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Investigation of the diffusion of dyes in agar gels
Journal of Food Engineering 111 (2012) 537–545
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Investigation of the diffusion of dyes in agar gels K. Samprovalaki ⇑, P.T. Robbins, P.J. Fryer Chemical Engineering, The University of Birmingham, Edgbaston, B15 2TT Birmingham, UK
a r t i c l e
i n f o
Article history: Received 17 June 2011 Received in revised form 21 February 2012 Accepted 17 March 2012 Available online 24 March 2012 Keywords: Diffusion Fick’s law Image analysis
a b s t r a c t An experimental set-up and a measurement technique were developed so that diffusion in model foods (gels of agar) could be visualised and quantified. The diffusion of aqueous solutions of varying concentrations of two dyes (rhodamine 6G and methylene blue) in gels of agar was followed in situ at three temperatures (30, 50, 70 °C) until equilibrium was reached. The nature of the diffusion process (in terms of the amount of dye diffused into the gel) was studied using image analysis techniques. The diffusion coefficient, D, was estimated using Fick’s second law of diffusion and found to be 1010 m2 s1. The effect of the size of the diffusing molecule, as well as that of the processing temperature on diffusion was investigated and found to be significant. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Diffusion of components in foods is an important process in the food industry which is not fully understood. The major diffusion processes used in the food industry are: (i) Leaching, the transfer of solutes from a solid to an adjacent liquid which is used to extract sugar, vegetable oils, coffee, tea and other food solutes (Schwartzberg, 1980). (ii) Infusion, the transfer of solutes from a liquid into a solid which is used to transfer colours, flavours, curing and conditioning agents into foods (Schwartzberg and Chao, 1982). In extraction processes the structure of the food must be destroyed. In other thermal processes (for example, pasteurisation of milk or fruit juice) the aim is the preservation of vitamins and aroma producing compounds with as little effect on the structure of food as possible. The enhancement of mass transfer within foods may well be of significant strategic advantage industrially, if there are diffusional problems that, when overcome, may lead to production of improved or so called ‘‘enriched’’ products. This will be beneficial to both the consumer and the manufacturer. A major limitation of infusion techniques is the lack of successful measurement techniques to quantify the depth of penetration and localisation of infused material. Kemp and Fryer (2007) developed a technique based on image analysis and optical microscopy to visualise diffusion into solids under conditions of elevated temperatures and in the presence of alternating electric fields. ⇑ Corresponding author. Current address: Epidavrou 100, 104 41 Athens, Greece. Tel.: +30 6934816867; fax: +30 2105143160. E-mail address: [email protected] (K. Samprovalaki). 0260-8774/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfoodeng.2012.03.024
Diffusion as a scientific term refers to the net movement of matter, attributed to the random movements of molecules, taking place from a region of high concentration to one of lower concentration (Agutter et al., 2000). Like most scientific concepts, diffusion theory should not be construed as a fixed body of terms, laws and data, but rather as an evolving one. It has a ‘‘classical’’ form and numerous variants of this classical form developed ad hoc for specific applications in different disciplines. The classical theory of diffusion has both a macroscopic, phenomenological aspect, and a microscopic, mechanistic aspect (Agutter et al., 2000). The mathematical theory of diffusion in isotropic substances is based on the hypothesis that the rate of transfer of diffusing substance through unit area of a section is proportional to the concentration gradient. This is commonly called Fick’s 1st Law and can be written as (Crank, 1975):
F ¼ DrC
ð1aÞ
which can be written for a 1D slab geometry as:
F ¼ D
@C @x
ð1bÞ
To find the concentration profile and the rate of transfer, a mass balance is combined with Fick’s first law to give:
@C ¼ rðDrCÞ @t
ð2aÞ
or for a 1D slab geometry with constant diffusion coefficient, D:
@C @2C ¼D 2 @t @x
ð2bÞ
This is often called Fick’s second law. Fick’s law of diffusion is useful in studies of the solid, where concentrations vary with position and time (Cussler, 1984).
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Nomenclature C D Deff F G L T X
concentration of diffusing substance [% w/v] diffusion coefficient [m2 s1] effective diffusion coefficient [m2 s1] rate of transfer per unit area of section [kg m2 s1] grey intensity thickness of the slab [m] dimensionless time dimensionless distance in the x direction
Quantitative measurements of the rate at which a diffusion process occurs are usually expressed in terms of a diffusion coefficient (D). In some cases, for example diffusion in dilute solutions, D can be taken as constant, while in others, for example diffusion in high molecular weight polymers, D depends very markedly on concentration. Formally, a system is said to be Fickian only when the diffusion coefficient is constant, i.e. independent of concentration. Analytical solutions of Eq. (2b) covering a range of specimen geometries can be found in Crank (1975). However, the application of these principles to food systems is usually difficult because of the complex and heterogeneous structure of foods, which may change during processing or storage. Exact analytical solutions cannot always be found due to: (i) non-constant diffusion coefficient, as D can be a function of concentration, temperature, moisture content, water activity, structure of food, etc. and (ii) the shape change of the food system. In a porous medium like food, the measured diffusion coefficient will be significantly smaller than the molecular diffusion coefficient because of tortuosity effects (the more tortuous the region the longer the route between two points) and because of the hindering effects of the surface of the pores on the molecule’s random oscillations. This is why a functional form of the diffusion coefficient called effective diffusion coefficient (Deff), that encompasses all ignorance of the pores’ geometry, and is a function of the local geometry as well as of the solvent and solute, is found by comparing Eq. (2b) to experimental data with relevant geometry and initial and boundary conditions. If an estimate of Deff is available, then a mass transfer simulation with the relevant initial and boundary conditions can be performed and the concentration profile within the food with time can be obtained. By using Deff, diffusion at the higher length scales investigated can be treated as one-dimensional although at a cellular level it is three-dimensional (Crank, 1975). This type of approximation often works well if the distances over which diffusion occurs are large compared to the size of the pores, as is the case for this work, where agar gel pore size is of nanometre order of magnitude (Narayanan et al., 2006). A number of papers have been published in which diffusion coefficients in foods are calculated or estimated. Allali et al. (2010) used an approximate solution of Fick’s second law for unsteady state mass transfer in spherical configuration to calculate the Deff of water and sucrose in the osmotic dehydration of strawberries and values were found to be in the range of 1010 m2 s1. The increase of temperature and initial sucrose concentration led to an enhancement of water and sugar diffusivities. Meinders and van Vliet (2011) showed that oscillatory sorption experiments of bread crusts could be very well described by a Fickian diffusion model applied to spherical particles, assuming a constant diffusion coefficient. Voogt et al. (2011) showed that experimentally observed dynamical water profiles of bread rolls that differed in crust permeability were predicted well by a Fickian diffusion model with a single Deff. Spiro and Chong (1997) calculated the intra-bean diffusion coefficients for caffeine (molecular weight 194.19 g mol1) by solving Fick’s second law of diffusion and found values in the
H Z t x y c n
dimensionless distance in the y direction grey intensity ratio time [s] distance in the x direction [m] distance in the y direction [m] dimensionless concentration number of series
order of 1010 m2 s1. Ruiz-López et al. (2012) developed an analytical model based on Fick’s second law of diffusion to evaluate mass transfer properties and describe drying kinetics of chayote slices. Water diffusivities were estimated as 4.44 1010, 6.00 1010, 7.04 1010 and 8.60 1010 m2 s1 for drying processes at 40, 50, 60 and 70 °C, respectively. El-Belghiti et al. (2005) studied the extraction of sugar from sugar beet slices following various pulsed electric field (PEF) treatments and calculated the diffusion coefficient using Fick’s diffusion equation to be 1.68 ± 0.40 1010 m2 s1. By heating the solution at mild temperatures of 30, 40 and 50 °C, the coefficient of diffusion was increased and the kinetics of extraction was enhanced. A time of about 70 min was needed to obtain a yield of 93% at ambient temperature and around 40 min to obtain the same yield at 50 °C. Jemai and Vorobiev (2002) investigated the effects of a moderate electric field pulse (MEFP) on the diffusion coefficient of soluble substances from apple slices and found that an electrical pretreatment with moderate temperature elevation (10–15 °C) combined with a low thermal treatment, significantly enhanced the diffusion coefficient (Del) compared with reference values. For instance, after a standard MEFP treatment (500 V cm1, 1000 pulses, 100 ls duration) Del was 3.9 1010 m2 s1 at 20 °C compared with reference values of 2.5 and 4.4 1010 m2 s1 for untreated and denatured samples, respectively, and 13.4 1010 m2 s1 at 75 °C compared with 10.2 1010 m2 s1 for thermally denatured samples. Values of diffusion coefficients in foods and agar gels are shown in Table 1. As there is no standardised method for estimating diffusivity, the published values present variability, due to both the diversity of experimental and calculation methods used in each case, and of product structure and composition (Boudhrioua et al., 2003). The aim of this work was to develop a measurement technique as well as a simple diffusional model to analyse dye diffusion in gels. In this work, two dye types and temperature effects are investigated, while in later work (Samprovalaki et al., 2012) the effect of electric fields on diffusion seen by Kemp and Fryer (2007) was studied in more detail. The objectives of this project were to study diffusion in gels, the effect of temperature and size of diffusing molecule on diffusion, and the application of Fickian theory on interpreting the results.
