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Density oscillations in precipitate domains of a propagating Cr(OH)3 ring
Chemical Physics Letters 374 (2003) 183–186 www.elsevier.com/locate/cplett
Density oscillations in precipitate domains of a propagating CrðOHÞ3 ring Nadeen Hilal, Rabih Sultan
*
Department of Chemistry, American University of Beirut, Bliss street, Beirut, Lebanon Received 31 October 2002; in final form 30 April 2003
Abstract A propagating CrðOHÞ3 band in 1D and a circular ring in 2D were observed and studied in PVA/GDA gels. In a recent simulation (Al-Ghoul and Sultan, J. Phys. Chem. A 107 (2003) 1095) using the model of M€ uller and Polezhaev, the precipitate density of the 1D pulse exhibited temporal oscillations at early times. We present here new experiments to verify this theoretical finding. Oscillations were indeed observed in the density and width of the precipitate ring, suggesting that the obtained pattern is that of a breathing pulse. The properties of the pulse studied in this Letter, capture the characteristics of a pulse of mineral deposit conjectured earlier (Earth-Sci. Rev. 29 (1990) 163) using a model of geochemical pattern formation. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction
Cr3þ ðaqÞ þ 3OH ðaqÞ ! CrðOHÞ3 ðsÞ
ð1Þ
Propagating precipitate zones exist in gelled media where ion diffusion couples to precipitation and precipitate re-dissolution recations [1–4]. In 1D, such zones consist of either a single band or a stratum of bands (Liesegang [5–7] patterns). The global propagation of the pattern results from a precipitation front at the leading edge and a re-dissolution front at the tail. In 2D, a propagating precipitate ring or a migrating set of rings is observed. In this Letter, we focus on the CrðOHÞ3 system, studied in 1D [2] and 2D [3], with the following precipitation and re-dissolution reactions:
CrðOHÞ3 ðsÞ þ OH ðaqÞ ! CrðOHÞ 4 ðaqÞ
ð2Þ
*
Corresponding author. Fax: +961-1365217. E-mail address: [email protected] (R. Sultan).
A single CrðOHÞ3 precipitate zone is shown to propagate in both 1D and 2D (a band or a ring, respectively). In a recent theoretical study [8] based on the model of M€ uller and Polezhaev [9], the propagation of a single band or a stratum of bands was simulated both in the absence and presence of an electric field. The conditions for obtaining either type of pattern are delineated by a recently developed criterion [10], based on the Lifshitz– Slyozov [11] instability. The simulations [8] capture the patterning features of a propagating stratum of CoðOHÞ2 bands [1,12,13], and a migrating single CrðOHÞ3 band [2,3], observed experimentally. The proper reaction scheme, with the exact stoichiometry, was used to model each
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00726-7
184
N. Hilal, R. Sultan / Chemical Physics Letters 374 (2003) 183–186
Fig. 1. Temporal oscillations in the density maximum of a 1D CrðOHÞ3 pulse obtained from a reaction–diffusion calculation on that particular system, with a stoichiometry represented by the scheme of Eqs. (1) and (2). This simulation was performed in [8]. Notably examine the region within the dashed frame.
system separately. One interesting result of the simulations in the single-band regime (like in the observed CrðOHÞ3 band), is that the peak of precipitate density (qmax ) exhibits temporal oscillations within an overall rectangular hyperbola type of curve, as shown in Fig. 1. In this Letter, we investigate the properties of the CrðOHÞ3 propagating ring in order to verify the oscillations predicted by the theoretical modeling. We carry out experiments in 2D (in specially designed dishes), as this allows for better measurements [3]. It is interesting to note here that a similar propagating pulse of a mineral deposit was conjectured using a geochemical model [14], based on the Ostwald [15] supersaturation–nucleation–depletion cycle. A comparison between the CrðOHÞ3 oscillations studied here and the profile of the aforementioned pulse is presented in Section 3.
