Marinated or pickled eggs are enjoyed in cultures across the world, and recipes for these eggs use diffusion to saturate the egg white of a hard-boiled egg with sauce. Inspired by the marinated egg, this experiment demonstrates diffusion in an easy and quantitative way. Peeled hard-boiled eggs are placed in a dye solution, and the penetration distance of the dye into the egg white is measured as a function of time using image processing. The penetration distances are consistent with the expected dependence on the square root of time, and the diffusion of the dye occurs faster at higher temperatures as expected. This experiment provides a new way to demonstrate diffusion-based mass transfer visually and to make a connection between physics and culture.

Cooking involves diverse aspects of science, and different cultures often have similar food recipes relying on the same scientific principle. For example, Korean soy sauce braised eggs (GyeRan JangJoRim) are prepared by simmering hard-boiled eggs in a soy-sauce-based marinade for a short period of time. Similarly, recipes for Japanese Ramen eggs (Aji Tamago) marinate hard-boiled eggs in a similar sauce for hours in a refrigerator. Western pickled eggs and Chinese salted eggs are also similar. The common physical principle of these marinated eggs from different cultures is that their cooking recipes rely on diffusion of sauces through egg whites.1 

Diffusion is an important mass transfer mechanism by which a substance is transported from a region of high concentration to a region of low concentration as a result of random molecular motion driven by thermal energy.2 Molecules move randomly (i.e., Brownian motion) driven by their thermal energy in a medium of identical molecules.3 This phenomenon called self-diffusion occurs even without any concentration gradients. When there is a concentration gradient of the molecules, the random motion of each individual molecule results in a directed flux in the direction of the concentration gradient. In this chemical or transport diffusion, the concentration distribution of the diffusing substance through an isotropic medium is described by Fick's second law of diffusion: ∂C/∂t = D2C. Here, C is concentration, t is time, and D is the diffusion coefficient. For diffusion into a semi-infinite medium, the following square-root relationship is valid if the initial concentration is zero and the surface concentration remains constant: the distance of penetration δ is proportional to the square root of time and the diffusion coefficient: δ ∝ (Dt)1/2. As D increases with temperature, δ increases with temperature.

A simple demonstration of diffusion is to release an odor in air, or a drop of dye in stationary water, and then to measure the traveling distance of the odor or dye over time by students' noses or eyes.4 The diffusion coefficient of the diffusing substance can be estimated by plotting the measured distances vs time and then curve fitting the plot. These demonstrations illustrate the phenomenon of diffusion, but it is challenging to measure the traveling distance of odor or dye accurately, eliminating the effect of advection on the transport of odor or dye, and to show the effect of temperature on diffusion.

We have developed a new demonstration of diffusion using peeled hard-boiled eggs and food dye, inspired by the aforementioned cooking methods for marinated eggs. The eggs sit in the dye solution for predetermined time intervals, and a single egg is taken out at different time points. The egg is sliced in half and then the cross-section is imaged. From the images, the penetration distance of the dye into the egg white is measured by image processing. Then, the square-root relationship (δt1/2) is confirmed by fitting it to the obtained data. The experiment is then repeated at different temperatures to demonstrate the effect of ambient temperature on diffusion.

A dye solution was prepared by mixing 25 ml of red food dye (Tone's red food color) with 1.25 l of water. The concentration of the dye had no noticeable effect on the diffusion behavior of the dye into egg whites, but too low concentration is not desirable for imaging. Peeled hard-boiled eggs, which were purchased at a local grocery store, were moved directly from the refrigerator into the dye solution. A container (190 mm in diameter and 90 mm in height) allowed 12–15 eggs to be fully immersed in the dye solution. The container with the eggs was covered with plastic wrap and aluminum foil to prevent contamination and evaporation and then stored at 4 °C (in a refrigerator), 21 °C (at room temperature), or 60 °C (in a convection oven).

