離開學術界前還是決定讓她重見天日...（二）

（上接《離開學術界前還是決定讓她重見天日...（一）》）

3.Results and Discussion

Figure 1 Dynamic profiles of the reactive components: simulation results in comparison with experimental results reported by Kalash et al[22], a. Optical image of precipitation band, noted that the first five bands spread about 1 cm long. b. Simulation results of precipitation phase P in the range of 1 cm. c. Concentration of chromite including CrO42- and aqueous CuCrO4 measured every 0.25 cm in day 1, 4, 7, data obtained from Kalash et al[22]. d. Concentration gradients calculated from the data in figure 1c. e. Simulation results of dynamic profiles of chromite concentration. The legend 500, 1000, …, 4000 are the number of iterations in calculation. Noted that graph figure 1e only shows the concentration profiles starting from the boundary where the concentration of chromite is 0.2 M. This boundary is moving according to experiments. f. Concentration of copper including CuSO4 and CuCrO4 (aq), also from Kalash. g. Simulation result of copper concentration compared with figure 1f. h. Simulation results of profiles of CuCrO4(aq), which is a propagating reaction front as iteration time increases. i. profiles of Cu2+, concentration of which stays 0.01 M indicates that reaction front has not propagated to that location yet and no Cu2+ is consumed there.

Among all the papers that report Liesegang bands, Kalash et al[22] is the first to measure the dynamic profiles of the reactive components in 1D experiments as indicated in Figure 1a. The tube was first filled with a homogeneous gel phase containing 0.01 M CuSO4 and gelatin as the inner electrolyte, after that 0.2 M solution of K2CrO4 as the outer electrolyte was gently poured into the tube on the top of the gel phase. The experimental setup was in a dark room A UV-Vis spectrophotometer was used to record the absorbance every 0.25 cm intervals. He also used a digital camera to take optical photos and scanned the photos to get a map of the precipitation bands.

In Kalash』s work, the concentration profiles of copper and chromite were presented in Day 1, Day 4, and Day 7. Conclusion can be drawn directly from the experimental results as follows:

1.The distinctly spaced precipitation bands are formed as figure 1a shows. This phenomenon is not that surprising any more since it is now well-known as Liesegang phenomenon.

2. The boundary where the concentration of chromite equals the bulk concentration in the solution is gradually moving into the gel phase as indicated in Figure 1c. This is easy to understand because of the diffusion of chromite.

However, there are some other subtle details that are not easily understood intuitively.

3. The concentration gradient of chromite is less and less steep from day1 to day4 and finally reached steady in day7. This trend can be simply calculating the slope of figure 1c. The concentration gradients are presented in figure 1d clearly for better illustrating ideas.

4. The dynamic profiles of copper concentration are very interesting but counterintuitive. As shown in figure 1f, small peaks of concentration are observed in all cases. The peak concentration is even higher than the primitive concentration of copper in the gel phase. It is abnormal in a way since we expect the concentration on the left should be below the initial concentration of copper so that copper ions can diffuse to left to meet chromite and form precipitate. But the concentration gradients shown in figure 1f does not support our expectations.

5. The peak concentration increases from day1 to day4, but decreases from day4 to day7. The boundary where the concentration of copper equals the bulk concentration in the gel phase is also gradually moving to right side as indicated in figure 1f.

The five features as summarized above can be completely reproduced at the same time using our reaction diffusion model based on the Oswald supersaturation theory combined with classical nucleation theory. Simulation results are shown in figure 1b, 1e, 1g, 1h, 1i.

To reproduce the experimental features, some thermodynamic parameters of the system are estimated accordingly. The solubility product of CuCrO4 is =3.6×10-6, thus the solubility of CuCrO4 in 0.01 M CuSO4 solution is hence calculated as below：

Then x=0.3479 mM, thus solubility ie , is adjusted to 3.479×10-4 M. The molar volume are calculated using molar mass and density of CuCrO4, . Dc is about to reproduce the experimental trend. The rest of the parameters are kept the same as reported in discussion part.

