Power law behavior in chemical reactions

Article abstract of DOI:10.1063/1.1827603

Reactions between metals and chloride solutions have been shown to exhibit magnetic field fluctuations over a wide range of size and time scales. Power law behavior observed in these reactions is consistent with models said to exhibit self-organized criticality. Voltage fluctuations observed during the dissolution of magnesium and aluminum in copper chloride solution are qualitatively similar to the recorded magnetic signals. In this paper, distributions of voltage and magnetic peak sizes, noise spectra, and return times are compared for both reactions studied.

Full text of DOI:10.1063/1.1827603

Power law behavior in chemical reactions
J. R. Claycomb, D. Nawarathna, V. Vajrala, and J. H. Miller
Citation: J. Chem. Phys. 121, 12428 (2004); doi: 10.1063/1.1827603
View online: http://dx.doi.org/10.1063/1.1827603
View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v121/i24
Published by the American Institute of Physics.
Additional information on J. Chem. Phys.
Journal Homepage: http://jcp.aip.org/
Journal Information: http://jcp.aip.org/about/about_the_journal
Top downloads: http://jcp.aip.org/features/most_downloaded
Information for Authors: http://jcp.aip.org/authors
Downloaded 01 Oct 2012 to 171.67.34.69. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

JOURNAL OF CHEMICAL PHYSICS
VOLUME 121, NUMBER 24
22 DECEMBER 2004
Power law behavior in chemical reactions
J. R. Claycomb
Department of Mathematics and Physics, Houston Baptist University, Houston, Texas 77074-3298
and Texas Center for Superconductivity and Advanced Materials, University of Houston, Houston,
Texas 77204-5005
D. Nawarathna, V. Vajrala, and J. H. Miller, Jr.
Department of Physics and Texas Center for Superconductivity and Advanced Materials,
University of Houston, Houston, Texas 77204-5005
͑
Received 20 July 2004; accepted 8 October 2004͒
Reactions between metals and chloride solutions have been shown to exhibit magnetic field
fluctuations over a wide range of size and time scales. Power law behavior observed in these
reactions is consistent with models said to exhibit self-organized criticality. Voltage fluctuations
observed during the dissolution of magnesium and aluminum in copper chloride solution are
qualitatively similar to the recorded magnetic signals. In this paper, distributions of voltage and
magnetic peak sizes, noise spectra, and return times are compared for both reactions studied.
©
2004 American Institute of Physics. ͓DOI: 10.1063/1.1827603͔
I. INTRODUCTION
II. EXPERIMENTAL PROCEDURE
The reactions studied include the replacement reactions
of magnesium or aluminum in aqueous cupric chloride solu-
tion,
Self-organized criticality ͑SOC͒ describes emergent be-
havior in certain physical, chemical, and biological systems.1
Externally driven systems exhibiting SOC are characterized
by the interaction of many degrees of freedom, where small
perturbations lead to disturbances, or avalanches, of all sizes.
SOC is observed in slowly driven interaction dominated
Mg͑s͒ϩCuCl →MgCl ͑aq͒ϩCu͑s͒,
͑1͒
͑2͒
2
2
2
␣
2Al͑s͒ϩ3CuCl ͑aq͒→2AlCl ͑aq͒ϩ3Cu͑s͒.
2
3
threshold systems along with 1/f noise, power law behav-
ior in avalanche size, and lifetime distributions and spatial
fractals with no characteristic length scale. Examples include
During each of these reactions, copper precipitates out of
solution while Mg or Al dissolves to form aqueous MgCl or
2
3
flux dynamics in superconductors, human encephalographic
AlCl . Electrochemical currents that generate voltage and
3
4
5
signals, solar flares, and earthquakes, to name a few. Sys-
tem size and avalanche quantization size limit the observed
power law ranges. For example, the smallest avalanche in the
magnetic fluctuations are driven by ionic reactions at the
metal surface ͑Fig. 1͒. Local reaction rates are affected by
concentration fluctuations, pitting dissolution of the solid and
the formation of the Cu precipitate layer, which tends to
inhibit the reaction. Breakdown of the copper layer can result
from the liberation of hydrogen, convective turbulence near
the reactive surface, and erosion of underlying metal. Film
1
Bak, Tang, Wiesenfeld ͑BTW͒ sand pile model corresponds
to the toppling of a single grain of sand while no avalanche
can occur larger that the sand pile itself.
Avalanche behavior in chemical reactions is manifested
as sporadic bursts of activity detected as peaks in the corre-
sponding current or voltage time series. We have previously
measured current fluctuations during the dissolution of met-
als in chloride solutions using a high-Tc superconducting
6
quantum interference device ͑SQUID͒ magnetometer and
have determined power law exponents for each reaction stud-
ied. Fractal patterns have been observed in copper dendrite
7
,8
␣
growth formations and 1/f noise has been measured in
various electrochemical noise reactions.9 Considerable re-
search has been focused on the study of chaos in electro-
–11
1
2
chemical reactions. A model of SOC in chemical reactions
has been reported, as well.13 In this paper we report voltage
measurements of avalanche behavior in reactions of metals
in chloride solutions. The resulting power law exponents are
compared with those obtained from magnetic measurements
of each reaction.
FIG. 1. Schematic of Mg dissolution and Cu deposition giving rise to cur-
rent and voltage fluctuations.
0021-9606/2004/121(24)/12428/3/$22.00
12428
© 2004 American Institute of Physics
Downloaded 01 Oct 2012 to 171.67.34.69. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

