In the Classroom edited by
JCE DigiDemos: Tested Demonstrations
Ed Vitz Kutztown University Kutztown, PA 19530
Double-Diffusive Convection: A Simple Demonstration submitted by:
Mario Markus Max-Planck-Institut für Molekulare Physiologie, Postfach 500247, 44202 Dortmund, Germany;
[email protected] checked by:
Richard W. Schaeffer Department of Chemistry, Kutztown University, Kutztown, PA 19530
A simple, yet elegant demonstration is described to show double-diffusion convection. The demonstration captures the students’ interest and is a perfect starting point to examine this interesting and ubiquitous phenomena. Simplest Version of the Experiment: A Demonstration A small spoonful of nondairy coffee-whitener (also called coffee creamer; examples of brands are Coffee-Mate and Cafita) is stirred with water; hand stirring is sufficient. Any mass proportion between 0.7% and 33% of whitener to water is possible, however better visualization is achieved with the larger percentages. The mixture is injected using an ordinary pipet on the bottom of a Petri dish containing water; height of the water should be 3–10 mm. The outer side of the pipet should be free of whitener (swift cleaning with the fingers suffices) to prevent whitener being released before the pipet reaches the bottom. A household cup may be used instead of a Petri dish. The injected quantity should lead, right after injection, to a circle of whitener with a diameter of 1–2 cm. After a few minutes, hundreds of fine “needles’’ grow radially, as illustrated in Figure 1.
Figure 1. Needles containing micelles observed after a few minutes. The needles appear white since the micelles scatter light shining from above. Diameter of the dish is 6 cm.
526
Journal of Chemical Education
•
The easiest demonstration consists in placing the dish horizontally on a black background and observing with the naked eye. For presentation with an overhead projector, optical contrast can be enhanced by placing a green transparent sheet between the dish and the light source. The clearest enlarged visualization is attained using a video camera that monitors light scattered vertically upwards by the needles.1 Explanation The phenomenon is explained by realizing that both the qualitative features of structure formation and the quantitative analysis (see the last section) are analogous to the results obtained with a surfactant兾glycerine兾water mixture reported previously (1). In fact, the product labels on coffee-whiteners indicate that they contain surfactants. In addition, they contain sugars, which can assume here the physicochemical role of glycerine. The behavior of the surfactant兾glycerine兾water mixture was explained (1) by double-diffusive convection (for reviews on this phenomenon, see refs 2–4 ). In general, this type of convection appears in a medium (M) containing a layer (L). L consists of a mixture of two diffusing substances, A and B, with the following properties: (i) the mixture of A and B is in a gravitationally stable position, that is, L floats if it is less dense than M, or else it stays on the bottom of M; (ii) A is gravitationally unstable in the absence of B; (iii) B is gravitationally stable in the absence of A; and (iv) B diffuses much faster than A. If the interface between L and M remains perfectly flat, nothing happens. However, if the unavoidable, microscopic fluctuations at this interface cause a sufficiently thin protrusion of the mixture L into M, then this protrusion is depleted by the rapidly diffusing B and the gravitationally unstable A enlarges the protrusion; that is, convective transport of A occurs. Let us now look at the scenario of double-diffusive convection in our particular case. The surfactant molecules, which are composed of a hydrophobic chain and a hydrophilic head, self-assemble into molecular aggregates called micelles (see e.g., ref 5). The hydrophobic chains occupy the interior of the aggregates, while the head groups are located at the micelle–water interface, as sketched in Figure 2. Sugar in the whitener (corresponding to glycerine and indicated by S in Figure 2) has a greater density than water, whereas the micelles are less dense than water. Altogether, the mixture is denser than water; that is, it is gravitationally stable at the
Vol. 81 No. 4 April 2004
•
www.JCE.DivCHED.org
In the Classroom
bottom. However, if microscopic fluctuations at the interface cause a sufficiently thin upwards protrusion (sketched in Figure 2), this protrusion will be rapidly depleted of S by diffusion (arrows in Figure 2),2 while it will retain most of the micelles. This is due to the fact that micelles are much larger and thus diffuse much slower than sugar molecules. Being depleted of S, the protrusion grows upwards because the micelles are less dense than water. In our case, the growing protrusions develop into extremely thin and straight structures, so that we refer to them as needles. As shown in ref 1, shortly after the upwards growth described above, the needles are oriented radially outwards. This radial orientation occurs because the original drop expands outward as a result of gravity, and exerts shearing forces that cause a radially outward flow of the surrounding water; the needles follow this flow near the bottom of the dish. The resulting quasi-two-dimensional confinement of the needles allows easy visualization if compared to other double-diffusive convective systems, such as those given in the next section.