2. Materials and methods Experiments were designed to investigate the diffusion of dyes into agar gels at three temperatures (30, 50 and 70 °C). The experimental set-up was designed to enable visualisation of diffusion and is represented schematically in Fig. 1. It consisted of the experimental cell, a camera with control box [Photonic Science High Sensitivity Mono CoolView camera (Photonic Science Ltd., UK)], ring lights, a Peltier stage (LTS 120 Stage with PE 94 Controller and Linksys software, Linkam Scientific Instruments Ltd., UK), a temperature logging interface, and computers with software for image
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Diffusant
Deff 1010 (m2 s1)
Temperature (°C)
Reference
Sugar beet Sugar beet Papaya Carrot Carrot Carrot Potato Potato Potato Grapes Mangoes Cassava Fish Cod (frozen) Cod (brined) Herring Mackerel Plaice Pork sausage Apple Apple Apple Gouda cheese White cheese Cheese 0.79% agar gel Agar gel Agar gel Agarose gel (4%) Agarose gel (3%) Agarose gel Agarose gel Agarose gel Agarose gel
Sucrose Water Sucrose Sucrose Water Water Water Water Water Water Water Water Water Water Water Water Water Water Water Water Sucrose Water Salt Salt Salt Sucrose Amaranth dye Sorbic acid Lysozyme Nisin Thiamine Glucose Acetic acid Chromate
2.5 0.4–1.3 28.95 14 7.5 9.4 3.8 8.3 2.6 2.2 8.6 4.9 5.2 2.5–70 0.3–2.3 0.1–1.9 1.1–2.2 2.9 47–57 8.1 0.79 0.9–28 230 510 1.0–5.5 2.47 0.6–4.9 6.7–7.3 90 19.2–81.4 4.3–16.7 4.5–13.2 7.5–22.0 9.0–11.10
23–25 40–84 35 80 60 60 60 60 60 60 60 60 60 5 to (30) 20 30 30 30 20 60 40–70 30–70 20 20 4–60 5 – – 25 5.4–22.3 20–80 – – 20–70
Schwartzberg and Chao (1982) Doulia et al. (2000) Fernandez et al. (1995) Mittal (1999) Mulet (1994), Ruiz et al. (1997) Zogzas and Maroulis (1996) Husain et al. (1972) Zogzas and Maroulis (1996) Mulet (1994) Simal et al. (1996) Hernandez et al. (2000) Hernandez et al. (2000) Balaban and Piggot (1988) Doulia et al. (2000) Doulia et al. (2000) Doulia et al. (2000) Doulia et al. (2000) Doulia et al. (2000) Doulia et al. (2000) Zogzas and Maroulis (1996) Simal et al. (1998) Doulia et al. (2000) Geurts et al. (1974) Turhan (1996) Doulia et al. (2000) Schwartzberg and Chao (1982) Doulia et al. (2000) Doulia et al. (2000) Mattisson et al. (2000) Sebti et al. (2004) Doulia et al. (2000) Doulia et al. (2000) Doulia et al. (2000) Doulia et al. (2000)
Data Logging Computer Camera Control Box
TC-08 Temperature Logging Interface
Camera
Ring Lights Peltier Stage Controller
Sample
Peltier Stage Thermocouples Fig. 1. Schematic representation of the experimental set-up used to perform diffusion experiments and measurements.
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40 mm Dye reservoirs
10 mm 25 mm
5mm
5 mm 10 mm Aluminium sheet
PVC coating
Gel
Thermocouples Fig. 2. Schematic representation of the experimental cell.
acquisition data collection [Image Pro-Plus 4.5 (Media Cybernetics, Inc., USA)]. The experimental cell is depicted in Fig. 2 and consists of an aluminium sheet 45 mm long, 30 mm wide and 1.5 mm thick covered on one side with a white PVC layer. Aluminium was chosen so that heat would be transferred quickly between the cell and the Peltier stage. A cavity of the shape and dimensions shown in Fig. 2 was bored into the plastic. Agar gel slabs were made by mixing agar powder (1.5% w/v) (Fluka 05039, UK, gelling temp. 33.8 °C) with calcium chloride dihydrate (3% w/v) (Sigma Aldrich Co. Ltd., UK, C-3881) and distilled water and heating at 80–90 °C until the powder dissolved. These gels are isotropic with no directional differences in their structure (Odake et al., 1990). Thus all the problems associated with the anisotropy of food could be excluded. CaCl2 was added so that electrical conductivity was adjusted. Agar gels used in the experiments were tested by differential scanning calorimetry (Micro DSC II, SETARAM Instrumentation); gels did not melt until they were reheated above 80 °C which is above the highest temperature used in these experiments, therefore no change in the physical state of the gels occurred during the experiments. Dye solutions of rhodamine 6G and methylene blue dyes (both from BDH chemical, Poole) were used for each experiment so that the effect of the mass of species on diffusion could be evaluated. Methylene blue is a dye of similar charge to rhodamine 6G (+1) but of lower molecular weight (319.85 and 479.02 g/mol, respectively). Stock solutions of dye were made by dilution in distilled water. Concentrations of rhodamine 6G used were: 0.002, 0.004, 0.006, 0.008, and 0.01% w/v. The concentration of methylene blue used was 0.01% w/v. CaCl2 was added in the dye solutions to the same concentration as that in the gel so that the electrical conductivities of the solid and liquid phases were identical, giving uniform field strength, this was important for later work when electric fields were applied (Samprovalaki et al., 2012). Gel samples set in the experimental cell so that the sample volume was controlled accurately. Gel solutions were kept warm, in a conical flask, on a hotplate until they were used to form gels. The experimental procedure is described below: (i) The two reservoirs were first covered with two pieces of plastic as shown in Fig. 2 so that only the cavity in the middle of the cell could receive the gel. (ii) A syringe and needle (Microlance 3, 19GA2) was used to take the solution out of the flask and extrude it into the cell so that air inclusion was minimised.
(iii) The cell was then immediately covered with a 10 mm diameter circular microscope cover glass (thickness No. 0, BDH, Merck Eurolab Ltd., Poole, UK) so that: (i) evaporation was avoided, (ii) the thickness of the sample was always the same and (iii) the dye was prevented from travelling on top of the gel slab. (iv) The gel solution was locked in the centre of the cell and allowed to set to form a gel of dimensions 10 mm long, 5 mm wide and 1 mm thick. (v) Once the gel had set, the two plastic pieces were removed and the cell was mounted on the Peltier cooling stage under the camera. Zinc oxide heat transfer paste was used to ensure good thermal contact between the cell and the stage. The cooling stage was already programmed at the required temperature before the cell was placed on it. (vi) The temperature and image acquisition software were then initialised. (vii) The dye, which was kept in a water bath at the same set temperature, was then added in the two reservoirs simultaneously. (viii) Diffusion progress was then visualised by capturing images every 10 min until the agar gel was saturated with dye and equilibrium was reached (approximately 2 h 30 min). Experiments were performed in triplicates. Image analysis software allows a colour image to be captured in monochrome and displayed in grey-scale as shown in Fig. 3. Resolution was 8-bit which means that there are 28 shades (256 shades, where value 255 stands for pure white and value 0 for black). A line profile is selected on this image, i.e. the computer calculates and displays a number for each pixel along a selected line depending on the shade of grey of that pixel. The system can measure over a length scale comparable to the pixel length. In all cases presented here 80 pixels correspond to 1 mm. The area which gave uniform grey intensity values and was used in the analysis is 100 pixels wide as shown in Fig. 3 (Samprovalaki and Fryer, 2011). For every pixel along the x-axis the software calculates the average grey intensity along the 100-pixel y-axis line. The variation in the mean value of grey intensity at fixed distances from the centre of the slab across the x-direction was then studied versus time. The change in grey intensity values measured during the experimental run can be due to two reasons: (i) the concentration of dye is changing, (ii) the optical properties of the image (e.g. shadowing, change in light source intensity, camera/software image correction).
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dye reservoirs y
x
Horizontal Centre line
100 pixels
Gel slab
X=-0.4
X=+0.4
X=-0.8
X=+0.8 Vertical Centre line X=0
Fig. 3. Area of analysis. The software calculates the average grey intensity along the 100-pixel y-axis line at selected distances from the centre of the slab along the x-axis and plots it versus time. The grey scale intensity values with time were used for the calculation of the diffusion coefficient of rhodamine 6G at five positions in the agar gel slab (i.e. centre 2 mm, centre 1 mm, centre, centre + 1 mm, centre + 2 mm which correspond to dimensionless distances X = 0.8, X = 0.4, X = 0, X = +0.4 and X = +0.8, respectively).