J. Chem. Phys., Vol. 121, No. 24, 22 December 2004
Power law behavior in chemical reactions
12429
FIG. 2. Experimental setup for measuring voltage time series during chemi-
cal reactions.
breakdown, or interfacial depinning, increases the local reac-
tion rate at freshly exposed metal surfaces resulting in cur-
rent and voltage fluctuations. The system thus contains some
essential features of dissipative SOC with one or more
activation-deactivation processes and energy dissipated as
heat.
FIG. 4. Voltage peaks recorded during the reaction of Al with CuCl2 ar-
ranged in decreasing magnitude showing symmetric positive and negative
distributions.
Figure 2 shows the experimental setup for the voltage
measurements. The reactions took place inside a 5ϫ5ϫ1.3
cm Teflon reaction vessel. A 0.4 g sample of Mg or Al was
3
and 3.1 for Al. Figure 6 shows a Zipf plot of peak voltage
magnitudes recorded during the Al reaction in descending
size. The linear slope of this curve on a log-log scale also
indicates power law behavior in the distribution of peak
sizes. Note that the deviations from linearity at the upper and
lower portions of this curve are most likely due to finite
system size and possibly avalanche size quantization. The
Zipf plot, obtained from a simple sorting of peak heights,
provides a clearer measure of the roll off points ͑and hence
range of the power law͒ compared to the histogram. Voltage
and magnetic return time distributions were also recorded for
each reaction ͑data not shown͒. The forward return time ⌬t
is defined as the time interval between the occurrence of a
voltage peak and a subsequent peak of greater than or equal
reacted with 20 ml of 1M aqueous CuCl . Two gold pin
2
electrodes separated by 1 cm surrounded the sample. Power
spectra were obtained from voltage output with a SR780
signal analyzer. Time series were recorded on a digital oscil-
loscope. Current fluctuations were detected by measuring
time-dependent chemomagnetic fields with a single channel
SQUID magnetometer as described in Ref. 6.
The duration of each reaction was ϳ2000 s. Voltage re-
cordings and noise measurements were made at least 100 s
after the electrolyte was introduced into the reaction vessel to
allow transient signals to subside. Figure 3 shows a voltage
time series recorded for 200 s during the aluminum reaction
with the experimental setup in Fig. 2. The voltage peaks are
due to sudden busts of activity or chemical avalanches. The
size of each chemical avalanche is determined by the mag-
magnitude. The distribution of return times D(⌬t) scales as
Ϫa
⌬
t
with aϭ0.9 for both reactions recorded electrically or
magnetically over at least 3 decades in ⌬t.
nitude of the voltage fluctuation sϭ͉Vp͉. Larger avalanches
Ϫ␣
Figure 7 shows voltage power spectra S(f)ϰf
re-
are rare compared to smaller, more frequent avalanches.
Similar time series were recorded during the magnesium re-
action. A double-sided histogram in Fig. 4 shows symmetric
distributions of positive and negative voltage fluctuations
due to ionic currents flowing in various directions during the
corded during reactions of Mg and Al with 1M CuCl up to
2
0
.5 kHz. The critical exponent ␣ϭ1.7 for the Mg reaction
and 3.0 for the Al reaction. These differ from the magnetic
power spectrum exponents, 1.8 for Mg and 2.0 for Al. Noise
Ϫ␶
power law ranges extend between ϳ1 and 10 kHz. Note that
Al reaction. The distribution of peak sizes D(s) scales as s
␣
1
/f noise is immediately observed after the chloride solu-
with ␶ϭ1.9 from the best line fit in Fig. 5. This histogram
consists of 20 bins of equal width on a logarithmic scale. The
power laws governing voltage and magnetic peak size distri-
butions extend over ϳ1.6 decades for each reaction. The
voltage peak size exponent is the same for each reaction,
while the magnetic peak size exponent differs, 2.5 for Mg
FIG. 5. Voltage peak size distribution D(s) recorded during the Al–CuCl2
FIG. 3. Voltage time series measured during the reaction of Al with CuCl2 .
reaction.
Downloaded 01 Oct 2012 to 171.67.34.69. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

