Modeling the controlled release of essential oils from a polymer matrix—A special case

Industrial Crops and Products 61 (2014) 23–30

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Modeling the controlled release of essential oils from a polymer matrix—A special case C. Tramón ∗ Agricultural Engineering Faculty, University of Concepción, Avda. Vicente Méndez 595, 3801061 Chillán, Chile

a r t i c l e

i n f o

Article history: Received 23 October 2013 Received in revised form 13 June 2014 Accepted 14 June 2014 Keywords: Essential oil Controlled release Diffusion coefficient Kirkendall effect Dusty Gas model Plant-based pesticides

a b s t r a c t Experimental evidence of a vacancy diffusion mechanism has been found in menthol/LDPE and thymol/LDPE solid-state diffusing systems, when samples are prepared by melting, mixing, and casting. The void pattern found in the cross-section of the samples after volatile release suggests the occurrence of a phenomenon similar to the Kirkendall effect. A Dusty Gas model, consistent with a Kirkendall system where no macroscopic volume change occurs, is proposed to describe the kinetics of the controlled release of the essential oils from the polymer matrix. The interdiffusion coefficient is predicted via a Free-Volume theory. The prediction of the diffusion coefficient and the simulation results are in good agreement with experimental gravimetric data. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Plant-based pesticides have been used for at least 2000 years in agricultural systems in China, Egypt, Greece and India (Ware, 1883). The intensive use of synthetic active principles in pest and disease control in agriculture began in the 1940s. In the past few years, these products have been restricted by regulatory governmental actions, due to health and environmental hazards. The need for new low-risk materials has boosted research and development concerning alternate pesticides, including those obtained from plant material (Regnault-Roger et al., 2005; Hütter, 2011). However, in spite of this intensive research on botanical pesticides, only two types have been successfully commercialized: those based on azadirachtin (neem seed extract), and essential oils. The latter are the focus of this work. The essential oils are mixtures of volatile chemical compounds, non-fatty and intensely aromatic, biosynthesized by plants for protection against disease and predator insects, or to attract beneficial pollinating insects. The denomination “essential oil” is equally applied to individual compounds, or fractions isolated from mixtures extracted from plant material, and to synthetic molecules of this nature. Technical and commercial opportunities for essential oil-based products in agriculture and other industries are addressed

∗ Tel.: +56 42 220 88 89; fax: +56 42 227 53 03. E-mail address: [email protected] http://dx.doi.org/10.1016/j.indcrop.2014.06.023 0926-6690/© 2014 Elsevier B.V. All rights reserved.

in several recent papers (Isman et al., 2011; Regnault-Roger et al., 2012; Shaaban et al., 2012). The new pesticides of biological origin must be rigorously standardized to a degree that ensures a certain level of effectiveness in the final use, in order to constitute a viable alternative to their synthetic counterparts (Isman, 2006). A well-designed controlled release system may contribute to enhance their performance, optimizing the action of the product and minimizing its residual impact. Controlled release systems based on polymer matrices are widely used because of their low cost and versatility. A broad range of release behaviors can be obtained (Risch and Reineccius, 1995); therefore, the development of controlled release predictive models becomes necessary. If a robust model is available, the behavior of different combinations of active principles and polymeric materials can be simulated at reduced cost, in order to achieve the desired performance. Modeling and simulation of the controlled release processes reduces the time and resources needed for experimental work in product and process development. Depending on the mass transfer mechanisms involved, release systems can be classified as: diffusion-controlled, erosion-controlled, swelling-controlled or combinations of these (Pothakamuri and Barbosa-Cánovas, 1995; Arifin et al., 2006), among others less frequent. Numerous models, covering diverse combinations of these mechanisms, have been reported for different applications. In some applications, an empirical or semi-empirical model is adequate; however, the advantages of a theoretical model based on the characteristics of the physical

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C. Tramón / Industrial Crops and Products 61 (2014) 23–30

Nomenclature a1 Ci C D0 ˜ D Di Di∗ Dij E

Fi Ji K1i , K2i Mi Ni N p R T Tgi V˜ 2j V˜ i Vˆ i Vˆ ∗

activity of the active principle molar concentration of component i total molar concentration constant pre-exponential factor interdiffusion coefficient intrinsic diffusion coefficient of component i self-diffusion coefficient of component i Maxwell–Stefan i–j pair diffusivity energy per mole that a molecule needs to overcome the attractive forces which constrains it to surrounding molecules body force acting per mol of component i diffusional flux of component i free-volume parameters for component i molecular weight of the component i total mass flux of component i net mass flux with respect to a stationary point system pressure universal gas constant absolute temperature glass transition temperature of component i molar volume of the polymer’s jumping unit molar volume of component i specific volume of component i specific critical hole free volume of component i

