Calculation of viscoelastic properties of edible films: application of three models

Cálculo de propriedades viscoelásticas de biofilmes: aplicação de três modelos

CALCULATION OF VISCOELASTIC PROPERTIES OF EDIBLE FILMS: APPLICATION OF THREE MODELS¹

Prabir K. CHANDRA², Paulo J. do A. SOBRAL²,^*

SUMMARY

The viscoelastic properties of edible films can provide information at the structural level of the biopolymers used. The objective of this work was to test three simple models of linear viscoelastic theory (Maxwell, Generalized Maxwell with two units in parallel, and Burgers) using the results of stress relaxation tests in edible films of myofibrillar proteins of Nile Tilapia. The films were elaborated according to a casting technique and pre-conditioned at 58% relative humidity and 22ºC for 4 days. The testing sample (15mm x 118mm) was submitted to tests of stress relaxation in an equipment of physical measurements, TA.XT2i. The deformation, imposed to the sample, was 1%, guaranteeing the permanency in the domain of the linear viscoelasticity. The models were fitted to experimental data (stress x time) by nonlinear regression. The Generalized Maxwell model with two units in parallel and the Burgers model represented the relaxation curves of stress satisfactorily. The viscoelastic properties varied in a way that they were less dependent on the thickness of the films.

Keywords: edible films; myofibrillar protein; linear viscoelasticity; mechanical properties.

RESUMO

CÁLCULO DE PROPRIEDADES VISCOELÁSTICAS DE BIOFILMES: APLICAÇÃO DE TRÊS MODELOS. As propriedades viscoelásticas de biofilmes podem fornecer informações à nível de estrutura dos biopolímeros usados. O objetivo deste trabalho foi testar três modelos simples, da teoria da viscoelasticidade linear (Maxwell, Maxwell generalizado com duas unidades em paralelo, e Burgers) em resultados de testes de relaxamento de tensão em biofilmes de proteínas miofibrilares de Tilápia do nilo. Os filmes foram elaborados segundo uma técnica tipo casting e condicionados em 58% de umidade relativa e 22°C por 4 dias, antes dos testes. Os corpos de prova (15mm x 118mm) foram submetidos aos testes de relaxamento em um equipamento de medidas físicas TA.XT2i. A deformação imposta na amostra foi de 1%, garantindo-se a permanência no domínio da viscoelasticidade linear. Os modelos foram ajustados aos dados experimentais (tensão x tempo) por regressão não linear. Os modelos de Maxwell generalizado com duas unidades em paralelo e o de Burgers representaram satisfatoriamente, as curvas de relaxamento de tensão. As propriedades viscoelásticas foram pouco dependentes da espessura dos filmes.

Palavras-chave: biofilmes; proteínas miofibrilares; viscoelasticidade linear; propriedades mecânicas.

1 — INTRODUCTION

The use of edible films for food packaging will depend, besides the questions like cost and availability, on their functional attributes such as mechanical properties, which are defined according to the test utilized in its characterization. The puncture force and deformation in the test of perforation [5, 13, 14, 19, 24]; modulus of elasticity, breaking stress and deformation in the test of traction [3, 7, 10, 17, 21]; and the Young’s modulus in the three-point bending test [1, 2, 22], are some examples encountered in the specialized literature.

These properties depend on the environmental conditions, notedly temperature and relative humidity, and also on the inherent characteristics: biopolymers, formulation, additives and the process of fabrication [11, 15, 27]. Another less studied parameter, which influences the properties of the edible films, is the thickness [16]. According to GENNADIOS, WELLER, TESTIN [12], the control of the film thickness is important for the uniformity of these materials, for the repeatability of the measurements of the properties and validity of the comparisons among the properties of the edible films.

CUQ et al [5] and SOBRAL [24, 25] conducted detailed studies about the influence of the thickness on the functional properties of edible films based on myofibrillar proteins and gelatin. They observed that the puncture force, in the perforation test, increased linearly with the thickness, while the deformation on the rupture remained practically constant, however, with elevated dispersion of data.