12430
J. Chem. Phys., Vol. 121, No. 24, 22 December 2004
Claycomb et al.
TABLE I. Peak size, return time, and power spectrum exponents obtained
from voltage and magnetic recordings during reactions of aluminum and
magnesium in chloride solutions.
Exponent
Peak size
Magnetic
Voltage
Al, ␶ϭ3.1
Mg, ␶ϭ2.5
Al, aϭ0.9
Mg, aϭ0.9
Al, ␣ϭ2.0
Mg, ␣ϭ1.8
Al, ␶ϭ1.9
Mg, ␶ϭ1.9
Al, aϭ0.9
Mg, aϭ0.9
Al, ␣ϭ3.0
Mg, ␣ϭ1.7
Ϫ␶
D(s)ϰs
Return time
Ϫa
D(⌬t)ϰ⌬t
Power spectrum
Ϫ␣
S(f)ϰf
law exponents in both reactions. The marked difference be-
tween the Mg and Al voltage power spectra may be attrib-
uted to the copper surface layer that progressively flakes off
during the Mg reaction but adheres to the metal surface
forming a fur coat during the Al reaction. A slower diffusion
of ions through the fur coat would also explain the overall
lower voltage and magnetic noise levels compared to the Mg
reaction.
FIG. 6. Zipf plot of voltage peak size magnitudes recorded during the reac-
tion of Al with CuCl2 .
tion is added while the peak size distribution is initially
Ϫ␶
2
rounded, scaling as s after ϳ10 s. Critical exponents for
each reaction are summarized in Table I.
Chemical reactions are probably the easiest systems with
which to study SOC-like behavior experimentally. By vary-
ing sample size and reactant concentration, parameter tuning
is possible that would not be as realizable in the study of
avalanche behavior in superconductors, earthquakes or rice
piles, for example. Investigation of power law behavior in
III. DISCUSSION
A key feature of models exhibiting SOC-like behavior is
the apparent nonrequirement of parameter tuning or, insensi-
tivity of power laws with respect to small changes in external
1
4
15
conditions. SOC-like behavior was observed in each reac-
combustion, electrochemical noise and corrosion reactions
tion for CuCl2 concentrations ranging from 0.125M to
may also shed light on the underlying physics of these pro-
cesses.
2.0M. The critical state, as evidenced by the power laws,
therefore, evolves without a careful adjustment of param-
eters. It remains to determine the precise time evolution,
range and dependence of the power laws governing various
reactions with different electrolyte concentrations. However,
no changes in the power law exponents were observed com-
paring 100 s windows side by side.
For slowly driven systems, the external driving occurs
on a much larger time scale than internal relaxation pro-
cesses. Here the reaction rate increases with the electrolyte
concentration and is slowed by the formation of the passive
copper layer on the metal surface. If the mechanisms gov-
erning the underlying dynamics in each of these reactions are
the same, then we should expect to measure similar power
ACKNOWLEDGMENTS
This work was supported by the Robert A. Welch Foun-
dation ͑Grant No. E-1221͒, the Texas Center for Supercon-
ductivity and Advanced Materials, and the Institute for Space
Systems Operations.
1
P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 ͑1987͒.
H. J. Jensen, Self-Organized Criticality, Emergent Complex Behavior in
2
Physical and Biological Systems ͑Cambridge University Press, Cam-
bridge, 1998͒.
S. Field, J. Witt, F. Nori, and X. Ling, Phys. Rev. Lett. 74, 1206 ͑1995͒.
3
4
Y. Georgelin, L. Poupard, R. Sart e` ne, and J. C. Wallet, Eur. Phys. J. B 2,
3
03 ͑1999͒.
5
6
P. Bak, How Nature Works ͑Oxford University Press, New York, 1997͒.
J. R. Claycomb, K. E. Bassler, J. H. Miller, Jr., M. Nersesyan, and D.
Luss, Phys. Rev. Lett. 87, 178303 ͑2001͒.
V. Fleury, Arbres de Pierre, la Croissance Fractale de la Mati e` re ͑Flam-
marion, Paris, 1998͒.
S. Zhong, Y. Wang, M. Wang, M.-Z. Zhang, X.-B. Yin, R.-W. Peng, and
N.-B. Ming, Phys. Rev. E 67, 061601 ͑2003͒.
F. Mansfeld and H. Xiao, J. Electrochem. Soc. 140, 2205 ͑1993͒.
C. Monticelli, G. Brunoro, A. Frignani, and G. Trabanelli, J. Electrochem.
Soc. 139, 706 ͑1992͒.
A. Legat and V. Dole cˇ ek, J. Electrochem. Soc. 142, 1851 ͑1995͒.
J. L. Hudson and T. T. Tsotsis, Chem. Eng. Sci. 49, 1493 ͑1994͒.
7
8
9
0
1
11
1
1
2
3
B. Gaveau, D. Latr e´ moli e` re, and M. Moreau, J. Chem. Phys. 118, 5754
͑
2003͒.
1
1
4
5
M. Stapleton, M. Dingler, and K. Christensen, arXiv:cond-mat/0306647,
004 ͑unpublished͒.
2
FIG. 7. Voltage power spectra recorded during reactions of Mg and Al with
CuCl2 .
V. N. Skokov, A. V. Reshetnikov, V. P. Koverda, and A. V. Vinogradov,
Physica A 293, 1 ͑2001͒.
Downloaded 01 Oct 2012 to 171.67.34.69. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions