A general approach to describe the antimicrobial agent release from highly swellable films intended for food packaging applications

Journal of Controlled Release 90 (2003) 97–107 www.elsevier.com / locate / jconrel

A general approach to describe the antimicrobial agent release from highly swellable films intended for food packaging applications G.G. Buonocore b , M.A. Del Nobile a , *, A. Panizza b , M.R. Corbo a , L. Nicolais b a

b

Istituto di Produzioni e Preparazioni Alimentari, University of Foggia, Via Napoli, 25 -71100 Foggia, Italy Department of Materials and Production Engineering, University of Naples ‘ Federico II’, P. le Tecchio, 80 -0125 Naples, Italy Received 17 December 2002; accepted 11 March 2003

Abstract A mathematical model able to describe the release kinetics of antimicrobial agents from crosslinked polyvinylalcohol (PVOH) into water is presented. The model was developed by taking into account the diffusion of water molecules into the polymeric film, the counter-diffusion of the incorporated antimicrobial agent from the film into water, and the polymeric matrix swelling kinetic. To validate the model the water sorption kinetics as well as the release kinetics of three antimicrobial agents (i.e., lysozyme, nisin and sodium benzoate, all approved to be used in contact with food) were determined at ambient temperature (25 8C). The three investigated active agents were entrapped in four films of PVOH with a different degree of crosslink. The model was successfully used to fit all the above sets of data, corroborating the validity of the hypothesis made to derive it.  2003 Elsevier Science B.V. All rights reserved. Keywords: Modeling; Swelling; Controlled release of antimicrobial agent; Moving boundary; Packaging

1. Introduction In the last years antimicrobic release systems have been used mainly in pharmaceutical applications [1– 4], while their use in food packaging is still restricted [5], although it is expected to grow due to the increase of the concept of ‘active packaging’. The aim of controlled release systems intended for food packaging applications is to transfer the antimicro*Corresponding author. Tel.: 139-881-589-233; fax: 139-881740-211. E-mail addresses: [email protected] (M.A. Del Nobile), [email protected] (M.A. Del Nobile).

bial agent from the polymeric carrier to the liquidlike food in order to maintain a predetermined concentration of the active compound in the packed food for a determined period of time. Drug release has been often studied in the past because of its importance in the pharmaceutical field. In fact, several studies are reported in the literature focused on the studying and modelling of drug delivery devices for pharmaceutical applications [6– 12]. As reported by Siepman and Peppas [13], the quantitative description of the phenomena controlling the drug release kinetics is highly desirable to properly design a new controlled drug delivery device. The practical benefit of a mathematical

0168-3659 / 03 / $ – see front matter  2003 Elsevier Science B.V. All rights reserved. doi:10.1016 / S0168-3659(03)00154-8

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model consists in the possibility to simulate the effects of the design parameters on the performance of the release system. In the review proposed by Siepmann and Peppas [13], different mathematical models are presented to describe the drug release kinetics. The simplest are the empirical and semiempirical models. Among them, the most used are that proposed by Higuchi [14], and the ‘power law’ equation. The simplicity of the above two models is counterbalanced by the fact that their applicability is restricted to very few cases, as also pointed out by Peppas and co-worker [15,16]. Besides the above approach, mechanistics theories were proposed, with the attempt to give a real insight into the underlying mechanisms controlling the drug release. Most of the mechanistics models take into account the diffusion of the penetrant into the slab, the swelling and dissolution of the polymer and the release of the drug into the external liquid solution. In the specific case of antimicrobial films intended for food packaging applications several studies can be found in the literature dealing with this subject [17–20]. However, the results obtained are quite limited because the study has been restricted to the case of release of the antimicrobial agents from hydrophobic or moderately hydrophilic polymeric matrices. Therefore, the developed models cannot be applied to the case of highly swellable polymeric matrices. The aim of this work is to develop a mathematical model able to describe the release kinetics of active compounds from a highly swellable films intended for food packaging applications. The proposed model consists of two parts: the first describing the water uptake kinetic, the second describing the release of the antimicrobial agent from the polymeric matrix. The mass balance for the penetrant and the active substance were numerically solved by means of the finite volume method.