The viscoelastic properties are also important, because they can provide information directly related to the conformation of the macromolecules and to the phenomenon of molecular relaxation [8]. Studies about viscoelasticity in food have been frequent [4, 23, 26], however, only one work dedicated to this subject has been found in the edible films specialized literature [6].

CUQ et al [6] developed a viscoelastic model, independent of the thickness, to determine the viscoelastic properties of Sardine myofibrillar protein-based films, based on the generalized model of Maxwell. However, the authors added an element of rupture in series with the unit of Maxwell, with the objective of using it in the nonlinear viscoelastic domain. The total number of parallel elements was directly correlated to the thickness of the films. The rheological parameters (proportion of parallel bifurcations including elastic element, modulus of elasticity and coefficient of viscosity) were determined by adjustments of the experimental data, which appeared independent of the thickness of the film, but sensible to the type of data utilized for the calculations (force-deformation, relaxation, or creep) and to the experimental conditions applied to obtain these data.

Practically, there are two types of tests to determine the viscoelastic properties: dynamic and static tests [8, 18, 26]. In the dynamic tests, a sinusoidal variation of the deformation is imposed to the material and a variation of the stress necessary is observed, which must also oscillate according to a sine curve, however, with a phase displacement of 0 to p/2, in relation to the deformation. If the material is purely elastic or liquid (Newtonian), the phase displacement will be 0 and p/2, respectively [18]. Among the viscoelastic properties determined in these tests, are the storage modulus (G’) and the loss modulus (G"). In the static tests, a deformation or a constant stress is imposed, and the variation of the stress or the deformation is observed, respectively, during the progress of time. The first test is called stress relaxation and the second, creep compliance. In these cases, the model utilized in the treatment of the data establishes the viscoelastic properties.

Generally, the linear viscoelastic models, for the static tests, are developed from two elements: a spring and a hydraulic dashpot. It is considered that the first one works according to Hooke’s law, and the second element obeys the Newton’s law. A spring and a dashpot in series (Figure 1A), constitutes the model of Maxwell, and in parallel (Figure 1B), the model of Kelvin (or Voigt) [8]. These are the simplest models based on the theory of linear viscoelasticity.

The equation of the model of Maxwell, for stress relaxation tests, is easily encountered in the literature (equation 1) and can be obtained, considering that the total deformation is the sum of the deformation of the spring and that of the dashpot [4, 8, 18]. In the case of the Kelvin model, the deformation is equal in its two elements , but the stress applied is the sum of the stress on the spring and that on the dashpot . The model of Kelvin, in its turn, is only applied to the tests of creep [18].

Another well-known model is the generalized Maxwell model, the equation for which is obtained by summing equation 1 applied to each model of Maxwell [4, 8, 9, 18]. Its most simple representation consists of two units in parallel (Figure 2), for which the expression is easily obtained, as equation 2.

Another model of four elements is that of Burgers, which can also be considered as a simple viscoelastic model (Figure 3). In this, Maxwell and Kelvin models are connected in series. The constitutive equation for the model of Burgers can be derived, considering the re-sponse of a strain under constant stress for each con-nected element in series, as shown in Figure 3 [9].

The total deformation at a given time (t) would be the sum of the deformations of the three elements (equation 3), where the spring and the dashpot, of the model of Maxwell are considered as two elements.

where, is the deformation of the spring , is the deformation of the dashpot , both from the model of Maxwell, and is the deformation according to the model of Kelvin (equation 4).

Substituting , and in equation 3, the following constitutive equation is obtained:

The equation 5 can be applied in tests of stress relaxation, imposing a known initial deformation . Applying the functions of Heaviside: and of Dirac: and , the equation 5 can be represented in the following form:

where,

Taking the Laplace transforms of equation 6, we get to equation 7.

