. Relationship Between Sensor Response and Vapor Pressure

By extracting the characteristic quantities from the obtained responses, we investigated the relationship between the sensor response and vapor pressure. In a previous study [17], parameters such as the adsorption process, desorption process, slope between the adsorption and desorption processes, and maximum response value were extracted, and kernel ridge regression was used to predict the alcohol content of the unknown alcohol. In this study, we applied the same approach and plotted the relationship between the three features ∆, t₉₀, and t₁₀, and the reference vapor pressure P_vap using common logarithms (Figure 8). Because the slope is negative, we applied common logarithms to the absolute values. All three parameters showed a linear correlation with the vapor pressure. The larger log₁₀|Δ|, the higher log₁₀P_vap, and the larger log₁₀t₉₀ and log₁₀t₁₀, the lower log₁₀P_vap. This reflects that, as the vapor pressure of the fragrance increases, condensation and evaporation in the pores of the nanoparticle thin film occur more rapidly, whereas in substances with lower vapor pressure, these processes occur more slowly. These results suggest that it may be possible to estimate the vapor pressure by comprehensively interpreting multiple parameters based on differences in the adsorption and desorption rates of fragrances obtained using this sensor within the vapor pressure range of 1–490 Pa. These findings suggest that the nanostructured silica surface influences the adsorption–desorption rate balance of fragrance molecules, thereby affecting the apparent vapor pressure estimated from the QCM responses.

While low vapor pressure generally slows desorption, the prediction error is not governed by vapor pressure alone. Compound-specific interfacial effects—such as molecular polarity, hydrogen-bonding ability to surface silanols, and surface tension that affects capillary condensation—also modulate the adsorption and desorption dynamics on the silica nanoparticle surface. These effects can shift the feature triplet (Δ, t₉₀, t₁₀) relative to the regression trend and thus influence the estimation accuracy.

The parameters Δ, t₉₀, and t₁₀ correspond to the apparent adsorption and desorption rates; specifically, t₉₀ and t₁₀ quantitatively represent the characteristic times of adsorption and desorption, respectively.

4.4. Cross-Validation

In this study, the apparent vapor pressure denotes the vapor pressure predicted from the QCM response features, whereas the reference vapor pressure refers to the literature-based values calculated from thermodynamic data. The regression model was trained to reproduce the reference vapor pressure using the apparent vapor pressure derived from QCM responses.

To investigate the applicability of this method for estimating the vapor pressure of unknown samples, a cross-validation was performed. LOGO-CV was used to evaluate regression models (

Table 2. Mean RMSE for each combination of features.

Symbol	Combination of Features	Regression Equation Used for Prediction	Mean RMSE
(a)	log10|Δ|	4	0.403 ± 0.243
(b)	log10t90	5	0.321 ± 0.201
(c)	log10t10	6	0.593 ± 0.237
(d)	log10|Δ|, log10t90	7	0.303 ± 0.178
(e)	log10|Δ|, log10t10	8	0.379 ± 0.192
(f)	log10t90, log10t10	9	0.317 ± 0.178
(g)	log10|Δ|, log10t90, log10t10	10	0.297 ± 0.175

). The prediction model was built based on the apparent vapor pressure values. First, when comparing cases with a single feature (Table 2 (a), (b), (c)), the RMSE was smallest for log₁₀t₉₀, followed by log₁₀|Δ|, and log₁₀t₁₀ in descending order. This indicated that the model using log₁₀t₉₀ demonstrated the highest prediction accuracy. Similarly, in Figure 8, the R² value was the highest for log₁₀t₉₀, followed by log₁₀|Δ| and log₁₀t₁₀, indicating that the model using log₁₀t₉₀ showed the highest correlation. These results were consistent. Next, when comparing the cases with two features (Table 2 (d), (e), (f)), the RMSE was smallest for “log₁₀|Δ|, log₁₀t₉₀,” followed by “log₁₀t₉₀, log₁₀t₁₀,” and “log₁₀|Δ|, log₁₀t₁₀.” In all two-variable models, the RMSE was smaller than that of the single-feature regression model, indicating an improvement in error. Furthermore, the regression model using all three features (log₁₀|Δ|, log₁₀t₉₀, and log₁₀t₁₀) (Table 2 (g)) showed a smaller RMSE than any of the single- and two-variable models. This suggests that combining multiple features contributes to improving the prediction accuracy. Next, the MAPE of the predicted values relative to reference vapor pressure values was calculated for the three-variable model that showed the highest prediction accuracy (

Table 3. Mean Absolute Percentage Error (MAPE) for each fragrance.

Test Data	MAPE
Undecanal	220
Phenethyl Alcohol	19.3
Ethyl Benzoate	279
Linalyl Acetate	51.6
Benzyl Acetate	46.5
Hexyl Alcohol	51.4
Hexyl Acetate	29.1
Benzaldehyde	95.5
p-Cymene	30.8
Limonene	49.6
Ethyl Hexanoate	22.0
Isoamyl Butyrate	60.8
Isobutyl Isobutyrate	41.8
α-Terpinene	69.9

). The MAPE was smallest for phenethyl alcohol (19.3%) and largest for ethyl benzoate (279%).

