overhead projector demonstrations A Classroom Demonstration of Electrodeposited Fractal Patterns Edelfredo Garcia and C. H. Liu Arizona State University Tempe, AZ 85287 Because of the increasing interest in teaching fractal geometry ( I ) , it is opportune to design practical student experiments where ihe complexity ofred systems can be examined. This is a n inexpensive laboratory experiment combining the recommended techniques for teaching fractal geometry in the classroom with the standard procedures for studvine electrochemical de~osition(ECD) of 72-81 in the regime oi low solution conramified centration and low applied constant driving force. The patterns in this regime are statistically self-similar fractals. The term fractal refers to a shape made of parts similar to the whole in some way1 "Statistically self-similar" means that the oattern retains its overall shaue at limited len&h scales ofobservation. Although a crude version of the experiment was outlined in a conference a few years ago (5). i n updated version is presented here. The kxperiGental set-UP does not consist of sophisticated instrumentation, where computer-based or di$tizing techniques (6, 7) may be required to obtain fractal dimension estimates of the patterns. The set-up is suitable for overhead demonstrations, can be adapted to physical science laboratory courses, and is an attractive way to introduce students to fractal growth phenomena and scaling concepts.
edited by
DORIS KOLB Bradley Univerrity Peoria, iL 61625
fractal geometry, the set-up can be modified for other methods that may improve the statistics of the estimates. The working grid is used in the following manner. One can visualize a single "box" N of a given length scale factor r a s c o n s i s t i n g o f l x l =1,2 x 2 = 4 , 4 x 4 = 1 6 , a n d 8 x 8 = 64 basic cells. This is equivalent to having four different grids, respectively, of 1 x 1/32 = 1132, 2 x 1132 = 1/16, 4 x 1/32 = 118, and 8 x 1/32 = 114 scale factors, with the advantage that the set-up can be focused and calibrated a t once during the entire experiment. The working grid is shown 16"
Experimental Procedure Apparatus
Figure 1. Working grid
Aworking grid of 112 in. x 112 in. basic cells is constructed by drawing fine straight lines on a piece of 16 in. x 16 in. white paper. The grid also can be constructed from quadrille paper sheets that conform to these dimensions. Because 1024 Le., 32 x 32) basic cells are provided, each cell scales the length of the grid by a factor of 1/32. This grid can be used to estimate the fractal dimension D of two-dimensionally projected images of two-dimensional patterns. Several counting methods can be used for this purpose. Because our goal here is to introduce students to basic scaling concepts, we prefer to use the simplest of these approaches: the box-counting method ( I ) . However, depending on the background of the target audience in h i s description apparently has been adopted as a practical working definition for fractals by B. 6. Mandelbrot, who pioneered fractal geometry in the early 70's.
F gLre 2 S ae vrew of the expermenla set-up tar CJ or PI w re calhb ana (e) ode. ( 0 ) CJ fol anoae,(c)P cxlq as plates, (a)r~bDort ~ ngs. Plexiglas pla!form. Metal clips and spacers between theoverhead and the plafforrn are not shown
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in Fiwre I ; representative boxes are shown in boldhce for claritv. Notice that the construction of the e n d bv" itselfcan be uskd in the classroom to illustrate the concepts of scaling, dimensionality, and scaled symmetry. The secondary grids also can be finely marked with ink of different colors to facilitate visualization. The grid can be placed on a projection screen or on any light-colored wall surface in the classroom and secured with tape. An electrochemical cdl of cir&lar geometry is fabricated by the following procedure. A 10-cm diameter circular hole is cut in the center of a 13 cm x 13 cm copper foil 0.005 in. in thickness. The foil is then cleaned by soakingin 0.01 M nitric acid for one to two minutes, rinsed several times with distilled water, and dried with paper towel. Two circular 114-in. thick, 12-cm diameter Plexiglas plates are fabricated. A0.5-mm diameter hole is drilled through the center of one plate. ' h o 4-in. segments of 114-in. vacuum rubber tubing are glued to the bottom of the other plate, near its edge and equally spaced, to provide a base. The rubber base alleviates possible external vibrations and temperature changes due to heating by the overhead projector lamp while the set-up is being illuminated. Depending on the lamp and overhead configurations used, the generated heat may still be a problem. In such case, a 114-in. thick, 12-in. x 12-in. Plexiglas plate can be used a s a platform between the cell and the overhead projector. Additionally, strips of cardboard can be used a s spacers to provide air space for ventilation between the overhead projector and the platform. (See Fig. 2.) The copper foil is placed between the cell plates and the plates are clamped tightly together with metal clips. In this way, the gap between the Plexiglas plates is of approximately the same width a s the thickness of the copper foil and may be taken for the cell thickness. However, the students should be reminded that the solution sample size, current density, and potential drop across the electrodes are affected directly by the effective cell thickness.
