Chemical and Mathematical Analysis of Fractal Copper Crystals Grown through Diffusion Limited Aggregation

 

By Jonathan Schwalbe

 

Abstract

There were three purposes to this project. The first was to design a more effective apparatus for the growth of copper aggregate crystals grown through diffusion limited aggregation. The second purpose was to determine the effect of varying molarities on the fractal properties of copper fractal crystals. The third purpose was to identify the chemical reactions involved. The results showed that the relationship between the area of the fractal crystal and the molarity of the solution in which it was grown was inverse. In the higher molarity solutions, the crystal grew more densely. More Cu2+ ions were in the vicinity of the cathode when the potential was applied. Therefore more ions will reduce at any given time, causing a more dense crystal to be formed. Further results showed that the fractal dimension was not dependent on molarity. When the molarity of the solution the crystals were grown in stayed constant during electrolysis, the fractal dimension was constant at 1.45 to 1.46. When different molarity solutions were used initially and the molarity of that solution remained constant throughout the electrolysis, the fractal dimension was still 1.45 to 1.46. Using the new apparatus it could be concluded that with a constant molarity, the value of the fractal dimension remained the same when other conditions in the trial remain constant (e.g. voltage and number of coulombs used).  Finally, chemical analyses showed chemical reactions at the anode were much more complex than originally thought. The mass change of the anode was most likely a result of oxidation of the anode.  However, since the mass change at the anode was less than the mass change at the cathode and a black solid was observed on the anode.

 

Introduction

The first purpose of this project was to design a more effective apparatus for the growth of copper aggregate crystals grown through diffusion limited aggregation than the one used in the first two years of the project. The second purpose of this project was to determine the effect of varying molarities on the fractal properties of the crystals. The third purpose was to identify the chemical reactions involved. The hypotheses were:

1) The relationship between the molarity of the solution and the area of the copper crystals would be inverse.

2) The apparatus would maintain constant molarity for each trial so the fractal dimension of the copper crystals would remain the same.

3) The chemistry at the anode was the reverse reaction of the cathode.

 

Background

Chaos theory states that all natural events and structural design, although they may appear to be random, actually exhibit patterns. Chaos theory extends from complex patterns like the veins in a leaf to the branches in a tree. Structures that have a pattern of self similarity, the repetition of a single pattern to form a more complex structure, are described as being fractal.1 First coined by Benoit B. Mandelbrot, the term fractal is short for fractional dimensional, meaning an object with  a non-integer dimension between one and three, for example 1.46, rather than a whole number dimension such as a line with dimension of one or a square with a dimension of two or a cube with dimension of three (1). 

 

Mandelbrot described three different fractal dimensions: box-counting, self similarity, and compass. The box-counting dimension was used in this project. It is a mathematically derived number which corresponds to the value of the slope of the graph relating the number of boxes needed to cover an image to the size of the respective boxes. Copper aggregate crystals exhibit a dimension that is not a whole number because they have fractal properties.

 

Chaos enters this project because of brownian motion, which is caused by collisions between the ions in solution in addition to the pull of the cathode. In this experiment copper ions were moving in solution until they were attracted by the pull of the cathode. A fractal pattern was created when the Cu2+ ions reduced to copper solid.

 

 

Previous studies on zinc aggregate crystals were done using a small electrolysis set up as seen in Figure 1 using a  DC power source, a petri dish with Zn(NO3)2 solution and a zinc-wire anode with a point cathode (1, 4). In the experiment either voltage was varied with constant concentration, or the concentration of the Zn(NO3)2 solution was varied, at constant voltage. The crystals were then scanned into a computer and analyzed with the program Fractal Dimension by Boston University. Fractal Dimension is a program that calculates the box-counting dimension of a fractal. In my initial project using this same setup, the fractal dimension of the crystals varied as an inverse relationship to the concentration of the solution, similar to the relationship between the area of the crystal versus the concentration of the solution (2).

  

My initial project, using Cu(SO4)2, had several uncontrolled variables. As the crystal grew closer to the anode, the area of solution between the crystal and the anode decreased, therefore the resistance in the circuit was decreasing. With decreased resistance, the current increased because of Ohm’s law, which states that current is directly proportional to voltage and indirectly proportional to resistance. In the experiment the voltage remained constant at 8.0 volts. With more current passing through the solution as the time increased, the number of copper ions being reduced in each trial varied since in each of the previous trials the time was constant. To accommodate for this, a calculator program was written which, with the additional use of a CBL, monitored the current which kept the number of coulombs that passed through the solution constant (see Appendix A). The other problem variable was the molarity of the solution, which was not remaining constant during the electrolysis.

 

Procedure and Methods

 

Fractal Analysis

The apparatus was designed as seen in Figures 2 and 3 and assembled from 0.25” plexiglass. (See Appendix B for detailed building instructions)   

 

Three trials were done with 0.650L of reagent grade copper (II) sulfate in the following molarities: 0.200M, 0.400M, 0.600M, 0.800M, and 1.00M using a constant potential of 8.0 volts. Electrolysis was regulated using a TI-83 and a CBL programmed to stop the trial when 320 coulombs had passed through the solution. A digital picture was taken of the crystal from a set distance and angle in each trail. Analysis of the crystal was done using the program Graphics Converter and Boston University’s Fractal Dimension (3).