S
S
S
S S
S
Related Systems Double-diffusive convection, as described above, is an ubiquitous phenomenon in science and technology. One dramatic example is relevant to chemical engineering: sewage assumed to be safely disposed at the bottom of the sea may contain—although altogether denser than seawater—a component that rises upwards, analogously to the micelles sketched in Figure 2. Another example is the transport of salt in oceans: upper regions are heated by the sun so that their salt density increases owing to evaporation. The situation is like that of Figure 2 but turning the picture upside down, replacing the micelles by salt ions, replacing S by heat and assuming that the more salty, hotter upper regions are altogether more dense than the lower regions. The structures in this case are called “salt fingers’’ and they are considered to lead to the dominant mixing processes in oceans (3). Note that salt fingers, drilling their way downwards into deeply disposed, less salty sewage, may push this sewage upwards, this being another mechanism for polluting upper sea layers (4). Additional examples are: transport of sluggish polymer molecules in the extracellular matrix of connective living tissues (6), banded metal inhomogeneities arising in alloys after crystallization in multicomponent melts (2), and “helium fingers’’ in the sun (2). Discussion The experimental procedure presented in this report is much easier to implement than hitherto reported arrangements showing double-diffusive convection (2–4, 7, 8). In fact, quasi-two-dimensional confinement of the needles (close to the bottom) is attained here without external action; in alternative systems, such a confinement has been attained by more complicated geometrical or hydrodynamical forcing procedures (7, 8). In addition, the micelles are large enough to scatter light, so as to appear milky; thus, in comparison with other arrangements, no dyes or devices indicating the refractive index are needed. The experiment described here can be done within minutes using household equipment, it involves totally harmless components and it costs only a few dollars. www.JCE.DivCHED.org
•
S S
Figure 2. Scheme explaining double-diffusive convection in the system presented here. The mixture (dark gray) of sugar (S) and micelles is denser than the water layer (light gray). An upwards protrusion is rapidly depleted of the denser S as it diffuses faster than the micelles. Since the micelles are less dense than water, the protrusion rises.
Expanding the Demonstration into a Laboratory Project The simple experiment described above is suitable for a fast demonstration during class. In this section more advanced aspects and procedures, involving quantitative analyses, which are suitable for a laboratory project, are described. If regarded three-dimensionally, the double-diffusion phenomenon is more complex than the sketch of a planar section shown in Figure 2. In fact, the growth of protrusions causes a depletion of micelles that inhibits growth in their surroundings. Thus, the protrusions will keep certain distances to each other in the horizontal direction, as revealed by a light microscope and illustrated in Figures 3A and 3B. The darkest spots in these figures correspond to the growing protrusions, while the lightest spots indicate maximum depletion of micelles. Such arising structures are called “cells” in the literature (9), the size of a cell being defined as the mean distance between minima or between maxima of light intensity. The time interval that cells appear in these experiments is a factor 2–3-fold smaller than the time interval at which radially oriented needles become first visible. In the previous work (1), which was performed with the surfactant兾glycerine兾water mixtures, measurements of the “emergence time’’ Tem of the cells versus the initial glycerine concentration G yielded,
Tem ∝ G − α
(1)
where α = 0.48. We assume that the same relationship holds
Vol. 81 No. 4 April 2004
•
Journal of Chemical Education
527
In the Classroom A
Tem / s
500
200
100
0.5
1
2
5
10
20
50
20
50
W (mass percent) B 1.0
0.5
λ / mm
Figure 3. (A, B) Observations (with a microscope) of cells appearing in the early stages of the experiment. In contrast to Figure 1, the dark regions here correspond to a high concentration of micelles since the latter absorb light coming from below, while they are observed from above. (A) Initial concentration W of whitener is 1.5% mass. (B) Smaller cell sizes at W of 24% mass. (C) Typical scheme obtained by Fourier analysis of an image using the algorithm in ref 10.