of a 4% water solution at 20 8C: 11–14 cps). The crosslinking agent was glyoxal (40% aqueous solution, Sigma–Aldrich) and the catalyser was HCl (37%, Sigma–Aldrich). The active compounds used were: lysozyme from chicken egg white (purchased from Sigma–Aldrich), nisin (donated by Danisco Innovation, 15 NorthStreet, Beaminster, Dorset DT8 3DZ, UK) and sodium benzoate (purchased from Sigma–Aldrich). Lysozyme consists of 129 amino acids and has a molecular weight of 14 000 Da. Nisin consists of 34 amino acids and has a molecular weight of 3400 Da. Sodium benzoate has a molecular weight of 121 g / mol.

2.2. Methods

2.1. Materials

2.2.1. Film preparation PVOH (6.5 g) was dissolved into 50 ml of distilled water by keeping the solution 30 min into an autoclave at 120 8C. The obtained solution was slowly cooled and crosslinked by adding first a known amount of glyoxal and, immediately after, 0.2 ml of a 37% aqueous solution of HCl. The obtained solution was cast onto a plate and dried at ambient conditions for 48 h. The obtained films had an average thickness of 120 mm. The thickness of the obtained films was measured by means of a Digimatic Micrometer (Mitutoyo, accuracy equal to 0.5 mm). The value of the film thickness was obtained by averaging 100 measurements. The active films were obtained by using the following procedure: PVOH (3.25 g) was dissolved in 25 ml of distilled water by keeping the solution for 30 min in an autoclave at 120 8C. The obtained solution was slowly cooled at room temperature. Later, 100 mg of active agent were added, the obtained solution was stirred at ambient temperature until the preservative was completely dissolved. The obtained solution was crosslinked adding first a known amount of glyoxal and, immediately after, 0.2 ml of a 37% aqueous solution of HCl. For the sake of simplicity the investigated films will be referred to as:

The films studied in this work were obtained crosslinking PVOH (Sigma catalog code P1763. MW570 000–100 000. Hot water soluble. Viscosity

2.2.1.1. Film without active compound. Film A: pure glyoxal: 7.7% (w / w) of PVOH; Film B: pure glyoxal: 2.0% (w / w) of PVOH; Film C: pure

2. Materials and methods

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glyoxal: 0.77% (w / w) of PVOH; Film D: pure glyoxal: 0.077% (w / w) of PVOH.

2.2.1.2. Film with the active compounds (100 mg). Film A L (lysozyme), Film A N (nisin), and A SB (sodium benzoate): pure glyoxal: 7.7% (w / w) of PVOH. Film B L (lysozyme), B N (nisin), and B SB (sodium benzoate): pure glyoxal: 2.0% (w / w) of PVOH. Film C L (lysozyme), C N (nisin), and C SB (sodium benzoate): pure glyoxal: 0.77% (w / w) of PVOH. Film D L (lysozyme), D N (nisin), and D SB (sodium benzoate): pure glyoxal: 0.077% (w / w) of PVOH. 2.2.2. Water sorption kinetics The produced films were cut in small pieces (131 cm) and desiccated under vacuum at least 48 h. They were weighed and left at ambient temperature in 30 ml of distilled water. The swelling kinetics were evaluated by periodically measuring the weight of the films by means of a micro-balance (Scaltec model SPB32, with an accuracy of 0.0001 g), after gently blotting the surface with a tissue, until the equilibrium condition was reached. The swelling ratio was evaluated as: g Film 2 g Dry Film S.r. 5 ]]]]]]. g Dry Film All data shown are the average value of three replicates.

2.2.2. Active compounds release kinetics The prepared active films (0.012 cm3200 cm 2 ) were put into a container and brought in contact with 250 ml of distilled water at ambient temperature, under moderate stirring. The active compounds release kinetics were evaluated by monitoring, by means of an HPLC, the antimicrobial concentration in the surrounding solution until an equilibrium value was reached. All data shown are the average value of three replicates.