after heating, 0.600 g of CrðNO3 Þ3 9H2 O were added to the solution (to obtain [Cr3þ 0 ¼ 0:0802 M), and the solution was stirred again until complete dissolution of the salt. Then, using a microsyringe, 0.06 ml of GDA cross-linking agent (50%, 5.6 M) were added with constant stirring. Finally, five drops of 3.0 M HCl were added to adjust the pH of the solution to 1.0. The mixture was poured into a special methacrylate glass Petri dish, whose cover is hollow at the center with a 4 cm long cylindrical tube of inner diameter 1.6 cm acting as a reservoir for the outer electrolyte (NaOH), shown in Fig. 2a. The Cr3þ gel was poured while hot through the tube, and spread evenly (without air bubbles) in the dish as a thin layer of 0.7 mm thickness. The dish was left overnight for gelation to complete. The next day, the small gel disk at the bottom of the central tube was removed, then the void circular cavity was cleaned and the tube filled with 2.50 M NaOH marking time t ¼ 0. The NaOH diffused slowly
2. Experimental 2.1. Chemicals and setup The gel solution of Cr3þ ions was prepared according to the procedure described in [3]. 0.64 g of PVA (degree of polymerization n ¼ 500) were added to 18.7 ml of double distilled water, and the mixture was stirred for 10 min using a magnetic stirring bar. The mixture was then vigorously heated with continuing stirring for another 10 min until complete dissolution occurred. Immediately
Fig. 2. Petri dish and hollow cover with a tube, for hosting the Cr3þ gel prepared as described in Section 2.1. The NaOH is filled in the tube after removing the gel from its bottom. (a) Diametric cross-section; (b) top view showing the CrðOHÞ3 precipitate ring.
N. Hilal, R. Sultan / Chemical Physics Letters 374 (2003) 183–186
185
into the Cr3þ gel solution causing the precipitation of a single light-green ring which became larger as time advanced (Fig. 2b).
and subsequently its width, as well as the area under the absorbance peak.
2.2. Measurements by image analysis
3. Results and discussion
A digital camera (Mavica FD90, Sony) was placed in a fixed position at a constant distance above the dish. The system was illuminated with a lamp placed directly underneath an opaque methacrylate glass board on which the dish was placed. The role of this board was to homogenize the intensity of incident light passing through the entire dish. The experiment was performed in a dark room to ensure that there is no light interference from outside. Pictures were taken repeatedly over a span of 8.2 h, first every 5 min, then the interval was gradually increased to 10, then 15 then 20 min. The origin (x ¼ 0) was set at the interface between the Cr3þ gel in the dish and the NaOH solution in the tube. Distance was calibrated relative to a 1.00 cm scale attached to the dish cover. Using the software SigmaGel [16], a horizontal line cut was drawn from the origin to a point beyond the CrðOHÞ3 precipitate domain. The precipitate pulse profile was thus mapped onto an absorbance versus distance plot (see Fig. 3). We also measured other parameters such as the position of the front and back edge of the band,
The results of the experiment are shown in Fig. 4. The diffusion profile (location of the leading edge of the ring d, versus time t) is shown in frame a. The thin curve is an interpolation between the experimental points, while the thick curve is a d ¼ at1=2 fit. Frames b, c and d display the time evolution of the ring width w, the density maximum qmax and the scaled precipitate mass m (measured by the area under the absorbance profile), respectively. All three properties exhibit temporal oscillations as predicted by the simulations of [8]. The overall trend in w and m (area) fits a curve of the rectangular hyperbola type around which the oscillations occur, just resembling the results of the simulations [8]. Note that a plot of the area versus time obtained from the simulations (not shown here), also displays temporal oscillations similar to those of qmax , shown in Fig. 1. Analysis of the w, qmax and m data reveals that the three quantities oscillate in phase (w m correlation ¼ 0.95, qmax m correlation ¼ 0.92, w qmax correlation ¼ 0.88). Thus, the height and the width of the pulse profile stretch and contract in
Fig. 3. Map of absorbance versus distance for the CrðOHÞ3 pulsed profile, obtained from digital pictures at times: (a) 1.87 h, (b) 2.03 h, (c) 2.37 h, (d) 2.70 h, (e) 3.03 h.