If changes in the optical properties of the image can be neglected, grey scale intensity, G, is a function of the concentration, C, of the dye in the gel. 3. Data analysis methods
1 ð1Þn C C0 4P ¼1 C1 C0 p n¼0 2n þ 1 ( ) Dð2n þ 1Þ2 p2 t ð2n þ 1Þpx cos exp 2 2l 4l
ð3Þ
3.1. Analysis of diffusion
The boundary conditions can be written in terms of the following dimensionless parameters:
To generate a diffusional model based around Eq. (2b) for comparison with experimental data there are four areas to consider:
c¼
(i) (ii) (iii) (iv)
geometry, boundary and initial conditions, functionality of the diffusion coefficient, D, conversion of the measured value.
To be able to generate an analytic solution, then a 1 dimensional (1D) geometry, a constant diffusion coefficient, D, the same initial condition at all points and a constant boundary value are desirable. For the experimental set up here the geometry is slab like and the material is isotropic, initial concentration in the gel is 0 and the boundaries are held at a constant value. This leaves if a 1D or 2D solution is necessary and if a constant diffusion coefficient gives a good fit to the experimental data.
T¼ X¼
C C0 C1 C0 Dt
ð4aÞ
ð4bÞ
L2 x L
ð4cÞ
Then Eq. (3) becomes (Crank, 1975):
c¼
C C0 C1 C0
¼1
( ) 1 ð1Þn 4P ð2n þ 1Þ2 p2 ð2n þ 1Þp exp T cos X p n¼0 2n þ 1 4 2
ð5Þ
with boundary conditions: T = 0 all X, c = 0 and T > 0 X = ±1, c = 1 3.1.1. Solution in one-dimension It is possible to obtain solutions of Eq. (2b) for one-dimensional diffusion in a medium bounded by two parallel planes. These solutions apply in practice to diffusion into a plane sheet of material so that effectively all the diffusing substance enters through the plane faces and a negligible amount through the edges. Very often the problem is symmetrical about the central plane of the sheet (as is this case), and the formulae are then most convenient if the centre is taken as x = 0 and the surfaces at x = ±L. The gel is approximated as an infinite slab of thickness 2L, with an initial concentration, C0 in the region L < x < L and a constant concentration of C1 at the boundary surfaces. Crank (1975) has given the solution in the following form:
3.1.2. Solution in two-dimensions It is possible to obtain solutions of Eq. (2) for diffusion in two dimensions using finite difference techniques but it is more easily solved in commercial finite element software. For this work, FEMLAB™ (now Comsol multiphysics™) was used. A characteristic picture of this analysis is shown in Fig. 4(a). The diffusion profile produced by the model is similar to that observed in the videos of the experiments as shown in Fig. 4(b). This confirmed the fact that two-dimensional diffusion is taking place. However, since the one-dimensional solution is easier to manipulate, it was considered useful to determine the range of conditions under which the assumption of one-dimensional diffusion can be used.
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Fig. 4. (a) Characteristic picture of the solution of the diffusion equation in two dimensions using finite element software (Femlab 2.3). The diffusion profile produced by the model is similar to that observed in the videos of the experiments as shown in (b).
3.2. Identification of appropriate range for 1D approximation The solutions of Eq. (5) for diffusion in one (1-D) and two dimensions (2-D) were compared and the results of this comparison are plotted in Fig. 5. Dimensionless concentration, c, is plotted against dimensionless time, T, for two dimensionless distances, one in the centre, X0, and one at X = 0.8, X0.8, as shown in Fig. 3. The graph in Fig. 5 shows that for dimensionless time T < 0.2, the lines of 1-D model and 2-D model coincide. In other words, if the run of the experiment lasts for T < 0.2, either the 1-D or 2-D model could be used for the analysis. The time t of the experiment corresponding to dimensionless time T can be calculated using Eq. (4b). Assuming a diffusion coefficient in the range of 1 1010 m2 s1 which is typical in food, as shown in Table 1, and 2L = 5 mm, Eq. (4b) gives:
t¼
TL2 0:2 ð2:5 103 Þ2 ¼ 12; 500 s 208 min ¼ D 1010
Therefore, for the analysis of the experimental data as long as the time does not exceed 3 h either 1-D or 2-D solutions can be used. 3.3. Estimation of the effective diffusion coefficient The grey scale intensity change with time and position was used for the calculation of the effective diffusion coefficient as all
measurements project the change of the local concentration of the dye within the gel with time and position. As the grey scale intensity varies linearly with concentration, for the range of dye concentrations (<0.01% w/v) used here (Samprovalaki and Fryer, 2011), then the concentration terms in Eq. (5) can be directly replaced by the grey intensity.
c¼Z¼ ¼1
G G0 G1 G0
( ) 1 ð1Þn 4P ð2n þ 1Þ2 p2 ð2n þ 1Þp exp T cos X p n¼0 2n þ 1 4 2
ð6Þ
Eq. (6) has summation terms which make direct calculation of Deff difficult. Therefore an estimation of the diffusion coefficient was attempted. Data fitting was carried out using Eq. (6) and the sum of the square of the differences technique (using a fitting routine programmed in Matlab 6.5). A value of D was guessed (1 1010 m2 s1); a solution for Z(x, t) was obtained and compared to the measured value of Z. The model line was adjusted by altering the diffusion coefficient until the value of the sum of the squares reached a minimum. This value of Deff, which minimises the least squares error and gives the best fit between the model and experimental curves is the estimated value of Deff. In all cases, there was an initial under-prediction of concentration (i.e. D was too low) followed by an over-prediction (i.e. D was too high),
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1
DIMENSIONLESS CONCENTRATION (c)
X=0.8 0.8 1D 0.6 2D 0.4 X=0
0.2
0
-0.2 0
0.1
0.2
0.3
0.4
0.5
0.6
2
DIMENSIONLESS TIME (Dt/L ) Fig. 5. Comparison of the solutions of the diffusion equation in one and two dimensions. Dimensionless concentration, c, is plotted against dimensionless time, T, for two dimensionless distances, one in the centre, X0 and one at X = 0.8 from the boundary. (+): 1-D diffusion model at X0, (s): 2-D diffusion model at X0, (x): 1-D diffusion model at X0.8, (e): 2-D diffusion model at X0.8.
example shown in Fig. 6. This shows that a constant diffusion coefficient is not the best model for the experimental data, but the best fit D value still gives some useful information as discussed below. 4. Results and discussion Eq. (6) was solved for five different positions in the slab (centre 2 mm, centre 1 mm, centre, centre + 1 mm, centre + 2 mm) and the mean Deff estimated for each dye solution concentration and temperature are tabulated in Table 2 for both rhodamine 6G and methylene blue. It is obvious that Deff changes with temperature and concentration of the diffusing solution. Two-way ANOVA was used to evaluate the difference with temperature and concentration which was found to be significant (at 95% confidence interval), with significant interaction between the two factors, for both rhodamine 6G and methylene blue experiments. From Fig. 6 and Table 2 it is obvious that Deff depends on concentration as it suddenly drops from a high value to a lower one. Such changes can occur for solute diffusion through gels where a type of partitioning can happen, as gel properties can be sensitive to the concentration of the diffusing species. The flux across the gel is constant (at steady state) and is proportional to the concentration gradient. If this concentration gradient is larger on one side, the diffusion coefficient on this side must be smaller to maintain the constant flux. This physically could be due to a change in pore size in the gel which in turn changes the effective diffusivity seen and gives non-Fickian diffusion. This type of effect is discussed in Crank (1975) along with showing that if the diffusion coefficient varies with concentration the value of D deduced from a measurement of the steady state flow is some kind of mean value over the range of concentrations involved. In later work (Samprovalaki et al., 2012) a different form of analysis was used to account for the non-Fickian diffusion as the constant D value was not suitable to differentiate in all situations. Non-Fickian diffusion behaviour with gels has been apparent in the literature for a long time, for in-
stance, Bigwood (1930) noted that not only is diffusion in gels highly dependent on the absolute concentration of diffusing substance (in contrast to the classical theory that diffusion depends only on concentration gradients), but that it is both slow and unpredictable. Taylor et al. (1936) noted that the diffusion coefficient (D) decreased as the concentration of the diffusing material increased in the gel. However a 1D, constant Deff model is the easiest model to manipulate and in this work it is clearly able to show differences in diffusion behaviour due to temperature and to the type of diffusing species. Diffusion coefficient values for the same concentration of dye solution changed significantly with type of dye and temperature with significant interaction between the two factors (Two-way ANOVA was used). The differences observed between values (the Deff for methylene blue was higher than the one for rhodamine 6G) could be easily explained by the size of diffusing molecules. Methylene blue has a lower molecular weight than rhodamine 6G. Schwartzberg and Chao (1982) and Giannakopoulos and Guilbert (1986) have reported that diffusivities decreased when molecular weight increased. In a study of diffusion of developing materials in gelatine thin layers, such as for photographic film, (Iwano, 1969) a significant decrease in the diffusion coefficient with increased molecular size was noted. Muhr and Blanshard (1982) gave an idealised picture of the structure of hydrogels to study theories of solute transport through them. The structure of hydrogels was described as a mesh, with the spaces between the polymer chains filled with water. Agar gel structure is heterogeneous, which means that capillary forces are responsible for solvent retention, there is a great deal of inter-polymer interaction, and polymer chains are virtually immobile at the molecular level so the openings between chains are constant in size and location. Solute transport within agar occurs primarily within the water-filled regions in the space delineated by the polymer chains. The movement of solute will be affected by factors which reduce the size of these spaces, such as the size of the solute in relation to the size of the openings between polymer chains.