99

Liao et al. [21]. In particular, lysozyme was determined by means of an HPLC (Agilent Mod. 1100). A C18 Reverse phase column was used (25034 mm, 5 mm) and a gradient elution with water–acetonitrile gradients (1 ml / min) containing 0.1% TFA was used. Typical gradient was 0–60% acetonitrile over 20 min, with lysozyme eluting at 10 min. The detection was made at a wavelength of 254 and 225 nm. The calibration curve was constructed for peak area against lysozyme concentration of standard solutions from 6 to 300 ppm, with five replicate samples for each lysozyme concentration.

2.2.3.2. Nisin determination. Quantitative determination of nisin into the water solution was made by slightly modifying the method proposed by Liu and Hansen [22]. In particular, nisin was determined by means of an HPLC (Agilent Mod. 1100). A C18 reversed-phase column was used (25034 mm, 5 mm) and a gradient elution with water–acetonitrile gradients (1 ml / min) containing 0,1% TFA was used. Typical gradient was 20–60% acetonitrile over 20 min, with nisin eluting at 10 min. The detection was made at a wavelength of 254 and 225 nm. The calibration curve was constructed for peak area against nisin concentration of standard solutions from 2.5 to 100 ppm, with five replicate samples for each nisin concentration. 2.2.3.3. Sodium benzoate determination. Quantitative determination of sodium benzoate into the water solution was made following the method proposed by Pylypiw and Grether [23]. In particular, sodium benzoate was determined by means of an HPLC (Agilent Mod. 1100) using a C18 Reverse phase column (25034 mm, 5 mm). Isocratic conditions of mobile phase with a flow of 0.8 ml / min were used, with sodium benzoate eluting at 15 min. The detection was made at a wavelength of 225 nm. The calibration curve was constructed for peak area against sodium benzoate concentration of standard solutions from 2.5 to 100 ppm, with five replicate samples for each sodium benzoate concentration.

2.2.3. HPLC active compounds assay 2.2.3.1. Lysozyme determination. Quantitative determination of lysozyme into the water solution was made by slightly modifying the method proposed by

2.2.3.4. Modeling. In the following a model to describe the release kinetics of an active compound from a swelling polymeric network is presented. The antimicrobial agent is initially entrapped into a

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highly hydrophilic dry polymer (PVOH). The release kinetics of an active compound from a highly hydrophilic polymer placed in contact with water depends on the following phenomena: (1) water diffusion; (2) macromolecular matrix relaxation kinetic; (3) diffusion of the active compound through the swollen polymeric network. For the sake of simplicity the description of the above phenomena will be presented separately in the following.

2.2.3.5. Water diffusion and macromolecular matrix relaxation. The sorption kinetics of low molecular weight compounds in polymers are generally governed by two simultaneously occurring phenomena. A substantially stochastic phenomenon (related to Brownian motion), where the penetrant flows exclusively driven by a concentration gradient and a relaxation phenomenon driven by the distance of the local system from the equilibrium [24]. When the mass transfer takes place in a substantially unperturbed polymeric matrix, as in the case of gas diffusion in rubbery polymers (or wherever the solvent-induced polymer swelling is negligible), the diffusion process is substantially controlled by a stochastic phenomenon. The other limiting behaviour is encountered when a very thin slab of polymer is put in contact with a swelling compound. In this case the characteristic diffusion time is much lower than the polymer relaxation time; hence, polymer relaxation becomes the limiting phenomenon controlling the solvent uptake kinetic [24]. In the case of water diffusion in hydrophilic polymers, such as in the case under investigation, the experimental observations range between these two limiting phenomena. To better illustrate these two distinct aspects of the water sorption kinetics, stochastic diffusion and polymer matrix relaxation will be presented separately in the following. Water diffusion related to Brownian motions is generally described by means of the Fick’s First Law. In the specific case of diffusion through a plane sheet, the Fick’s model reduces to the following expression: →

W ≠C W → J 5 2 D F ? ]] ?x ≠x

(1)

Several approaches are reported in the literature to describe solvent-induced polymer relaxation [24].