It should be noted that in our simulation, we force the concentration of CrO42- at x=0 to be the initial concentration 0.2 M at all time. This boundary condition will lead to a flux of CrO42- diffusing to the left. Moreover, it means that all the simulation results after x=0 represents the profiles of the system after the boundary location where the concentration of CrO42- equals 0.2 M in real experiments. For example, in day1, that boundary location is at about 3 cm, but 5 cm and 8 cm in day4 and day7 respectively.

In comparison with experiments, we first plot the profiles of precipitation phase CuCrO4 in figure 1b. Obviously, the simulation successfully reproduces the distinctly spaced precipitation band. The band that forms later is wider than the earlier one as also indicated in figure 1a. The time needed for each band to appear becomes longer and longer which is also known as one of the features of Liesegang system[10, 21].

Figure 1e depicts the dynamic profiles of chromite concentration in comparison with Figure 1c and 1d. As you can see in figure 1e, the simulated concentration of chromite (including CrO42- and CuCrO4(aq)) is not only decreasing but also decreased more and more slowly and tend to reach steady. The trend perfectly suits what observed in the experimental data in Figure 1b and 1c. We need to explain that we cannot observe the moving boundary because we set the left boundary condition as a constant concentration of 0.2 M. This boundary condition means a flux of chromite will continuously diffuse into the gel phase. As a result, we can only compare the simulation results with the experimental results that starts from the location where the concentration of chromite is 0.2 M.

Figure 1g depicts the profiles of copper (including CuCrO4(aq) and Cu2+) while figure 1h depicts the profiles of aqueous CuCrO4 and figure 1i shows the profiles of Cu2+. The concentration profiles of CuCrO4(aq) is like a propagating reaction front as you can see in Figure 1h, the superposition of figure 1h and figure 1i produces figure 1g which shows a peak distribution of the total copper concentration. This peak has also been observed by the UV-Vis spectrophotometer as shown in figure 1f because what UV-Vis spectrophotometer measured is the total copper concentration and cannot distinguish aqueous CuCrO4 and Cu2+.

As for Cu2+, Figure 1i clearly shows that there is a concentration gradient along the vertical direction of the gel phase. And it is the concentration gradient that can drive the Cu2+ to diffuse to meet the CrO42- specie and produces CuCrO4(aq). Once CuCrO4(aq) exceeds its solubility and gets supersaturated enough, the aqueous CuCrO4 molecules can then nucleate and grow up. The nucleation and growth of the crystals in return will remarkably rapidly decrease the local concentration of CuCrO4(aq). After this, the supersaturation is then decreased to a low level which then slows down the nucleation and growth process of crystals. The feedback between supersaturation and nucleation/growth produces an oscillation in vicinity of the spots where nucleation happens. And this oscillation information can be spread out if coupled with diffusion which is ubiquitous in a space dependent system.

Figure 2 Influence of diffusion coefficients of A B C on the distribution of precipitation phase P. 『111』 means DA=DB=Dc=10-5 cm2·s-1, and 『112』, 』113』, 『122』, 『212』 are analogous, and 『220.5』 means DA=DB2×10-5 cm2·s-1, DC=0.5×10-5 cm2·s-1, 『221.5』 means DA=DB=2×10-5 cm2·s-1, DC=1.5×10-5 cm2·s-1.

Figure 1 have demonstrated the success of our method in simulating the Liesegang phenomenon. But during our simulation we also realized that, the geometry of Liesegang bands can be controlled by adjusting the diffusion coefficients of the reactive species. The significance for the parameter study is on one hand to explain why under different conditions the Liesegang band have different geometries[22, 38, 39] and on the other hand to suggest the way to design and create the patterns as we wanted. The latter is more meaningful in the aspect of potential application in material science and engineering.

In Figure 2, the effect of diffusion coefficients was investigated. Comparing (1), (2), (3), which are when the diffusion coefficient of C is larger than A and B under all conditions, as the diffusion coefficient of C increases and A, B stay the same, the propagation of the precipitation phase P slows down. Less and less precipitation band formed within a given time. However, the width of the precipitation band become wider and wider.

Comparing (4), (5), (7), which aims at exploring the relative importance of A, B, C. The pattern formation is individually special, in (4) when DA is the smallest, the concentration of different peaks is almost the same. In (5) when DB is the smallest, the propagation of the precipitation band is the fastest while in (7) when Dc is the smallest, only several peaks form and precipitation band becomes flat. It should be noted that choosing different species to do Liesegang ring experiments could lead to very different formation of precipitation patterns.

Comparing (6), (7), (8), (9), which aims at exploring the transitionary behavior of Liesegang system. When C diffuses slower than A and B under all conditions, the oscillation behavior of precipitation phase P is gradually diminishing as the diffusion coefficient of C decreases from 2×10-5 cm2*s-1 to 0.5×10-5 cm2*s-1. This phenomenon might be quite important because it tells us that Liesegang ring pattern or other ring patterns existed in diffusion and reaction system sometimes may not appear due to the variation of an important process under different experimental conditions. Here in this work, a slight variation of the diffusion coefficient of C will lead to distinct pattern formation. Furthermore, in such systems, apparently different patterns can be observed even under the same experimental condition in the view of the experimenter. However, as we proved using this model here, a slight change of certain process can lead to very different results. This instability and chaotic behavior is the intrinsic property of the system. Often a lot of researchers tend to simply attribute this instability into randomness of the experiments especially when they cannot explain their experimental result scientifically and totally mix chaotic system with random system together.

Unfortunately, chaotic system is ubiquitous in nature. What we can do in our research is decoupling different parameters in the system using control variable method. After carefully investigating each variable clearly enough, we can combine all information together to propose a mechanism and build up a proper model based on the experimental observations and existing knowledge. The model should not only be able to reconcile experimental results that have been done but also predict new experiments or provide useful advice for material engineering.

Figure 3 Predictions on the crystal morphology by our simulation a. distribution of precipitation phase b. Average radius of the precipitated crystals c. the particle density distribution d. schematic of crystal distribution along the gel phase.

Based on the pre-nucleation mechanism of periodic precipitation, the kinetic model can not only reproduce the periodic distribution of crystal morphology (Figure 3a) but also can predict the particle density or number of nucleus (Figure 3b) and also the average radius of the crystals (Figure 3d). Figure 3a shows the precipitation phase discretely distributed along the gel media which is known as the Liesegang band. The nucleation flux in the diffusion and reaction model actually leads to a repeating cycle of supersaturation, nucleation, and depletion. Taking CuCrO4 Liesegang ring as an example, first of all, the local CrO42- concentration is increased by the diffusion from outer electrolyte into inner gel reaction medium, and then the reaction between Cu2+ and CrO42- continuously produces CuCrO4. As a result, CuCrO4 gets supersaturated, the supersaturation can then overcome the activation energy needed for decreasing the entropy from individual disordered molecules into a relatively well-organized nucleus and overcoming the surface tension of the nucleus. After that, each nucleus can undergo crystal growth controlled by the local supersaturation of CuCrO4 constituent. If the supersaturation is below a threshold value, the crystal can dissolve and thus its radius will decrease. If the supersaturation is above the threshold value, the crystal can grow and thus its radius will increase starting from the radius of an initial nucleus. The nucleation and crystal growth process can extremely rapidly deplete the constituents in that location and finally lower the ion concentrations in its close vicinity. As a result, within that vicinity, the number of nucleus will decrease, but due to quasi-equilibrium between crystals growth and dissolution, the radius of the crystals will be much larger than in the location where particle density is large but average radius is small as shown by Figure 3b and Figure 3d. Noted that if quasi-equilibrium is reached, from Gibbs-Thompson relationship,

higher supersaturation corresponds to smaller radius. As a result, in the location where supersaturation is high, large quantities of nucleation will form and the average radius will be low. Lower supersaturation will cause opposite outcome. Figure 3c depicts the visualized idea of what the crystal will look like in reality under microscope or SEM. The primary morphology of Liesegang ring experiments is the distinctly spaced precipitation band which can be easily observed using bare eye. But the secondary morphology of the crystals predicted here by our model can be verified using a proper experimental set up to observe the micro morphology of the precipitation bands and the space between two bands. It will be very interesting to see such type of patterns created by microcrystals. Because it apparently makes the crystals look like they have a life and sense to organize themselves elegantly. The predictions on the particle density and average radius can be verified by microscope since the radius of the crystals are to the order of micrometers.

Figure 4 Corrosion is also a nonlinear reaction-diffusion system. a. copper rectangle coupons corroded in 200 μL water droplets (pH 9.0, aerated) under gamma radiation after 72 h. b. SEM image of the morphology between two precipitation bands on the surface of copper. c. microscope image of the local geometry of landscape shows that the center of the rings has a pitting hole there. d. schematic mechanism proposed by us to explain the copper corrosion morphology are corresponding to the 2D Liesegang ring experiment. e. Open circuit potential of Carbon Steel at Ph 7.5. Red lines and stars are drawn to investigate the scaling laws of the curve f. Scaling laws of the time points obtained in figure 4e. g. Scaling laws of the band width in figure e.

In our recent study, we realized that periodic chemical wave can also appear in metal corrosion process since the process of oxide formation involved with nucleation and crystal growth. Figure 4 demonstrates our recent experimental observations about copper and carbon steel corrosion that might have Liesegang phenomenon in them. This project aims at predicting the corrosion of copper and carbon steel which are the components of the containers used for the long-term disposal of the spent fuel from nuclear reactor[40]. Figure 4a is the optical image of corroded rectangular copper coupon by a droplet with an initial pH 9. The coupon is placed in a gamma cell for 72 hours to mimic the irradiation condition of the used fuel container (UFC). The reason we choose a droplet is that the droplet on the surface of the copper coupon can get supersaturated much faster due to a small solution volume than using a bulk solution. Astonishing patterns which looks like fingerprint appeared. A general question is how can these precipitated oxides formed in such a similar manner to plane wave. As far as our knowledge, no other research group has ever reported the chemical wave formed on a metal surface. The morphology of the precipitation bands was characterized by Scanning Electron Microscope (SEM) as shown in Figure 4b. It is amazing to find distinct morphology in precipitation bands and the space between them. In the location of precipitation band, the particle density is small while the average radius of the particles is large. And for the space between two bands, the particle density is large while the average radius is small. This type of pattern exactly fits our predictions for Liesegang bands or rings.

As a matter of fact, there is always a focal point in the center for a set of ring pattern. In most of the case, the center is a badly corroded hole that collapsed down as indicated by Figure 4c, this type of hole and ring geometry was also reported by another separated group[41, 42]. But they only reported one or two rings. The experimental details of this work will be explained in another paper later in the future. Figure 4d represents the schematic of our proposed mechanism for the formation of the chemical wave of cuprous oxide. The initial Cu(OH)2 species comes from the deposition of Cu2+ in the solution. The copper hydroxide colloid is a gelatinous phase as is known in corrosion field[29, 30]. The gelatin could be very important to stabilize the crystals in a similar manner to the effect of gelatin medium used in Liesegang ring experiment. The mechanism is very analogous to a 2D Liesegang ring experiment where the center is the source of the Cu2+ flux from corrosion.

Even though the mechanism we proposed has not been verified yet, the primary morphology (the discontinuous precipitation band) and the secondary morphology (inverse relationship between average radius and particle density) can both be simulated in the same time using the same mechanism for Liesegang rings. The geometry in figure 4c characterized by us and other research group[41, 42] also indicates that the mechanism described in figure 4d is almost right, maybe not perfect yet.

Is the copper corrosion the only case that remind us of Liesegang phenomenon?

No. Figure 4e which looks like curves of Liesegang band simulation are actually Open Circuit Potential (OCP) measured for carbon steel corroded in pH 7.5 buffer solution at ambient temperature. In Liesegang phenomenon, there are some scaling laws discovered and proved by lots of experimental and theoretical work[10, 14, 21]. Figure 4f and 4g are obtained from figure 4e to elucidate the scaling laws in this oscillation OCP. The first scaling law of Liesegang phenomenon is that , where represent the location in which the nth band forms. The second scaling law of Liesegang phenomenon is the information of band width, increases linearly with . To use the two scaling laws to test if the OCP curve are related to Liesegang phenomenon, a series of data points are gained from figure 4e by drawing a straight red line across each band. The red points on the curve were used to create figure 4f and figure 4g. represents the time point on nth band in the curve. represents the band width by calculating the time difference of two neighbor points on the pulse. It clearly shows that the OCP curve fits the two scaling laws that only existed in Liesegang phenomenon.

However, under pH 7 and pH 8, the OCP is like the curves we normally see in most cases. We only observed this periodic oscillation behavior in pH 7.5. A slight variation of pH will not produce any pulse like that. This chaotic behavior has been addressed in our discussion for Liesegang phenomenon in Figure 2. Just as in Figure 2 (6)(7)(8)(9), the oscillation curve can only be obtained under certain values of diffusion coefficients. The transitionary states between smooth curve like (6) and oscillation curve like (8)(9) are a few oscillation peaks (1 or 2) as indicated in Figure 2(7). It means in a 2D reaction diffusion system, it is possible to observe only one or two rings around the focal point. That explained in some different conditions for copper corrosion, we observed only one or two ring patters instead of neither a set of ring patterns nor common continuous pattern. That the oscillation and instability only happen under certain condition is the feature of a nonlinear and chaotic system[36]. In many other experiments of metal corrosion which are not shown here, we continuously observed oscillation behaviors, like oscillation of the dissolved metal concentration measured by inductively coupled plasma mass spectrometry(ICP-MS), oscillation of pH and some other interesting patterns. These inexplicable experimental results indicated that conventional understanding of metal corrosion like mix potential theory[43] alone is not enough to understand all the behavior of corrosion. Specifically, without taking nucleation and crystal growth process into account, mix potential theory can never predict a OCP curve as measured in experiment in Figure 4e. Apparently, a new perspective of understanding corrosion process is needed to fulfil the theory in this field.

4.Conculsion

The simulation results not only successfully fit the experimental observations in many details, but also predicts the micro morphology of the precipitates although it has not been observed and verified yet. The model clearly shows the reaction front of the intermedium species propagating along the space slower and slower. This explained why the time needed for later precipitation bands to appear is much longer than earlier ones. Furthermore, only under certain conditions, can the Liesegang phenomenon shows up. A slight variation of one physical process or chemical parameter might lead to very different pattern formation. it implies that Liesegang experiment system is a chaotic system.

Combining the simulation work and our recent experimental findings, it is prospective to conclude that the cycle of supersaturation, nucleation, depletion in a very nonlinear feedback way can produce many unexpected experimental observations like interesting patterns, oscillation of concentrations or potentials and any other type of chaotic behavior in a diffusion-reaction system. New perspective of understanding corrosion process is needed to fulfil the theory in this field.

Acknowledgment

This research is funded by National Natural Science Foundation of China (21377122) and China Scholarship Council Visiting Research Program in 2016. (Do remember to add other programs involved in Wren』s Lab)

Reference

[1] GRZYBOWSKI B A. Chemistry in motion: reaction-diffusion systems for micro-and nanotechnology [M]. John Wiley & Sons, 2009.

[2] CHOI K-S. Shape control of inorganic materials via electrodeposition [J]. Dalton Transactions, 2008, 40): 5432-8.