 

Chemical Analysis

To determine the chemistry involved in growing the crystals, the apparatus in Fig. 1 was setup. Separate tests for pH using bromthyml blue and phenolphthalien were done during diffusion limited aggregation of  0.500M solutions of Zn(NO3)2, MgSO4, CuCl2, Cu(NO3)2, and MnSO4.  In each case new copper-wire anodes and cathodes were used. The possible electrochemical reactions with these molecules are seen in Appendix D. Using the apparatus as seen in Figures 2 and 3, readings of the pH were taken around the anode after 0, 5, and 10 minutes.

 

To determine if anode mass was lost during electrolysis the apparatus in Figure 1 was used. The mass of the anode was recorded before and after the electrolysis. To test for changes on a larger scale (a change larger than one milligram), the apparatus as seen in Figure 4 was assembled. The copper foil anode was massed before electrolysis and after the two hour electrolysis.

 

 

Results

In Figure 5, the area of the crystals versus the molarity of the solutions is graphed. A power model was fit to the data. As can be seen, the area of the crystals is inversely proportional to the molarity of the solution they were grown in. Since the number of copper atoms in each crystal is the same, decrease in area for higher molarity crystals indicates an increase in crystal density. Area values for lower molarities do not fit the model as seen in Figure 5 as well as the areas for higher molarities.

 

Figure 5  Area of the crystals versus the molarity of the solutions

 

The program Fractal Dimension was used to determine if there was more deviation in the area calculations at lower molarities. The program analyzed pictures of the crystals grown from solutions with molarity 0.200M, and 0.800M copper fractal crystals (Pictures of the these can be seen in Appendix C). Because each crystal is different, the program may calculate a different number for the area depending on where the program user initially places the first analysis box. The program calculated the area of the crystal ten different times. The standard deviation of the area in the higher molarity crystal (s2 = 8.16) was shown to be statically greater than the standard deviation from the lower molarity crystal (s2=4.00). Therefore, the power model of the data, which has a p value of 0.889, is the proper model to represent the area versus the molarity. Since the deviation of the 0.200M test was over the curve, and the deviation of the 0.400M test was under the curve, the variance between the actual values and the model is random. The actual values miss the model, but they miss in a random way.

   

Figure 6 shows the fractal dimension versus the molarity of the solution. The line of best fit through the points is one with a slope of virtually zero. This relationship supports that the fractal dimension is not dependent on the molarity of the solution. Any variations are not significant (p=.899). So, the null hypothesis that the fractal dimension and the molarity are not related was accepted.

 

Figure 6 Fractal dimension versus the molarity of the solution

  

 

As seen in Table 1 tests with phenolphthalein and bromthymol blue showed no color change for Zn(NO3)2, Cu(NO3)2, MnSO4, and CuCl2 indicating no pH change at either the anode or the cathode. MgSO4 showed a color change at the cathode with phenolphthalein and a color change at the anode with the bromthyml blue.

 

As seen in Table 1, with  MgSO4 and MnSO4, a bubble formed at the cathode. With the Zn(NO3)2, blue coloration appeared at the anode from copper metal from the anode going to Cu2+ in solution. With Zn(NO3)2, Cu(NO3)2, and CuCl2, an aggregation formed at the cathode, and with CuCl2, corrosion occurred on the anode wire.

 

Table 1: Results of Electrolysis Experiments to Test for Chemical Reactions

 

   

Table 2 shows the masses of the cathode and the anode from the test using the apparatus as seen in Figure 4 . The masses of the anode and cathode were taken before and after electrolysis. There was 0.0309g of copper not accounted for.

 

Table 2: Mass of Anode and Cathode

 

 

Table 3 below shows the pH of the solution taken with a probe around the anode with respect to time during a particular trial using the apparatus in Figures 2 and 3. As time progressed, the pH goes down around the anode.

 

 

Table 3: pH of Solution around Anode

 

 

Discussion

Supporting the hypothesis, the relationship between the area of the fractal crystal and the molarity of the solution in which it was grown was inverse. Because the number of coulombs was controlled, the number of Cu2+ ions reduced to form the copper atoms that made up each crystal were the same. However, in the higher molarity solutions, the crystal grew more densely. More Cu2+ ions were in the vicinity of the cathode when the potential was applied. Therefore more ions will reduce at any given time, causing a more dense crystal to be formed.

 

In initial experiments during this project using the apparatus in Figure 1, the molarity of the solution during each trial did not remain constant, and the fractal dimension did change.