Among them, one of the simplest is that proposed by Long and Richman [25]. They assumed that when the external water activity is suddenly increased (i.e., the film is placed in contact with water), the solvent concentration at the polymer surface does not reach instantaneously its equilibrium value (as it would be if polymer matrix relaxation was negligible). On the contrary, it first suddenly increases to a value lower than the equilibrium one. At a later stage, it gradually increases until the system reaches the equilibrium. The rate at which the water concentration at the boundaries gradually increases is directly related to the relaxation of the macromolecular matrix. In this investigation the following empirical expression [26] is proposed to describe the boundary condition relaxation rate: dastd ]] 5 ha1 ?Œ] astd j ? h1 2 exp f 2s1 2 astdd g j dt

(2)

a (t) is the normalised water volume fraction at the boundaries of the film at time t. It spans from zero to one, and represents the driving force of the macromolecular matrix relaxation phenomenon. As reported in Eq. (2), dastd / dt is expressed as the product of two terms. The first one, which prevails at the early stage of the hydration process, is the kinetic constant of the polymer relaxation phenomenon; therefore, it is an increasing function of polymer macromolecular mobility. The second term, which prevails at the later stage of the hydration process, accounts for the fact that dastd / dt has to decrease as the concentration at the boundary of the film approaches its equilibrium value (i.e., as a (t) levels off to one); therefore it is a decreasing function of a (t). 2.2.3.6. Determination of the water sorption kinetics. The evolution of the water concentration profile inside the polymeric matrix was calculated by solving a set of differential equations obtained by writing the integral water mass balance equation for each of the ‘n’ elements in which the film was divided. For each of the ‘n’ elements it is possible to write the following equation: d ] dt

E srC d dV 5 E FD

Vstd

W

W F

≠ →→ ]s r CWd ?x n ≠x

G dS

(3)

Sstd

It is worth noting that by considering elements with a

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volume that changes during hydration (the volume of each element changes in such a way that it contains always the same amount of the polymeric matrix, Lagrangian approach) instead of considering elements with a fixed volume (Eulerian approach), it is possible to neglect the convective mass flux. The set of ‘n’ equations, obtained by writing Eq. (3) for the ‘n’ elements, were solved numerically using the following initial and boundary conditions: CW 5 0 ⇒ t 5 0; 0 , x , L

5

B CW 5 C W std ⇒ ; t; x 5 L(t) ≠s r CWd ]]] 5 0 ⇒ ; t; x 5 0 ≠x

(4)

2.2.3.7. Determination of the active compound release kinetics. The amount of antimicrobial agent released into water was calculated by solving simultaneously a set of differential equations obtained by writing the integral mass balance equation for both water (Eq. (3)) and the active substance for each of the ‘n’ elements in which the film was divided: d ] dt

E srC

d dV 5

AC

Vstd

E FD

AC F

≠ →→ ] s r CACd x n ≠x

G dS

(5)

Sstd

Eqs. (3) and (5) were solved numerically using Eq. (4) for water and the following initial and boundary conditions for the active compound: In. CAC 5 C AC ⇒ t 5 0; 0 , x , L

5

CAC 5 C AC (t) ⇒ ; t; x 5 L(t) ≠s r CACd ]]] 5 0 ⇒ ; t; x 5 0 ≠x

101

(5) were fitted to the antimicrobial agent release kinetics data to test the entire model and to determine the values of the diffusion coefficient and the partition coefficient of the active agent. The second fitting was conducted by fixing the values of D W F , C BW (0) and a1 to that found in the first fitting. It is worth noting that the approach proposed to validate the model is more severe than the classical approach, which consists in fitting Eqs. (3) and (5) to active compound release kinetics data to determine all the B AC models parameters (i.e., D W and F , C W (0), a1 , D F KAC ). Figs. 1 and 2 show the amount of water sorbed into the polymer plotted as function of hydration time. As shown in the above figures, the amount of water sorbed at equilibrium increases as the degree of crosslink decreases. In the same figures the curves obtained by fitting Eq. (3) to the experimental data are also shown. The values of the model’s parameters obtained are listed in Table 1. As expected, the value of the water diffusion coefficient increases as the degree of crosslink decreases. The accuracy of fit was evaluated by means of the relative percent difference [28]. In the case under investigation the calculated values of E¯ % for the four sets of data are listed in Table 2. From the curves shown in Figs. 1 and 2, and from the values E¯ % reported above, it can be inferred that the first part of the model satisfactorily fits the data, corroborating the hypothesis made to derive it.