Fig. 4. Time variation of: (a) distance travelled, (b) pulse width, (c) maximum density, and (d) precipitate mass for the CrðOHÞ3 pulse. The latter three properties exhibit temporal oscillations as predicted by the simulations of [8].
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N. Hilal, R. Sultan / Chemical Physics Letters 374 (2003) 183–186
concerted modes. This suggests that the propagating pattern is that of a breathing pulse, which expands and shrinks periodically, as can be distinctly inferred from the profiles of Fig. 3. Note that each location shown in Fig. 4a. corresponds to an average of two measurements taken on the left and the right portions of the ring along a horizontal cut, to correct for imperfections in the ring centering. The w, qmax and m measurements were performed on the right portion only, as the latter parameters are not affected by the centering. A pulse of geological mineral deposit was obtained by Sultan et al. [14], using a geochemical simulation model involving diffusion, infiltration flow, precipitation and dissolution, thus resembling the dynamics governing the pulse formation here. An analytical expression for a steady deposition pulse was obtained. Using numerical simulations, the pulse was shown to develop internal spatial structure as a rate coefficient was increased. Transition to a ÔwiggleÕ structure (with an undulatory density profile) was experienced, followed by the onset of distinct bands separated by clear domains. The wiggle pattern exhibited temporal oscillations in a breather mode fashion, just like the behavior of the CrðOHÞ3 pulse obtained here (compare Fig. 7 of [14] with Fig. 3 in the present Letter). Thus we see that a great similarity exists between precipitate patterning in gelled media and geochemical self-organization [17,18] phenomena. This study unveils the validity and robustness of the theoretical model used [8–10], as the latter suggested minute experimental refinements leading to rich patterning features, not detected in the original experiments [2,3].
Acknowledgements This work was supported by a ÔTeaching and Advising WorkshopÕ grant, (URB, American University of Beirut). The authors thank Prof. Mazen Al-Ghoul for his valuable feedback.
References [1] R.F. Sultan, Phys. Chem. Chem. Phys. (PCCP) 4 (2002) 1253. . Smidr [2] M. Zrınyi, L. Galfi, E oczki, Z. Racz, F. Horkay, J. Phys. Chem. 95 (1991) 1618. [3] R. Sultan, Sh. Panjarian, Phys. D 157 (2001) 241. [4] I. Das, A. Pushkarna, N.R. Argawal, J. Phys. Chem. 93 (1989) 7269. [5] R.E. Liesegang, Chemische Fernwirkung, Lieseg. Photograph. Arch. 37 (1896) 305; continued in 37 (1896) 321. [6] R.E. Liesegang, Naturewiss. Wochenschr. 11 (1896) 353. [7] K.H. Henisch, Crystals in Gels and Liesegang Rings, Cambridge University Press, Cambridge, 1988. [8] M. Al-Ghoul, R. Sultan, J. Phys. Chem. A 107 (2003) 1095. [9] S.C. M€ uller, A.A. Polezhaev, Chaos 4 (1994) 631. [10] M. Al-Ghoul, R. Sultan, J. Phys. Chem. A 105 (2001) 8053. [11] I.M. Lifshitz, V.V. Slyozov, J. Phys. Chem. Solids 19 (1961) 35. [12] V. Nasreddine, R. Sultan, J. Phys. Chem. A 103 (1999) 2934. [13] R. Sultan, R. Halabieh, Chem. Phys. Lett. 332 (2000) 331. [14] R. Sultan, P. Ortoleva, F. DePasquale, P. Tartaglia, EarthSci. Rev. 29 (1990) 163. [15] Wi. Ostwald, Lehrbuch der Allgemeinen Chemie, 2. Aufl., Band II, 2. Teil: Verwandtschaftslehre, Engelmann, Leipzig, 1899, p.779. [16] SigmaGel, Gel Analysis Software, Jandel Scientific, 1995. [17] P. Ortoleva, Geochemical Self-Organization, Oxford University Press, New York, 1994. [18] J.H. Kruhl (Ed.), Fractals and Dynamic Systems in Geoscience, Springer, Berlin, 1994.