Notably, phenethyl alcohol, which has a similarly low vapor pressure (1.44 Pa) but stronger polarity and hydrogen-bonding ability, exhibited a small MAPE, whereas undecanal and ethyl benzoate, which are less polar, showed larger deviations. This suggests that interfacial interactions, in addition to vapor pressure, affect the dynamic response.

4.5. Comparison with Existing Vapor Pressure Measurement Methods

The method used in this study was compared with existing vapor pressure measurement methods, including the static method, boiling point method, isoteniscope method, gas saturation method, DSC method, and triple expansion method, in terms of accuracy, measurement time, required liquid volume, and the necessity of cleaning the apparatus. First, in terms of accuracy, the reported values for the existing methods were approximately 1% for the static method [24],3% for the boiling point method [25], 5–10% for the isoteniscope method [26], 10–30% for the gas saturation method [26], less than 1% for the DSC method [27], and approximately 2% for the triple-expansion method (e.g., MINIVAP VPL Vision (Grabner Instruments, Vienna, Austria) specifications) [28]. On the other hand, the method used in this study yielded an MAPE of 19.3–220%, indicating a lower accuracy compared to existing methods. Regarding the measurement time, in the static method, it has been reported that it took over 6 h for the pressure to reach equilibrium because of the diffusion of volatile impurities in the liquid during measurements of tetrakisethylmethylaminohafnium [29]. In the boiling point method, it is estimated that approximately 30 min to 1 h are required to heat the sample to the target temperature, approximately 15 to 30 min to confirm equilibrium, and approximately 15 to 30 min for the measurement itself. Therefore, although there is no clear indication of the required time in the literature, considering these factors, the total measurement time was estimated to be approximately 1 to 2 h. In the isoteniscope method, there are reports indicating that measurements were performed within 1 to 2 h, including heating and equilibration procedures [30]. In the gas saturation method, when measuring eicosane, there are reports of continuously flowing gas for approximately seven days [31]. In the DSC method, there are reports of approximately 2 h for a series of measurements of decane, dodecane, and tetradecane [32]. The triple expansion method takes at least approximately 5 min (e.g., according to the specifications of the MiniVap VPL Vision) [28]. However, the method used in this study requires only a few minutes for the measurement itself, significantly reducing the measurement time compared with existing methods. Regarding the required liquid volume, the static method requires approximately 0.5 mL [24], the boiling point method requires approximately 150 mL [33], the isoteniscope method requires approximately 10 mL [33], and the gas saturation method requires approximately 1 g [33]. The DSC method requires approximately 1 g [33], and the triple expansion method requires 3.2 mL, including the cleaning process (e.g., according to the specifications of the MiniVap VPL Vision) [28]. In contrast, the method used in this study can measure with 200 μL of liquid, making it highly convenient in terms of sample consumption. Regarding the necessity of cleaning after measurement, in all methods—the static method [31], boiling point method [34], isoteniscope method [35], gas saturation method [31], DSC method [36], and triple expansion method [28]—cleaning is often required because of sample residue or contamination inside the apparatus, which requires time and effort when switching between measurements. However, the method used in this study does not require cleaning before or after the measurement and allows consecutive measurements of multiple types of samples. This was expected to significantly improve the overall efficiency of the experiment.

5. Conclusions

In this study, we aimed to develop a method for quickly and easily estimating vapor pressure based on the characteristics related to vapor pressure, which is an important indicator of the volatility of fragrances. Fragrance vapors were measured using a QCM sensor modified with a thin film of silica nanoparticles. When measuring fragrances with different vapor pressures, clear differences in the sensor response waveforms (rise and fall) were observed for each fragrance, quantitatively demonstrating the influence of the vapor pressure on the sensor response. In particular, for fragrances with a high vapor pressure, condensation and evaporation occurred more rapidly in the nanoscale pores, which was reflected in the response and recovery times. A regression model using multiple features (Δ, t₉₀, and t₁₀) extracted from the response waveforms showed smaller prediction errors than when using a single feature, contributing to improved vapor pressure estimation accuracy. This method also demonstrated a better response for low-polarity fragrances that were difficult to measure using impedance-type sensors. This method is significant as a new, simple measurement technique because it enables nondestructive evaluation in a shorter time and with less sample volume than existing vapor pressure measurement methods for the same measurement targets. In fact, this method requires only a few minutes of measurement time, 200 µL of liquid, and no cleaning of the device, making it useful as a rapid screening method in the development of perfumes and air fresheners. The results also suggest that the nanostructured silica surface may influence the interfacial adsorption–desorption behavior of fragrance molecules, thereby affecting their apparent vapor pressure characteristics. Furthermore, by applying this method to volatile substances other than fragrances, it is expected to be applicable to a wide range of fields, such as estimating the vapor pressure of harmful substances and assessing environmental risks. This method may show reduced accuracy for highly polar or charged compounds (e.g., amines and carboxylic acids) because of strong hydrogen-bonding or electrostatic interactions with surface silanol groups, as well as for heavy molecules (>200 Da) that diffuse and desorb slowly from the nanoporous layer. Nevertheless, for most neutral organic compounds within the tested volatility range, the method provides reproducible and consistent vapor pressure estimations.