One egg was taken out of the solution at predetermined times and imaged. Time intervals were chosen for each of the different temperatures to ensure that a representative diffusion curve could be obtained. For imaging, the egg was first washed under cold water briefly and then dried using Kim wipes. The egg was then sliced in half along a longitudinal axis using an egg slicer (Beaverve Egg Slicer with stainless steel cutting wires). Care was taken to ensure that the part of the egg touching the container in the dye solution was not exposed in the sliced cross section and that the egg slicer was thoroughly cleaned to remove any dyed remnants from previous eggs.

Egg images were taken on a LED backlight (LuxPad22, Nanguang) in a photo light box (Samtian) to obtain consistent images of the eggs. In the light box, the LED backlight was placed to face upward, and a circular stand holding the egg was placed at the center of the LED backlight. The stand was smaller than the egg, so that it did not show up in images. A small bubble level was used to ensure that the cross section of the egg was parallel to the ground.

A digital camera (Canon EOS Rebel T3) with a zoom lens (Tamron AF28-75 mm f/2.8 Macro) was placed above the light box using a tripod. A bubble level was used to position the camera such that the camera was level with the ground, ensuring that the camera's optical axis was perpendicular to the cross-sectional surface of the egg. The camera was set to a shutter speed of 1/100 s, an ISO value of 100, and a lens aperture of f8 in the manual mode. The image was taken using the camera timer to avoid any disturbances from the shutter button being pressed. Then, the process was repeated for the other half of the egg. An example image can be seen in Fig. 1(a).

Fig. 1.

Image processing steps. (a) Original image of an egg immersed at 21 °C for 5 h. (b) Grayscale image with intensity adjustment. (c) Black and white or binary image. (d) Processed image with the identified boundaries of the dye band. In the enlarged inset, blue lines show the inner and outer boundaries of the egg white area penetrated by the red food dye. For each pixel on the inner boundary, the minimum distance to the outer boundary was found, and a selection of these distances is illustrated with green lines in the figure. The penetration distance was the average of all these distances. In this case, it was 0.99 mm.

Fig. 1.

Image processing steps. (a) Original image of an egg immersed at 21 °C for 5 h. (b) Grayscale image with intensity adjustment. (c) Black and white or binary image. (d) Processed image with the identified boundaries of the dye band. In the enlarged inset, blue lines show the inner and outer boundaries of the egg white area penetrated by the red food dye. For each pixel on the inner boundary, the minimum distance to the outer boundary was found, and a selection of these distances is illustrated with green lines in the figure. The penetration distance was the average of all these distances. In this case, it was 0.99 mm.

Close modal

To maintain consistency across all images and all experimental trials, the light intensities of the lightbox and the backlight were not changed across any of the images or any of the experimental trials, and the position of the tripod, lightbox, backlight, and circular stand were marked and meticulously aligned. Also, eggs were imaged with a reference ruler (Fig. 1(a)) so that the pixel size of each image could be measured. To evaluate the uncertainty of the demonstration, the experiment was performed twice for the lowest and highest temperatures.

Images were processed using matlab to measure the penetration distance of the dye into the egg white. The image processing steps are summarized in Fig. 1. First, the original color image (Fig. 1(a)) was converted to a grayscale image using “rgb2gray,” and the intensity of the grayscale image was adjusted for maximum contrast using “imadjust,” as shown in Fig. 1(b). Second, the grayscale image was transformed into a black-and-white or binary image using “imbinarize” with a threshold of 0.3 as shown in Fig. 1(c). Because the images were taken under the same conditions, the threshold value was held constant across all images. Any speckles were removed from the binary image using “bwareafilt.” As shown in Fig. 1(c), the area of the egg white penetrated by the dye, or the dye band, appeared white. Third, the inner and outer boundaries of the dye band were determined by using “bwtraceboundary” as shown by the blue lines in Fig. 1(d). Finally, the penetration distance of the dye was determined as follows. For each point of the inner boundary, its minimum distance with respect to the outer boundary was found, and the found distances were averaged. The sampling of the distances, shown as green lines in Fig. 1(d), shows quite uniform spacing between the boundaries. Accordingly, the standard deviation of the minimum distances (0.06 mm) was small compared to the average value (0.99 mm). Also, the average and standard deviation of the minimum distances of the other egg half (0.99 ± 0.10 mm) were very close. This average minimum distance was the penetration distance δ of the red dye into the egg white.