And, solving the equation 7 for , we obtain:

Expanding the equation 8 by partial fractions and applying the inverse transforms of Laplace, the equation of stress relaxation of the Burgers model, can be written as follows:

where,

The knowledge of the parameters of equation 9 allows the calculation of the viscoelastic properties according to the model of Burgers, in the test of stress relaxation, using the set of equations 10.

Considering the interest in models of easy utilization in the domain of the linear viscoelasticity theory, the objective of this work was the utilization of models of Maxwell (equation 1), generalized Maxwell with two units in parallel (GM2P) (equation 2) and Burgers (equation 9) in the tests of stress relaxation of the films of myofibrillar proteins of Nile Tilapia with three different thicknesses.

2 — MATERIAL AND METHODS

2.1 – Elaboration of edible films

The edible films were elaborated with myofibrillar proteins, prepared from fillet (pre rigor mortis) of Nile Tilapia (Oreochromis niloticus), produced in the campus of the University of São Paulo (USP) at Pirassununga, Brazil. The characteristics of these proteins, as well as the preparation of the myofibrillar proteins, are described in the literature [20].

Edible films were prepared according to a casting technique, which consists of preparing a filmogenic solution (FS), followed by application in a support and drying. Initially, the proteins were hydrated with water at room temperature, in a beaker. Next, the rest of the water and the plasticizer were added. Then the FS was treated thermically at 40ºC for 30 minutes. The control of temperature (±0.5ºC) was obtained in a water-bath with digital control of temperature (Tecnal, TE184). The FS of Myofibrillar Protein of Nile Tilapia (MPNT) was prepared in the following condition [19,25]: protein, 1g/100g of FS; plasticizer, 45g of glycerine/100g of MPNT, and pH maintained at 2.7 with glacial acetic acid, controlled by a testing pH-meter (Tecnal, TEC-2). Weighing (±0,0001g) of the proteins and plasticizers was made in an analytical balance (Scientech, SA210). All of the reagents used had analytical grade (Synth).

In the plates of plexiglas (11.8cm x 11.8cm) prepared previously, 27, 57 and 83g of FS were applied, to obtain films with different thicknesses. The control of weight (±0.01g) of the FS was made in a semi-analytical balance (Marte, AS2000). The filmogenic solutions were dehydrated in an oven with circulation and renovation of air (Marconi, MA037), with PI control (±0.5°C) of temperature at 35°C and room relative humidity, for 18-20 hours. Three to four test samples per plate were prepared, cutting the films in rectangles of 15mm x 118mm [7]. These samples were then conditioned at 22°C and 58% relative humidity for 4 days in a desiccators containing a saturated solution of NaBr. After this period, the thickness of the test sample was measured in nine different points with a digital micrometer (±0.001 mm), with a sensor of 6.4mm in diameter. The thickness of the test sample was taken as an arithmetic average of nine readings. All the characterizations were made under room conditions climatically adjusted to a temperature of 22°C and relative humidity between 55 and 65%.

2.2 – Stress Relaxation tests

The stress relaxation tests were conducted with an instrument for physical measurements, TA.XT2i (SMS). The initial length of the test samples was 80mm. The control program, "Texture Expert" V. 1.15 (SMS), imposed a deformation of 1% which was maintained constant for 70 seconds. The computer registered the stress, which decreased with time, necessary for the maintenance of the deformation. The value of deformation was chosen to guarantee that the behavior of the material was within the linear viscoelastic domain.

2.3 – Treatment of Data

The calculation of the viscoelastic properties was made by a Pentium microcomputer, using the equations 1, 2 and 9. The program Statistica was used to make adjustment of the models to the data. The sum of squares of the residuals was used for testing the accu-racy of the models. The correlation coefficient was also calculated for fitting experimental data to the models.