B

(6)

The value of C BAC (t) was determined assuming that the outer solution can be treated as a well stirred solution [27]. The expression used was:

r C BAC (t) 5 KAC rW C W AC (t)

(7)

3. Results and discussions The model validation was conducted in two subsequent steps. First Eq. (3) was used to fit to the water sorption kinetics data. In this way the first part of the model was tested, and the values of D W F , C BW (0) and a1 , were determined. Later, Eqs. (3) and

Fig. 1. Swelling ratio plotted as a function of time. (j) Film A, (s) Film B, (———) best fit of the proposed model to the experimental data.

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102

Fig. 2. Swelling ratio plotted as a function of time. (j) Film C, (s) Film D, (———) best fit of the proposed model to the experimental data.

Fig. 3. Amount of lysozyme released plotted as a function of time. (j) Film A, (s) Film B, (———) best fit of the proposed model to the experimental data.

Table 1 B Values of D W F , C W (0) and a1 obtained by fitting Eq. (3) to the water sorption kinetics data Sample

2 DW F (cm / s)

B CW (0) (g H 2 O / g dry polymer)

a1 (1 / s)

Film Film Film Film

5.41310 28 1.124310 27 1.804310 27 3.863310 27

0.336 0.722 1.363 1.473

1.997310 24 1.419310 24 1.637310 24 2.359310 24

A B C D

Figs. 3–8 show the antimicrobial agent release kinetics for the four investigated films. As shown in the above figures, the amount of the active compound released at equilibrium always decreases as the degree of crosslink of the polymeric matrix increases. Considering that the degree of crosslink slightly influence the partition coefficient of the

Fig. 4. Amount of lysozyme released plotted as a function of time. (j) Film C, (s) Film D, (———) best fit of the proposed model to the experimental data.

Table 2 Calculated values of E¯ % for both the water sorption kinetics and the active compound release kinetics Sample

Film Film Film Film

A B C D

E¯ % Water

Lysozyme

Nisin

Sodium benzoate

3.5 5.9 3.3 5.3

12 9.1 8.7 8.9

16 7.8 6.4 9.5

5.6 7.0 6.5 6.6

Fig. 5. Amount of nisin released plotted as a function of time. (j) Film A, (s) Film B, (———) best fit of the proposed model to the experimental data.

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103

Fig. 6. Amount of nisin released plotted as a function of time. (j) Film C, (s) Film D, (———) best fit of the proposed model to the experimental data.

Fig. 8. Amount of sodium benzoate released plotted as a function of time. (j) Film C, (s) Film D, (———) best fit of the proposed model to the experimental data.

active agent, it is reasonable to hypothesize that part of the preservative loaded into the film is chemically bonded to the polymer backbone via the crosslinking reaction. In fact, one of the active site of the crosslinking agent could react with one of the sites on the polymer backbone, attaching the crosslinking agent to the matrix, while the other site could react with the active compound, bonding the preservative to the matrix. Therefore, as the quantity of the crosslinking agent increases, also the number of the above unreacted sites increases, leading to an increase of the active compound molecules bonded to the PVOH backbone. Figs. 3–8 also point out that

