Multi-scale mechanics and structure of semi-hard cheese T.J. Faber a,b, P.J.Schreurs b, J.M.J.G. Luyten a, H.E.H.Meijer b a FrieslandCampina Research, Deventer, The Netherlands (timo.faber@frieslandcampina.com)) b Eindhoven University of Technology, Eindhoven, The Netherlands (t.j.faber@tue.nl) ABSTRACT This paper shows the first results of deformation simulations on a composite cheese model, in which fat is a filler in a water-protein matrix. The work is part of multidisciplinary project in which multiscale structuretexture models for semi-hard cheese will be developed, and subsequenly used in product and process redesign. In the composite, linear elastic properties for both fat and the water-protein matrix are assumed. Virtual representative volume elements (RVE) are generated, in which fat volume fraction, fat diameter, and amount of interphase material between fat and the matrix are varied. In finite element simulations a shear is imposed on the RVE and the resulting shear modulus and Young s modulus are calculated. Simulations show good correspondence with experimental results from literature. Temperature and fat volume fraction are the most dominant factors in determining the Young s modulus of the composite. Changing spatial distribution of the filler in the composite by changing the particle diameter had no effect on the Young s modulus. A soft interphase between filler and matrix lowers the Young s modulus, but a specific interface effect was not demonstrated. Keywords: structure-texture relations; multi-scale modeling; finite element analysis; food; cheese; INTRODUCTION Foods created by food processing, such as bread, chocolate and cheese are artificially structured products [1] that consist of structural elements spanning a wide range of length scales. The sensory texture of these food materials, depend strongly on the arrangement, mechanical behaviour, and interactions of these structural elements [2]. For many non-food materials, like polymers [3] and ceramics [4], the link between both structure and mechanical properties and process and structure is well established, using multi-scale models and computational methods. In most food materials these structure-property and process-structure relations are still lacking. As a consequence, product and process design in the food industry often still remains a very empirical activity, where ingredients and processing conditions are altered and the properties of the resulting food samples, e.g. sensory texture, are determined in retrospect. The goal of this recently started, multidisciplinary project is to derive quantitative structure-sensory texture and process-structure relations for semi-hard cheese (Figure 1) and use these relations in product and process redesign. Figure 1. Structure-property and process structure relations used for food product and process design.
We will focus on the attributes of sensory texture that become apparent at the start of the oral processing. They have a more mechanical nature and can be measured with fundamental rheological methods, also referred to as instrumental texture (underline in Figure 1). This has the advantage that they can be linked to theories and models that take colloidal or molecular mechanisms into account [5]. Corresponding rheological and fracture quantities are Young s modulus, storage and loss modulus, shear modulus, yield stress, fracture stress, fracture energy and strain at fracture [6]. These quantities are strain rate dependent. Since we are mainly interested in mechanical behavior under oral processing conditions we will look at loading durations of seconds. Reported strain rates for the first bite are 1 s -1 [7]. For the model development and simulations we will benefit from the advances made in polymer and material science. We will adopt material models describing cheese mechanical behavior at different length scales and use the appropriate simulation tools and measuring techniques to explore these models and understand the cheese mechanics. Figure 2 gives an overview of these models ranked by their length scale (clockwise from top left). In this paper we will remain at the continuum and composite level and present the first results of simple deformation simulations varying fat (filler) amount and dimensions in a protein-water matrix. Figure 2. Material models describing cheese at different length scales. MATERIALS & METHODS Simulations have been performed using a commercial FEM package. A representative volume element (RVE) is modeled in two dimensions, i.e. deformations in the third dimension are assumed to be zero (plane strain). The dimensions are 50 50 μm. The thickness of the sample is 20 μm. The material consists of a matrix (protein/water) with fillers (fat) with or without a soft interphase between matrix and filler. This interphase serves to simulate either high cohesion (no interphase) or low cohesion between matrix and filler, making the filler inert. In practice phase separated serum between the fat globule and the protein-water matrix may act as such an interphase [8]. On the opposite, a strong cohesion between the milk fat globule membrane and the protein in the matrix has been suggested [9]. The material behavior of matrix, filler and interphase, is assumed to be linear elastic. Material properties of these components are listed in Table 1. At 26ºC filler is softer and at 14ºC stiffer than the matrix as a result of the liquid and solid state of the fat. To investigate the effect of spatial distribution of fat through the matrix, irrespective to the change in filler modulus, we assume the Young s modulus of the filler to be independent of filler particle size. Normally this would scale with 1/diameter for fat globules consisting of liquid fat.