Because the tests of stress relaxation involved more than fourteen thousand pairs of data (stress and time), a macro in Visual Basic for Excel2000 was developed for the reduction of the number of data to 14-16 pairs of stress and time, by making an average of 50 points on either side of the point considered, thus maintaining the precision. To check this precision, the reduced points were superimposed on the original data (not shown).

3 — RESULTS AND DISCUSSION

An example of the curve obtained in the tests of stress relaxation can be observed in Figure 4. Initially, an ascendant curve obtained by the imposition of constant deformation was noted. Thereafter, it was observed that the force necessary for the maintenance of the deformation decreased with time. This behavior is characteristic of viscoelastic materials, like biofilms [6], gels [26] and some foods [4, 23].

In Figure 5, some stress relaxation data for the MPNT films, selected by the macro, are plotted. The curves shown in this figure were obtained by values calculated by the equations 1, 2 and 9, using nonlinear regressions. It was observed that, only the GM2P and Burgers models predicted satisfactorily the behavior of stress relaxation of the edible films, with excellent correlation coefficients (R > 0.99) and values of the sum of squares of the residuals (»0.025). In spite of an acceptable correlation coefficient (R » 0.85) and the sum of squares of the residuals (»0.674), the model of Maxwell did not allow a satisfactory calculation of the values of stress as a function of time (Figure 5). In this context it is necessary to make a note that both the models of generalized Maxwell and Burgers are expressed by the same regression equation, the only difference being that the coefficients of the regression equation do not convey the same interpretations for the two models.

The viscoelastic properties of the edible films, as the average of 7 to 8 replications calculated by the nonlinear regressions, are shown in Figure 6. The coefficients of viscosity and the modulus of elasticity (Y1) are plotted in Figures 6A and 6B, respectively for the first mechanical elements of the models of Maxwell, GM2P and Burgers. The properties and Y2 of the second mechanical elements of the models of GM2P and Burgers are plotted in Figures 6C and 6D, respectively.

Independent of the three models used in this study, the coefficients of viscosity were seen to be more sensitive to the thickness, than to the modulus of elasticity. The first coefficient of viscosity for the Burgers model, for example, which was in the order of 725 MPa, for a thickness of 0.024mm, increased a little, when the thickness was increased to 0.052mm, decreasing to values in the order of 626 MPa in thicker films (0.093mm), while the values of the respective modulus of elasticity (Y1), remained around 3.8 MPa. In the second set of elements, the coefficient of viscosity and the modulus of elasticity (Y2) decreased from 46.5 to 34.7 MPa, and from 10.1 to 8.3 MPa, respectively, when the thickness increased from 0.024 to 0.093. The viscosity decreased 34%, while the modulus of elasticity decreased only 21.7%. This behavior was seen to be practically identical to the other two models studied.

According to CUQ et al [5] and MAHMOUD and SAVELLO [16], the mechanical properties of edible films are influenced by the type and density of the molecular interaction among the biopolymer chains and the thickness of the film. The viscoelastic properties of myofibrillar protein-based films, calculated by CUQ et al [6], are apparently independent of the thickness (between 10 and 60mm). However, a considerable dispersion of the experimental points was noted in their work for a given thickness, which was more pronounced than the variations observed in coefficient of viscosity and modulus of elasticity, in this work. On the other side, when conducting tests that allowed the calculation of an applied stress, the thickness was included in the calculation in such a way that it would be reasonable to think that the effect of the thickness could be minimized. But, the results presented, did not support this hypothesis. More specific studies would be necessary to elucidate this abnormal behavior.

Comparing the values of the viscoelastic properties among the three models, (considering the results of Maxwell model for didactical reason), it was observed that the coefficient of viscosity of the first element , increased in average by 75%, from the model of Maxwell to that of Burgers, and that these values calculated by the GM2P model differed very little from the values determined by the model of Burgers. The modulus of elasticity (Y1), in its turn, showed a less important increase, of 19.8% in average, between the Models of Maxwell and Burgers, however, with a slight reduction in the respective values determined by the generalized model of Maxwell.