V˜ ci wi xi

molar critical volume of component i weight fraction of component i mole fraction of component i

i

Greek symbols i volume fraction of component i  overlap factor, introduced because the same free volume is available to more than one molecule i activity coefficient of component i  ratio of molar volumes for the active principle and polymer jumping units viscosity of component i i i molar chemical potential of component i Subscripts 1 active principle polymer 2

system are clear. Diffusional mass transport is predominant in many applications in pest and disease control; although, in the open literature, experimental data for physical and transport properties of essential oils are scarce. A polymer-based controlled release device for essential oils is a solid-state, asymmetrical (relatively small molecules of one or more active principles, mixed with large polymer molecules), diffusing system. Solid-state diffusion-controlled release systems, particularly in drug delivery, are often represented by Fick’s model, subject to appropriate boundary conditions. Matrix systems, with an initial solute concentration below its solubility, are often represented as monolithic solutions; see, for example, the review article by Siepmann and Siepmann (2012) on drug release. Given the relative immiscibility of essential oils in many commercial polymers, the equilibrium occurs at low concentrations of the active principle and, in practice, many diffusing systems operate above the solubility limit, corresponding to disperse mass transfer systems. Reservoir devices and dispersed systems are usually divided in

two regions: the “core” (non-diffusive, undissolved solute), and the dissolved region, where diffusion occurs (Arifin et al., 2006). Several models for calculating the solute’s self-diffusion coefficient are available; for instance, those based on the Free-Volume (FV) theory. Fujita (1961) and Vrentas and Duda (1977a,b) are among the best known FV-based estimating methods. However, in any case, this solution/diffusion type of model requires knowledge about the system’s thermodynamics, specially about equilibrium properties and the relation between self- and mutual- (or multicomponent) diffusion coefficients, which are not yet available, accurately measured or easily estimated, for most naturally derived chemicals. Diffusion in solids may occur through different mechanisms. Solute atoms which are considerably smaller than solvent atoms are incorporated to the interstitial sites of the host lattice, diffusing through an interstitial mechanism. Solute atoms similar in size to the host atoms usually form substitutional solutions, in which vacancy, indirect interstitial, and interstitial-substitutional exchange may be the mechanisms involved (Mehrer, 2007). In solid-state diffusion, the phenomenon denominated “Kirkendall effect” (Smigelskas and Kirkendall, 1947) is seen as the most explicit evidence of a vacancy mechanism. Kirkendall’s experiment was designed to demonstrate that diffusion in solids involves atoms jumping into vacant sites. Consider a solid diffusing system consisting in a mixture of species (1) and (2). The diffusion occurs through inert markers placed in a plane in the solid. If the diffusion rates of the species are different, there will exist a net flow of matter through the inert markers, causing the solid to shift bodily with respect to them. The flow of matter is matched by an equal and opposite flow of vacant sites. This can only happen if diffusion occurs through a vacancy mechanism. Mechanisms based on place exchange between atoms would not allow the diffusion rates of the species to be different. The original article described the phenomenon in crystalline metals but, since then, the Kirkendall effect has been widely observed in other solids as well as immiscible fluids. Although the denomination had not been adopted yet, the same effect was observed by Robinson (1946) with inert markers in polymer–solvent systems, where the large polymer molecules diffuse more slowly than the small solvent molecules. Robinson’s work is cited by Balluffi et al. (2005) as an example of the Kirkendall effect, in their derivation of a model for the diffusion of substitutional particles in a chemical concentration gradient, where the difference in the fluxes of two substitutional species requires a net flux of vacancies. Recently, new evidence about vacancy mechanisms in solidstate diffusion in polymeric systems has been published. For instance, Thompson (2010) has found evidence of the Kirkendall effect in solvent-free poly(methylmethacrylate) lithography. Also, Gao et al. (2013) developed a “one-pot” method for the fabrication of poly(styrene-co-acrylic acid)/inorganic hybrid hollow spheres, and proposed a Kirkendall-type mechanism to explain their formation. While the full nature of these phenomena is yet to be elucidated, the evidence of a vacancy diffusion mechanism in polymeric systems is undeniable. Fluid transport through polymers has also been studied in the context of membrane separation operations, in which taking into account the nature of the membrane is crucial to a successful modeling. Membranes can be porous or dense (non-porous); therefore, besides the classic solution–diffusion model, transport mechanisms may also include viscous flow, Knudsen flow or molecular sieving. An interesting approach to pervaporation modeling is given by Schaetzel et al. (2001). In their work, a Dusty Gas (DG) membrane model is considered. The term “Dusty Gas” was coined by Mason et al. (1967) to describe flow and diffusion of gases in porous media. According to this model, the polymer is visualized as a matrix of large molecules held stationary in space,