Table 1. Properties of materials in the composite Property 1 Symbol Matrix Filler 2 Interphase Young s modulus (26ºC) [N/m 2 ] E 26 1*10 5 2*10 3 1*10 1 Young s modulus (14ºC) [N/m 2 ] E 14 1*10 5 6*10 5 1*10 1 Poisson ratio [-] μ 0.45 0.45 0.45 1 All numbers for matrix and filler are taken from [10] (strain rate = 2.8 x 10 2 s 1 ). The RVE is fixed at the bottom and loaded with a prescribed (5 μm) horizontal displacement of the top edge, which cannot move in vertical direction. The force, needed to prescribe the deformation can be calculated and provides the shear resistance (shear modulus) G of the RVE. For isotropic materials this can be converted into a Young s modulus E using the equation: ( 1+ ) G E = 2 μ, in which μ is the Poisson s ratio. We assume that the Poisson s ratios for the composite and individual components are identical. Composite morphologies are generated with a Matlab code. The filler particles are drawn from a normal distribution with a standard deviation of 10% of the mean until a volume fraction limit is exceeded. Particles are placed randomly in the RVE. The thickness of the interphase is calculated using a prescribed interphase volume fraction. The particle picking procedure leads to higher filler and interphase volume fractions than prescribed. On the opposite, particles might overlap as a result of the random placement, or partially fall of the RVE boundary, which leads to lower volume fractions. There is no correction for these events, therefore the actual volume fraction of filler and interphase will deviate from the prescribed value. With all combinations of the input parameter values listed in Table 2 and the two Young s moduli for the filler deformation simulations are performed. For the filler Young s modulus at 14ºC only the minimum and maximum of the input parameters are evaluated. For each combination the number of analysis is 10, resulting in 10 different morphologies. Of each analysis the actual filler volume fraction, mean diameter and interphase volume fraction is calculated and will be denoted with the subscript o. Input parameters are indicated with the subscript i. Table 2. Values for the input parameters used in the simulations. Inputparameter symbol min max steps E f,26 steps E f,14 filler mean particle size [μm] d i 1 10 3 9 filler volume fractiono [-] f i 0.05 0.35 0.10 0.30 interphase volume fraction [-] i i 0 0.06 0.02 0.03 RESULTS & DISCUSSION Figure 3 shows a selection of graphics from the finite element analyses for each combination of minimum and maximum value of the input parameters. The first and fourth columns show images of simulated composites. Column two and five show images of the calculated Von Mises stress after deformation of these composites with the soft filler particle (E f,26 ), column three and six with the stiff filler particles (E f,14 ). The overlapping of particles is clearly visible in Figure 3p and 3o. In practice at higher temperatures overlapping will lead to coalescence, resulting in more spherical, larger particles (see Figure 2, top right picture). Only in the case of large particles and low fat volume fraction the interphase completely covers the fat globule. Stress distribution is more homogeneous with smaller fat particles (column two and three). The concentration of stress towards the middle of the RVE in these pictures is probably a result of the applied boundary conditions. Column five and six show that with softer particles, the stress concentrates between the particles, and with stiffer particles within the particles. This might influence the fracture pattern of the cheese.
f i i i d i = 1 E f,26 E f,14 d i =10 E f,26 E f,14 0 0.05 a b c d e f 0.06 g h i j k l 0 0.35 m n o p q r 0.06 s t u v w x Figure 3. Graphics from the finite element analyses for each combination of minimum and maximum value of the input parameters. Accuracy and precision of morphology generating procedure The accuracy of an output parameter of the morphology generating procedure is defined as the absolute value of the relative difference of the mean input and output value, with all input variables kept constant. The precision is the standard deviation of the output value of a variable, given constant input values. Smaller values indicate a higher accuracy and precision. The accuracy for d o is smaller than 2% (n=10) and the precision smaller than 7% for all parameter combinations. The accuracy for f o is 8-20%, the precision is 1-3% for d i =1 and 6-21% for d i =10. The accuracy for i o is 2% for f i =0.05 and d i =10 and 33% for f i =0.35 and d i =10. Precision for i o is between 3-21%, with higher precision for d i =1. The lower accuracy for f o and i o with higher filler concentrations and larger particle diameters can be explained by the increasing probability of overlapping particles. This reduces the actual amount of filler and interphase. The precision of the morphology generating procedure is good enough to draw conclusion from the deformation experiments. The accuracy will be accounted for by showing the output values of the variables in Figure 4, 6 and 7. Young s modulus of the composite Figure 4 shows the Young s modulus of the composite E composite on a logarithmic y-axis, as a function the Young s modulus of the filler at two temperatures, 26ºC and 14ºC. These different Young s moduli are represented by a temperature scale on the x-axis. Note that temperature itself is not a variable in these simulations. The two linear regression lines connect the results from simulations with minimum and maximum filler volume fractions respectively. Figure 4 shows good agreement with Figure 5, which is taken from literature [10] and are real measurements on full fat cheese (33% fat) and low fat cheese (5% fat). At 14 ºC E composite increases with increasing fat volume fraction, which can be explained by the fat being stiffer than the matrix. The opposite is true at 26 ºC. This results in the two lines crossing at approximate 20 ºC, the temperature at which E matrix and E filler are equal and filler concentration has no effect on E composite. Figure 4 and 5 also show that the temperature efect on the Young s modulus of the composite decreases with decreasing fat content.
5,5 10 log Ecomposite [N/m 2 ] 5 4,5 filler volume fraction = 0,06 filler volume fraction = 0,30 4 14 20 26 Temperature Figure 4. Log E composite as a function of filler Young s modulus (E filler ) and fat volume fraction (f o ). Simulations with E filler,14 =6*10 5 N/m 2 and E filler,26 =2*10 3 N/m 2 are represented by a temperature scale on the x-axis. : f o = 0.06; : f o = 0.30. Particle diameter d o =1μm, interphase volume fraction i o =0. Lines are from linear regression. Figure 5. Taken from [10]. Log E composite as a function of temperature. These are experimental results from compression tests on low fat cheese (open circles, f=0.05) and full fat cheese (crosses, f=0.3). 5,5 5,5 f=0,055 i=0 10 log Ecomposite [N/m 2 ] 5 4,5 f=0,055 d=1 f=0,30 d=1 f=0,060 d=10 f=0,28 d=10 10 log Ecomposite [N/m 2 ] 5 4,5 f=0,30 i=0 f=0,055 i=0,067 f=0,30 i=0,053 4 14 20 26 4 14 20 26 Temperature Temperature Figure 6. Log E composite as a function of filler Young s modulus (E filler ) and particle diameter (d o ).,, from Figure 4, d o =1 μm. f o = 0.060, d o =10μm; : f o = 0.28 d o =10μm. --- linear regression lines for d i =10μm. Interphase volume fraction i o =0. Figure 7. Log E composite as a function of filler Young s modulus (E filler ) and interphase volume fraction (i o ).,, from Figure 4, d o =1 μm. f o = 0.055, i o =0.067, : f o = 0.30 i o =0.053. --- linear regression lines for i i =0.06μm. Particle diameter d o =.1 μm. Figure 6 and 7 show the effect of the remaining variables: the particle diameter and the interphase volume fraction. The solid lines and solid data points are from Figure 4. The dotted lines and open data points are new values for particle diameter (Figure 6) or interphase volume fraction (Figure 7). Figure 6 shows a minor effect from particle diameter, but this is an artifact of the morphology generating procedure: the output value of the filler volume fraction changed when changing the particle diameter. Thus merely spreading the fat over the matrix by decreasing particle size has no effect on E composite.. As stated before, E filler and therefore E composite, changes with particle size when the filler is liquid, but this was not taken into account in the simulations. Figure 7 indicates that the presence of a soft interphase lowers the Young s modulus of the composite. A simulation with the interphase distributed as small or large filler particles in the protein matrix with a volume fraction of 0.06 resulted in a decrease of the Young s modulus of loge matrix -loge composite = 0.2. This of the same order of magnitude as the difference between the solid and dotted lines in Figure 8. This suggests that in our simulations there is no specific effect from the location of the interphase between the filler and the matrix on the Young s modulus of the composite.
Outlook In these simulations we have only evaluated the linear region of the stress strain curves and fracture was out of scope. Next steps would be to include non-linear material models [11], damage models, larger deformations and appropriate loading geometries (e.g. indentation, compression) in the simulations and validate the models with experiments. CONCLUSION A composite model of cheese has been applied in FEM simulations. Simulations show a good correspondence with experimental results from literature. Temperature and fat volume fraction are the most dominant factors in determining the Young s modulus of the composite. Changing spatial distribution of the filler in the composite by changing the particle diameter had no effect on the Young s modulus. A soft interphase between filler and matrix lowers the Young s modulus but a specific interface effect was not demonstrated. REFERENCES [1] Van der Sman, R. & Van der Goot, A. 2009. The science of food structuring. Soft Matter, 5, 501-510. [2] Aguilera, J. 2005. Why food micro structure? Journal of Food Engineering, 67, 3-11. [3] Meijer, H.E.H., Govaert, L.E. 2005. Mechanical performance of polymer systems: The relation between structure and properties. 2005. Progress in Polymer Science, 30 (8-9), 915-938. [4] Wyss, H.; Tervoort, E. & Gauckler, L. 2005. Mechanics and microstructures of concentrated particle gels. Journal of The American Ceramic Society, 88, 2337-2348. [5] Foegeding, E., Brown, J.; Drake, M. & Daubert, C. 2003. Sensory and mechanical aspects of cheese texture. Intermational Dairy Journal, 13, 585-591. [6] International Dairy Federation. 1991. Rheological and fracture properties of cheese. IDF Bulletin. [7] Agrawal, K. & Lucas, P. 2003. The mechanics of the first bite. Proceedings of the Royal Society B: Biological Sciences, 270, 1277-1282. [8] Hassan, A. & Awad, S. 2005. Application of exopolysaccharide-producing cultures in reduced-fat Cheddar cheese: Cryo-scanning electron microscopy observations. Journal of Dairy Science, 88, 4214-4220. [9] Everett, D. & Auty, M. 2008. Cheese structure and current methods of analysis. Intermational Dairy Journal, 18, 759-773. [10]Luyten, H. 1988. The rheological and fracture properties of Gouda cheese. PhD Thesis, Wageningen University. [11]Goh, S.; Charalambides, M. & Williams, J. 2005. Characterization of the nonlinear viscoelastic constitutive properties of mild cheddar cheese from indentation tests. Journal of Texture Studies, 36, 459-477.