In relation to the second set of elements, it was observed that the GM2P model underestimated sensitively the values of viscosity and modulus of elasticity. The values of second coefficient of viscosity calculated by the model of Burgers were from 10 to 12 times larger than the respective values calculated by the GM2P model, while, in the case of the modulus of elasticity (Y1), this relation varied between 6.9 and 8.6 times. It must be observed that, though the differences of the values of the properties calculated with the GM2P and Burgers models, the coefficient of correlation, in both of the cases, remained practically equal. This occurred because, from the mathematical point of view, the two equations happened to be similar.

According to MIDOUX [18], the relation between the coefficient of viscosity and the modulus of elasticity, in the Maxwell model, is called relaxation time , which corresponds to the duration of stress to reduce to 1/e of its initial value. The knowledge of this time allows the calculation of the number of Deborah , where t₀ is the time of experimental observation: Db << 1 is the index of a viscous fluid; Db >> 1 is the index of an elastic solid; and Db » 1 is the index of viscoelastic behavior. By this form, using the data of Maxwell model (Figures 6A and 6B), an average value of t in the order of 125 s can be calculated, which implicates in Db = 1.8. This value allows to suggest then that the edible films of MPNT behave as viscoelastic solids. However, this is only a suggestion, because the model of Maxwell was not useful to predict satisfactorily, the curves of stress relaxation. The calculation of the relaxation time is more complicated and involves calculation of averages by integration [9], when models of more than two mechanical elements are used.

Unfortunately, there are not correlated works in the literature specialized in edible films. For this reason, it was not possible to make comparison with other results of viscoelastic properties. CUQ et al [6], the unique authors who worked with this theme, conducted tests similar to perforation with stress relaxation (force, in reality). Therefore, these authors determined values in Newton (measurement of force), and not in Mega Pascal (measurement of stress), being however, difficult to compare with the results presented in this work. For curiosity, however, we can remark that the values of the modulus of elasticity calculated in this work are lower, in the order of a magnitude of 2, than the modulus of elas-ticity determined by the tests of rupture by traction, with films of whey protein isolate and sodium dodecyl sulfate [7]. BUT, ARVANOTOYANNIS et al [2], conducting simi-lar tests as the authors above, determined the values for modulus of elasticity comparable to those shown in Figure 6B,D: 8.7 MPa (32.5% gelatin + 32.5% starch + 30% sorbitol + 5% water) and 9.8 MPa (34.5% gelatin + 34.5% starch + 26% sucrose + 5% water).

4 — CONCLUSIONS

The generalized model of Maxwell with two units in parallel and that of Burgers, in spite of being apparently simple, showed in a satisfying manner, the results of the tests of stress relaxation in films of myofibrillar proteins of Nile Tilapia. These models allowed the calculation of the viscoelastic properties of the films: coefficient of viscosity and modulus of elasticity in two elements. Al-though the variation of these properties with the thick-ness of the films is less important than the variations observed in the literature, it deserves more specific studies to explain such behavior.

More studies are also necessary for the physical in-terpretation of each of the four viscoelastic properties calculated. For this reason, the choice of the model must be done with caution.

5 — REFERENCES

6 — ACKNOWLEDGMENTS

The authors would like to acknowledge CAPES/COFECUB (205/97), FAPESP (95/9315-4, 98/12397-0) and CNPq for the financial support and fellowships.

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1

Recebido para publicação em 02/03/00. Aceito para publicação em 11/09/00.

² ZAZ-FZEA-USP, CP 23, 13630-000, Pirassununga (SP).

* A quem a correspondência deve ser enviada.

Publication Dates

• Publication in this collection
  20 Apr 2001
• Date of issue
  Aug 2000

History

• Accepted
  11 Sept 2000
• Received
  02 Mar 2000