[3] PENG X, MANNA L, YANG W, et al. Shape control of CdSe nanocrystals [J]. Nature, 2000, 404(6773): 59-61.

[4] MANN S. Biomineralization: principles and concepts in bioinorganic materials chemistry [M]. Oxford University Press on Demand, 2001.

[5] LIESEGANG R. A-Linien [J]. Photographisches Archiv, 1896, 321-6.

[6] LIESEGANG R. Ueber einige eigenschaften von gallerten [J]. Naturwissenschaftliche Wochenschrift, 1896, 10(30): 353-62.

[7] LIESEGANG R. Chemische fernwirkung [J]. Photographisches Archiv, 1897, 801): 331-6.

[8] VARMA K K, ALLAHABAD S G. *Liesegang Rings of Silver Chromate .2. [J]. Kolloid Z Z Polym, 1953, 133(1): 34-7.

[9] VARMA K K, GHOSH S. *Liesegang Rings of Silver Chromate .1. [J]. Kolloid Z Z Polym, 1953, 133(1): 32-4.

[10] AL-GHOUL M, AMMAR M, AL-KAYSI R O. Band Propagation, Scaling Laws and Phase Transition in a Precipitate System. I: Experimental Study [J]. J Phys Chem A, 2012, 116(18): 4427-37.

[11] SULTAN R F, ABDEL-RAHMAN A F M. On dynamic self-organization: examples from magmatic and other geochemical systems [J]. Lat Am J Solids Stru, 2013, 10(1): 59-73.

[12] NEEDHAM A E. The Development of Integumental Colour-Patterns [M]. The Significance of Zoochromes. Springer. 1974: 315-21.

[13] MALAGABAY. https://malagabay.wordpress.com/2013/10/31/Liesegang-rings-4-biology/Liesegang-biology/ [M]. 2013.

[14] DEE G T. Patterns Produced by Precipitation at a Moving Reaction Front [J]. Phys Rev Lett, 1986, 57(3): 275-8.

[15] CARTWRIGHT J H E, GARCIA-RUIZ J M, VILLACAMPA A I. Pattern formation in crystal growth: Liesegang rings [J]. Comput Phys Commun, 1999, 121(411-3.

[16] KRUG H J, BRANDTSTADTER H. Morphological characteristics of Liesegang rings and their simulations [J]. J Phys Chem A, 1999, 103(39): 7811-20.

[17] LAGZI I, VOLFORD A, BUKI A. Effect of geometry on the time law of Liesegang patterning [J]. Chem Phys Lett, 2004, 396(1-3): 97-101.

[18] LAGZI I, KOWALCZYK B, GRZYBOWSKI B A. Liesegang Rings Engineered from Charged Nanoparticles [J]. J Am Chem Soc, 2010, 132(1): 58-+.

[19] WITHERS N. LIESEGANG RINGS Nanoparticles ring the changes [J]. Nat Chem, 2010, 2(3): 160-.

[20] SMOUKOV S K, LAGZI I, GRZYBOWSKI B A. Independence of Primary and Secondary Structures in Periodic Precipitation Patterns [J]. J Phys Chem Lett, 2011, 2(4): 345-9.

[21] MANSOUR A A, AL-GHOUL M. Band Propagation, Scaling Laws, and Phase Transition in a Precipitate System. 2. Computational Study [J]. J Phys Chem A, 2015, 119(35): 9201-9.

[22] KALASH L, FARAH H, EDDIN A Z, et al. Dynamical profiles of the reactive components in direct and revert Liesegang patterns [J]. Chem Phys Lett, 2013, 590(69-73.

[23] OSTWALD W. Studien über die Bildung und Umwandlung fester K?rper [J]. Zeitschrift für physikalische Chemie, 1897, 22(1): 289-330.

[24] LEVAN M E, ROSS J. Measurements and a Hypothesis on Periodic Precipitation Processes [J]. J Phys Chem-Us, 1987, 91(24): 6300-8.