C. Tramón / Industrial Crops and Products 61 (2014) 23–30

through which the permeate molecules move. Schaetzel et al. (2001) coupled the DG model with the Maxwell–Stefan diffusion equation and either the Flory–Huggins activity model or Freundlich’s empirical equation, to obtain a model which accurately represents the behavior of an ideal diffusion system. In the controlled release of small molecules from a polymer matrix, usually, the polymer is not considered stationary, and therefore, a DG model does not apply. But a Kirkendall-type active principle/polymer system can be seen as a DG system, in which the polymer molecules are held stationary in space. In this work, a DG model is proposed for a particular case of controlled release of essential oils from a polymer matrix, in which the experimental evidence supports the hypothesis of a vacancy mechanism. 2. Theory

The total flux N1 of the active principle with respect to a stationary point in space is: N1 = J1 + x1 N

N = N1 + N2

n+1

(Ci Fi − wi k=1 Ck Fk ) xi ( − wi ) ∇ i − i ∇p + RT CRT CRT n+1  (xj Ni − xi Nj )

=

CDij

j=1

,

i = 1, n + 1

(4)

j= / i where Fi is the body force acting per mol of component i, and Dij is the Maxwell–Stefan i–j pair diffusivity. As previously stated, the DG model is an application of these relations: the polymer is considered a pore wall, consisting of big molecules (“dust” of infinite molar mass) of concentration spatially uniform. The dust is motionless; therefore, Nn+1 = 0, and an external force Fn+1 must be exerted on the dust molecules to prevent them from moving in response to pressure gradients. Under the assumption of no external forces acting on the components 1 to n, for a binary volatile/polymer system, and assuming that the pressure gradients are negligible, the first of Eq. (4) becomes: −

x1 x N ∇ 1 = 2 1 RT CD12

(5)

The activity coefficients may be introduced to express the left member of Eq. (5):



−

1 + x1

∂ ln 1 ∂x1



∇ x1 =

x2 N1 CD12

(6)

Combining Eqs. (2), (3) and (6) gives:



J1 = −

1 + x1

∂ ln 1 ∂x1



CD12 ∇ x1

(7)

Comparison of Eq. (7) with Fick’s law, Eq. (1), yields the following relationship between the Maxwell–Stefan diffusivity and Fick’s intrinsic diffusion coefficient:



1 + x1

∂ ln 1 ∂x1



D12 = D1

(8)

Then, for ideal mixtures: D12 = D1

(9)

and therefore, for an ideal DG system: N1 =

−CD1 ∇ x1 (1 − x1 )

(10)

According to Eq. (10), the interdiffusion can be characterized by a diffusion coefficient defined by:

Fick’s law has traditionally been the basis for modeling ideal diffusional processes. According to Fick’s approach, the diffusional flux J1 of the active principle with respect to a plane within the solid system can be represented in terms of the intrinsic diffusion coefficient, D1 :

˜ = D

J1 = −CD1 ∇ x1

˜ = D1 D

(1)

(3)

where N2 is the total flux of the polymer. The Maxwell–Stefan formulation provides a more general approach for describing mass transport, accounting for thermodynamic non-idealities and external force fields. The Maxwell–Stefan relations for diffusion under the influence of external body forces are (Krishna and Wesselingh, 1997):

2.1. Determination of diffusion coefficients

2.2. Mass transfer model

(2)

In Eq. (2), the net mass flux N with respect to the stationary point is:

−

In spite of the numerous methods related to mass transfer phenomena described in scientific literature, there is no standardized method to determine the diffusivity of volatiles in polymer matrices. Also, the effective or apparent diffusivity, and not the intrinsic diffusion coefficient, is generally measured. The main method that can be used to determine the apparent diffusivity of small molecules through solid or semi-solid polymer matrices is the determination of sorption or desorption kinetics. This gravimetric method is based on the measurement of mass gain, or loss, of volatile compounds from the polymer matrix, until equilibrium, in precisely defined experimental conditions. The effective diffusivity is calculated from the non-steady part of the sorption/desorption curve, assuming a Fick’s model mechanism and appropriate boundary conditions. This method is widely used because of its versatility, even though the usual assumption of constant apparent diffusivity and other simplifications (system geometry and homogeneity, temperature, existence of a boundary layer) may cause the calculated diffusion coefficient varying in several orders of magnitude (Cava et al., 2005), depending on experimental conditions, which are not always included in the computations. Other available method is the determination of concentration profiles, in one-dimensional diffusion, measured from their distance to the interface as a function of time. This procedure is generally destructive, as it requires slicing of the samples. The permeation method is also widely used for solid and dense interfaces, and nuclear magnetic resonance can be used to determine the self-diffusion coefficient by using radioactively labeled molecules (Cayot et al., 2008). Besides gravimetric determination, Fourier transform infrared (FTIR) spectroscopy has been used for the study of sorption/desorption of volatiles from polymers; particularly, attenuated total reflection (ATR) allows the quantification of the effective diffusion coefficient based on the evolution of its intensity ratio with time (Fieldson and Barbari, 1993), incorporating the data to a Fick’s model. Diffusivity values obtain by this method and those obtained by gravimetry are in accordance (Cava et al., 2004).