Density oscillations in precipitate domains of a propagating CrðOHÞ3 ring Nadeen Hilal, Rabih Sultan
*
Department of Chemistry, American University of Beirut, Bliss street, Beirut, Lebanon Received 31 October 2002; in final form 30 April 2003
Abstract A propagating CrðOHÞ3 band in 1D and a circular ring in 2D were observed and studied in PVA/GDA gels. In a recent simulation (Al-Ghoul and Sultan, J. Phys. Chem. A 107 (2003) 1095) using the model of M€ uller and Polezhaev, the precipitate density of the 1D pulse exhibited temporal oscillations at early times. We present here new experiments to verify this theoretical finding. Oscillations were indeed observed in the density and width of the precipitate ring, suggesting that the obtained pattern is that of a breathing pulse. The properties of the pulse studied in this Letter, capture the characteristics of a pulse of mineral deposit conjectured earlier (Earth-Sci. Rev. 29 (1990) 163) using a model of geochemical pattern formation. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction
Cr3þ ðaqÞ þ 3OH ðaqÞ ! CrðOHÞ3 ðsÞ
ð1Þ
Propagating precipitate zones exist in gelled media where ion diffusion couples to precipitation and precipitate re-dissolution recations [1–4]. In 1D, such zones consist of either a single band or a stratum of bands (Liesegang [5–7] patterns). The global propagation of the pattern results from a precipitation front at the leading edge and a re-dissolution front at the tail. In 2D, a propagating precipitate ring or a migrating set of rings is observed. In this Letter, we focus on the CrðOHÞ3 system, studied in 1D [2] and 2D [3], with the following precipitation and re-dissolution reactions:
CrðOHÞ3 ðsÞ þ OH ðaqÞ ! CrðOHÞ 4 ðaqÞ
ð2Þ
*
Corresponding author. Fax: +961-1365217. E-mail address: [email protected] (R. Sultan).
A single CrðOHÞ3 precipitate zone is shown to propagate in both 1D and 2D (a band or a ring, respectively). In a recent theoretical study [8] based on the model of M€ uller and Polezhaev [9], the propagation of a single band or a stratum of bands was simulated both in the absence and presence of an electric field. The conditions for obtaining either type of pattern are delineated by a recently developed criterion [10], based on the Lifshitz– Slyozov [11] instability. The simulations [8] capture the patterning features of a propagating stratum of CoðOHÞ2 bands [1,12,13], and a migrating single CrðOHÞ3 band [2,3], observed experimentally. The proper reaction scheme, with the exact stoichiometry, was used to model each
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00726-7
184
N. Hilal, R. Sultan / Chemical Physics Letters 374 (2003) 183–186
Fig. 1. Temporal oscillations in the density maximum of a 1D CrðOHÞ3 pulse obtained from a reaction–diffusion calculation on that particular system, with a stoichiometry represented by the scheme of Eqs. (1) and (2). This simulation was performed in [8]. Notably examine the region within the dashed frame.
system separately. One interesting result of the simulations in the single-band regime (like in the observed CrðOHÞ3 band), is that the peak of precipitate density (qmax ) exhibits temporal oscillations within an overall rectangular hyperbola type of curve, as shown in Fig. 1. In this Letter, we investigate the properties of the CrðOHÞ3 propagating ring in order to verify the oscillations predicted by the theoretical modeling. We carry out experiments in 2D (in specially designed dishes), as this allows for better measurements [3]. It is interesting to note here that a similar propagating pulse of a mineral deposit was conjectured using a geochemical model [14], based on the Ostwald [15] supersaturation–nucleation–depletion cycle. A comparison between the CrðOHÞ3 oscillations studied here and the profile of the aforementioned pulse is presented in Section 3.