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(a)
180 exp model 160
140
120
100
80
60
40
0
50
100
150
200
250
TIME (min)
(b)
200 exp model
180 160 140 120 100 80 60 40 20
0
20
40
60
80
100
120
140
160
180
200
TIME (min) Fig. 6. Model fit was performed for measurements at the centre of the sample (X = 0), using Matlab 6.5. Grey intensity experimental data (e) and simulated data (–) (model line profiles) are plotted versus time for 1.5% agar with 3% CaCl2 with (a) rhodamine 6G 0.01% (w/v) at 50 °C and (b) methylene blue 0.01% (w/v) at 50 °C.
Table 2 Effective diffusion coefficient values (Deff 1010 m2 s1) in 1.5% agar gel with 3% CaCl2. Dye
Concentration (g/100 ml)
Temperature (°C) 30
50
70
Rhodamine 6G
0.002 0.004 0.006 0.008 0.01
2.89 2.70 2.61 2.89 4.88
5.36 3.75 4.51 3.27 5.17
4.60 5.45 7.63 6.31 11.22
Methylene blue
0.01
5.74
6.39
12.35
The diffusion coefficient of rhodamine 6G in ethanol has been found by Bilenberg et al. (2003) to be 1010 m2 s1. The diffusion coefficient of rhodamine 6G in water has been investigated by photon burst analysis by Schuster et al. (2000) and was found to be 2.5 (±1.7) 1010 m2 s1 at room temperature. Gendron et al. (2008)
calculated the effective diffusion coefficients of rhodamine 6G to be 4 1010 m2 s1. Veilleux and Coulombe (2010) reported an average diffusion coefficient of 3.3 1010 m2 s1 for rhodamine 6G in deionised water. Zhang et al. (2011) reported a mean diffusion coefficient of 4.0 1010 m2 s1 for rhodamine 6G in water. A similar order of magnitude is seen in this work although Westrin (1991) noted that the effective diffusion coefficient in a gel (Deff) is lower than the corresponding diffusion coefficient in water (Daq). There are two main reasons for this. First the polymer reduces the available volume (or area) to a fraction e of the total. This is referred to as the exclusion effect. Second, the impermeable segments of polymer molecules increase the path length for a diffusing solute. This is referred to as the obstruction effect. The increase in the path length is sometimes represented by the tortuosity factor s by analogy with the porous materials. Various models have been used to explain diffusion phenomena in gels. Models based on obstruction theory were more consistent with the experimental data for diffusion in heterogeneous hydrogels such as agar gels.
K. Samprovalaki et al. / Journal of Food Engineering 111 (2012) 537–545
These models assume that the presence of impenetrable polymer chains causes an increase in the path length for diffusive transport. The polymer chains act as a sieve allowing passage of a solute molecule only if it can pass between the polymer chains. In models based on this theory, the polymer chains are considered to be immobile and randomly embedded in a uniformly permeable continuous phase (Amsden, 1998a,b). 5. Conclusions Diffusion of dyes (rhodamine 6G and methylene blue) in gels of agar was investigated at three temperatures (30, 50, 70 °C) until equilibrium was reached. Ways to analyse and model the experimental data were presented and the diffusion coefficients of the dyes were estimated using Fick’s second law of diffusion. The calculated values of the diffusion coefficients were in correlation with values reported in the literature. The effect of temperature and size of the diffusing molecule was investigated. The estimated diffusion coefficient for methylene blue was higher than the one for rhodamine 6G, as expected, since methylene blue has a lower molecular weight than rhodamine 6G. Analysis of results suggests that the diffusion coefficient is not truly Fickian, and has variability with temperature and concentration of the dye solution. A different method of analysis of experimental results is presented elsewhere (Samprovalaki et al., 2012). Acknowledgements The authors thank Unilever Research Colworth, Bedford, UK for the financial support. Marianna Grammatika is acknowledged for the solution of the diffusion equation in two dimensions using finite element software (Femlab 2.3). References Agutter, P.S., Malone, P.C., Wheatley, D.N., 2000. Diffusion theory in biology: a relic of mechanistic materialism. Journal of the History of Biology 33, 71–111. Allali, H., Marchal, L., Vorobiev, E., 2010. Blanching of strawberries by ohmic heating: effects on the kinetics of mass transfer during osmotic dehydration. Food Bioprocess Technology 3, 406–414. Amsden, B., 1998a. Diffusion in hydrogels: mechanisms and models. Macromolecules 31, 8382–8395. Amsden, B., 1998b. Solute diffusion in hydrogels. An examination of the retardation effect. Polymer Gels and Networks 6, 13–43. Balaban, M., Pigott, G.M., 1988. Mathematical model of simultaneous heat and mass transfer in food with dimensional changes and variable transport parameters. Journal of Food Science 53 (3), 935–939. Bigwood, E.J., 1930. The distribution of ions in gels. Transactions of the Faraday Society 26, 704–719. Bilenberg, B., Kutter, J.P., Kristensen, A. (2003). Improved microfluidic design of an on-chip tunable dye laser. 7th International Conference on Miniaturised Chemical and Biochemical Analysis Systems, October 5–9, 2003, Squaw Valley, California, USA. Boudhrioua, N., Bonazzi, C., Daudin, J.D., 2003. Estimation of moisture diffusivity in gelatin-starch gels using time-dependent concentration-distance curves at constant temperature. Food Chemistry 82, 139–149. Crank, J., 1975. The Mathematics of Diffusion. Oxford University Press, New York. Cussler, E.L., 1984. Diffusion, Mass Transfer in Fluid Systems. Cambridge University Press, New York. Doulia, K., Tzia, K., Gekas, V., 2000. A knowledge base for the apparent mass diffusion coefficient (Deff) of foods. International Journal of Food Properties 3 (1), 1–14. El-Belghiti, K., Rabhi, Z., Vorobiev, E., 2005. Kinetic model of sugar diffusion from sugar beet tissue treated by pulsed electric field. Journal of the Science of Food and Agriculture 85, 213–218. Fernandez, D., Velezmoro, C., Zapata, C., 1995. Determination of sucrose diffusivity in papaya (Carica papaya L.) immersed in a sugar solution. Ciencia e Tecnologia de Alimentos 15, 246–250. Gendron, P.-O., Avaltroni, F., Wilkinson, K.J., 2008. Diffusion coefficients of several rhodamine derivatives as determined by pulsed field gradient – nuclear magnetic resonance and fluorescence correlation spectroscopy. Journal of Fluorescence 18, 1093–1101.