25

D1 (1 − x1 )

(11)

For low volatile concentrations, Eq. (11) becomes: (12)

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Furthermore, the relation between the intrinsic and the selfdiffusion coefficient D1∗ is: D1 = D1∗

V˜ ∂ ln a1 V˜ 2 ∂ ln x1

Table 1 Low-density polyethylene (LDPE) physical property data.a

(13)

∂C1 ∂ = ∂t ∂z



D1∗

∂C1 ∂z



(15)

Experimental data can be incorporated to this model to obtain the value of the diffusion coefficient. Boundary conditions for Eq. (15) are set according to each experiment. Consider a polymer matrix in the shape of a slab of thickness L, at constant and uniform temperature T, and the controlled release of the volatile from the slab to a fluid in motion, with tangent velocity v above the slab. The concentration of the volatile in the bulk medium is zero. Initially, the model compound is uniformly distributed in the polymer matrix. C1 (z, t) is the time-dependent concentration profile of the volatile compound within the slab. A uniform concentration profile is assumed as the initial condition of the system: C1 (z, t = 0) = C10 (z) = C10

(16)

The first boundary condition is given by zero-flux condition in the bottom of the slab:

 ∂C1  ∂z 

Reference

Glass transition temperature Specific critical hole free volume Williams–Landel–Ferry parameter Williams–Landel–Ferry parameter

Mostafa et al. (2009)

−1

10−3 m3 kg

WLF C12 = 40.7 K WLF C22

(14)

Therefore, using Eqs. (9), (10), (12) and (14), the balance equation for the active principle’s one-dimensional mass transfer through the slab is:

Description

Tg2 = 175 ± 0.5 K Vˆ 2∗ = 1.036 ×

For an ideal mixture and low concentrations of the active principle, and combining with Eq. (9), the equivalence between the Maxwell–Stefan diffusivity and the self-diffusion coefficient of the active principle is demonstrated: D12 = D1∗

Property

a

= 13.1 K

Ritums et al. (2007)

Ritums et al. (2007) Ritums et al. (2007)

Subscript 2: polymer.

Free-Volume theories, originally applied to liquids (Cohen and Turnbull, 1959), state that the volume occupied by a substance consists of two parts: the volume occupied by the molecules themselves and the empty space between the molecules. Diffusion occurs due to local density fluctuations. These fluctuations generate empty spaces, toward which the diffusing molecules move. Therefore, molecular transport is ruled by the probable occurrence of two events: a vacant site of sufficient size appears next to a molecule, and the molecule’s energy is enough to jump into the vacancy. Free-Volume theories are known to represent accurately diffusion data for polymer systems, at different temperatures and over most of the concentration range, although the calculation of some of the polymer’s parameters is usually done through empirical correlations (Zielinski and Duda, 1992) and a fully predictive model remains a challenge. Duda et al. (1982) presented the following expression for the self-diffusion coefficient of small molecules in polymers: D1∗ = D0 exp



−E RT



 exp



−(w1 Vˆ 1∗ + w2  Vˆ 2∗ ) w1 (K11 /)(K21 − Tg1 + T ) + w2 (K12 /)(K22 − Tg2 + T )

(19)

=0

(17)

z=0

If perfect sink conditions are assumed for the release medium, and the convective mass transfer resistance is negligible (Sh → ∞), then the second boundary condition is given by: C1 (z = L, t) = C1L (t) = 0

(18)

This mathematical model describes the controlled release of the volatile from a polymer matrix, given the physical conditions and the ideality assumptions stated above. If a relation D1∗ (C) is available (for instance, from FV theory), Eqs. (15)–(18) can be solved, for example, via numerical simulation, using orthogonal collocation in z-domain, and a Runge–Kutta method in t-domain (Villadsen and Michelsen, 1978). The calculation of diffusion coefficients can also be carried out, via non-linear regression, from Eqs. (15)–(18) and experimental data generated according to any gravimetric or spectroscopic methodology, under the assumption of constant interdiffusion coefficient. 2.3. Prediction of the diffusion coefficient using Free-Volume theory There are several reviews of diffusion coefficient prediction models in open literature (Crank, 1968). Selecting the most appropriate model for this study depends on several factors: compatibility with the mass transfer model (in this case, a DG model), performance quality in the low-concentration range for the volatile compound (corresponding to most of the applications in pest and disease control), and predictive capabilities (model parameters can be obtained without using diffusion data).