the shape of the active compound release kinetics curve strongly depends on the molecular weight of the antimicrobial agent. In fact, as the molecular weight of the active compound decreases and / or the degree of crosslink decreases the release kinetics curve shows an overshoot. As in the case of nisin entrapped in samples C and D, and in the case of sodium benzoate entrapped in film A, B, C and D. In fact, the concurrence of the following can lead to an inversion in the sign of the active compound concentration gradient during the release process: (a) the preservative concentration at the boundary increases as the amount of active compound released increases; (b) there is a superposition between water penetration into the polymeric matrix and diffusion of the active agent in the outer water solution. The curves shown in Figs. 3–8 were obtained by fitting Eqs. (3) and (5) to the experimental data, the values of the diffusion coefficient and partition coefficient of the active compound obtained are listed in Table 3. As one would expect, the value of the active compound diffusion coefficient always increases as the degree of crosslink decreases. Also in this case the accuracy of fit was evaluated by means of E¯ %; the values obtained are listed in Table 2. As it can be inferred from the data shown in Figs. 3–8 and from the calculated values of E¯ %, the proposed model satisfactorily fits the experimental data corroborating the hypothesis used to derive it. It is worth noting that regardless the presence of an

Fig. 7. Amount of sodium benzoate released plotted as a function of time. (j) Film A, (s) Film B, (———) best fit of the proposed model to the experimental data.

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104

Table 3 Values of the diffusion coefficient and partition coefficient of the active agent obtained by fitting Eqs. (5) and (6) to the antimicrobial agent release kinetics data Sample

Lysozyme D LF (cm 2 / s)

Film Film Film Film

211

A B C D

3.83310 2.45310 210 2.10310 29 9.98310 29

Nisin KL 431.37 45.62 15.06 6.33

Sodium benzoate

D FN (cm 2 / s)

KN

210

3.01310 3.16310 29 6.24310 29 8.61310 29

overshoot the proposed model satisfactorily fits the antimicrobial agent release kinetics.

152.80 67.58 34.57 26.73

C iW C iAC

4. Conclusion

C BW (t)

In this work a model able to describe the release kinetics of active compounds from crosslinked PVOH was presented. The model was developed by taking into account the phenomena involved during the release of the active compound from a polymeric swelling matrix; i.e., water diffusion, macromolecular matrix relaxation and antimicrobial agent diffusion through the polymeric matrix. Model validation was conducted by first fitting Eq. (3) to the water sorption kinetic data. Subsequently, Eqs. (3) and (5), developed to describe the antimicrobial agent release kinetics, were fitted to the active compound release data. From the obtained results, it can be inferred that the proposed models satisfactorily fit the experimental data, corroborating the hypothesis made to derive it. The adopted approach has a potential relevance on practical aspects of packaging because the modeling of the antimicrobial release from food packaging material can be considered as a useful tool needed to simulate and therefore to predict the behavior of the release device in the real working condition.

C BAC (t)

5. List of symbols

E¯ %

CW

→

CAC

the local water concentration expressed as g water / g dry polymer the active compound concentration expressed as g active compound / g dry polymer

CW AC (t) C In. AC C BW (0)

DW F

i

DW F

i 11

D AC F DW F D LF D NF SB DF

J KAC KL KN

D SB (cm 2 / s) F 28

1.25310 4.20310 28 2.55310 28 2.54310 28

KSB 55.82 60.01 25.96 21.39

the water concentration at the center of the element ‘i’ the active compound concentration at the center of the element ‘i’ the water concentration at the film boundary at time t the active compound concentration at the film boundary at time t the active compound concentration into the outer water at time t the initial active compound concentration into the film the water concentration at the film boundary at time zero, it represents the instantaneous response of the system to the increase of the external water activity the water diffusion coefficient at the interface between the element ‘i’ and the element ‘i21’ the water diffusion coefficient at the interface between the element ‘i 11’ and the element ‘i’ the active compound diffusion coefficient the water diffusion coefficient the lysozyme diffusion coefficient the nisin diffusion coefficient the sodium benzoate diffusion coefficient the mean relative deviation modulus defined by the following equation: N exp uM i 2 M p u 100 ]] E¯ % 5 ] N exp ? o i 51 Mi the diffusive mass flux the active compound partition coefficient the lysozyme partition coefficient the nisin partition coefficient

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KSB L(t) m ip ML (t)

MN (t)

MSB (t)