The experiment and analysis were conducted by three undergraduate students in engineering, who are the first three authors of this study. Figure 2 compares the penetration of the red food dye into the white of peeled hard-boiled eggs at three different temperatures (4 °C, 21 °C, and 60 °C) at five time points (1, 3, 5, 8, and 24 h). As shown in Fig. 3, the dye diffused deeper into the egg whites with time and temperature. When plotted with respect to t1/2 (Fig. 3, the inset), measured penetration distances show a reasonably linear relationship with t1/2, in accordance with the expected square-root relationship [i.e., δ(t) = kt1/2]. The R2 values for the least squares fit of the relationship were 0.97, 0.94, and 0.86 for 4 °C, 21 °C, and 60 °C, respectively.

Fig. 2.

Comparison of dye penetration into the egg white among 4 °C, 21 °C, and 60 °C at 1, 3, 5, 8, and 24 h.

Fig. 2.

Comparison of dye penetration into the egg white among 4 °C, 21 °C, and 60 °C at 1, 3, 5, 8, and 24 h.

Close modal
Fig. 3.

Penetration distance of the dye vs time at (a) 4 °C, (b) 21 °C, and (c) 60 °C with the square-root relationship fitted against the entire dataset of each case. Error bars in (a) and (c): standard deviation (N =2).

Fig. 3.

Penetration distance of the dye vs time at (a) 4 °C, (b) 21 °C, and (c) 60 °C with the square-root relationship fitted against the entire dataset of each case. Error bars in (a) and (c): standard deviation (N =2).

Close modal

It should be noted that the results shown in Fig. 3 were not from one single egg for each temperature, but from multiple eggs immersed at the same time. There could be variations in the structure and permeability of the egg white among eggs. The standard deviations of the penetration distance were found for time points with two data points. While examining more eggs marinated under the same conditions would be required to conclusively support the t1/2 dependence, the fact that most data points seem to lie within a standard deviation of the best fit line provides additional support for the dependence.

The diffusion of the dye occurred more rapidly at higher temperatures.5,6 The penetration distance increased with the temperature at each of the time points in Fig. 2, and the value of k was found to be 0.41, 0.52, and 1.64 for 4 °C, 21 °C, and 60 °C, respectively, in Fig. 3. Since k is proportional to D1/2, the increasing k values suggest that the diffusion coefficient of the dye through the egg white increased with the temperature.

The proposed experiment using peeled hard-boiled eggs and food dye can demonstrate the physics of diffusion well, and it can be done easily in various educational settings. When employed at school, the proposed experiment can be modified as follows. A common cooking apparatus such as a slow cooker or instant pot can be used instead of a convection oven to demonstrate diffusion at a high temperature. Eggs can be immersed in a dye solution at staggered times in advance of the experiment in order to allow a class of students to make all the measurements at the same time. The image processing part of the experiment can be replaced by students' manual measurements of the penetration distance. Different types of dye can be used to demonstrate how diffusion is affected by the molecule size of dyes. Soy sauce or marinade of students' choice could be used instead of the food dye solution so that students could “taste” the differences in diffusion. Since the proposed demonstration is inspired by cooking methods to marinate or pickle hardboiled eggs, students can learn how to relate their learning of diffusion to daily life or the real world through context-based learning.

This study was supported by the University of Nebraska Collaboration Initiative Grant. C.E. and H.B. were supported by the Undergraduate Creative Activities and Research Experience (UCARE) program of the University of Nebraska-Lincoln.

The authors have no conflicts of interest to disclose.

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