Eq. (19) has been used in this work for the prediction of the selfdiffusion coefficient in thymol/LDPE and menthol/LDPE systems. 3. Experimental 3.1. Chemicals Low-density polyethylene (LDPE) was selected as a model polymer matrix, due to its extensive use as packaging material and to the wide availability of physical property data. Gravimetrical data has been obtained in this study for the compounds thymol (C10 H14 O, 2-isopropyl-5-methylphenol) and menthol (C10 H20 O, [1R,2S,5R]-2-isopropyl-5methylcyclohexanol). These compounds have been selected due to their high biological activity: as a fungicide the former (Shaaban et al., 2012), and as an insecticide the latter (Regnault-Roger et al., 2012). The LDPE, type DOW 641S in pellets, was obtained from Dow Petrochemicals. Physical properties used in calculations are presented in Table 1. Thymol (melting point 49–51 ◦ C) and menthol (melting point 36–38 ◦ C), both U.S.P. grade (≥99% purity), were obtained from Merck Chemicals. 3.2. Sorption/desorption experiments The LDPE samples (15 cm2 of surface area and 20 ␮m thickness) were equilibrated (constant weight during successive measurements) inside the desired volatile compound at 55 ◦ C (above the melting points of the essential oils). Then, samples were wiped off to remove excess volatile, and placed into a measuring chamber

C. Tramón / Industrial Crops and Products 61 (2014) 23–30

Tangential inert flow

According to Zielinski and Duda (1992), K12 / and K22 were obtained from the WLF (Williams–Landel–Ferry) parameters for LDPE:

One-dimensional mass transfer C1 (z ,t )

27

L

z

casting mold

 Vˆ 2∗ K12

= 2.303C12 C22

(20)

K22 = C22

(21)

Fig. 1. Scheme of the diffusion system.

under tangential inert flow, as shown in Fig. 1, also at 55 ◦ C (time zero). The weight of each sample was measured during desorption in an analytical balance. All experiments were carried out in triplicate.

The specific critical hole free volume of the active principles (Vˆ 1∗ ) was estimated using Biltz’s group contribution method (Biltz, 1934), as suggested by most of the consulted literature on the Vrentas–Duda FV model. The ratio of the molar volume of the active principle jumping unit to the molar volume of a polymer jumping unit is represented by the parameter  (polymer branches are assumed to have a certain flexibility, which allows them to be reoriented or “jump”).

3.3. Determination of concentration profiles Samples of 0.012 m thickness, with a nominal initial essential oil concentration of 10% (w/w), were obtained by melting of the essential oil/polymer mix, followed by casting in metallic cylindrical molds. A fraction of the volatiles was lost during the melting/casting process; therefore, in practice, the volatile concentration of the samples never exceeded 9% (w/w). The samples were kept in the metallic molds to ensure one-dimensional diffusion, and placed into the measuring chamber under tangential inert flow at 55 ◦ C. The samples were removed from the chamber at 30 days, cooled at room temperature, and then sliced with the maximum spatial resolution (3 mm thick slices). Each slice was weighed, fit into a cylindrical metallic mold and placed back into the measuring chamber at 55 ◦ C, until constant weight, which was considered an indication of the complete release of volatiles. Each experiment was carried out in triplicate.

=

M1 Vˆ 1∗ V˜ 2j

(22)

The average size of a polymer jumping can be estimated from empirical correlations, as a function of the polymer’s glass transition temperature, Tg2 (Hong, 1995): V˜ 2j = 0.0925Tg2 + 69.47 if Tg2 < 295 K

This remains one of the limitations of the predictive capabilities of FV theory. The Dullien equation (Dullien, 1972) is used to predict selfdiffusion coefficients for liquids. In their work, Zielinski and Duda equated the Vrentas–Duda expression for the self-diffusion coefficient (Vrentas and Duda, 1977a) and the Dullien equation, in the pure active principle limit (w1 → 1). This combination gives a relation between viscosity and FV parameters for the active principle:

 3.4. Controlled release experiments Samples with nominal initial active principle concentrations of 1, 5 and 10% (w/w) were prepared by the melting–casting method. Again, a fraction of the volatiles was lost during the process. Cylinders of 0.006 m and 0.012 m thickness were manufactured into metallic molds, and placed into the measuring chamber at 55 ◦ C under tangential inert flow. The controlled release kinetics were determined by comparative weight loss measurements of the samples, until constant weight, in triplicate.

3.5. Statistical analysis In order to determine differences among diffusivities calculated from every dataset, results of each experiment were analyzed by analysis of variance (ANOVA) with the SAS (statistical analysis system) program and the LSD (least significant difference) test (P < 0.5).

4. Prediction of the interdiffusion coefficient In the proposed mass transfer model, the self-diffusion coefficient, D1∗ , also corresponds to the interdiffusion coefficient that describes the mass transfer rate of the volatile through the polymer, as shown in Eqs. (9), (12) and (14). To calculate D1∗ from Eq. (19), twelve independent parameters need to be obtained. Some of them can be grouped, and therefore only ten parameters need to be determined: D0 , E, Vˆ 1∗ , Vˆ 2∗ , , K11 /, K21 − Tg1 , K12 /, K22 , and Tg2 . All of these parameters have physical significance and are not simply adjustable parameters.

