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Table 3 Example of application of the mathematical models for release rate of repellents from different device systems and results descriptions obtained by authors in their studies

From: Mosquito‐repellent controlled‐release formulations for fighting infectious diseases

Equation models Previous results description References
Higuchi, Avrami’s or Weibull and Korsmeyer-Peppas equation models The Higuchi model was employed to investigate the kinetic study of release of citronella oil from tamarind gum (TG) and carboxymethylated tamarind gum (CTG) microcapsules where the non-Fickian and Fickian diffusion mechanisms controlled the oil release. Furthermore, the use of Avrami’s model in those systems of release of citronella oil exhibited diffusion coefficient n < 1, indicating the Fickian diffusion mechanism that governed the systems. Finally, the Korsmeyer-Peppas model was also used to evaluate the release of oil from microcapsules. The prediction data from this model was fitted well with the experimental data. The parameter R2 was between 0.7642 to 0.9885, with a diffusion coefficient that demonstrated that the oil loaded into microcapsules was controlled by mechanism of Fickian and non - Fickian diffusion. [29]
Higuchi zero-order, Higuchi and Korsmeyer-Peppas models The mechanism of release of citronella oil from microcapsules was evaluated with three models known as Higuchi zero-order, Higuchi and Korsmeyer-Peppas. With Higuchi model it was possible to obtain a high parameter R2 of 0.9820 and the diffusion coefficient was close to 0.5, indicating that the oil loaded intro microcapsules device was governed by Fickian diffusion mechanism. While by use of Korsmeyer-Peppas model the parameter R2 was 0.9800 and the diffusion coefficient was close to 1, demonstrating that the release of oil from microcapsule was controlled by non-Fickian diffusion (anomalous diffusion) mechanism. Finally, the Higuchi zero-order model was a non-significant influence in release of citronella oil from the microcapsules. [31]
Korsmeye-Peppas model The release rate kinetic of neem oil from polymer microcapsules was investigated with the Ritger–Peppas model. The parameter R2 of neem oil into polymer microcapsules was linear. Further, the value of “n” obtained by use of model, showed that the release of neem oil from microcapsule is governed by Fickiam diffusion mechanism. [28]
Higuchi, Korsmeyer-Peppas and Weibull models Korsmeyer-Peppas model showed that the release of DEET from microcapsules was controlled by the Fickian diffusion mechanism. The prediction data obtained by the Peppa’s and Weibull models were in agreement to the experimental data of release of DEET from microcapsules. With Higuchi the constant R2 was lower than R2 obtained by other models. [44]
Higuchi and Korsmeyer-Peppas models The best correlation coefficient R2 equal to 0.9547 was obtained by use of the Korsmeyer-Peppas model for release of citronella oil loaded cotton microcapsules. The “n” value was equal to 0.5833, indicating that the system is controlled by anomalous non-Fickian diffusional mechanism. While for the release of oil of citronella loaded into polyester microcapsules, the best parameters R2 and “n” were 0.9477 and 0.3177, respectively. Therefore, the Fickian diffusion mechanism was observed. [43]
Korsmeyer-Peppas model The kinetic study of release of Satureja hortensis essential oil (SEO) from the alginate matrix was evaluated with the Korsmeyer-Peppas model. The predicted data obtained by the model was fitted with the experimental data with correlation coefficients R2 of three microparticles higher than 0.9. Furthermore, the parameter “n” was between 0.408 to 0.498, demonstrating that the mechanism of release rate of oil from microparticles was by Fickian diffusion. [218]
Semi-empirical power law or Korsmeyer-Peppas model The kinetic of release rate of the geraniol-to-zein ξ = 3 system, at different temperatures was determined. The results, in the range where 5 to 95 wt% of geraniol was evaporated, were fitted with the semi-empirical power law model. For a reservoir system with a spherical geometry with Fickian diffusion through the wall rate-limiting, the best fit value for the release exponent for the present data set was n = 0.80. [219]
Mapossa model This is a simple implicit mechanistic model used to predict the release rate of DEET and Icaridin from the microporous LLDPE strands that are covered by a skin like membrane that controls the release rate. In all cases, the model employed was a reasonable fit to the experimental data. This model assumes quasi-steady state diffusion and is based on the assumptions of a dimensionally stable and inert solid scaffold. This means that it will break down if the polymer absorbs and swells in the presence of the repellent. In this case, polyethylene is non-polar polymer, therefore, it was appropriate with this model. [14]
Avrami’s equation The Avrami’s equation was used to estimate the release rate of the limonene from nanomelusions systems. Results demonstrated that the release rate of repellent was controlled by the diffusion mechanism. The experimental and prediction data were fitted. [179]
Avrami’s equation
Higuchi model
To investigate the release of limonene oil from nanoemulsion, Avrami’s equation was employed. The results showed that the values of “n” for both homogenizations were almost in the same range of 0.6 to 1.0, suggesting that the release rate of limonene occurred through diffusion mechanism.
The kinetic study of release rate of citronella oil from nanoemulsion was investigated by Higuchi’s model where the predicted data obtained by this equation was well fitted with the experimental data. Results from Higuchi’s equation, showed that the “n” value was equal to 0.5, suggesting that the release of citronella oil from nanoemulsion was controlled by diffusion mechanism.
Higuchi and Korsmeyer-Peppas models The Higuchi and Korsmeyer-Peppas models were employed to evaluate the citronella oil release kinetics from β-cyclodextrin. Among those models, the Korsmeyer-Peppas, R2 = 0.9877 and n = 0.6166 ± 0.0275, when compared to the Higuchi model (R2 = 0.9751) showed better correlation and demonstrated a good fit between predicted data with the experimental data. The parameter “n” proved that the mechanism of release of oil from β-cyclodextrin was controlled by anomalous diffusion (0.5 < n < 1). [88]
Korsmeyer-Peppas model The Korsmeyer-Peppas model was employed to investigate the thyme oil release kinetic from β-cyclodextrin. The correlation coefficient for cotton fabrics treated with MCT- β -CD loaded with thyme oil was R2 = 0.9657 and parameter “n” was equal to 0.5444 demonstrating that the mechanism of release of oil was through anomalous diffusion mechanisms. [188]
Korsmeyer-Peppas model DEET was released slowly from the nanosphere systems. The study was governed by the diffusion mechanisms (Fickian diffusion and polymer relaxation). The device system maintained effective release rate of DEET, which has ensured performance activity times for more than nine hours. The prediction data obtained with the Korsmeyer-Peppas model was fitted well with experimental data. [169]
Korsmeyer-Peppas model The kinetic study of the amount of DEET released from polyurethane and polyurea microcapsules was evaluated using the Korsmeyer-Peppas model. The polyurethane microcapsules controlled the release rate of DEET well compared to the polyurea microcapsule. The mechanism of DEET release from polyurethane demonstrated “n” equal to 0.2120 while for DEET released from polyurea exhibited “n” equal to 0.2762. The results suggest that the non-Fickiam including diffusion as well as polymer relaxation mechanism was achieved. [185]