N → n S S(t) Vi V V(t)

x→ r a (t)

a1 n BW (t) n eq. W di din dy dz ri

r

rP rW rAC Dt

the sodium benzoate partition coefficient the half-thickness of the film at time t the initial mass of the element ‘i’ amount of lysozyme released into water at time t expressed as g lysozyme / g dry polymer amount of nisin released into water at time t expressed as g nisin / g dry polymer amount of sodium benzoate released into water at time t expressed as g sodium benzoate / g dry polymer the number of time integration steps the versor normal to the surface the surface of the volume V the surface of the volume V(t) the volume of the element ‘i’ an arbitrary volume fixed in space the volume that contains an amount of macromolecular matrix equal to the initial one the axial coordinate the axial versor is defined by the following equation: a (t) 5 n BW (t) /n eq. W constant to be evaluated by fitting the model to the experimental data the water volume fraction at the film boundary at time t the equilibrium water volume fraction sorbed in the film the side length of the element ‘i’ the initial side length of the sample the length of the y-side of the sample the length of the z-side of the sample the average macromolecular matrix density, it was evaluated by means of the following expression: m ip /V i the density of the polymeric matrix defined as the ratio between the weight of the matrix and the volume of the mixture (i.e., water plus matrix) ex3 pressed as g matrix / cm hydrated matrix the initial density of the polymer the water density the active compound density the time step used to integrate Eqs. (3) and (4)

Dx i

105

the difference between the inner and outer axial coordinate of the element ‘i’

Acknowledgements The work was funded under the MURST ‘Piani di Potenziamento della Ricerca Scientifica e Tecnologica’.

Appendix A The polymeric film was discretized as shown in the following figure:

Eq. (3) was approximated by means of the following expression [29–31]: m ip ?sC iWssN 1 1d Dtd 2 C iWsN Dtdd 5 DtSsN Dtd

F

W i 11

DF

sN Dtd

i 11

i 11

i

i

r sN Dtd ? C W sN Dtd 2 r sN Dtd ? C WsN Dtd ? ]]]]]]]]]]]]] 1 0.5sDx i 11sN Dtd 1 Dx isN Dtdd i 2 DW F sN Dtd i

i

i 21

i 21

r sN Dtd ? C WsN Dtd 2 r sN Dtd ? C W sN Dtd ? ]]]]]]]]]]]]] 0.5sDx isN Dtd 1 Dx i 21sN Dtdd

G (A1)

Eq. (5), related to active substance mass balance, was approximated exactly in the same way of Eq. AC (3), substituting D W F with D F and C W with CAC . i To evaluate D , it was assumed that the water concentration changes linearly between two adjacent elements [32]. The following expression was used to evaluate r i ((N 1 1) Dt):

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106

r issN 1 1d Dtd 1 5 ]]]]]]]]]]] i C WssN 1 1d Dtd C iACssN 1 1d Dtd 1 ] 1 ]]]]] 1 ]]]]] rp rW rAC

(A2)

The following expression was used to evaluate V i ((N 1 1) Dt): m ip C iWssN 1 1d Dtd ? m pi V ssN 1 1d Dtd 5 ] 1 ]]]]]] rp rW i

C iACssN 1 1d Dtd ? m pi 1 ]]]]]] rAC

(A3)

The following expression was used to evaluate Dx i 11 ((N 1 1) Dt): i 11

Dx

i 11

V ssN 1 1d Dtd ssN 1 1d Dtd 5 ]]]]] SssN 1 1d Dtd

(A4)

The evolution of the polymer surface was evaluated by considering an isotropic swelling of the wet polymer. The above approximation is reasonable, since experiments show that the elastic modulus of the polymer does not vary significantly after water sorption. Since S((N 1 1) Dt) 5 dy ((N 1 1) Dt) ? dz ((N 1 1) Dt), we need an equation for dy ((N 1 1) Dt) and another for dz ((N 1 1) Dt. The two equations have the same form, so we can write a single equation in terms of the generic symbol d ((N 1 1) Dt). Assuming the additivity upon mixing of the water and macromolecular matrix volumes, we get: ]]]] V issN 1 1d Dtd d issN 1 1d Dtd 5 3 ]]]] V is0d ? d iin.ssN 1 1d Dtd (A5)

œ

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