(23)

ln 1 = ln

+

0.124 × 10−7 V˜ c1 RT 2/3

V˜ 1

− ln D0

Vˆ 1∗ E + RT (K11 /)(K21 − Tg1 + T )

(24)

Considering constant temperature throughout the release experiments, E was set to zero and energy effects were included in the pre-exponential parameter D0 . Free-volume parameters for the active principle, K11 / and (K21 − Tg1 ), were obtained from Eq. (24), through linear regression from viscosity data. Pre-exponential parameter D0 was calculated from Eq. (24) at the experimental temperature. Eq. (24) requires the input of the following data for the active principle: viscosity, 1 (T); molar critical volume, V˜ c1 , and molar volume, V˜ 1 at the experimental temperature. These properties are not available in open literature for neither of the studied compounds, and therefore they were estimated from auxiliary methods. To minimize the need for other physical property data, group contribution methods were preferred. The method presented by Nannoolal (2006) was selected for the estimation of V˜ c1 ; the revised GCVOL method (Ihmels and Gmehling, 2003) was used to calculate V˜ 1 , and the Sastri–Rao method (Sastri and Rao, 1992) was used for the estimation of 1 . Besides the molecular structure, these auxiliary methods require only the normal boiling point of the volatile to be known. This data is available from the manufacturer or from literature for each selected compound. Table 2 shows the calculated FV parameters for the selected essential oil/LDPE systems.

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C. Tramón / Industrial Crops and Products 61 (2014) 23–30

Table 2 Free-volume parameters for the Vrentas–Duda model.a Essential oil

Vˆ 1∗ × 103 (m3 kg−1 )



K11 / × 107 (m3 kg−1 K−1 )

K21 − Tg1 (K)

D0 (m2 s−1 )

Menthol Thymol

0.852 0.901

1.495 1.643

3.27 2.96

−40.6 −40.6

1.40 × 10−6 3.50 × 10−6

K12 /: polymer first free-volume parameter to overlap factor ratio = 0.844 × 10−6 m3 kg−1 K−1 . K22 : polymer second free-volume parameter = 13.1 K. Vˆ 1∗ : essential oil specific critical hole free volume. : ratio of molar volumes for the active principle and polymer jumping units. K11 /: essential oil first free-volume parameter to overlap factor ratio. K21 − Tg1 : essential oil second free-volume parameter minus the essential oil glass transition temperature. D0 : constant pre-exponential factor. a Eq. (19).

5. Results and discussion 5.1. Mass uptake The mass of volatile compounds sorbed into the LDPE films at equilibrium condition is shown in Table 3, as well as the equilibrium concentration of each volatile. All desorption and sustained release experiments in this work were carried out at concentrations below these solubility limits, even considering the standard deviation of the data. Therefore, at the desired temperature, the studied essential oil/polymer pairings are not dispersed systems, and could, in principle, be represented by monolithic models. 5.2. Interdiffusion coefficients Non-linear regression was used to obtain the interdiffusion coefficient for both menthol/LDPE and thymol/LDPE systems, from Eqs. (15)–(18), under the assumption of constant diffusion coefficient over the concentration interval of each experiment. Different sets of gravimetric data were considered separately: the concentration profile, determined from the samples at 30 days of volatile release; the controlled release kinetics of samples of different size and initial concentration, and the desorption experiments. The results are shown in Table 4. Experimental conditions are specified in each case. There was a statistically significant difference (P < 0.05) between most datasets, which was expected, due to the assumption of constant diffusivity. The values obtained from the concentration profile method were statistically equal to those obtained from the controlled release kinetic data, starting from the same volatile concentration (10%). However, there appears to be no clear tendency of the results as a function of initial concentration or sample size. An imperfect mixing of the molten samples before casting could favor the accumulation of the active principle in specific zones of the cylinder. The determination of the initial concentration profiles would clarify this aspect. In spite of this, for each volatile, the results obtained from different datasets were of the same order of magnitude, with the evident exception of the results obtained from sorption/desorption experiments. In those cases, the diffusion coefficients were two orders of magnitude smaller than the results obtained from the melting/casting samples. As expected, the polymers molecular dynamics were affected by processing. The magnitude of the impact shows the utmost relevance of selecting an appropriate elaboration process to produce