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Figure 3. Projected image of a typlcai ECD fractal pattern. Grid backaround contrast was added artificiallv. ExDerimental conditions are described in the text.
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(a) y = 4.6e-2
+1.2~
Set-up Calibration Prior to use for fractal mowth. the set-UDmust be moobf erly focused and calibrated against a fractal known fractal dimension, D. We select here a triadic Koch curve a s the calibrating device because its expected D value is well known, and because that value has been estimated with the box-counting method by others (Ref. I, Vol. 1, Chapter 3). The curve is photocopied onto a clear transparency, which is then placed between the cell plates, centered on the overhead, and illuminated. The resultant image is projected on the grid, and its fractal dimension is estimated with the box-counting method described below. For a given length scale factor r, the boxes N(r) required to cover or "occu~v"the entire image are counted. For strictlv selfsimila? and statisticall; self-similar fractals, the ~ollow~ . h is a Droin= Dower law is obeved: N(r) = h (. ~ l.i -, )where poxionality constant. ~ h u sa, log-log plot of N(r) vefsus llr will give a straight line with positive slope equal to D and intercept equal to log h. When using the working grid, however, it may happen that part of a n image appears to overlap the edge of a box, and thus a decision must be made a s to whether or not that box is "occupied". Consistency is here the key to success; i.e., once a counting method is adopted, it should be applied to all length scales. We have found the following 'Unlike statisticallv self-similar fractals. strictlv self-similar fractals -. presewe the r shapes at at engtn sca es'of oo;ewat ons 31noeed 1tne goa of tne expenment s the constr~cton of pnase d agrams w tn concenlratlon and appl~edor v ng forceas tne parame-
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Journal of Chemical Education
log l/r Figure 4. Results from the box-counting method analysis for: (a) the calibration process using a triadic Koch curve (solid circles); (b) a Koch cuwe accordingly to ref 1, Vol. I . pp 102 and 127 (clear circles); and (c)the image of the ECD pattern shown in Figure 3 (solid triangles). method useful for minimizing uncertainty during counting without overestimating the results. If the edge of a box just overlaps part of a n image, that box is counted as "occupied". Electrodeposition The box-counting method is applied to the image of a n ECD pattern a t the end of electrolysis. Usually, . a pattern . is electrodcposit(!d under galvanostatic conditions, at room temperature, and without supportmg electrolyte. Ilepending upon salt cnncentrntion (0.001 to 0.1 molar, and applied current 11 to 10 mA 10 to 20 min may be requinvi to mow a att tern of suitable size. Nitrate and sulfate salt solutions bf copper can be used. Of course, the deposition of other metals from their salt solutions is possible also. However, one should not use dissolving anodes that may interfere with the growth of the electrodeposited metal. For example, when electrolyzing zinc salt solutions, copper
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anodes should not be used because of copper codeposition. In that case the anode can be made from zinc foil or preferably from an inactive metal such as platinum. 3 To run an exoeriment. an aliauot of the solution to be electrolyzed is purged for about 50 min with nitrogen gas to remove dissolved oxveen. The cell is centered on the overhead, and 1mL of the prepurged solution is slowly introduced through the small hole in the upper plate with a hypodermic syringe. A slight excess can be injected to insure that there are no empty spaces in the cell. The total sample size to be used depends on the particular cell configurations employed. Acopper or platinum wire is now introduced through the orifice a t the center of the upper plate to act as the cathode of the cell. The cathode is connected to the negative terminal of a constant current generator, while the copper foil, acting as the anode, is connected to the positive terminal. During electrolysis, the cell is illuminated and the image of the growing pattern is projected on the working grid.
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Results Figure 3 shows the image of an ECD pattern obtained from 0.04 M CuS04 solution after applying 3 mA for 20 min. The results from the box-counting method are shown in Figure 4. Here the data represented by solid circles correspond to the calibration process while the data in clear circles have been taken from reference 1,Vol. 1, pp 102 and 127. Both sets of data correspond to the triadic Koch curve. Althoueh onlv four and six leneth scales.. res~ectivelv. . ".were used, the results are almost the same, i.e., D = 1.2, as can be seen from the corres~ondinereeression eauations. The expected value for the tiiadic ~oc