[25] BUKI A, KARPATISMIDROCZKI E, ZRINYI M. Computer-Simulation of Regular Liesegang Structures [J]. J Chem Phys, 1995, 103(23): 10387-92.

[26] BUKI A, KARPATINESMIDROCZKI E, ZRINYI M. Experimental and theoretical studies on the Liesegang phenomenon. [J]. Magy Kem Foly, 1995, 101(12): 524-38.

[27] BUKI A, KARPATISMIDROCZKI E, ZRINYI M. 2-Dimensional Chemical-Pattern Formation in Gels - Experiments and Computer-Simulation [J]. Physica A, 1995, 220(3-4): 357-75.

[28] B KI A. The Liesegang phenomenon http://www.insilico.hu/Liesegang/index.html [M]. 2007.

[29] DURANDKEKLIKIAN L, MATIJEVIC E. Needle-Type Colloidal Copper (Ii)Hydroxide Particles [J]. Colloid Polym Sci, 1990, 268(12): 1151-8.

[30] HENRY M, BONHOMME C, LIVAGE J. Synthesis and characterisation of copper(II) hydroxide gels [J]. J Sol-Gel Sci Techn, 1996, 6(2): 155-67.

[31] AGLADZE K, STEINBOCK O. Waves and vortices of rust on the surface of corroding steel [J]. J Phys Chem A, 2000, 104(44): 9816-9.

[32] NAKOUZI E, STEINBOCK O. Self-organization in precipitation reactions far from the equilibrium [J]. Science advances, 2016, 2(8): e1601144.

[33] LEE K J, MCCORMICK W D, OUYANG Q, et al. Pattern-Formation by Interacting Chemical Fronts [J]. Science, 1993, 261(5118): 192-4.

[34] LEE K J, MCCORMICK W D, PEARSON J E, et al. Experimental-Observation of Self-Replicating Spots in a Reaction-Diffusion System [J]. Nature, 1994, 369(6477): 215-8.

[35] GY RGYI L, FIELD R J. A three-variable model of deterministic chaos in the Belousov–Zhabotinsky reaction [J]. Nature, 1992, 355(6363): 808-10.

[36] SAGUES F, EPSTEIN I R. Nonlinear chemical dynamics [J]. Dalton Transactions, 2003, 7): 1201-17.

[37] THANH N T K, MACLEAN N, MAHIDDINE S. Mechanisms of Nucleation and Growth of Nanoparticles in Solution [J]. Chem Rev, 2014, 114(15): 7610-30.

[38] RAJURKAR N, AMBEKAR B. Studies on Liesegang rings of cobalt hydroxide in 1% agar gel medium [J]. J Mol Liq, 2015, 204(205-9.

[39] LAGZI I, UEYAMA D. Pattern transition between periodic Liesegang pattern and crystal growth regime in reaction-diffusion systems [J]. Chem Phys Lett, 2009, 468(4-6): 188-92.

[40] ROSBORG B, WERME L. The Swedish nuclear waste program and the long-term corrosion behaviour of copper [J]. J Nucl Mater, 2008, 379(1-3): 142-53.

[41] BJORKBACKA A, JOHNSON C M, LEYGRAF C, et al. Role of the Oxide Layer in Radiation-Induced Corrosion of Copper in Anoxic Water [J]. J Phys Chem C, 2016, 120(21): 11450-5.

[42] BJORKBACKA A, HOSSEINPOUR S, LEYGRAF C, et al. Radiation Induced Corrosion of Copper in Anoxic Aqueous Solution (vol 15, pg C5, 2012) [J]. Electrochem Solid St, 2012, 15(6): S5-S.

[43] ANDERSEN T N, GHANDEHARI M H, EYRING H. Limitation to Mixed Potential Concept of Metal Corrosion - Copper in Oxygenated Sulfuric-Acid Solutions [J]. J Electrochem Soc, 1975, 122(12): 1580-5.