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Effect of moderate electric field pulses on the diffusion coefficient of soluble substances from apple slices. International Journal of Food Science and Technology 37, 73–86. Kemp, M.R., Fryer, P.J., 2007. Enhancement of diffusion through foods using alternating electric fields. Innovative Food Science and Emerging Technologies 8 (1), 143–153. Mattisson, C., Roger, P., Jönsson, B., Axelsson, A., Zacchi, G., 2000. Diffusion of lysozyme in gels and liquids. A general approach for the determination of diffusion coefficients using holographic laser interferometry. Journal of Chromatography 743, 151–167. Meinders, M.B.J., van Vliet, T., 2011. Oscillatory water sorption dynamics of bread crust. Food Research International 44, 2814–2821. Mittal, G.S., 1999. Mass diffusivity of food products. Food Reviews International 15 (1), 19–66. Muhr, A.H., Blanshard, J.M.V., 1982. Diffusion in gels. Polymer 23, 1012–1026. Mullet, A., 1994. Drying modelling and water diffusivity in carrots and potatoes. Journal of Food Engineering 22, 329–348. Narayanan, J., Xiong, J.-Y., Liu, X.-Y., 2006. Determination of agarose gel pore size: absorbance measurements vis a vis other techniques. Journal of Physics: Conference Series 28, 83–86. Odake, S., Hatae, K., Shimada, A., Iibuchi, S., 1990. Apparent diffusion coefficient of sodium-chloride in cubical agar-gel. Agricultural and Biological Chemistry 54 (11), 2811–2817. Ruiz, M.A., Salgado, M.A., Waliszewski, K.N., García, M.A., 1997. The effect of path diffusion on the effective moisture diffusivity in carrot slabs. Drying Technology 15 (1), 169–181. Ruiz-López, I.I., Ruiz-Espinosa, H., Arellanes-Lozada, P., Bárcenas-Pozos, M.E., García-Alvarado, M.A., 2012. Analytical model for variable moisture diffusivity estimation and drying simulation of shrinkable food products. Journal of Food Engineering 108, 427–435. Samprovalaki, K., Fryer, P.J., 2011. Development of an experimental procedure for real time investigation of diffusion in foods. Sensing and Instrumentation for Food Quality and Safety 5 (2), 78–89. Samprovalaki, K., Robbins, P.T., Fryer, P.J., 2012. A study of diffusion of dyes in model foods using a visual method. Journal of Food Engineering 110, 441–447. Schuster, J., Cichos, F., Wrachtrup, J., von Borczyskowski, C., 2000. Diffusion of single molecules close to interfaces. Single molecules 1, 299. Schwartzberg, H.G., 1980. Continuous counter-current extraction in the food industry. Chemical Engineering Progress 76 (4), 67–85. Schwartzberg, H.G., Chao, R.Y., 1982. Solute diffusivities in leaching processes. Food Technology 36 (2), 73–86. Sebti, I., Blanc, D., Carnet-Ripoche, A., Saurel, R., Coma, V., 2004. Experimental study and modelling of nisin diffusion in agarose gels. Journal of Food Engineering 63, 185–190. Simal, S., Mulet, A., Catalá, P.J., Cañellas, J., Rosello, C., 1996. Moving boundary model for simulating moisture movement in grapes. Journal of Food Science 61 (1), 157–160. Simal, S., Benedito, J., Sanchez, E.S., Rosello, C., 1998. Use of ultrasound to increase mass transport rates during osmotic dehydration. Journal of Food Engineering 36, 323–336. Spiro, M., Chong, Y.Y., 1997. The kinetics and mechanism of caffeine infusion from coffee: the temperature variation of the hindrance factor. Journal of the Science of Food and Agriculture 74, 416–420. Taylor, R.L., Herrmann, D.B., Kemp, A.R., 1936. Diffusion of water through insulating material. Industrial Engineering Chem. 28, 1011. Turhan, M., 1996. Modelling of salt transfer in white cheese during short initial brining. Netherlands Milk and Dairy Journal 50, 541–550. Veilleux, J., Coulombe, S., 2010. Mass diffusion coefficient measurements at the microscale: imaging a transient concentration profile using TIRF microscopy. International Journal of Heat and Mass Transfer 53, 5321–5329. Voogt, J.A., Hirte, A., Meinders, M.B.J., 2011. Predictive model to describe water migration in cellular solid foods during storage. Journal of the Science of Food and Agriculture 91, 2537–2543. Westrin, 1991. Diffusion in gels containing immobilised gels. Biotechnology and Bioengineering 38, 439–446. Zhang, Z., Nadezhina, E., Wilkinson, K.J., 2011. Quantifying diffusion in a biofilm of Streptococcus mutans. Antimicrobial Agents and Chemotherapy 55 (3), 1075– 1081. Zogzas, N.P., Maroulis, Z.B., 1996. Effective moisture diffusivity estimation from drying data. A comparison between various methods of analysis. Drying Technology 14 (7&8), 1543–1573.
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Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng
Investigation of the diffusion of dyes in agar gels K. Samprovalaki ⇑, P.T. Robbins, P.J. Fryer Chemical Engineering, The University of Birmingham, Edgbaston, B15 2TT Birmingham, UK
a r t i c l e
i n f o
Article history: Received 17 June 2011 Received in revised form 21 February 2012 Accepted 17 March 2012 Available online 24 March 2012 Keywords: Diffusion Fick’s law Image analysis
a b s t r a c t An experimental set-up and a measurement technique were developed so that diffusion in model foods (gels of agar) could be visualised and quantified. The diffusion of aqueous solutions of varying concentrations of two dyes (rhodamine 6G and methylene blue) in gels of agar was followed in situ at three temperatures (30, 50, 70 °C) until equilibrium was reached. The nature of the diffusion process (in terms of the amount of dye diffused into the gel) was studied using image analysis techniques. The diffusion coefficient, D, was estimated using Fick’s second law of diffusion and found to be 1010 m2 s1. The effect of the size of the diffusing molecule, as well as that of the processing temperature on diffusion was investigated and found to be significant. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Diffusion of components in foods is an important process in the food industry which is not fully understood. The major diffusion processes used in the food industry are: (i) Leaching, the transfer of solutes from a solid to an adjacent liquid which is used to extract sugar, vegetable oils, coffee, tea and other food solutes (Schwartzberg, 1980). (ii) Infusion, the transfer of solutes from a liquid into a solid which is used to transfer colours, flavours, curing and conditioning agents into foods (Schwartzberg and Chao, 1982). In extraction processes the structure of the food must be destroyed. In other thermal processes (for example, pasteurisation of milk or fruit juice) the aim is the preservation of vitamins and aroma producing compounds with as little effect on the structure of food as possible. The enhancement of mass transfer within foods may well be of significant strategic advantage industrially, if there are diffusional problems that, when overcome, may lead to production of improved or so called ‘‘enriched’’ products. This will be beneficial to both the consumer and the manufacturer. A major limitation of infusion techniques is the lack of successful measurement techniques to quantify the depth of penetration and localisation of infused material. Kemp and Fryer (2007) developed a technique based on image analysis and optical microscopy to visualise diffusion into solids under conditions of elevated temperatures and in the presence of alternating electric fields. ⇑ Corresponding author. Current address: Epidavrou 100, 104 41 Athens, Greece. Tel.: +30 6934816867; fax: +30 2105143160. E-mail address: [email protected] (K. Samprovalaki). 0260-8774/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfoodeng.2012.03.024
Diffusion as a scientific term refers to the net movement of matter, attributed to the random movements of molecules, taking place from a region of high concentration to one of lower concentration (Agutter et al., 2000). Like most scientific concepts, diffusion theory should not be construed as a fixed body of terms, laws and data, but rather as an evolving one. It has a ‘‘classical’’ form and numerous variants of this classical form developed ad hoc for specific applications in different disciplines. The classical theory of diffusion has both a macroscopic, phenomenological aspect, and a microscopic, mechanistic aspect (Agutter et al., 2000). The mathematical theory of diffusion in isotropic substances is based on the hypothesis that the rate of transfer of diffusing substance through unit area of a section is proportional to the concentration gradient. This is commonly called Fick’s 1st Law and can be written as (Crank, 1975):
F ¼ DrC
ð1aÞ
which can be written for a 1D slab geometry as:
F ¼ D
@C @x
ð1bÞ
To find the concentration profile and the rate of transfer, a mass balance is combined with Fick’s first law to give:
@C ¼ rðDrCÞ @t
ð2aÞ
or for a 1D slab geometry with constant diffusion coefficient, D:
@C @2C ¼D 2 @t @x
ð2bÞ
This is often called Fick’s second law. Fick’s law of diffusion is useful in studies of the solid, where concentrations vary with position and time (Cussler, 1984).
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Nomenclature C D Deff F G L T X
concentration of diffusing substance [% w/v] diffusion coefficient [m2 s1] effective diffusion coefficient [m2 s1] rate of transfer per unit area of section [kg m2 s1] grey intensity thickness of the slab [m] dimensionless time dimensionless distance in the x direction
Quantitative measurements of the rate at which a diffusion process occurs are usually expressed in terms of a diffusion coefficient (D). In some cases, for example diffusion in dilute solutions, D can be taken as constant, while in others, for example diffusion in high molecular weight polymers, D depends very markedly on concentration. Formally, a system is said to be Fickian only when the diffusion coefficient is constant, i.e. independent of concentration. Analytical solutions of Eq. (2b) covering a range of specimen geometries can be found in Crank (1975). However, the application of these principles to food systems is usually difficult because of the complex and heterogeneous structure of foods, which may change during processing or storage. Exact analytical solutions cannot always be found due to: (i) non-constant diffusion coefficient, as D can be a function of concentration, temperature, moisture content, water activity, structure of food, etc. and (ii) the shape change of the food system. In a porous medium like food, the measured diffusion coefficient will be significantly smaller than the molecular diffusion coefficient because of tortuosity effects (the more tortuous the region the longer the route between two points) and because of the hindering effects of the surface of the pores on the molecule’s random oscillations. This is why a functional form of the diffusion coefficient called effective diffusion coefficient (Deff), that encompasses all ignorance of the pores’ geometry, and is a function of the local geometry as well as of the solvent and solute, is found by comparing Eq. (2b) to experimental data with relevant geometry and initial and boundary conditions. If an estimate of Deff is available, then a mass transfer simulation with the relevant initial and boundary conditions can be performed and the concentration profile within the food with time can be obtained. By using Deff, diffusion at the higher length scales investigated can be treated as one-dimensional although at a cellular level it is three-dimensional (Crank, 1975). This type of approximation often works well if the distances over which diffusion occurs are large compared to the size of the pores, as is the case for this work, where agar gel pore size is of nanometre order of magnitude (Narayanan et al., 2006). A number of papers have been published in which diffusion coefficients in foods are calculated or estimated. Allali et al. (2010) used an approximate solution of Fick’s second law for unsteady state mass transfer in spherical configuration to calculate the Deff of water and sucrose in the osmotic dehydration of strawberries and values were found to be in the range of 1010 m2 s1. The increase of temperature and initial sucrose concentration led to an enhancement of water and sugar diffusivities. Meinders and van Vliet (2011) showed that oscillatory sorption experiments of bread crusts could be very well described by a Fickian diffusion model applied to spherical particles, assuming a constant diffusion coefficient. Voogt et al. (2011) showed that experimentally observed dynamical water profiles of bread rolls that differed in crust permeability were predicted well by a Fickian diffusion model with a single Deff. Spiro and Chong (1997) calculated the intra-bean diffusion coefficients for caffeine (molecular weight 194.19 g mol1) by solving Fick’s second law of diffusion and found values in the
H Z t x y c n
dimensionless distance in the y direction grey intensity ratio time [s] distance in the x direction [m] distance in the y direction [m] dimensionless concentration number of series
order of 1010 m2 s1. Ruiz-López et al. (2012) developed an analytical model based on Fick’s second law of diffusion to evaluate mass transfer properties and describe drying kinetics of chayote slices. Water diffusivities were estimated as 4.44 1010, 6.00 1010, 7.04 1010 and 8.60 1010 m2 s1 for drying processes at 40, 50, 60 and 70 °C, respectively. El-Belghiti et al. (2005) studied the extraction of sugar from sugar beet slices following various pulsed electric field (PEF) treatments and calculated the diffusion coefficient using Fick’s diffusion equation to be 1.68 ± 0.40 1010 m2 s1. By heating the solution at mild temperatures of 30, 40 and 50 °C, the coefficient of diffusion was increased and the kinetics of extraction was enhanced. A time of about 70 min was needed to obtain a yield of 93% at ambient temperature and around 40 min to obtain the same yield at 50 °C. Jemai and Vorobiev (2002) investigated the effects of a moderate electric field pulse (MEFP) on the diffusion coefficient of soluble substances from apple slices and found that an electrical pretreatment with moderate temperature elevation (10–15 °C) combined with a low thermal treatment, significantly enhanced the diffusion coefficient (Del) compared with reference values. For instance, after a standard MEFP treatment (500 V cm1, 1000 pulses, 100 ls duration) Del was 3.9 1010 m2 s1 at 20 °C compared with reference values of 2.5 and 4.4 1010 m2 s1 for untreated and denatured samples, respectively, and 13.4 1010 m2 s1 at 75 °C compared with 10.2 1010 m2 s1 for thermally denatured samples. Values of diffusion coefficients in foods and agar gels are shown in Table 1. As there is no standardised method for estimating diffusivity, the published values present variability, due to both the diversity of experimental and calculation methods used in each case, and of product structure and composition (Boudhrioua et al., 2003). The aim of this work was to develop a measurement technique as well as a simple diffusional model to analyse dye diffusion in gels. In this work, two dye types and temperature effects are investigated, while in later work (Samprovalaki et al., 2012) the effect of electric fields on diffusion seen by Kemp and Fryer (2007) was studied in more detail. The objectives of this project were to study diffusion in gels, the effect of temperature and size of diffusing molecule on diffusion, and the application of Fickian theory on interpreting the results.
2. Materials and methods Experiments were designed to investigate the diffusion of dyes into agar gels at three temperatures (30, 50 and 70 °C). The experimental set-up was designed to enable visualisation of diffusion and is represented schematically in Fig. 1. It consisted of the experimental cell, a camera with control box [Photonic Science High Sensitivity Mono CoolView camera (Photonic Science Ltd., UK)], ring lights, a Peltier stage (LTS 120 Stage with PE 94 Controller and Linksys software, Linkam Scientific Instruments Ltd., UK), a temperature logging interface, and computers with software for image
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K. Samprovalaki et al. / Journal of Food Engineering 111 (2012) 537–545 Table 1 Diffusion coefficient values in food and agar gels as reported in literature. Product
Diffusant
Deff 1010 (m2 s1)
Temperature (°C)
Reference
Sugar beet Sugar beet Papaya Carrot Carrot Carrot Potato Potato Potato Grapes Mangoes Cassava Fish Cod (frozen) Cod (brined) Herring Mackerel Plaice Pork sausage Apple Apple Apple Gouda cheese White cheese Cheese 0.79% agar gel Agar gel Agar gel Agarose gel (4%) Agarose gel (3%) Agarose gel Agarose gel Agarose gel Agarose gel
Sucrose Water Sucrose Sucrose Water Water Water Water Water Water Water Water Water Water Water Water Water Water Water Water Sucrose Water Salt Salt Salt Sucrose Amaranth dye Sorbic acid Lysozyme Nisin Thiamine Glucose Acetic acid Chromate
2.5 0.4–1.3 28.95 14 7.5 9.4 3.8 8.3 2.6 2.2 8.6 4.9 5.2 2.5–70 0.3–2.3 0.1–1.9 1.1–2.2 2.9 47–57 8.1 0.79 0.9–28 230 510 1.0–5.5 2.47 0.6–4.9 6.7–7.3 90 19.2–81.4 4.3–16.7 4.5–13.2 7.5–22.0 9.0–11.10
23–25 40–84 35 80 60 60 60 60 60 60 60 60 60 5 to (30) 20 30 30 30 20 60 40–70 30–70 20 20 4–60 5 – – 25 5.4–22.3 20–80 – – 20–70
Schwartzberg and Chao (1982) Doulia et al. (2000) Fernandez et al. (1995) Mittal (1999) Mulet (1994), Ruiz et al. (1997) Zogzas and Maroulis (1996) Husain et al. (1972) Zogzas and Maroulis (1996) Mulet (1994) Simal et al. (1996) Hernandez et al. (2000) Hernandez et al. (2000) Balaban and Piggot (1988) Doulia et al. (2000) Doulia et al. (2000) Doulia et al. (2000) Doulia et al. (2000) Doulia et al. (2000) Doulia et al. (2000) Zogzas and Maroulis (1996) Simal et al. (1998) Doulia et al. (2000) Geurts et al. (1974) Turhan (1996) Doulia et al. (2000) Schwartzberg and Chao (1982) Doulia et al. (2000) Doulia et al. (2000) Mattisson et al. (2000) Sebti et al. (2004) Doulia et al. (2000) Doulia et al. (2000) Doulia et al. (2000) Doulia et al. (2000)
Data Logging Computer Camera Control Box
TC-08 Temperature Logging Interface
Camera
Ring Lights Peltier Stage Controller
Sample
Peltier Stage Thermocouples Fig. 1. Schematic representation of the experimental set-up used to perform diffusion experiments and measurements.
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40 mm Dye reservoirs
10 mm 25 mm
5mm
5 mm 10 mm Aluminium sheet
PVC coating
Gel
Thermocouples Fig. 2. Schematic representation of the experimental cell.
acquisition data collection [Image Pro-Plus 4.5 (Media Cybernetics, Inc., USA)]. The experimental cell is depicted in Fig. 2 and consists of an aluminium sheet 45 mm long, 30 mm wide and 1.5 mm thick covered on one side with a white PVC layer. Aluminium was chosen so that heat would be transferred quickly between the cell and the Peltier stage. A cavity of the shape and dimensions shown in Fig. 2 was bored into the plastic. Agar gel slabs were made by mixing agar powder (1.5% w/v) (Fluka 05039, UK, gelling temp. 33.8 °C) with calcium chloride dihydrate (3% w/v) (Sigma Aldrich Co. Ltd., UK, C-3881) and distilled water and heating at 80–90 °C until the powder dissolved. These gels are isotropic with no directional differences in their structure (Odake et al., 1990). Thus all the problems associated with the anisotropy of food could be excluded. CaCl2 was added so that electrical conductivity was adjusted. Agar gels used in the experiments were tested by differential scanning calorimetry (Micro DSC II, SETARAM Instrumentation); gels did not melt until they were reheated above 80 °C which is above the highest temperature used in these experiments, therefore no change in the physical state of the gels occurred during the experiments. Dye solutions of rhodamine 6G and methylene blue dyes (both from BDH chemical, Poole) were used for each experiment so that the effect of the mass of species on diffusion could be evaluated. Methylene blue is a dye of similar charge to rhodamine 6G (+1) but of lower molecular weight (319.85 and 479.02 g/mol, respectively). Stock solutions of dye were made by dilution in distilled water. Concentrations of rhodamine 6G used were: 0.002, 0.004, 0.006, 0.008, and 0.01% w/v. The concentration of methylene blue used was 0.01% w/v. CaCl2 was added in the dye solutions to the same concentration as that in the gel so that the electrical conductivities of the solid and liquid phases were identical, giving uniform field strength, this was important for later work when electric fields were applied (Samprovalaki et al., 2012). Gel samples set in the experimental cell so that the sample volume was controlled accurately. Gel solutions were kept warm, in a conical flask, on a hotplate until they were used to form gels. The experimental procedure is described below: (i) The two reservoirs were first covered with two pieces of plastic as shown in Fig. 2 so that only the cavity in the middle of the cell could receive the gel. (ii) A syringe and needle (Microlance 3, 19GA2) was used to take the solution out of the flask and extrude it into the cell so that air inclusion was minimised.
(iii) The cell was then immediately covered with a 10 mm diameter circular microscope cover glass (thickness No. 0, BDH, Merck Eurolab Ltd., Poole, UK) so that: (i) evaporation was avoided, (ii) the thickness of the sample was always the same and (iii) the dye was prevented from travelling on top of the gel slab. (iv) The gel solution was locked in the centre of the cell and allowed to set to form a gel of dimensions 10 mm long, 5 mm wide and 1 mm thick. (v) Once the gel had set, the two plastic pieces were removed and the cell was mounted on the Peltier cooling stage under the camera. Zinc oxide heat transfer paste was used to ensure good thermal contact between the cell and the stage. The cooling stage was already programmed at the required temperature before the cell was placed on it. (vi) The temperature and image acquisition software were then initialised. (vii) The dye, which was kept in a water bath at the same set temperature, was then added in the two reservoirs simultaneously. (viii) Diffusion progress was then visualised by capturing images every 10 min until the agar gel was saturated with dye and equilibrium was reached (approximately 2 h 30 min). Experiments were performed in triplicates. Image analysis software allows a colour image to be captured in monochrome and displayed in grey-scale as shown in Fig. 3. Resolution was 8-bit which means that there are 28 shades (256 shades, where value 255 stands for pure white and value 0 for black). A line profile is selected on this image, i.e. the computer calculates and displays a number for each pixel along a selected line depending on the shade of grey of that pixel. The system can measure over a length scale comparable to the pixel length. In all cases presented here 80 pixels correspond to 1 mm. The area which gave uniform grey intensity values and was used in the analysis is 100 pixels wide as shown in Fig. 3 (Samprovalaki and Fryer, 2011). For every pixel along the x-axis the software calculates the average grey intensity along the 100-pixel y-axis line. The variation in the mean value of grey intensity at fixed distances from the centre of the slab across the x-direction was then studied versus time. The change in grey intensity values measured during the experimental run can be due to two reasons: (i) the concentration of dye is changing, (ii) the optical properties of the image (e.g. shadowing, change in light source intensity, camera/software image correction).
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dye reservoirs y
x
Horizontal Centre line
100 pixels
Gel slab
X=-0.4
X=+0.4
X=-0.8
X=+0.8 Vertical Centre line X=0
Fig. 3. Area of analysis. The software calculates the average grey intensity along the 100-pixel y-axis line at selected distances from the centre of the slab along the x-axis and plots it versus time. The grey scale intensity values with time were used for the calculation of the diffusion coefficient of rhodamine 6G at five positions in the agar gel slab (i.e. centre 2 mm, centre 1 mm, centre, centre + 1 mm, centre + 2 mm which correspond to dimensionless distances X = 0.8, X = 0.4, X = 0, X = +0.4 and X = +0.8, respectively).
If changes in the optical properties of the image can be neglected, grey scale intensity, G, is a function of the concentration, C, of the dye in the gel. 3. Data analysis methods
1 ð1Þn C C0 4P ¼1 C1 C0 p n¼0 2n þ 1 ( ) Dð2n þ 1Þ2 p2 t ð2n þ 1Þpx cos exp 2 2l 4l
ð3Þ
3.1. Analysis of diffusion
The boundary conditions can be written in terms of the following dimensionless parameters:
To generate a diffusional model based around Eq. (2b) for comparison with experimental data there are four areas to consider:
c¼
(i) (ii) (iii) (iv)
geometry, boundary and initial conditions, functionality of the diffusion coefficient, D, conversion of the measured value.
To be able to generate an analytic solution, then a 1 dimensional (1D) geometry, a constant diffusion coefficient, D, the same initial condition at all points and a constant boundary value are desirable. For the experimental set up here the geometry is slab like and the material is isotropic, initial concentration in the gel is 0 and the boundaries are held at a constant value. This leaves if a 1D or 2D solution is necessary and if a constant diffusion coefficient gives a good fit to the experimental data.
T¼ X¼
C C0 C1 C0 Dt
ð4aÞ
ð4bÞ
L2 x L
ð4cÞ
Then Eq. (3) becomes (Crank, 1975):
c¼
C C0 C1 C0
¼1
( ) 1 ð1Þn 4P ð2n þ 1Þ2 p2 ð2n þ 1Þp exp T cos X p n¼0 2n þ 1 4 2
ð5Þ
with boundary conditions: T = 0 all X, c = 0 and T > 0 X = ±1, c = 1 3.1.1. Solution in one-dimension It is possible to obtain solutions of Eq. (2b) for one-dimensional diffusion in a medium bounded by two parallel planes. These solutions apply in practice to diffusion into a plane sheet of material so that effectively all the diffusing substance enters through the plane faces and a negligible amount through the edges. Very often the problem is symmetrical about the central plane of the sheet (as is this case), and the formulae are then most convenient if the centre is taken as x = 0 and the surfaces at x = ±L. The gel is approximated as an infinite slab of thickness 2L, with an initial concentration, C0 in the region L < x < L and a constant concentration of C1 at the boundary surfaces. Crank (1975) has given the solution in the following form:
3.1.2. Solution in two-dimensions It is possible to obtain solutions of Eq. (2) for diffusion in two dimensions using finite difference techniques but it is more easily solved in commercial finite element software. For this work, FEMLAB™ (now Comsol multiphysics™) was used. A characteristic picture of this analysis is shown in Fig. 4(a). The diffusion profile produced by the model is similar to that observed in the videos of the experiments as shown in Fig. 4(b). This confirmed the fact that two-dimensional diffusion is taking place. However, since the one-dimensional solution is easier to manipulate, it was considered useful to determine the range of conditions under which the assumption of one-dimensional diffusion can be used.
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Fig. 4. (a) Characteristic picture of the solution of the diffusion equation in two dimensions using finite element software (Femlab 2.3). The diffusion profile produced by the model is similar to that observed in the videos of the experiments as shown in (b).
3.2. Identification of appropriate range for 1D approximation The solutions of Eq. (5) for diffusion in one (1-D) and two dimensions (2-D) were compared and the results of this comparison are plotted in Fig. 5. Dimensionless concentration, c, is plotted against dimensionless time, T, for two dimensionless distances, one in the centre, X0, and one at X = 0.8, X0.8, as shown in Fig. 3. The graph in Fig. 5 shows that for dimensionless time T < 0.2, the lines of 1-D model and 2-D model coincide. In other words, if the run of the experiment lasts for T < 0.2, either the 1-D or 2-D model could be used for the analysis. The time t of the experiment corresponding to dimensionless time T can be calculated using Eq. (4b). Assuming a diffusion coefficient in the range of 1 1010 m2 s1 which is typical in food, as shown in Table 1, and 2L = 5 mm, Eq. (4b) gives:
t¼
TL2 0:2 ð2:5 103 Þ2 ¼ 12; 500 s 208 min ¼ D 1010
Therefore, for the analysis of the experimental data as long as the time does not exceed 3 h either 1-D or 2-D solutions can be used. 3.3. Estimation of the effective diffusion coefficient The grey scale intensity change with time and position was used for the calculation of the effective diffusion coefficient as all
measurements project the change of the local concentration of the dye within the gel with time and position. As the grey scale intensity varies linearly with concentration, for the range of dye concentrations (<0.01% w/v) used here (Samprovalaki and Fryer, 2011), then the concentration terms in Eq. (5) can be directly replaced by the grey intensity.
c¼Z¼ ¼1
G G0 G1 G0
( ) 1 ð1Þn 4P ð2n þ 1Þ2 p2 ð2n þ 1Þp exp T cos X p n¼0 2n þ 1 4 2
ð6Þ
Eq. (6) has summation terms which make direct calculation of Deff difficult. Therefore an estimation of the diffusion coefficient was attempted. Data fitting was carried out using Eq. (6) and the sum of the square of the differences technique (using a fitting routine programmed in Matlab 6.5). A value of D was guessed (1 1010 m2 s1); a solution for Z(x, t) was obtained and compared to the measured value of Z. The model line was adjusted by altering the diffusion coefficient until the value of the sum of the squares reached a minimum. This value of Deff, which minimises the least squares error and gives the best fit between the model and experimental curves is the estimated value of Deff. In all cases, there was an initial under-prediction of concentration (i.e. D was too low) followed by an over-prediction (i.e. D was too high),
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1
DIMENSIONLESS CONCENTRATION (c)
X=0.8 0.8 1D 0.6 2D 0.4 X=0
0.2
0
-0.2 0
0.1
0.2
0.3
0.4
0.5
0.6
2
DIMENSIONLESS TIME (Dt/L ) Fig. 5. Comparison of the solutions of the diffusion equation in one and two dimensions. Dimensionless concentration, c, is plotted against dimensionless time, T, for two dimensionless distances, one in the centre, X0 and one at X = 0.8 from the boundary. (+): 1-D diffusion model at X0, (s): 2-D diffusion model at X0, (x): 1-D diffusion model at X0.8, (e): 2-D diffusion model at X0.8.