the desired release behavior. Also, it opens possibilities regarding new products design, but raises several challenges in modeling mass transfer in polymeric systems. During the determination of concentration profiles and controlled release kinetics, no measurable macroscopic volume change in the samples was observed, although some degree of shrinkage was expected. This led to the hypothesis of a vacancy diffusion mechanism. It is not a commonly reported phenomenon in polymers, although there are precedents (as described in Section 1 of this work). Before moving forward with the prediction of the diffusion coefficients, further evidence was collected. Fig. 2 presents an SEM photography of the cross section of an LDPE sample, after the complete release of volatiles (in this case, menthol). The SEM photographs of Kirkendall voids are available in the open literature (to the best of this author’s knowledge) only for metallic compounds. The void pattern in Fig. 2 resembles that of metallic systems with moderate to low void formation. The presence of voids in the polymer matrices, after the release of volatiles, supports the hypothesis of a vacancy mechanism. Furthermore, Balluffi et al. (2005) demonstrated that, when there is no change in the total specimen volume, the overall diffusion in a Kirkendall-type system can be described using only the interdiffusivity, measured in a single reference frame. This is consistent with Eqs. (14) and (15) of the DG-type model derived for these volatile/polymer systems. Eqs. (19)–(24) were used for the prediction of the self-diffusion coefficient, which, according to the DG model, also corresponds to the interdiffusion coefficient. Results are shown in Table 4. Note that Eq. (19) is a function of concentration. Predicted results are given for the lower (infinite dilution) and the upper limit of this concentration range (10%), in each case. The predicted values, for both menthol and thymol, are in good agreement with the coefficients obtained from melting/casting data. Besides the sui generis nature of the proposed model (constitutive equations and assumptions), there are several other potential error sources in the prediction by Eq. (19), introduced by the estimation of physical property data for the volatile: specific critical hole free volume (Vˆ 1∗ ), viscosity, 1 (T); molar critical volume, V˜ c1 ,

Table 3 Mass uptake of menthol and thymol in LPDE.a

a

Essential oil

% mass uptake (g/100 g polymer)

Equilibrium concentration (% (w/w))

Menthol Thymol

32.3 ± 8.1 12.0 ± 2.1

24.3 ± 4.6 10.7 ± 1.7

Sorption experiments. Temperature: 55 ◦ C, film thickness: 20 ␮m.

Fig. 2. SEM photograph of Kirkendall-type voids in LDPE samples after release of menthol (10% initial concentration).

C. Tramón / Industrial Crops and Products 61 (2014) 23–30

29

Table 4 Experimental and predicted diffusion coefficients for binary systems essential oil/LDPE. Essential oil

Experimental conditions

Nominal initial essential oil concentration (%)

Menthol

Melting/casting method Concentration profile experiments Melting/casting method Controlled release experiments

10

0.012

10

0.012 0.006 0.012 0.006 0.012 0.006 20 × 10−6

5 1

Thymol

Sorption/desorption method

Equilibrium

Melting/casting method Concentration profile experiments Melting/casting method Controlled release experiments

10

0.012 0.006 0.012 0.006 0.012 0.006 20 × 10−6

5 1 Equilibrium

D (gravimetry) (m2 s−1 )

D (predicted value) (m2 s−1 )

1.67 ± 0.08 × 10−11 c

1.89–2.07 × 10−11

Average value: 1.63 ± 0.96 × 10−11 1.94 ± 0.16 × 10−11 1.17 ± 0.12 × 10−11 2.33 ± 0.18 × 10−11 0.67 ± 0.49 × 10−11 3.05 ± 0.57 × 10−11 0.60 ± 0.22 × 10−11 1.32 ± 0.23 × 10−13

bc d b de a e f

2.10 ± 0.23 × 10−11 c

0.012

10

Sorption/desorption method

Sample thickness (m)

2.24–2.58 × 10−11

Average value: 2.26 ± 0.91 × 10−11 2.24 ± 0.12 ·10−11 c 1.53 ± 0.21 × 10−11 e 2.77 ± 0.07 × 10−11 b 1.85 ± 0.08 × 10−11 d 3.09 ± 0.12 × 10−11 a 1.29 ± 0.12 × 10−11 f 1.20 ± 0.31 × 10−13 g

Means followed by different letters in the same column, for the same volatile, are statistically different (LSD test, P < 0.05).

Menthol concentration, [w/w%]

and molar volume, V˜ 1 . Measurement of the essential oil’s physical properties, particularly the viscosity, is relevant for successful modeling and simulation. Also, Dullien’s equation, which describes the relation between the viscosity of the active principle and the self-diffusion coefficient, and is the basis for Eq. (24), needs to be validated for complex molecules and interactions.