0.2
for Tem versus the initial concentration S of sugar in the experiments described here. Considering that the whitener is only diluted without changing its composition, S is proportional to the initial concentration of whitener W . Thus, we write:
T em ∝ W
−α
(2)
Tem is defined as the time at which the minima of the intensity of the light absorbed by the system (light regions in Figures 3A and 3B) are indistinguishable from the intensity measured with a dish containing just the water in a control experiment. Calculations for the surfactant兾glycerine兾water system (1) yielded a dependence of the mean size of cells λ (at time Tem) as a function of G , which is described by
λ ∝ G − β
(3)
A theoretical treatment (1) results in the value β = 0.37. Following the same arguments that lead to eq 2, we write
λ ∝ W
−β
(4)
for the whitener兾water mixtures considered in the present report. For the determination of Tem and of λ, experiments should be performed using a smaller dish to avoid growth of cells by expansion of the drop as a result of gravity; a convenient dish diameter is 2 cm if the volume of the mixture is 40 mL. The formation of cells should be monitored with video equipment. The emergence time Tem is then determined by observing the light intensity on the monitor. At t = Tem, λ can be determined by measuring a number of cell sizes on 528
Journal of Chemical Education
•
0.1 0.5
1
2
5
10
W (mass percent) Figure 4. (A) Experimentally determined cell emergence time Tem versus the initial concentration of the coffee-whitener. (B) Mean cell size λ (at t = Tem) versus W.
the monitor with a ruler, taking the average, and considering the corresponding amplification factor. If possible, however, the image should be stored in a computer, so that λ could be determined by two-dimensional Fourier analysis. For this, one may use an algorithm available in the Internet (10), which yields—for each image—a diagram like that exemplified in Figure 3C. Note that for each coordinate pair (m, n) of Figure 3C (dimensions of m and n are length᎑1), the dimensionless input (n1, n2) to the program should be n1 = mX and n2 = nY, where X and Y are the lengths of the sides of the images. The amplitudes of the output of the algorithm (complex numbers) are represented by grey levels in Figure 3C. The nearly circular shape of the diagram in this figure indicates a satisfactorily defined mean distance λ = 2π兾r, where r is the mean radius of the diagram. The log–log plots of Tem and of λ versus W, as measured for a large range of dilutions of the coffee-whitener are shown in Figure 4. The error bars are given by the standard deviation of the values obtained from 20 video images. These bars show that the statistical scattering of values is low, so that if students or teachers are only able to evaluate one image (or a
Vol. 81 No. 4 April 2004
•
www.JCE.DivCHED.org
In the Classroom
few) they would still obtain fairly significant results. The straight lines fitted in Figure 4 correspond to eqs 2 and 4; the slopes of these lines are α = 0.51 and β = 0.39. Note that the values given after eqs 1 and 3, which correspond to a system with clear evidence for double-diffusive convection (1), are amazingly close to the values found here with the coffee-whitener. Notes 1. To obtain a clear video image I recommend: (i) using an ordinary lamp with a household light bulb (around 60 W), directed to the dish from above; (ii) avoiding other comparatively strong light sources; (iii) putting the lamp roughly one-half meter away from the dish; and (iv) directing the light beam at the smallest possible angle with respect to the dish’s bottom, watching that there is no substantial shadow of the dish’s border. 2. There is certainly diffusion of S everywhere out of the mix-
www.JCE.DivCHED.org
•
ture, but the depletion is much faster out of the protrusion because there the surface to volume ratio is largest.
Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Kötter, K.; Markus, M. Europhys. Lett. 2001, 55, 807. Turner, J. S. Ann. Rev. Fluid Mech. 1985, 17, 11. Chen, C. F.; Johnson, D. H. J. Fluid. Mech. 1984, 138, 405. Huppert, H. E.; Turner, J. S. J. Fluid Mech. 1981, 106, 299. Dionisio, M.; Sotomayor, J. J. Chem. Educ. 2000, 77, 59. Harper, G. S.; Comper, W. D.; Preston, B. N. J. Biol. Chemistry 1984, 259, 10582. Taylor, J.; Veronis, G. Science 1986, 231, 39. Linden, P. F. Geophys. Fluid Dynam. 1974, 6, 1. Shirtcliffe, T. G. L.; Turner, J. S. J. Fluid Mech. 1970, 41, 707. Numerical Recipes Books On-Line Home Page. http://libwww.lanl.gov/numerical/bookcpdf/c12-4.pdf (accessed Jan 2004) Chapter 12.4
Vol. 81 No. 4 April 2004
•
Journal of Chemical Education
529