example shown in Fig. 6. This shows that a constant diffusion coefficient is not the best model for the experimental data, but the best fit D value still gives some useful information as discussed below. 4. Results and discussion Eq. (6) was solved for five different positions in the slab (centre 2 mm, centre 1 mm, centre, centre + 1 mm, centre + 2 mm) and the mean Deff estimated for each dye solution concentration and temperature are tabulated in Table 2 for both rhodamine 6G and methylene blue. It is obvious that Deff changes with temperature and concentration of the diffusing solution. Two-way ANOVA was used to evaluate the difference with temperature and concentration which was found to be significant (at 95% confidence interval), with significant interaction between the two factors, for both rhodamine 6G and methylene blue experiments. From Fig. 6 and Table 2 it is obvious that Deff depends on concentration as it suddenly drops from a high value to a lower one. Such changes can occur for solute diffusion through gels where a type of partitioning can happen, as gel properties can be sensitive to the concentration of the diffusing species. The flux across the gel is constant (at steady state) and is proportional to the concentration gradient. If this concentration gradient is larger on one side, the diffusion coefficient on this side must be smaller to maintain the constant flux. This physically could be due to a change in pore size in the gel which in turn changes the effective diffusivity seen and gives non-Fickian diffusion. This type of effect is discussed in Crank (1975) along with showing that if the diffusion coefficient varies with concentration the value of D deduced from a measurement of the steady state flow is some kind of mean value over the range of concentrations involved. In later work (Samprovalaki et al., 2012) a different form of analysis was used to account for the non-Fickian diffusion as the constant D value was not suitable to differentiate in all situations. Non-Fickian diffusion behaviour with gels has been apparent in the literature for a long time, for in-
stance, Bigwood (1930) noted that not only is diffusion in gels highly dependent on the absolute concentration of diffusing substance (in contrast to the classical theory that diffusion depends only on concentration gradients), but that it is both slow and unpredictable. Taylor et al. (1936) noted that the diffusion coefficient (D) decreased as the concentration of the diffusing material increased in the gel. However a 1D, constant Deff model is the easiest model to manipulate and in this work it is clearly able to show differences in diffusion behaviour due to temperature and to the type of diffusing species. Diffusion coefficient values for the same concentration of dye solution changed significantly with type of dye and temperature with significant interaction between the two factors (Two-way ANOVA was used). The differences observed between values (the Deff for methylene blue was higher than the one for rhodamine 6G) could be easily explained by the size of diffusing molecules. Methylene blue has a lower molecular weight than rhodamine 6G. Schwartzberg and Chao (1982) and Giannakopoulos and Guilbert (1986) have reported that diffusivities decreased when molecular weight increased. In a study of diffusion of developing materials in gelatine thin layers, such as for photographic film, (Iwano, 1969) a significant decrease in the diffusion coefficient with increased molecular size was noted. Muhr and Blanshard (1982) gave an idealised picture of the structure of hydrogels to study theories of solute transport through them. The structure of hydrogels was described as a mesh, with the spaces between the polymer chains filled with water. Agar gel structure is heterogeneous, which means that capillary forces are responsible for solvent retention, there is a great deal of inter-polymer interaction, and polymer chains are virtually immobile at the molecular level so the openings between chains are constant in size and location. Solute transport within agar occurs primarily within the water-filled regions in the space delineated by the polymer chains. The movement of solute will be affected by factors which reduce the size of these spaces, such as the size of the solute in relation to the size of the openings between polymer chains.
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(a)
180 exp model 160
140
120
100
80
60
40
0
50
100
150
200
250
TIME (min)
(b)
200 exp model
180 160 140 120 100 80 60 40 20
0
20
40
60
80
100
120
140
160
180
200
TIME (min) Fig. 6. Model fit was performed for measurements at the centre of the sample (X = 0), using Matlab 6.5. Grey intensity experimental data (e) and simulated data (–) (model line profiles) are plotted versus time for 1.5% agar with 3% CaCl2 with (a) rhodamine 6G 0.01% (w/v) at 50 °C and (b) methylene blue 0.01% (w/v) at 50 °C.
Table 2 Effective diffusion coefficient values (Deff 1010 m2 s1) in 1.5% agar gel with 3% CaCl2. Dye
Concentration (g/100 ml)
Temperature (°C) 30
50
70
Rhodamine 6G
0.002 0.004 0.006 0.008 0.01
2.89 2.70 2.61 2.89 4.88
5.36 3.75 4.51 3.27 5.17
4.60 5.45 7.63 6.31 11.22
Methylene blue
0.01
5.74
6.39
12.35
The diffusion coefficient of rhodamine 6G in ethanol has been found by Bilenberg et al. (2003) to be 1010 m2 s1. The diffusion coefficient of rhodamine 6G in water has been investigated by photon burst analysis by Schuster et al. (2000) and was found to be 2.5 (±1.7) 1010 m2 s1 at room temperature. Gendron et al. (2008)
calculated the effective diffusion coefficients of rhodamine 6G to be 4 1010 m2 s1. Veilleux and Coulombe (2010) reported an average diffusion coefficient of 3.3 1010 m2 s1 for rhodamine 6G in deionised water. Zhang et al. (2011) reported a mean diffusion coefficient of 4.0 1010 m2 s1 for rhodamine 6G in water. A similar order of magnitude is seen in this work although Westrin (1991) noted that the effective diffusion coefficient in a gel (Deff) is lower than the corresponding diffusion coefficient in water (Daq). There are two main reasons for this. First the polymer reduces the available volume (or area) to a fraction e of the total. This is referred to as the exclusion effect. Second, the impermeable segments of polymer molecules increase the path length for a diffusing solute. This is referred to as the obstruction effect. The increase in the path length is sometimes represented by the tortuosity factor s by analogy with the porous materials. Various models have been used to explain diffusion phenomena in gels. Models based on obstruction theory were more consistent with the experimental data for diffusion in heterogeneous hydrogels such as agar gels.
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These models assume that the presence of impenetrable polymer chains causes an increase in the path length for diffusive transport. The polymer chains act as a sieve allowing passage of a solute molecule only if it can pass between the polymer chains. In models based on this theory, the polymer chains are considered to be immobile and randomly embedded in a uniformly permeable continuous phase (Amsden, 1998a,b). 5. Conclusions Diffusion of dyes (rhodamine 6G and methylene blue) in gels of agar was investigated at three temperatures (30, 50, 70 °C) until equilibrium was reached. Ways to analyse and model the experimental data were presented and the diffusion coefficients of the dyes were estimated using Fick’s second law of diffusion. The calculated values of the diffusion coefficients were in correlation with values reported in the literature. The effect of temperature and size of the diffusing molecule was investigated. The estimated diffusion coefficient for methylene blue was higher than the one for rhodamine 6G, as expected, since methylene blue has a lower molecular weight than rhodamine 6G. Analysis of results suggests that the diffusion coefficient is not truly Fickian, and has variability with temperature and concentration of the dye solution. A different method of analysis of experimental results is presented elsewhere (Samprovalaki et al., 2012). Acknowledgements The authors thank Unilever Research Colworth, Bedford, UK for the financial support. Marianna Grammatika is acknowledged for the solution of the diffusion equation in two dimensions using finite element software (Femlab 2.3). References Agutter, P.S., Malone, P.C., Wheatley, D.N., 2000. Diffusion theory in biology: a relic of mechanistic materialism. Journal of the History of Biology 33, 71–111. Allali, H., Marchal, L., Vorobiev, E., 2010. Blanching of strawberries by ohmic heating: effects on the kinetics of mass transfer during osmotic dehydration. Food Bioprocess Technology 3, 406–414. Amsden, B., 1998a. Diffusion in hydrogels: mechanisms and models. Macromolecules 31, 8382–8395. Amsden, B., 1998b. Solute diffusion in hydrogels. An examination of the retardation effect. Polymer Gels and Networks 6, 13–43. Balaban, M., Pigott, G.M., 1988. Mathematical model of simultaneous heat and mass transfer in food with dimensional changes and variable transport parameters. Journal of Food Science 53 (3), 935–939. Bigwood, E.J., 1930. The distribution of ions in gels. Transactions of the Faraday Society 26, 704–719. Bilenberg, B., Kutter, J.P., Kristensen, A. (2003). Improved microfluidic design of an on-chip tunable dye laser. 7th International Conference on Miniaturised Chemical and Biochemical Analysis Systems, October 5–9, 2003, Squaw Valley, California, USA. Boudhrioua, N., Bonazzi, C., Daudin, J.D., 2003. Estimation of moisture diffusivity in gelatin-starch gels using time-dependent concentration-distance curves at constant temperature. Food Chemistry 82, 139–149. Crank, J., 1975. The Mathematics of Diffusion. Oxford University Press, New York. Cussler, E.L., 1984. Diffusion, Mass Transfer in Fluid Systems. Cambridge University Press, New York. Doulia, K., Tzia, K., Gekas, V., 2000. A knowledge base for the apparent mass diffusion coefficient (Deff) of foods. International Journal of Food Properties 3 (1), 1–14. El-Belghiti, K., Rabhi, Z., Vorobiev, E., 2005. Kinetic model of sugar diffusion from sugar beet tissue treated by pulsed electric field. Journal of the Science of Food and Agriculture 85, 213–218. Fernandez, D., Velezmoro, C., Zapata, C., 1995. Determination of sucrose diffusivity in papaya (Carica papaya L.) immersed in a sugar solution. Ciencia e Tecnologia de Alimentos 15, 246–250. Gendron, P.-O., Avaltroni, F., Wilkinson, K.J., 2008. Diffusion coefficients of several rhodamine derivatives as determined by pulsed field gradient – nuclear magnetic resonance and fluorescence correlation spectroscopy. Journal of Fluorescence 18, 1093–1101.
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