after heating, 0.600 g of CrðNO3 Þ3 9H2 O were added to the solution (to obtain [Cr3þ 0 ¼ 0:0802 M), and the solution was stirred again until complete dissolution of the salt. Then, using a microsyringe, 0.06 ml of GDA cross-linking agent (50%, 5.6 M) were added with constant stirring. Finally, five drops of 3.0 M HCl were added to adjust the pH of the solution to 1.0. The mixture was poured into a special methacrylate glass Petri dish, whose cover is hollow at the center with a 4 cm long cylindrical tube of inner diameter 1.6 cm acting as a reservoir for the outer electrolyte (NaOH), shown in Fig. 2a. The Cr3þ gel was poured while hot through the tube, and spread evenly (without air bubbles) in the dish as a thin layer of 0.7 mm thickness. The dish was left overnight for gelation to complete. The next day, the small gel disk at the bottom of the central tube was removed, then the void circular cavity was cleaned and the tube filled with 2.50 M NaOH marking time t ¼ 0. The NaOH diffused slowly
2. Experimental 2.1. Chemicals and setup The gel solution of Cr3þ ions was prepared according to the procedure described in [3]. 0.64 g of PVA (degree of polymerization n ¼ 500) were added to 18.7 ml of double distilled water, and the mixture was stirred for 10 min using a magnetic stirring bar. The mixture was then vigorously heated with continuing stirring for another 10 min until complete dissolution occurred. Immediately
Fig. 2. Petri dish and hollow cover with a tube, for hosting the Cr3þ gel prepared as described in Section 2.1. The NaOH is filled in the tube after removing the gel from its bottom. (a) Diametric cross-section; (b) top view showing the CrðOHÞ3 precipitate ring.
N. Hilal, R. Sultan / Chemical Physics Letters 374 (2003) 183–186
185
into the Cr3þ gel solution causing the precipitation of a single light-green ring which became larger as time advanced (Fig. 2b).
and subsequently its width, as well as the area under the absorbance peak.
2.2. Measurements by image analysis
3. Results and discussion
A digital camera (Mavica FD90, Sony) was placed in a fixed position at a constant distance above the dish. The system was illuminated with a lamp placed directly underneath an opaque methacrylate glass board on which the dish was placed. The role of this board was to homogenize the intensity of incident light passing through the entire dish. The experiment was performed in a dark room to ensure that there is no light interference from outside. Pictures were taken repeatedly over a span of 8.2 h, first every 5 min, then the interval was gradually increased to 10, then 15 then 20 min. The origin (x ¼ 0) was set at the interface between the Cr3þ gel in the dish and the NaOH solution in the tube. Distance was calibrated relative to a 1.00 cm scale attached to the dish cover. Using the software SigmaGel [16], a horizontal line cut was drawn from the origin to a point beyond the CrðOHÞ3 precipitate domain. The precipitate pulse profile was thus mapped onto an absorbance versus distance plot (see Fig. 3). We also measured other parameters such as the position of the front and back edge of the band,
The results of the experiment are shown in Fig. 4. The diffusion profile (location of the leading edge of the ring d, versus time t) is shown in frame a. The thin curve is an interpolation between the experimental points, while the thick curve is a d ¼ at1=2 fit. Frames b, c and d display the time evolution of the ring width w, the density maximum qmax and the scaled precipitate mass m (measured by the area under the absorbance profile), respectively. All three properties exhibit temporal oscillations as predicted by the simulations of [8]. The overall trend in w and m (area) fits a curve of the rectangular hyperbola type around which the oscillations occur, just resembling the results of the simulations [8]. Note that a plot of the area versus time obtained from the simulations (not shown here), also displays temporal oscillations similar to those of qmax , shown in Fig. 1. Analysis of the w, qmax and m data reveals that the three quantities oscillate in phase (w m correlation ¼ 0.95, qmax m correlation ¼ 0.92, w qmax correlation ¼ 0.88). Thus, the height and the width of the pulse profile stretch and contract in
Fig. 3. Map of absorbance versus distance for the CrðOHÞ3 pulsed profile, obtained from digital pictures at times: (a) 1.87 h, (b) 2.03 h, (c) 2.37 h, (d) 2.70 h, (e) 3.03 h.