 

This was a source of error in the experiment since the molarity during each trial did not remain constant. Another source of error was that as the crystal grew the distance between the anode and cathode decreased, resulting in a decrease in resistance. At this point a new apparatus was designed to control the molarity during each trial and to provide a 650 times larger crystal-growth area. The final source of error in the original assembly was that the current was not regulated, therefore a TI-83 program was written to control the number of coulombs used in each trial. When the molarity of the solution the crystals were grown in stays constant during the electrolysis, the fractal dimension was constant at 1.45 to 1.46. When different molarity solutions were used initially, and the molarity of that solution remained constant throughout the electrolysis, the fractal dimension was still 1.45 to 1.46. Using the new apparatus it can be concluded that with a constant molarity, the value of the fractal dimension will remain the same if the other conditions in the trial remain constant (e.g. voltage and number of coulombs used). The results supported the hypothesis that the fractal dimension is not dependent on molarity.

   

The chemical analysis did not fully support the hypothesis since there were at least two reactions occurring at the anode. The mass lost by the anode was less than the mass gained by the cathode and there were both pH and non-pH related color changes. The mass change of the anode was most likely a result of oxidation of the anode. Support for this reaction at the anode was substantiated by the formation of  blue color at the anode when the colorless electrolyte Zn(NO3)2 was used. This color was not a result of pH changed. The mass gained by the reduction of Cu2+ ions to Cu(s) at the cathode should have equaled the mass lost by the anode when Cu(s) atoms oxidized to Cu2+. However, mass change at the anode was less than the mass change at the cathode, and a black solid was observed on the anode. The extra mass was most likely the copper (II) oxide (CuO(s)). The oxygen required for the formation of CuO(s) originated as water in solution. It is the oxidation of this water that gives the copper the oxygen needed to form the CuO(s).  Further evidence for equation 2 is that the pH decreased at the anode indicating that hydronium ions (H+) were formed. These ions originated in the oxidation of the water and of the copper to form CuO(s).

 

Conclusions

 Further study could look at the effect of varying the voltage in each trial. It is believed that this would change the fractal dimension of the fractal crystal and possibly the area. Since copper(II) oxide coats the anode, if this form of copper (II) oxide does not conduct electricity, it would limit the size of the crystal since eventually the anode would become totally covered in copper(II) oxide. Therefore by using a smaller anode, the critical point where the anode no longer is conductive, due to copper (II) oxide coating, could be reached more readily.

 

Acknowledgments

I received help on my project by the following people.

Mr. Stefan Anderson introduced the idea of chaos and fractals to me. He was my advisor for the initial stages of the experiment. He formulated the original procedure and designed the apparatus as seen in Figure 1.

Mr. Brad Peterson worked with me on the statistical analysis of the data. He instructed me on the different uses of the statistical methods used in this project.

Mrs. Lois Fruen, gave me advise in the preparation and development of this paper.

Dr. David Boyd allowed me to use his laboratory at the University of St. Thomas over the summer of 2001. He provided advise in understanding the chemistry of the project.

 

 

Works Cited

1. Peitgen, Jurgens, and Dietmar Saupe. Chaos and Fractals: New Frontiers of Science. New York: Springer, 1992.

2. Schwalbe, Jonathan. "A Study of Copper Crystals Grown through Diffusion Limited Aggregation." Breck School, 2000. (unpublished paper)

3. Boston, University Of. “Activities, Experiments, and Programs.” On Growth and Form 1996: 7. Online. Internet. 10 Jun. 2001. Available: http://polymer.bu.edu.

4. Fruen, Lois. The Real World of Chemistry ed. 5. Dubuque: Kendall/Hunt, 2001.

 

Bibliography

1. Gleick, James. Chaos: Making a New Science. New York: Penguin, 1987.

2. Schroeder, Manfred. Fractals, Chaos, Power Laws. New York: W. H. Freeman, 1991.

3. Barnsley, Michael. Fractals Everywhere. Cambridge: Academic Press, 1993.

 

 

Appendix A

A program was written to collect the data from the CBL. Data coming from the CBL is in the form of amps. This data is automatically multiplied by the time between trials (30 seconds). This leaves the units in the form of coulombs. This number is then stored in List 2. The program then sums List 2 and stores that in List 3. This number is displayed on the calculator each time it changes (every 30 seconds). This is the number of coulombs passed through the circuit at that time. When this number exceeds the stopping value, in my case 320, the calculator displays “STOP”. Telling the user to stop the trial.

 

Appendix B

To construct the apparatus the following pieces of 0.25” plexiglass are needed.

 20.00cm x 20.00 cm piece      (2) (Growth plates)

 28.28cm x 28.28cm piece       (1) (Base)

 7cm supports for base           (4)

 28.28cm x 10cm piece            (2) (Wall)

 27.55cm x 10cm piece            (2) (Wall)

 Copper weights                     (4) (Attach to the top growth plate at the corners)

 

Adhere these pieces in the configuration as seen in Figures 2 and 3 with silicon sealant. Drill a hole the size of the tubing used to enclose the cathode wire through the bottom of the apparatus in the middle until half way through the bottom growth plate. Stop and attach  new size bit to match the size of the cathode wire and continue drilling until through the bottom growth plate which is attached to the bottom of the base. The top growth plate rests on the top of the bottom one during electrolysis.