10 9 8 7 6 5 4 3 2 1 0

The calculated FV parameters (Table 2) were incorporated to the simulation of the controlled release of each active principle from the polymer matrix. To solve the controlled release model, Eqs. (15)–(18), eight total collocation points were used. The resulting ordinary differential equations system was solved by a fourth order Runge–Kutta method, obtaining the concentration profile of the active principle in the slab, C1 (z, t), as a discrete function. Average active principle concentration for each discrete sampling time, C¯ 1i , was obtained via numerical integration of these results. Average absolute deviation (AAD) of the simulation from experimental data was calculated for each release curve according to the following equation:

 1   ¯ C1isim − C¯ 1iexp  n n

AAD =

(25)

i=1

0

10

20

30

40

50

60

70

80

t [days]

Thymol concentration, [w/w%]

5.3. Controlled release kinetics

10 9 8 7 6 5 4 3 2 1 0

where n, number of measurements; sim, simulated value; and exp, average value of the experimental triplicates. The AAD for both menthol/LDPE and thymol/LDPE systems, considering in each case all datasets, is 0.23% (w/w). The predicted controlled release kinetic curves are presented in Fig. 3, against the experimental data. The macroscopic accuracy achieved by the simulation can be considered satisfactory for many applications in pest and disease management. 6. Conclusion

0

10

20

30

40

50

60

70

80

t [days] Fig. 3. Controlled release kinetic curves:  experimental, — simulated (DG-type model, FV self-diffusion coefficient, Biltz/Sastri–Rao auxiliary methods).

The experimental evidence gathered from macroscopic observations and SEM photography supports the hypothesis of a vacancy diffusion mechanism in the studied systems, when samples are prepared by the melting/casting method. The void patterns found in the cross-section of the polymer samples after volatile release are similar to those found in metallic compounds and alloys where the Kirkendall effect is present. Although the formation of voids need further documentation, and the full diffusion mechanism is yet to be elucidated, the assumption of a Kirkendall-type system constitutes a first approach to a more complex modeling.

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C. Tramón / Industrial Crops and Products 61 (2014) 23–30

The proposed DG model is consistent with Kirkendall-type systems when there is no total volume change. The interdiffusion coefficient predicted via the DG model is in good agreement with the gravimetric data, with the exception of the datasets obtained by the sorption/desorption method. The molecular dynamics of these systems are affected by processing, and further evidence is needed to achieve a full understanding of the measured differences. Also, some effort is yet needed in order to improve the predictive capabilities of the FV theory, but other methods for predicting the self-diffusion coefficient of essential oils in polymers may be coupled with the DG model, when they become available. The prediction of the controlled release kinetics is qualitatively good. No mass transfer data has been used in the simulations. This shows an advantage of the DG model, when applicable, over the traditional solution/diffusion model: fewer physical properties of the volatile compounds, and no information about mutual or multicomponent diffusion coefficients, are needed. Furthermore, even though the DG model cannot predict phase separation or particle ripening, it does produce a good macroscopic prediction, since the diffusion still occurs in the direction of the concentration gradient. Therefore, although the gravimetric data in this study was obtained for volatile concentrations below saturation, a DG model could also be applied to disperse systems to obtain a qualitative prediction of their behavior. Acknowledgments The author acknowledges the valuable collaboration of Dr. Cecilia Fuentealba at the Technology Development Unit of the University of Concepción, in polymer processing. References Arifin, D.Y., Lee, L.Y., Wang, C.H., 2006. Mathematical modelling and simulation of drug release from microspheres: implications to drug delivery systems. Adv. Drug Deliv. Rev. 58, 1274–1325. Balluffi, R.W., Allen, S.M., Carter, W.C., 2005. Kinetics of Materials. John Wiley & Sons, New Jersey. Biltz, W., 1934. Rauchemie der festenstoffe. Voss, Leipzig. Cava, D., Lagarón, J.M., López-Rubio, A., Catalá, R., Gavara, R., 2004. On the applicability of FT-IR spectroscopy to test aroma transport properties in polymer films. Polym. Test. 23, 551–557. Cava, D., Catalá, R., Gavara, R., Lagarón, J.M., 2005. Testing limonene diffusion through food contact polyethylene by FT-IR spectroscopy: film thickness, permeant concentration and outer medium effects. Polym. Test. 24, 483–489. Cayot, N., Dury-Brun, C., Karbowiak, T., Savary, G., Voilley, A., 2008. Measurement of transport phenomena of volatile compounds: a review. Food Res. Int. 41, 349–362. Cohen, M.H., Turnbull, D., 1959. Molecular transport in liquids and glasses. J. Chem. Phys. 31 (5), 1164–1169. Crank, J., 1968. Diffusion in Polymers. London Academic Press, London. Duda, J.L., Vrentas, J.S., Ju, S.T., Liu, H.T., 1982. Prediction of diffusion coefficients for polymer–solvent systems. AIChE J. 28 (2), 279–285. Dullien, F.A.L., 1972. Predictive equations for self-diffusion in liquids: a different approach. AIChE J. 18, 62–70. Fieldson, G.T., Barbari, T.A., 1993. The use of FTIR-ATR spectroscopy to characterize penetrant diffusion in polymers. Polymer 34, 1146–1153. Fujita, H., 1961. Diffusion in polymer–diluent systems. Fortschr. Hochpolym. Forsch. 3, 1–47.

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