Fig. 4. Time variation of: (a) distance travelled, (b) pulse width, (c) maximum density, and (d) precipitate mass for the CrðOHÞ3 pulse. The latter three properties exhibit temporal oscillations as predicted by the simulations of [8].
186
N. Hilal, R. Sultan / Chemical Physics Letters 374 (2003) 183–186
concerted modes. This suggests that the propagating pattern is that of a breathing pulse, which expands and shrinks periodically, as can be distinctly inferred from the profiles of Fig. 3. Note that each location shown in Fig. 4a. corresponds to an average of two measurements taken on the left and the right portions of the ring along a horizontal cut, to correct for imperfections in the ring centering. The w, qmax and m measurements were performed on the right portion only, as the latter parameters are not affected by the centering. A pulse of geological mineral deposit was obtained by Sultan et al. [14], using a geochemical simulation model involving diffusion, infiltration flow, precipitation and dissolution, thus resembling the dynamics governing the pulse formation here. An analytical expression for a steady deposition pulse was obtained. Using numerical simulations, the pulse was shown to develop internal spatial structure as a rate coefficient was increased. Transition to a ÔwiggleÕ structure (with an undulatory density profile) was experienced, followed by the onset of distinct bands separated by clear domains. The wiggle pattern exhibited temporal oscillations in a breather mode fashion, just like the behavior of the CrðOHÞ3 pulse obtained here (compare Fig. 7 of [14] with Fig. 3 in the present Letter). Thus we see that a great similarity exists between precipitate patterning in gelled media and geochemical self-organization [17,18] phenomena. This study unveils the validity and robustness of the theoretical model used [8–10], as the latter suggested minute experimental refinements leading to rich patterning features, not detected in the original experiments [2,3].
Acknowledgements This work was supported by a ÔTeaching and Advising WorkshopÕ grant, (URB, American University of Beirut). The authors thank Prof. Mazen Al-Ghoul for his valuable feedback.
References [1] R.F. Sultan, Phys. Chem. Chem. Phys. (PCCP) 4 (2002) 1253. . Smidr [2] M. Zrınyi, L. Galfi, E oczki, Z. Racz, F. Horkay, J. Phys. Chem. 95 (1991) 1618. [3] R. Sultan, Sh. Panjarian, Phys. D 157 (2001) 241. [4] I. Das, A. Pushkarna, N.R. Argawal, J. Phys. Chem. 93 (1989) 7269. [5] R.E. Liesegang, Chemische Fernwirkung, Lieseg. Photograph. Arch. 37 (1896) 305; continued in 37 (1896) 321. [6] R.E. Liesegang, Naturewiss. Wochenschr. 11 (1896) 353. [7] K.H. Henisch, Crystals in Gels and Liesegang Rings, Cambridge University Press, Cambridge, 1988. [8] M. Al-Ghoul, R. Sultan, J. Phys. Chem. A 107 (2003) 1095. [9] S.C. M€ uller, A.A. Polezhaev, Chaos 4 (1994) 631. [10] M. Al-Ghoul, R. Sultan, J. Phys. Chem. A 105 (2001) 8053. [11] I.M. Lifshitz, V.V. Slyozov, J. Phys. Chem. Solids 19 (1961) 35. [12] V. Nasreddine, R. Sultan, J. Phys. Chem. A 103 (1999) 2934. [13] R. Sultan, R. Halabieh, Chem. Phys. Lett. 332 (2000) 331. [14] R. Sultan, P. Ortoleva, F. DePasquale, P. Tartaglia, EarthSci. Rev. 29 (1990) 163. [15] Wi. Ostwald, Lehrbuch der Allgemeinen Chemie, 2. Aufl., Band II, 2. Teil: Verwandtschaftslehre, Engelmann, Leipzig, 1899, p.779. [16] SigmaGel, Gel Analysis Software, Jandel Scientific, 1995. [17] P. Ortoleva, Geochemical Self-Organization, Oxford University Press, New York, 1994. [18] J.H. Kruhl (Ed.), Fractals and Dynamic Systems in Geoscience, Springer, Berlin, 1994.